Finite Group Databasebeta

← #29697245

Subgroups — #29697245

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839762C211yesmaximal, characteristic
2#21448692C211yesmaximal, characteristic, complemented
3#21448648C211yesmaximal, characteristic
4#21446512C211yesmaximal, characteristic, complemented
5#21446492C211yesmaximal, characteristic, complemented
6#21446462C211yesmaximal, characteristic, complemented
7#21446077C211yesmaximal, characteristic, complemented
8#21446069C211yesmaximal, characteristic, complemented
9#21445904C211yesmaximal, characteristic, complemented
10#21445849C211yesmaximal, characteristic, complemented
11#21445758C211yesmaximal, characteristic
12#21445647C211yesmaximal, characteristic, complemented
13#21445633C211yesmaximal, characteristic, complemented
14#21445601C211yesmaximal, characteristic, complemented
15C61×#1927C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22#16839762C418yescomplemented
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839669C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839646—21no—
36#16839644—21no—
37#16839640—21no—
38#16839640—21no—
39#16839624—21no—
40#16839621—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758198—21no—
45C4×#11758198C2211yescharacteristic
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C2×#11758226—22no—
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—41no—
64C2×#11758226—41no—
65C2×#11758226C2×C414yes—
66C22×D244—21no—
67C22×D244C2311yescharacteristic
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C2×#11758189—41no—
74C2×#11758189—41no—
75C2×#11758189C2×C414yescomplemented
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C4×D244—22no—
79C4×D244—41no—
80#14519739—22no—
81#14519739—41no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61C2×C414yescomplemented
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198D411yescharacteristic
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98#14519711—41no—
99#14519701—41no—
100#14519701C2×C414yescomplemented
101#14519699—22no—
102#14519699C2×C414yes—
103#14519697—22no—
104#14519697C2×C414yes—
105#14519689—22no—
106#14519688—41no—
107C8×Dic61—41no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×Q8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125#106×C61—41no—
126#103×C61—41no—
127#103×C61C2×C414yescomplemented
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131C23×D61—21no—
132C23×D61C2×D411yescharacteristic
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61C22×C412yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758226—28no—
153#11758226—44no—
154#11758226—44no—
155#11758226—44no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—24no—
160C2×D244—42no—
161C2×D244—42no—
162C2×D244C22×C412yes—
163C2×D244#4112yes—
164C2×D244#4112yes—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—24no—
170C2×C4×D61—24no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—81no—
178C2×C4×D61—81no—
179C2×C4×D61—81no—
180C2×C4×D61—81no—
181C2×C4×D61C22×C412yes—
182C2×C4×D61#4112yes—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61C22×C412yes—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61C22×C412yes—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—81no—
198#11758198—22no—
199#11758198—41no—
200#11758198—42no—
201#11758198—42no—
202#11758198—42no—
203#11758198—81no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61C22×C412yes—
210#11758196—24no—
211#11758196—81no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—81no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C22×C412yes—
222#11758189—44no—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61C4×D412yes—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C23×C411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61C4×D412yes—
283C2×Dic61#13712yes—
284D244—22no—
285D244—22no—
286D244—22no—
287D244—48no—
288D244#11714yes—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—48no—
294C4×D61—84no—
295C4×D61—84no—
296C4×D61—84no—
297C8×C61—22no—
298C8×C61—41no—
299#92418—22no—
300#92418—41no—
301#22578—611nomaximal, Sylow
302C22×C61C2×C4×D411yescharacteristic
303C22×C61#54611yescharacteristic
304C22×C61#54611yescharacteristic
305C22×C61C2×#13711yescharacteristic
306C22×C61#57511yescharacteristic
307C22×C61#57611yescharacteristic
308C22×C61#57711yescharacteristic
309C2×D61—22no—
310C2×D61—24no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—41no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—41no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—41no—
321C2×D61—42no—
322C2×D61—41no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—41no—
327C4×C61—81no—
328C4×C61—81no—
329C4×C61C2×#11711yescharacteristic
330C4×C61#51811yescharacteristic
331C4×C61#55111yescharacteristic
332C4×C61#55111yescharacteristic
333Dic61—22no—
334Dic61—22no—
335Dic61—42no—
336Dic61—41no—
337Dic61—81no—
338Dic61—81no—
339C2×#417—611no—
340#2345—611no—
341#2339—611no—
342#1935—611no—
343#1933—611no—
344#1928—611no—
345#1927D6111yescharacteristic, complemented
346#1864—611no—
347#1862—611no—
348#1823—611no—
349#1809—611no—
350#1788—611no—
351#1762—611no—
352C4×#102—611no—
353#1753—611no—
354C2×C61#234211yescharacteristic
355C2×C61C4×#13711yescharacteristic
356C2×C61#294611yescharacteristic
357C2×C61#330711yescharacteristic
358C2×C61#331411yescharacteristic
359C2×C61#332211yescharacteristic
360C2×C61#333111yescharacteristic
361D61—42no—
362D61—44no—
363C22×#41—611no—
364C22×M16—611no—
365C2×C4×D4—611no—
366C2×C4×D4—611no—
367C22×C4⋊C4—611no—
368C2×#107—611no—
369C2×#106C2×D6111yescharacteristic
370#417—618no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×C22⋊C8—611no—
380C2×#97—611no—
381C2×C4×C8—611no—
382#391—611no—
383#391C2×D6111yescharacteristic
384#389—611no—
385#389—611no—
386#387—1221no—
387#387—1221no—
388#386—1221no—
389#386—1221no—
390#381—1221no—
391#380—611no—
392#380—611no—
393#380—1221no—
394C4×C4⋊C4—611no—
395C4×C4⋊C4C2×D6111yescharacteristic
396C4×C22⋊C4—611no—
397C4×C22⋊C4—1221no—
398C2×#95—1221no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340C2×D6111yescharacteristic
405#336—611no—
406#336C2×D6111yescharacteristic
407C61#2257811yescharacteristic, complemented, Sylow
408C2×#41—614no—
409C2×#41—1222no—
410C2×#41—1222no—
411C2×#41—1222no—
412C2×#41—2441no—
413C2×#41—2441no—
414C22×Q8C22×D6111yescharacteristic
415C22×D4—611no—
416C22×D4—1221no—
417C23×C4—611no—
418C23×C4—1221no—
419C23×C4—1221no—
420C23×C4—1221no—
421C23×C4—1221no—
422C2×M16—614no—
423C2×M16—2441no—
424C2×M16—2441no—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8C22×D6111yescharacteristic
428C22×C8C22×D6111yescharacteristic
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—2441no—
432C4×D4—2441no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—611no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454#107—2441no—
455#106—41no—
456#103—41no—
457#103—614no—
458#103—2441no—
459#103C4×D6114yescomplemented
460#102—614no—
461#102—614no—
462#102—1222no—
463#102—1222no—
464C22⋊C8—1222no—
465#97—2441no—
466C4×C8—2441no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—611no—
480C24—1221no—
481#41—1228no—
482#41—2444no—
483#41—2444no—
484#41—2444no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8C2×C4×D6112yes—
491C2×D4—612no—
492C2×D4—612no—
493C2×D4—612no—
494C2×D4—1224no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—2442no—
499C2×D4—2442no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—611no—
509C22×C4—1224no—
510C22×C4—1224no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—4881no—
519C22×C4—4881no—
520C22×C4—4881no—
521C22×C4—4881no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4D4×D6111yescharacteristic
529C22×C4D4×D6111yescharacteristic
530C22×C4C23×D6111yescharacteristic
531M16—2444no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—612no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8C2×C4×D6112yes—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—612no—
550C4⋊C4—1224no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—4881no—
557C4⋊C4—4881no—
558C4⋊C4C2×C4×D6112yes—
559C22⋊C4—1222no—
560C22⋊C4—4881no—
561C22⋊C4—2441no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C42—21no—
566C42—21no—
567C42—81no—
568C42—1221no—
569C42—1221no—
570C42—4881no—
571C23—612no—
572C23—1222no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1222no—
578C23—1221no—
579C23—1222no—
580C23—1221no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23—2441no—
585C23C2×D4×D6111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590D4—614no—
591D4—1222no—
592D4—1222no—
593D4—1222no—
594D4—2448no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—4884no—
628C2×C4—4884no—
629C2×C4—4884no—
630C2×C4—2448no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4C22×C4×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#1451982211yescharacteristic
644C2×C4#1451982311yescharacteristic
645C8—22no—
646C8—41no—
647C8—1222no—
648C8—2441no—
649C22—1222no—
650C22—1222no—
651C22—1222no—
652C22—1224no—
653C22—2442no—
654C22—2442no—
655C22—2442no—
656C22—2441no—
657C22—2441no—
658C22—2442no—
659C22—2441no—
660C22—2442no—
661C22—2442no—
662C22—2441no—
663C22C4×D4×D6111yescharacteristic
664C22#1684040411yescharacteristic
665C22#1684040911yescharacteristic
666C22#137×D6111yescharacteristic
667C22#1684056111yescharacteristic
668C22#1684056411yescharacteristic
669C22#1684056811yescharacteristic
670C4—42no—
671C4—41no—
672C4—41no—
673C4—42no—
674C4—81no—
675C4—81no—
676C4—1222no—
677C4—1222no—
678C4—4881no—
679C4—4881no—
680C4—2442no—
681C4—2441no—
682C4#117×D6111yescharacteristic
683C4#1684023411yescharacteristic
684C4#1684043711yescharacteristic
685C4#1684044211yescharacteristic
686C2—2442no—
687C2—2444no—
688C2#2144866311yescharacteristic
689C2#2145463511yescharacteristic
690C2#2145464611yescharacteristic
691C2#2145780211yescharacteristic
692C2#2145790711yescharacteristic
693C2#2145801811yescharacteristic
694C2#2145813311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1927C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
cosocleC22×C4×C61C24
hypercenter#1927D61
hypocenterC61#22578
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21448692C2semidirect
4. #21446512C2semidirect
5. #21446492C2semidirect
6. #21446462C2semidirect
7. #21446077C2semidirect
8. #21446069C2semidirect
9. #21445904C2semidirect
10. #21445849C2semidirect
12. #21445647C2semidirect
13. #21445633C2semidirect
14. #21445601C2semidirect
15. C61×#1927C2semidirect
22. #16839762C4semidirect
75. C2×#11758189C2×C4semidirect
89. C4⋊C4×D61C2×C4semidirect
100. #14519701C2×C4semidirect
127. #103×C61C2×C4semidirect
345. #1927D61semidirect
407. C61#22578semidirect
459. #103C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168397622
#214486922
#214486482
#214465122
#214464922
#214464622
#214460772
#214460692
#214459042
#214458492
#214457582
#214456472
#214456332
#214456012
C61×#19272
#2257861