Finite Group Databasebeta

← #29718828

Subgroups — #29718828

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839918C211yesmaximal, characteristic
2C2×#16839823C211yesmaximal, characteristic
3#21451616C211yesmaximal, characteristic
4#21447214C211yesmaximal, characteristic, complemented
5#21447200C211yesmaximal, characteristic, complemented
6#21447141C211yesmaximal, characteristic, complemented
7#21447141C211yesmaximal, characteristic, complemented
8#21446689C211yesmaximal, characteristic, complemented
9#21446679C211yesmaximal, characteristic, complemented
10#21446512C211yesmaximal, characteristic, complemented
11#21446498C211yesmaximal, characteristic, complemented
12#21446131C211yesmaximal, characteristic
13#21446018C211yesmaximal, characteristic
14#21445754C211yesmaximal, characteristic
15C61×#2310C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19#16839918—24no—
20#16839823—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34C2×C22⋊#92418C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839700—21no—
38#16839693—21no—
39#16839690—21no—
40#16839690—21no—
41#16839668C2211yescharacteristic
42#16839668C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839622C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C23×Q8×C61C2211yescharacteristic
51C2×#117×C61C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C61×#391—21no—
54C61×#391—21no—
55C61×#391C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#381—21no—
60C61×#381—21no—
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—41no—
67C2×#11758202—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519756—22no—
73#14519756—22no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82#14519710—41no—
83#14519708—41no—
84C4⋊#92418—22no—
85C4⋊#92418—22no—
86#14519701—22no—
87#14519701—22no—
88#14519701—22no—
89#14519701—22no—
90#14519701—41no—
91#14519701—41no—
92#14519700—41no—
93#14519700—41no—
94C22⋊#92418—22no—
95C22⋊#92418D414yes—
96#14519688—22no—
97C4×#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61—22no—
106C22×Q8×C61—22no—
107C22×Q8×C61—21no—
108C22×Q8×C61—22no—
109C22×Q8×C61—22no—
110C22×Q8×C61—41no—
111C22×Q8×C61—41no—
112C22×Q8×C61C2311yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116#117×C61—22no—
117#117×C61—22no—
118#117×C61—41no—
119#117×C61—41no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133#95×C61—21no—
134#95×C61—21no—
135#95×C61—21no—
136#95×C61—21no—
137#95×C61—21no—
138C24×C61C2×D411yescharacteristic
139C2×Q8×C61—24no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—44no—
151C2×Q8×C61—44no—
152C2×Q8×C61—44no—
153C2×Q8×C61—44no—
154C2×Q8×C61—42no—
155C2×Q8×C61—42no—
156C2×Q8×C61#4112yes—
157C2×Q8×C61#4112yes—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—24no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—42no—
168C22×C4×C61—42no—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D412yes—
175C22×C4×C61#4112yes—
176C22×C4×C61C2411yescharacteristic
177C22×Dic61—21no—
178C22×Dic61—21no—
179C22×Dic61—21no—
180C22×Dic61—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—42no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—42no—
201C22⋊Dic61—42no—
202C22⋊Dic61—81no—
203#11758202—44no—
204#11758202—44no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211#11758196—42no—
212#11758196—81no—
213#11758196—81no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418C2×D412yes—
225C2×#92418C2×D412yes—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61#12112yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—48no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—48no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61#12112yes—
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61H(5,𝔽2)11yescharacteristic
280C2×C4×C61H(5,𝔽2)11yescharacteristic
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297#92418—22no—
298#92418—22no—
299#92418—41no—
300#92418—41no—
301#24514—611nomaximal, Sylow
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61C2×#12111yescharacteristic
310C22×C61#52511yescharacteristic
311C22×C61#53511yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57811yescharacteristic
314C22×C61#57811yescharacteristic
315C22×C61#57811yescharacteristic
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61#54611yescharacteristic
326C4×C61#54611yescharacteristic
327C4×C61#55111yescharacteristic
328C4×C61#55111yescharacteristic
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333C2×#451—611no—
334C2×#433—611no—
335#2625—611no—
336#2310D6111yescharacteristic
337#2068—611no—
338#2066—611no—
339#2056—611no—
340#2056—611no—
341#1971—611no—
342#1969—611no—
343#1935—611no—
344#1933—611no—
345#1873—611no—
346#1854—611no—
347#1787—611no—
348C2×C61—42no—
349C2×C61#241111yescharacteristic
350C2×C61#305911yescharacteristic
351C2×C61#306311yescharacteristic
352C2×C61#312111yescharacteristic
353C2×C61#312711yescharacteristic
354C2×C61#322311yescharacteristic
355C2×C61#322511yescharacteristic
356C23×Q8C2×D6111yescharacteristic
357C22×Q16—611no—
358C22×Q16—611no—
359C2×C22⋊Q8—611no—
360C2×#117C2×D6111yescharacteristic
361C2×#117C2×D6111yescharacteristic
362#451—1224no—
363#433—1224no—
364C2×#107—611no—
365C2×#106—611no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8—611no—
378C2×#97—611no—
379C2×C4×C8—611no—
380#400—1221no—
381#398—1221no—
382#397—1221no—
383#397—1221no—
384#391—21no—
385#391—21no—
386#391—611no—
387#391—611no—
388#391C2×D6111yescharacteristic
389#391C2×D6111yescharacteristic
390#391C2×D6111yescharacteristic
391#391C2×D6111yescharacteristic
392#389—611no—
393#389—611no—
394#381—21no—
395#381—21no—
396#380—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401C61#2451411yescharacteristic, complemented, Sylow
402C22×Q8—22no—
403C22×Q8—22no—
404C22×Q8—21no—
405C22×Q8—22no—
406C22×Q8—22no—
407C22×Q8—41no—
408C22×Q8—41no—
409C22×Q8—611no—
410C22×Q8C22×D6111yescharacteristic
411C22×Q8C22×D6111yescharacteristic
412C23×C4—21no—
413C23×C4C22×D6111yescharacteristic
414C2×Q16—1222no—
415C2×Q16—1222no—
416C2×Q16—1222no—
417C2×Q16—1222no—
418C2×Q16—2441no—
419C2×Q16—2441no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8—611no—
423C22×C8—611no—
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426#117—22no—
427#117—22no—
428#117—41no—
429#117—41no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C4⋊C4C22⋊D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4C22⋊D6111yescharacteristic
444C2×C22⋊C4C22⋊D6111yescharacteristic
445C2×C42—21no—
446C2×C42C22×D6111yescharacteristic
447C2×C42C22×D6111yescharacteristic
448#107—2441no—
449#106—2441no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#103—1222no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—2441no—
457#103—2441no—
458#103—2441no—
459#103—2441no—
460C22⋊C8—614no—
461C22⋊C8—1222no—
462#97—1222no—
463C4×C8—1222no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24C2×C22⋊D6111yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8#1175822212yes—
504C2×Q8#1175822212yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4D4×D6112yes—
521C22×C4#1175822412yes—
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C2×C22⋊D6111yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528Q16—2444no—
529Q16—2444no—
530C2×C8—612no—
531C2×C8—612no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—42no—
560C22⋊C4—4881no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1451974912yes—
574C23C22×C22⋊D6111yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—22no—
579Q8—48no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584Q8—2441no—
585Q8—2441no—
586Q8—2441no—
587Q8—2441no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—48no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4#1451974912yes—
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C2×C4C2×#1175822211yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C8—1222no—
639C8—1222no—
640C8—2441no—
641C8—2441no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1451974911yescharacteristic
650C22#1684027311yescharacteristic
651C22#1684032911yescharacteristic
652C22#1684057011yescharacteristic
653C22#1684057711yescharacteristic
654C22#1684057711yescharacteristic
655C22#1684057711yescharacteristic
656C4—22no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—42no—
663C4—81no—
664C4—81no—
665C4—4881no—
666C4—4881no—
667C4—4881no—
668C4—4881no—
669C4#1684040211yescharacteristic
670C4#1684040211yescharacteristic
671C4#1684043811yescharacteristic
672C4#1684043811yescharacteristic
673C2—42no—
674C2#2144920411yescharacteristic
675C2#2145547111yescharacteristic
676C2#2145550611yescharacteristic
677C2#2145604811yescharacteristic
678C2#2145611911yescharacteristic
679C2#2145690211yescharacteristic
680C2#2145692811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2310C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2310D61
hypocenterC61#24514
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. #21447214C2semidirect
5. #21447200C2semidirect
6. #21447141C2semidirect
7. #21447141C2semidirect
8. #21446689C2semidirect
9. #21446679C2semidirect
10. #21446512C2semidirect
11. #21446498C2semidirect
401. C61#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#168399182
C2×#168398232
#214516162
#214472142
#214472002
#214471412
#214466892
#214466792
#214465122
#214464982
#214461312
#214460182
#214457542
C61×#23102
#2451461