Finite Group Databasebeta

← #29729411

Subgroups — #29729411

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840121C211yesmaximal, characteristic, complemented
2C2×#16840043C211yesmaximal, characteristic, complemented
3#21450791C211yesmaximal, characteristic
4#21448908C211yesmaximal, characteristic, complemented
5#21447212C211yesmaximal, characteristic, complemented
6#21447205C211yesmaximal, characteristic, complemented
7#21447056C211yesmaximal, characteristic, complemented
8#21447051C211yesmaximal, characteristic
9#21446748C211yesmaximal, characteristic, complemented
10#21446744C211yesmaximal, characteristic
11#21446540C211yesmaximal, characteristic, complemented
12#21446462C211yesmaximal, characteristic, complemented
13#21446441C211yesmaximal, characteristic, complemented
14#21445647C211yesmaximal, characteristic, complemented
15C61×#1983C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519766C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20#16840121—24no—
21#16840043—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839678—21no—
36#16839678—21no—
37#16839669C2211yescharacteristic
38#16839666—21no—
39#16839666—21no—
40#16839665C2211yescharacteristic
41#16839660—21no—
42#16839658C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839638—21no—
45#16839638—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4×C4⋊Dic61—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×C22⋊D61—21no—
64C22×C22⋊D61—21no—
65C22×C4×D61—21no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777D414yescomplemented
70#14519773—22no—
71#14519773—41no—
72#14519766—22no—
73#14519766—41no—
74C4×D244—22no—
75C4×D244—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519711—22no—
95#14519706—41no—
96#14519706—41no—
97#14519701—22no—
98#14519701—22no—
99#14519699—22no—
100#14519699—22no—
101#14519698—22no—
102#14519698—22no—
103#14519697—22no—
104#14519697—22no—
105#14519688—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×Q8×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118#124×C61—22no—
119#124×C61—22no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127#103×C61—22no—
128#103×C61—41no—
129#102×C61—22no—
130#102×C61—41no—
131#95×C61—21no—
132C24×C61—21no—
133C23×D61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—42no—
153C2×C22⋊D61—42no—
154C2×C22⋊D61—42no—
155C2×C22⋊D61—42no—
156C2×C22⋊D61—81no—
157C2×C22⋊D61—81no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244#4112yes—
171C2×C4×D61—42no—
172C2×C4×D61—81no—
173C2×C4×D61—81no—
174C2×Dic122—24no—
175C2×Dic122—81no—
176C2×Dic122—81no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61#4112yes—
187C22⋊C4×C61—42no—
188C22⋊C4×C61—42no—
189C42×C61—42no—
190C42×C61C2×D412yes—
191C42×C61#4112yes—
192C22⋊Dic61—81no—
193C22⋊Dic61—81no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197C4⋊Dic61—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61C2×D412yes—
205C4⋊Dic61C2×D412yes—
206#11758196—81no—
207#11758196—81no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—81no—
213C4×Dic61—81no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61C22×D411yescharacteristic
228C22×D61—22no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235Q8×C61—41no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
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#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C22⋊D61—84no—
265C22⋊D61—84no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61#13712yes—
287D244—22no—
288D244—22no—
289D244—41no—
290D244—41no—
291C4×D61—84no—
292Dic122—84no—
293C8×C61—41no—
294C8×C61—41no—
295#92418—41no—
296#92418—41no—
297#25285—611nomaximal, Sylow
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61C2×#12111yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61#54811yescharacteristic
308C22×C61C2×#13711yescharacteristic
309C22×C61#57511yescharacteristic
310C22×C61#57611yescharacteristic
311C22×C61#57611yescharacteristic
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61#52911yescharacteristic
325C4×C61#54211yescharacteristic
326C4×C61#54811yescharacteristic
327C4×C61#55011yescharacteristic
328Dic61—22no—
329Dic61—42no—
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
333Dic61—41no—
334Dic61—81no—
335C2×#489—611no—
336C2×#474—611no—
337#2544—611no—
338#2369—611no—
339#2068—611no—
340#2067—611no—
341#2042—611no—
342#2041—611no—
343#1983—611no—
344#1983D6111yescharacteristic, complemented
345#1982—611no—
346#1940—611no—
347#1928—611no—
348#1925—611no—
349#1762—611no—
350C2×C61—161no—
351C2×C61#242611yescharacteristic
352C2×C61#303711yescharacteristic
353C2×C61#304111yescharacteristic
354C2×C61#331411yescharacteristic
355C2×C61#331911yescharacteristic
356C2×C61#337011yescharacteristic
357C2×C61#337511yescharacteristic
358D61—82no—
359C2×C4⋊Q8—611no—
360C2×#127—611no—
361C2×#124—611no—
362C2×#124C2×D6111yescharacteristic
363C2×C4×D4—611no—
364#489—1224no—
365#474—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×C4⋊C8—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8C2×D6111yescharacteristic
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×#102—611no—
380C2×#102—611no—
381C2×#102C2×D6111yescharacteristic
382C2×#102C2×D6111yescharacteristic
383C2×#97—611no—
384#394—1221no—
385#394—1221no—
386#391—611no—
387#391—1221no—
388#391—1221no—
389#390—611no—
390#389—611no—
391#389—611no—
392#389—1221no—
393#386—1221no—
394#386—1221no—
395C4×C4⋊C4—611no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—611no—
398C4×C4⋊C4—1221no—
399C4×C4⋊C4C2×D6111yescharacteristic
400#340—611no—
401#340—611no—
402#336—611no—
403#336—611no—
404C61#2528511yescharacteristic, complemented, Sylow
405C22×Q8—1221no—
406C22×Q8C22×D6111yescharacteristic
407C22×D4—611no—
408C22×D4—1221no—
409C22×D4—1221no—
410C22×D4C22×D6111yescharacteristic
411C23×C4—1221no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—614no—
417C4⋊Q8—1222no—
418#127—1222no—
419#127—2441no—
420#124—22no—
421#124—22no—
422#124—1222no—
423#124—2441no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—21no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—1222no—
450#107—1222no—
451C4⋊C8—22no—
452C4⋊C8—2441no—
453C4⋊C8—2441no—
454#103—22no—
455#103—41no—
456#103—1222no—
457#103—1222no—
458#102—22no—
459#102—41no—
460#102—1222no—
461#102—1222no—
462#102—1222no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#97—1222no—
467#95—21no—
468#95—611no—
469#95—611no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—1221no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—1224no—
485C2×Q8—4881no—
486C2×Q8—4881no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—612no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—4881no—
500C2×D4—4881no—
501C2×D4—2442no—
502C2×D4—2442no—
503C2×D4—2442no—
504C2×D4—2442no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—4881no—
516C22×C4—4881no—
517C22×C4—2442no—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—1222no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—4881no—
545C4⋊C4—4881no—
546C4⋊C4—2441no—
547C4⋊C4#1175822412yes—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—4881no—
551C22⋊C4—4881no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C42—42no—
556C42—1221no—
557C42—1221no—
558C42—1221no—
559C42—1221no—
560C42—4881no—
561C42—4881no—
562C42#1175822212yes—
563C42D4×D6112yes—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—1222no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23C2×D4×D6111yescharacteristic
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—4884no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—1222no—
589D4—1222no—
590D4—4884no—
591D4—4884no—
592D4—2441no—
593D4—2441no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—612no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—4884no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4#1451975012yes—
630C2×C4#1451975012yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C2×C4C2×#1175822211yescharacteristic
635C2×C4#1451982211yescharacteristic
636C2×C4#1451982411yescharacteristic
637C8—41no—
638C8—41no—
639C8—2441no—
640C8—2441no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—4881no—
649C22—4881no—
650C22—4881no—
651C22—4881no—
652C22—2442no—
653C22—2442no—
654C22—2442no—
655C22C2×#1451975011yescharacteristic
656C22#1684040911yescharacteristic
657C22#1684041811yescharacteristic
658C22#137×D6111yescharacteristic
659C22#1684055811yescharacteristic
660C22#1684056411yescharacteristic
661C22#1684056411yescharacteristic
662C4—22no—
663C4—22no—
664C4—42no—
665C4—81no—
666C4—81no—
667C4—1222no—
668C4—4881no—
669C4—2441no—
670C4—2441no—
671C4—2442no—
672C4—2442no—
673C4—2441no—
674C4#1684029411yescharacteristic
675C4#1684037711yescharacteristic
676C4#1684042011yescharacteristic
677C4#1684043211yescharacteristic
678C2—161no—
679C2—4882no—
680C2#2144938611yescharacteristic
681C2#2145529011yescharacteristic
682C2#2145532511yescharacteristic
683C2#2145790711yescharacteristic
684C2#2145798211yescharacteristic
685C2#2145856911yescharacteristic
686C2#2145862811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1983C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
hypercenter#1983D61
hypocenterC61#25285
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
1. C2×#16840121C2semidirect
2. C2×#16840043C2semidirect
4. #21448908C2semidirect
5. #21447212C2semidirect
6. #21447205C2semidirect
7. #21447056C2semidirect
9. #21446748C2semidirect
11. #21446540C2semidirect
12. #21446462C2semidirect
13. #21446441C2semidirect
14. #21445647C2semidirect
15. C61×#1983C2semidirect
69. #14519777D4semidirect
344. #1983D61semidirect
404. C61#25285semidirect

Maximal subgroups

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

subgroupindex
C2×#168401212
C2×#168400432
#214507912
#214489082
#214472122
#214472052
#214470562
#214470512
#214467482
#214467442
#214465402
#214464622
#214464412
#214456472
C61×#19832
#2528561