Finite Group Databasebeta

← #30827165

Subgroups — #30827165

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570217C211yesmaximal, characteristic
2#22569310C211yesmaximal, characteristic, complemented
3#22559111C211yesmaximal, characteristic, complemented
4#22559097C211yesmaximal, characteristic, complemented
5#22559085C211yesmaximal, characteristic, complemented
6#22559066C211yesmaximal, characteristic, complemented
7#22559050C211yesmaximal, characteristic, complemented
8#22559045C211yesmaximal, characteristic, complemented
9#22559021C211yesmaximal, characteristic, complemented
10C8×#11761616C211yesmaximal, characteristic, complemented
11#22558990C211yesmaximal, characteristic, complemented
12#22558681C211yesmaximal, characteristic, complemented
13#22558600C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic, complemented
15C67×#2974C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863726C411yescharacteristic
18#16863726C411yescharacteristic
19#16863686C411yescharacteristic
20#16863686C411yescharacteristic
21#16863601C411yescharacteristic
22#16863601C411yescharacteristic
23#16863547C411yescharacteristic
24#16863547C411yescharacteristic
25#16863526C2211yescharacteristic, complemented
26#16863526C2211yescharacteristic, complemented
27#16863523—21no—
28C8×Dic134C2211yescharacteristic, complemented
29C8×Dic134C2211yescharacteristic, complemented
30#16863520—21no—
31C4×#11761616C2211yescharacteristic
32C4×#11761616C2211yescharacteristic
33C4×#11761616C2211yescharacteristic
34C4×#11761616C2211yescharacteristic
35#16863454—21no—
36#16863454—21no—
37#16863449C2211yescharacteristic
38#16863449C2211yescharacteristic
39#16863449C2211yescharacteristic
40#16863148C2211yescharacteristic, complemented
41#16863148C2211yescharacteristic, complemented
42#16863146C2211yescharacteristic, complemented
43#16863146C2211yescharacteristic, complemented
44#16863126C2211yescharacteristic, complemented
45#16863126C2211yescharacteristic, complemented
46#16863126C2211yescharacteristic, complemented
47#16863126C2211yescharacteristic, complemented
48D4⋊(C67⋊C8)C2211yescharacteristic
49D4⋊(C67⋊C8)C2211yescharacteristic
50D4⋊(C67⋊C8)C2211yescharacteristic
51D4⋊(C67⋊C8)C2211yescharacteristic
52#16863118C2211yescharacteristic, complemented
53#16863118C2211yescharacteristic, complemented
54#16863117C2211yescharacteristic, complemented
55C8×C67⋊C8C2211yescharacteristic, complemented
56C2×C4×D4×C67C2211yescharacteristic
57#436×C67C2211yescharacteristic
58#436×C67C2211yescharacteristic
59#435×C67—21no—
60#435×C67—21no—
61C8×D4×C67C2211yescharacteristic
62C8×D4×C67C2211yescharacteristic
63#432×C67C2211yescharacteristic
64#431×C67C2211yescharacteristic
65#407×C67—21no—
66C2×C22⋊C8×C67—21no—
67C2×#11761616C2×C411yescharacteristic
68C2×#11761616C2×C411yescharacteristic
69C2×#11761616C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71#14524043C2×C411yescharacteristic
72#14524043C2×C411yescharacteristic
73#14524035—21no—
74#14524022—21no—
75#14524022C2×C411yescharacteristic
76#14524022C2×C411yescharacteristic
77C4×Dic134—21no—
78C4×Dic134C2311yescharacteristic
79C4×Dic134C2311yescharacteristic
80#14524002—21no—
81#14523974C2×C411yescharacteristic
82#14523974C2×C411yescharacteristic
83#14523974C2×C411yescharacteristic
84#14523974C2×C411yescharacteristic
85C4⋊(C67⋊C8)—21no—
86C4⋊(C67⋊C8)C2×C411yescharacteristic
87C4⋊(C67⋊C8)C2×C411yescharacteristic
88C4⋊(C67⋊C8)C2311yescharacteristic
89C4⋊(C67⋊C8)C2311yescharacteristic
90Dic67⋊C8—21no—
91Dic67⋊C8—21no—
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94#14523966C2×C411yescharacteristic
95#14523966C2×C411yescharacteristic
96#14523966C2×C411yescharacteristic
97#14523966C2×C411yescharacteristic
98#14523964C2×C411yescharacteristic
99#14523964C2×C411yescharacteristic
100#14523964C2×C411yescharacteristic
101#14523964C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103C22⋊(C67⋊C8)—21no—
104C22⋊(C67⋊C8)C2×C411yescharacteristic
105C22⋊(C67⋊C8)C2×C411yescharacteristic
106#14523953—21no—
107#14523953—21no—
108C4×C67⋊C8—21no—
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C8×Dic67—21no—
114C8×Dic67—21no—
115C22×D4×C67C2×C411yescharacteristic
116C23×C4×C67—21no—
117C2×M16×C67—21no—
118C2×M16×C67—21no—
119C22×C8×C67—21no—
120C22×C8×C67—21no—
121C4×D4×C67—21no—
122C4×D4×C67—21no—
123C4×D4×C67C2×C411yescharacteristic
124C4×D4×C67C2×C411yescharacteristic
125C4×D4×C67C2311yescharacteristic
126C4×D4×C67C2311yescharacteristic
127C2×C4⋊C4×C67C2×C411yescharacteristic
128C2×C22⋊C4×C67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67D411yescharacteristic
131C4⋊C8×C67D411yescharacteristic
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67—21no—
137C22⋊C8×C67—21no—
138C22⋊C8×C67—21no—
139C22⋊C8×C67D411yescharacteristic
140C22⋊C8×C67D411yescharacteristic
141#97×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×Dic134C22×C411yescharacteristic
147#11761616—22no—
148#11761616—22no—
149#11761616—41no—
150#11761616—41no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8C22×C411yescharacteristic
156C2×C67⋊C8C22×C411yescharacteristic
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C4⋊Dic67—21no—
160C4⋊Dic67—21no—
161C4⋊Dic67—21no—
162C4⋊Dic67C22×C411yescharacteristic
163C4⋊Dic67C22×C411yescharacteristic
164Dic67⋊C4—21no—
165Dic67⋊C4—21no—
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168C22⋊Dic67—21no—
169C22⋊Dic67—21no—
170C4×Dic67—21no—
171C4×Dic67—21no—
172C24×C67—21no—
173C2×D4×C67—22no—
174C2×D4×C67—21no—
175C2×D4×C67—41no—
176C2×D4×C67C22×C411yescharacteristic
177C2×D4×C67C22×C411yescharacteristic
178C22×C4×C67—21no—
179C22×C4×C67—21no—
180C22×C4×C67—21no—
181C22×C4×C67—21no—
182C22×C4×C67—21no—
183C22×C4×C67—81no—
184C22×C4×C67—81no—
185C22×C4×C67C22×C411yescharacteristic
186C22×C4×C67C2×D411yescharacteristic
187C22×C4×C67#4111yescharacteristic
188M16×C67—81no—
189M16×C67—81no—
190C2×C8×C67—21no—
191C2×C8×C67—21no—
192C2×C8×C67—81no—
193C2×C8×C67—81no—
194C2×C8×C67C2×D411yescharacteristic
195C2×C8×C67C2×D411yescharacteristic
196C2×C8×C67C2×D411yescharacteristic
197C2×C8×C67C2×D411yescharacteristic
198C4⋊C4×C67—21no—
199C4⋊C4×C67C22×C411yescharacteristic
200C4⋊C4×C67C22×C411yescharacteristic
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C22⋊C4×C67—21no—
204C22⋊C4×C67—21no—
205C42×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67—21no—
211C23×C67—42no—
212C23×C67—42no—
213C23×C67—81no—
214C23×C67C4×D411yescharacteristic
215D4×C67—21no—
216D4×C67—21no—
217D4×C67—21no—
218D4×C67—42no—
219D4×C67#13112yes—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—41no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—41no—
230C2×C4×C67—41no—
231C2×C4×C67—81no—
232C2×C4×C67—81no—
233C2×C4×C67—81no—
234C2×C4×C67C4×D411yescharacteristic
235C2×C4×C67C4×D411yescharacteristic
236C2×C4×C67C4×D411yescharacteristic
237C2×C4×C67C23×C411yescharacteristic
238C2×C4×C67C22×D411yescharacteristic
239C2×C4×C67C2×#4111yescharacteristic
240C2×Dic67—21no—
241C2×Dic67—21no—
242C2×Dic67—21no—
243C2×Dic67—21no—
244Dic134—21no—
245Dic134—21no—
246Dic134—21no—
247Dic134—21no—
248Dic134—42no—
249C8×C67—21no—
250C8×C67—21no—
251C8×C67—21no—
252C8×C67—21no—
253C8×C67—41no—
254C8×C67—41no—
255C67⋊C8—21no—
256C67⋊C8—21no—
257C67⋊C8—21no—
258C67⋊C8—21no—
259C67⋊C8—41no—
260C67⋊C8—41no—
261C67⋊C8—41no—
262C67⋊C8—41no—
263C22×C67—21no—
264C22×C67—22no—
265C22×C67—21no—
266C22×C67—21no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67—41no—
270C22×C67—41no—
271C22×C67—41no—
272C22×C67—84no—
273C22×C67C2×C4×D411yescharacteristic
274C4×C67—21no—
275C4×C67—21no—
276C4×C67—21no—
277C4×C67—21no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67C2×#13111yescharacteristic
281C4×C67#56811yescharacteristic
282C4×C67#57711yescharacteristic
283C4×C67#57711yescharacteristic
284Dic67—41no—
285Dic67—41no—
286Dic67—41no—
287Dic67—41no—
288#18028—671nomaximal, Sylow
289C2×C67—41no—
290C2×C67—42no—
291C2×C67—81no—
292C2×C67#293311yescharacteristic
293C2×C67#294911yescharacteristic
294C2×C67#295511yescharacteristic
295#3101—671no—
296#2986—671no—
297#2974D6711yescharacteristic
298#1592—671no—
299#1590—671no—
300#1588—671no—
301#1585—671no—
302#1583—671no—
303#1582—671no—
304SD16⋊C8—671no—
305C8×SD16—671no—
306#1572—671no—
307#1497—671no—
308#1478—671no—
309#1454—671no—
310C67#1802811yescharacteristic, complemented, Sylow
311#533—671no—
312C2×C4×D4C2×D6711yescharacteristic
313#483—671no—
314#483—671no—
315#476—671no—
316#476—671no—
317#461—671no—
318#461—671no—
319#450—671no—
320#450—671no—
321#446—671no—
322#446—671no—
323#446—1341no—
324C8×Q8—671no—
325C8×Q8—671no—
326C8×Q8—1341no—
327C4×SD16—671no—
328C4×SD16—671no—
329C4×SD16—671no—
330C4×SD16—671no—
331#436C2×D6711yescharacteristic
332#436C2×D6711yescharacteristic
333#435—21no—
334#435—21no—
335C8×D4C2×D6711yescharacteristic
336C8×D4C2×D6711yescharacteristic
337#433—1341no—
338#433—1341no—
339#432C2×D6711yescharacteristic
340#431—671no—
341#431—671no—
342#431—671no—
343#431C2×D6711yescharacteristic
344#407—21no—
345C2×C22⋊C8—21no—
346#335—671no—
347#335—671no—
348#334—671no—
349#334—671no—
350Q8⋊C8—671no—
351Q8⋊C8—671no—
352Q8⋊C8—671no—
353Q8⋊C8—671no—
354D4⋊C8—671no—
355D4⋊C8—671no—
356D4⋊C8—671no—
357D4⋊C8—671no—
358#322—671no—
359#322—671no—
360#322—671no—
361C82—671no—
362C22×D4C4×D6711yescharacteristic
363C23×C4—21no—
364C2×SD16—671no—
365C2×SD16—671no—
366C2×SD16—671no—
367C2×SD16—671no—
368C2×M16—21no—
369C2×M16—21no—
370C22×C8—21no—
371C22×C8—21no—
372C4⋊Q8—671no—
373C4⋊Q8—671no—
374#125—1341no—
375C22⋊Q8—671no—
376C22⋊Q8—671no—
377C22⋊Q8—1341no—
378C4×Q8—671no—
379C4×Q8—671no—
380C4×Q8—1341no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4C4×D6711yescharacteristic
384C4×D4C4×D6711yescharacteristic
385C4×D4C22×D6711yescharacteristic
386C4×D4C22×D6711yescharacteristic
387#117—1341no—
388C2×C4⋊C4C4×D6711yescharacteristic
389C2×C22⋊C4—21no—
390C2×C42C22×D6711yescharacteristic
391#106—671no—
392#106—671no—
393#106—671no—
394#106—671no—
395C4⋊C8—671no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—671no—
399C4⋊C8—1341no—
400C4⋊C8—1341no—
401C4⋊C8—1341no—
402C4⋊C8—1341no—
403C4⋊C8—1341no—
404C4⋊C8C22⋊D6711yescharacteristic
405C4⋊C8C22⋊D6711yescharacteristic
406C4⋊C8C22×D6711yescharacteristic
407C4⋊C8C22×D6711yescharacteristic
408#103—671no—
409#103—671no—
410#103—671no—
411#103—671no—
412#102—671no—
413#102—671no—
414#102—671no—
415#102—671no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8—21no—
419C22⋊C8—21no—
420C22⋊C8—21no—
421C22⋊C8—671no—
422C22⋊C8—671no—
423C22⋊C8—1341no—
424C22⋊C8—1341no—
425C22⋊C8C22⋊D6711yescharacteristic
426C22⋊C8C22⋊D6711yescharacteristic
427#97—21no—
428#97—1341no—
429#97—1341no—
430C4×C8—671no—
431C4×C8—671no—
432C4×C8—671no—
433C4×C8—671no—
434C4×C8—1341no—
435C4×C8—1341no—
436C4×C8—1341no—
437C4×C8C22×D6711yescharacteristic
438C4×C8C22×D6711yescharacteristic
439C24—21no—
440C2×Q8—671no—
441C2×Q8—671no—
442C2×Q8—1341no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—41no—
446C2×D4C2×C4×D6711yescharacteristic
447C2×D4C2×C4×D6711yescharacteristic
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4C2×C4×D6711yescharacteristic
456C22×C4C2×C22⋊D6711yescharacteristic
457C22×C4#1176162811yescharacteristic
458SD16—2681no—
459SD16—2681no—
460SD16—1342no—
461SD16—1342no—
462M16—81no—
463M16—81no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—81no—
467C2×C8—81no—
468C2×C8—671no—
469C2×C8—671no—
470C2×C8—671no—
471C2×C8—671no—
472C2×C8—1341no—
473C2×C8—1341no—
474C2×C8—1341no—
475C2×C8—1341no—
476C2×C8C2×C22⋊D6711yescharacteristic
477C2×C8C2×C22⋊D6711yescharacteristic
478C2×C8C2×C22⋊D6711yescharacteristic
479C2×C8C2×C22⋊D6711yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—671no—
482C4⋊C4—671no—
483C4⋊C4—1341no—
484C4⋊C4—1341no—
485C4⋊C4—1341no—
486C4⋊C4—1341no—
487C4⋊C4—1341no—
488C4⋊C4—1341no—
489C4⋊C4—1341no—
490C4⋊C4C2×C4×D6711yescharacteristic
491C4⋊C4C2×C4×D6711yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—1341no—
497C22⋊C4—1341no—
498C42—1341no—
499C42—1341no—
500C42C2×C4×D6711yescharacteristic
501C42C2×C22⋊D6711yescharacteristic
502C42#1176162811yescharacteristic
503C42C23×D6711yescharacteristic
504C23—21no—
505C23—21no—
506C23—42no—
507C23—42no—
508C23—81no—
509C23C4×C22⋊D6711yescharacteristic
510Q8—2682no—
511Q8—1341no—
512Q8—1341no—
513Q8—1341no—
514Q8—1341no—
515D4—21no—
516D4—21no—
517D4—21no—
518D4—42no—
519D4#1452405112yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—1341no—
535C2×C4—1341no—
536C2×C4—1341no—
537C2×C4—1341no—
538C2×C4C4×C22⋊D6711yescharacteristic
539C2×C4C4×C22⋊D6711yescharacteristic
540C2×C4C4×C22⋊D6711yescharacteristic
541C2×C4C22×C4×D6711yescharacteristic
542C2×C4C22×C22⋊D6711yescharacteristic
543C2×C4C2×#1176162811yescharacteristic
544C8—21no—
545C8—21no—
546C8—21no—
547C8—21no—
548C8—41no—
549C8—41no—
550C8—2681no—
551C8—2681no—
552C8—2681no—
553C8—2681no—
554C8—1341no—
555C8—1341no—
556C8—1341no—
557C8—1341no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—84no—
568C22C2×C4×C22⋊D6711yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—41no—
574C4—41no—
575C4—2681no—
576C4—2681no—
577C4—2681no—
578C4—2681no—
579C4C2×#1452405111yescharacteristic
580C4#1686416211yescharacteristic
581C4#1686420711yescharacteristic
582C4#1686420711yescharacteristic
583C2—41no—
584C2—42no—
585C2—81no—
586C2#2256893411yescharacteristic
587C2#2256906611yescharacteristic
588C2#2256911011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2974C2
socleC22×C67C2×C4×D4
prime radicalC22×D4×C67C2×C4
cosocleC42×C67C24
hypercenter#2974D67
hypocenterC67#18028
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #22569310C2semidirect
3. #22559111C2semidirect
4. #22559097C2semidirect
5. #22559085C2semidirect
6. #22559066C2semidirect
7. #22559050C2semidirect
8. #22559045C2semidirect
9. #22559021C2semidirect
10. C8×#11761616C2semidirect
11. #22558990C2semidirect
12. #22558681C2semidirect
14. #22558509C2semidirect
25. #16863526C22semidirect
26. #16863526C22semidirect
28. C8×Dic134C22semidirect
29. C8×Dic134C22semidirect
40. #16863148C22semidirect
41. #16863148C22semidirect
42. #16863146C22semidirect
43. #16863146C22semidirect
44. #16863126C22semidirect
45. #16863126C22semidirect
46. #16863126C22semidirect
47. #16863126C22semidirect
52. #16863118C22semidirect
53. #16863118C22semidirect
54. #16863117C22semidirect
55. C8×C67⋊C8C22semidirect
310. C67#18028semidirect

Maximal subgroups

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

subgroupindex
#225702172
#225693102
#225591112
#225590972
#225590852
#225590662
#225590502
#225590452
#225590212
C8×#117616162
#225589902
#225586812
#225586002
#225585092
C67×#29742
#1802867