Finite Group Databasebeta

← #30825784

Subgroups — #30825784

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570221C211yesmaximal, characteristic
2C8×#11761628C211yesmaximal, characteristic, complemented
3#22559085C211yesmaximal, characteristic, complemented
4#22559078C211yesmaximal, characteristic, complemented
5#22559071C211yesmaximal, characteristic, complemented
6#22559064C211yesmaximal, characteristic, complemented
7#22559031C211yesmaximal, characteristic, complemented
8#22559026C211yesmaximal, characteristic, complemented
9#22559021C211yesmaximal, characteristic, complemented
10#22559019C211yesmaximal, characteristic, complemented
11#22558990C211yesmaximal, characteristic, complemented
12#22558641C211yesmaximal, characteristic, complemented
13#22558617C211yesmaximal, characteristic, complemented
14#22558501C211yesmaximal, characteristic, complemented
15C67×#2966C211yesmaximal, characteristic, complemented
16C4×#11761628C2211yescharacteristic
17#16863734C411yescharacteristic
18#16863717C411yescharacteristic
19#16863700C411yescharacteristic
20#16863693C411yescharacteristic
21#16863615C411yescharacteristic
22#16863608C411yescharacteristic
23#16863556C411yescharacteristic
24#16863539C411yescharacteristic
25C8×Dic134C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C4×#11761618C2211yescharacteristic
28C4×#11761615C2211yescharacteristic
29C4×#11761616C2211yescharacteristic
30C4×#11761613C2211yescharacteristic
31C8×D268C2211yescharacteristic
32C8×D268C2211yescharacteristic
33C8×C22⋊D67—21no—
34C8×C22⋊D67—21no—
35#16863449C2211yescharacteristic
36#16863449C2211yescharacteristic
37#16863449C2211yescharacteristic
38#16863447—21no—
39C4×C8×D67—21no—
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46#16863126C2211yescharacteristic
47#16863126C2211yescharacteristic
48D4⋊(C67⋊C8)C2211yescharacteristic
49D4⋊(C67⋊C8)C2211yescharacteristic
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863118C2211yescharacteristic
55#16863118C2211yescharacteristic
56#529×C67C2211yescharacteristic
57#446×C67C2211yescharacteristic
58C8×Q8×C67C2211yescharacteristic
59#436×C67C2211yescharacteristic
60#435×C67—21no—
61C8×D4×C67—21no—
62C8×D4×C67C2211yescharacteristic
63#433×C67—21no—
64#432×C67C2211yescharacteristic
65#405×C67—21no—
66C2×C4×C8×C67C2211yescharacteristic
67C2×#11761628C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761616C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761613C2×C411yescharacteristic
72C2×C8×D67—21no—
73C2×C8×D67—21no—
74C4×Dic134C2×C411yescharacteristic
75C4×Dic134C2311yescharacteristic
76C4×D268C2×C411yescharacteristic
77C4×D268C2311yescharacteristic
78C4×C22⋊D67—21no—
79C4×C22⋊D67—21no—
80#14524002C2×C411yescharacteristic
81#14524000—21no—
82C42×D67—21no—
83#14523976C2×C411yescharacteristic
84#14523976C2×C411yescharacteristic
85#14523974C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87C4⋊(C67⋊C8)—21no—
88C4⋊(C67⋊C8)C2×C411yescharacteristic
89C4⋊(C67⋊C8)C2×C411yescharacteristic
90C4⋊(C67⋊C8)C2311yescharacteristic
91C4⋊(C67⋊C8)C2311yescharacteristic
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94#14523967C2×C411yescharacteristic
95#14523967C2×C411yescharacteristic
96#14523966C2×C411yescharacteristic
97#14523966C2×C411yescharacteristic
98#14523964C2×C411yescharacteristic
99#14523964C2×C411yescharacteristic
100#14523963C2×C411yescharacteristic
101#14523963C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103C22⋊(C67⋊C8)—21no—
104C22⋊(C67⋊C8)C2×C411yescharacteristic
105C22⋊(C67⋊C8)C2×C411yescharacteristic
106(C2×D67)⋊C8—21no—
107(C2×D67)⋊C8—21no—
108C4×C67⋊C8—21no—
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C8×Dic67—21no—
114C8×Dic67—21no—
115C2×M16×C67—21no—
116C22×C8×C67—21no—
117C22×C8×C67D411yescharacteristic
118C22×C8×C67D411yescharacteristic
119#126×C67—21no—
120#125×C67C2×C411yescharacteristic
121#124×C67C2×C411yescharacteristic
122#123×C67—21no—
123C22⋊Q8×C67C2×C411yescharacteristic
124#121×C67C2×C411yescharacteristic
125C4×Q8×C67C2311yescharacteristic
126C4×D4×C67—21no—
127C4×D4×C67C2311yescharacteristic
128#117×C67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67—21no—
137#97×C67—21no—
138#97×C67—21no—
139C4×C8×C67—21no—
140C4×C8×C67D411yescharacteristic
141C4×C8×C67D411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#11761628—22no—
145#11761628—22no—
146C2×Dic134C22×C411yescharacteristic
147C2×D268C22×C411yescharacteristic
148C2×C22⋊D67—21no—
149C2×C4×D67—21no—
150C2×C4×D67—21no—
151C2×C4×D67—21no—
152#11761618—22no—
153#11761616—22no—
154#11761615—22no—
155#11761613—22no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C8×D67—42no—
165C8×D67—42no—
166C4⋊Dic67C22×C411yescharacteristic
167C4⋊Dic67C22×C411yescharacteristic
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170C22⋊Dic67—21no—
171(C2×D67)⋊C4—21no—
172(C2×D67)⋊C4—21no—
173C4×Dic67—21no—
174C4×Dic67—21no—
175C4×Dic67—21no—
176C2×Q8×C67C22×C411yescharacteristic
177C2×D4×C67—21no—
178C2×D4×C67C22×C411yescharacteristic
179C22×C4×C67—21no—
180C22×C4×C67C22×C411yescharacteristic
181C22×C4×C67C2×D411yescharacteristic
182C22×C4×C67#4111yescharacteristic
183M16×C67—42no—
184C2×C8×C67—21no—
185C2×C8×C67—21no—
186C2×C8×C67—41no—
187C2×C8×C67—41no—
188C2×C8×C67—42no—
189C2×C8×C67C2×D411yescharacteristic
190C2×C8×C67C2×D411yescharacteristic
191C2×C8×C67C2×D411yescharacteristic
192C2×C8×C67C2×D411yescharacteristic
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67C22×C411yescharacteristic
198C4⋊C4×C67C22×C411yescharacteristic
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C22⋊C4×C67—21no—
204C42×C67—21no—
205C42×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67C4×D411yescharacteristic
211C22×D67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214D4×C67—21no—
215D4×C67—21no—
216D4×C67—42no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—21no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67—41no—
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C4×D411yescharacteristic
229C2×C4×C67C4×D411yescharacteristic
230C2×C4×C67C23×C411yescharacteristic
231C2×C4×C67C22×D411yescharacteristic
232C2×C4×C67C2×#4111yescharacteristic
233C22⋊D67—24no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236C2×Dic67—21no—
237D268—21no—
238D268#13112yes—
239C4×D67—24no—
240C4×D67—42no—
241C4×D67—41no—
242C4×D67—41no—
243Dic134—21no—
244Dic134#13112yes—
245C8×C67—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—41no—
250C8×C67—41no—
251C67⋊C8—21no—
252C67⋊C8—21no—
253C67⋊C8—21no—
254C67⋊C8—21no—
255C67⋊C8—41no—
256C67⋊C8—41no—
257C67⋊C8—41no—
258C67⋊C8—41no—
259C22×C67—21no—
260C22×C67—21no—
261C22×C67—21no—
262C22×C67—41no—
263C22×C67—41no—
264C22×C67—41no—
265C22×C67C2×C4×D411yescharacteristic
266C2×D67—22no—
267C2×D67—41no—
268C2×D67—41no—
269C4×C67—21no—
270C4×C67—21no—
271C4×C67—21no—
272C4×C67—21no—
273C4×C67—41no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67C2×#13111yescharacteristic
277C4×C67C2×#13111yescharacteristic
278C4×C67#57511yescharacteristic
279C4×C67#57511yescharacteristic
280Dic67—22no—
281Dic67—41no—
282Dic67—41no—
283#17909—671nomaximal, Sylow
284C2×C67—41no—
285C2×C67—81no—
286C2×C67#292911yescharacteristic
287C2×C67#294311yescharacteristic
288C2×C67#295511yescharacteristic
289D67—42no—
290#3102—671no—
291#2966D6711yescharacteristic, complemented
292C8×#41—671no—
293#1588—671no—
294#1587—671no—
295#1586—671no—
296#1585—671no—
297#1580—671no—
298#1579—671no—
299SD16⋊C8—671no—
300SD16⋊C8—671no—
301#1572—671no—
302#1488—671no—
303#1482—671no—
304#1452—671no—
305C67#1790911yescharacteristic, complemented, Sylow
306#529C2×D6711yescharacteristic
307C4×#41—671no—
308#485—671no—
309#482—671no—
310#479—671no—
311#478—671no—
312#464—671no—
313#463—671no—
314#452—671no—
315#449—671no—
316#446C2×D6711yescharacteristic
317C8×Q8—671no—
318C8×Q8—671no—
319C8×Q8C2×D6711yescharacteristic
320C4×Q16—671no—
321C4×SD16—671no—
322C4×SD16—671no—
323C4×D8—671no—
324#436C2×D6711yescharacteristic
325#435—21no—
326C8×D4—21no—
327C8×D4—671no—
328C8×D4—671no—
329C8×D4—1341no—
330C8×D4—1341no—
331C8×D4C2×D6711yescharacteristic
332#433—21no—
333#432C2×D6711yescharacteristic
334#431—671no—
335#431—671no—
336#431—671no—
337#431—1341no—
338#405—21no—
339C2×C4×C8—1341no—
340C2×C4×C8C2×D6711yescharacteristic
341#335—671no—
342#335—671no—
343#334—671no—
344#334—671no—
345Q8⋊C8—671no—
346Q8⋊C8—671no—
347Q8⋊C8—671no—
348Q8⋊C8—671no—
349D4⋊C8—671no—
350D4⋊C8—671no—
351D4⋊C8—671no—
352D4⋊C8—671no—
353#322—671no—
354#322—671no—
355#322—671no—
356#322—671no—
357C2×#41—671no—
358C2×Q16—671no—
359C2×SD16—671no—
360C2×SD16—671no—
361C2×D8—671no—
362C2×M16—21no—
363C22×C8—21no—
364C22×C8—1341no—
365C22×C8—1341no—
366C22×C8C22⋊D6711yescharacteristic
367C22×C8C22⋊D6711yescharacteristic
368#126—21no—
369#125C4×D6711yescharacteristic
370#124C4×D6711yescharacteristic
371#123—21no—
372C22⋊Q8C4×D6711yescharacteristic
373#121C4×D6711yescharacteristic
374C4×Q8—671no—
375C4×Q8—671no—
376C4×Q8C22×D6711yescharacteristic
377C4×D4—21no—
378C4×D4—671no—
379C4×D4—671no—
380C4×D4—1341no—
381C4×D4—1341no—
382C4×D4C22×D6711yescharacteristic
383#117—21no—
384#117—671no—
385#117—1341no—
386C2×C42—1341no—
387C2×C42C22×D6711yescharacteristic
388#107—671no—
389#107—671no—
390#106—671no—
391#106—671no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—671no—
395C4⋊C8—671no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—1341no—
399C4⋊C8—1341no—
400C4⋊C8—1341no—
401C4⋊C8C22×D6711yescharacteristic
402C4⋊C8C22×D6711yescharacteristic
403#103—671no—
404#103—671no—
405#103—671no—
406#103—671no—
407#102—671no—
408#102—671no—
409#102—671no—
410#102—671no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—671no—
415C22⋊C8—671no—
416C22⋊C8—1341no—
417C22⋊C8—1341no—
418C22⋊C8—1341no—
419C22⋊C8—1341no—
420#97—21no—
421#97—21no—
422C4×C8—21no—
423C4×C8—671no—
424C4×C8—671no—
425C4×C8—671no—
426C4×C8—671no—
427C4×C8—1341no—
428C4×C8—1341no—
429C4×C8—1341no—
430C4×C8C22⋊D6711yescharacteristic
431C4×C8C22⋊D6711yescharacteristic
432C4×C8C22×D6711yescharacteristic
433C4×C8C22×D6711yescharacteristic
434#41—1342no—
435#41—1342no—
436C2×Q8—671no—
437C2×Q8C2×C4×D6711yescharacteristic
438C2×D4—21no—
439C2×D4—671no—
440C2×D4—1341no—
441C2×D4C2×C4×D6711yescharacteristic
442C22×C4—21no—
443C22×C4—1341no—
444C22×C4—1341no—
445C22×C4—1341no—
446C22×C4C2×C4×D6711yescharacteristic
447C22×C4C2×C22⋊D6711yescharacteristic
448C22×C4#1176162811yescharacteristic
449Q16—1342no—
450SD16—1342no—
451SD16—1342no—
452D8—1342no—
453M16—42no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—2682no—
460C2×C8—2682no—
461C2×C8—671no—
462C2×C8—671no—
463C2×C8—671no—
464C2×C8—671no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8C2×C22⋊D6711yescharacteristic
470C2×C8C2×C22⋊D6711yescharacteristic
471C2×C8C2×C22⋊D6711yescharacteristic
472C2×C8C2×C22⋊D6711yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—671no—
478C4⋊C4—671no—
479C4⋊C4—1341no—
480C4⋊C4—1341no—
481C4⋊C4C2×C4×D6711yescharacteristic
482C4⋊C4C2×C4×D6711yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1341no—
489C22⋊C4—1341no—
490C22⋊C4—1341no—
491C42—21no—
492C42—1341no—
493C42—1341no—
494C42—1341no—
495C42C2×C4×D6711yescharacteristic
496C42C2×C22⋊D6711yescharacteristic
497C42#1176162811yescharacteristic
498C42C23×D6711yescharacteristic
499C23—21no—
500C23—1341no—
501C23C4×C22⋊D6711yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—672no—
505Q8—1341no—
506D4—21no—
507D4—21no—
508D4—42no—
509D4—672no—
510D4—1344no—
511D4—1341no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—2682no—
523C2×C4—2681no—
524C2×C4—2681no—
525C2×C4—1344no—
526C2×C4—1341no—
527C2×C4—1341no—
528C2×C4—1341no—
529C2×C4C4×C22⋊D6711yescharacteristic
530C2×C4C4×C22⋊D6711yescharacteristic
531C2×C4C4×C22⋊D6711yescharacteristic
532C2×C4C22×C4×D6711yescharacteristic
533C2×C4C22×C22⋊D6711yescharacteristic
534C2×C4C2×#1176162811yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—2681no—
542C8—2681no—
543C8—2681no—
544C8—2681no—
545C8—1341no—
546C8—1341no—
547C8—1341no—
548C8—1341no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—2681no—
556C22—2681no—
557C22—1342no—
558C22C2×C4×C22⋊D6711yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—2681no—
567C4—2681no—
568C4—1342no—
569C4#131×D6711yescharacteristic
570C4#131×D6711yescharacteristic
571C4#1686419411yescharacteristic
572C4#1686419411yescharacteristic
573C2—41no—
574C2—81no—
575C2—2682no—
576C2#2256890111yescharacteristic
577C2#2256900711yescharacteristic
578C2#2256911011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2966C2
socleC22×C67C2×C4×D4
prime radical#16863539C4
cosocleC42×C67C24
hypercenter#2966D67
hypocenterC67#17909
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. C8×#11761628C2semidirect
3. #22559085C2semidirect
4. #22559078C2semidirect
5. #22559071C2semidirect
6. #22559064C2semidirect
7. #22559031C2semidirect
8. #22559026C2semidirect
9. #22559021C2semidirect
10. #22559019C2semidirect
11. #22558990C2semidirect
12. #22558641C2semidirect
13. #22558617C2semidirect
14. #22558501C2semidirect
15. C67×#2966C2semidirect
291. #2966D67semidirect
305. C67#17909semidirect

Maximal subgroups

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

subgroupindex
#225702212
C8×#117616282
#225590852
#225590782
#225590712
#225590642
#225590312
#225590262
#225590212
#225590192
#225589902
#225586412
#225586172
#225585012
C67×#29662
#1790967