Finite Group Databasebeta

← #30827696

Subgroups — #30827696

Order 17152.

588 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570816C211yesmaximal, characteristic
2#22569156C211yesmaximal, characteristic, complemented
3#22559111C211yesmaximal, characteristic, complemented
4#22559104C211yesmaximal, characteristic, complemented
5#22559085C211yesmaximal, characteristic, complemented
6#22559080C211yesmaximal, characteristic, complemented
7#22559057C211yesmaximal, characteristic, complemented
8#22559045C211yesmaximal, characteristic, complemented
9#22559026C211yesmaximal, characteristic, complemented
10C8×#11761616C211yesmaximal, characteristic, complemented
11#22558962C211yesmaximal, characteristic, complemented
12#22558629C211yesmaximal, characteristic, complemented
13#22558622C211yesmaximal, characteristic
14#22558512C211yesmaximal, characteristic, complemented
15C67×#2969C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863806C411yescharacteristic
18#16863791C411yescharacteristic
19#16863766C411yescharacteristic
20#16863755C411yescharacteristic
21#16863652C411yescharacteristic
22#16863652C411yescharacteristic
23#16863640C411yescharacteristic
24#16863640C411yescharacteristic
25#16863526C2211yescharacteristic
26#16863526C2211yescharacteristic
27#16863524—21no—
28#16863524—21no—
29C8×Dic134C2211yescharacteristic
30C8×Dic134C2211yescharacteristic
31C4×#11761618C2211yescharacteristic
32C4×#11761618C2211yescharacteristic
33C4×#11761616C2211yescharacteristic
34C4×#11761616C2211yescharacteristic
35#16863424—21no—
36C2×C4⋊(C67⋊C8)C2211yescharacteristic
37#16863354—21no—
38C4×#11761612C2211yescharacteristic
39C4×#11761612C2211yescharacteristic
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46#16863126C2211yescharacteristic
47#16863126C2211yescharacteristic
48#16863126C2211yescharacteristic
49#16863126C2211yescharacteristic
50D4⋊(C67⋊C8)C2211yescharacteristic
51D4⋊(C67⋊C8)C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863117C2211yescharacteristic
55C8×C67⋊C8C2211yescharacteristic
56#529×C67C2211yescharacteristic
57#446×C67C2211yescharacteristic
58C8×Q8×C67C2211yescharacteristic
59#436×C67—21no—
60#436×C67C2211yescharacteristic
61#435×C67—21no—
62C8×D4×C67C2211yescharacteristic
63#433×C67—21no—
64#431×C67C2211yescharacteristic
65#405×C67—21no—
66C4×M16×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761616C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71C2×#11761612—21no—
72C2×#11761612C2×C411yescharacteristic
73C2×#11761612C2×C411yescharacteristic
74C22×C67⋊C8—21no—
75#14524043C2×C411yescharacteristic
76#14524043C2×C411yescharacteristic
77#14524035—21no—
78#14524022—21no—
79#14524022C2×C411yescharacteristic
80#14524022C2×C411yescharacteristic
81C4×Dic134—21no—
82C4×Dic134C2311yescharacteristic
83C4×Dic134C2311yescharacteristic
84#14524002—21no—
85#14523976C2×C411yescharacteristic
86#14523976C2×C411yescharacteristic
87#14523974C2×C411yescharacteristic
88#14523974C2×C411yescharacteristic
89C4⋊(C67⋊C8)—21no—
90C4⋊(C67⋊C8)C2311yescharacteristic
91C4⋊(C67⋊C8)C2311yescharacteristic
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94Dic67⋊C8—21no—
95Dic67⋊C8—21no—
96#14523967C2×C411yescharacteristic
97#14523967C2×C411yescharacteristic
98#14523966C2×C411yescharacteristic
99#14523966C2×C411yescharacteristic
100#14523966C2×C411yescharacteristic
101#14523966C2×C411yescharacteristic
102#14523964C2×C411yescharacteristic
103#14523964C2×C411yescharacteristic
104#14523954—21no—
105#14523954C2×C411yescharacteristic
106#14523954C2×C411yescharacteristic
107#14523953—21no—
108#14523953—21no—
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C8×Dic67—21no—
114C8×Dic67—21no—
115C2×M16×C67—21no—
116C2×M16×C67D411yescharacteristic
117C2×M16×C67D411yescharacteristic
118C22×C8×C67—21no—
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×C67D411yescharacteristic
139#97×C67D411yescharacteristic
140C4×C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×Dic134C22×C411yescharacteristic
147#11761618—22no—
148#11761618—41no—
149#11761616—22no—
150#11761616—41no—
151#11761612—22no—
152#11761612—41no—
153#11761612—42no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—42no—
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C4⋊Dic67—21no—
164C4⋊Dic67—21no—
165C4⋊Dic67—21no—
166C4⋊Dic67C22×C411yescharacteristic
167C4⋊Dic67C22×C411yescharacteristic
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171Dic67⋊C4—21no—
172C22⋊Dic67—21no—
173C22⋊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—22no—
184M16×C67—41no—
185M16×C67—42no—
186C2×C8×C67—21no—
187C2×C8×C67—21no—
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
211Q8×C67—21no—
212Q8×C67—21no—
213D4×C67—21no—
214D4×C67—21no—
215D4×C67—81no—
216C2×C4×C67—21no—
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—41no—
224C2×C4×C67—41no—
225C2×C4×C67—41no—
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C4×D411yescharacteristic
229C2×C4×C67C23×C411yescharacteristic
230C2×C4×C67C22×D411yescharacteristic
231C2×C4×C67C2×#4111yescharacteristic
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236Dic134—21no—
237Dic134—21no—
238Dic134—21no—
239Dic134—21no—
240Dic134—81no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—41no—
245C8×C67—41no—
246C8×C67#13712yes—
247C67⋊C8—22no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8#13112yes—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67C2×C4×D411yescharacteristic
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67C2×#13111yescharacteristic
270C4×C67#56811yescharacteristic
271C4×C67C2×#13711yescharacteristic
272C4×C67#57511yescharacteristic
273Dic67—41no—
274Dic67—41no—
275Dic67—41no—
276Dic67—41no—
277#18077—671nomaximal, Sylow
278C2×C67—41no—
279C2×C67—81no—
280C2×C67#293511yescharacteristic
281C2×C67C4×#13711yescharacteristic
282C2×C67#295611yescharacteristic
283#3172—671no—
284#2969D6711yescharacteristic
285#2962—671no—
286#1592—671no—
287#1591—671no—
288#1588—671no—
289#1587—671no—
290#1584—671no—
291#1582—671no—
292#1579—671no—
293C8×SD16—671no—
294#1565—671no—
295#1485—671no—
296#1483—671no—
297#1455—671no—
298C67#1807711yescharacteristic, complemented, Sylow
299#533—671no—
300#529C2×D6711yescharacteristic
301#501—671no—
302#497—671no—
303#491—671no—
304#489—671no—
305#470—671no—
306#470—671no—
307#468—671no—
308#468—671no—
309#446—671no—
310#446—671no—
311#446—1341no—
312#446—1341no—
313#446C2×D6711yescharacteristic
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8C2×D6711yescharacteristic
317C4×Q16—671no—
318C4×Q16—671no—
319C4×SD16—671no—
320C4×SD16—671no—
321#436—21no—
322#436C2×D6711yescharacteristic
323#435—21no—
324C8×D4C2×D6711yescharacteristic
325#433—21no—
326#431C2×D6711yescharacteristic
327#424—1341no—
328C2×C4⋊C8—671no—
329#405—21no—
330#405—1341no—
331C4×M16—671no—
332C4×M16—671no—
333C4×M16C2×D6711yescharacteristic
334#335—671no—
335#335—671no—
336#334—671no—
337#334—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344D4⋊C8—671no—
345D4⋊C8—671no—
346#322—671no—
347#322—671no—
348#322—671no—
349C82—671no—
350C2×Q16—671no—
351C2×Q16—671no—
352C2×SD16—671no—
353C2×SD16—671no—
354C2×M16—21no—
355C2×M16—671no—
356C2×M16—671no—
357C2×M16—1341no—
358C2×M16C22⋊D6711yescharacteristic
359C2×M16C22⋊D6711yescharacteristic
360C22×C8—21no—
361C22×C8—1341no—
362C4⋊Q8—671no—
363C4⋊Q8—671no—
364#126—21no—
365#125—1341no—
366#125C4×D6711yescharacteristic
367#124C4×D6711yescharacteristic
368#123—21no—
369C22⋊Q8—671no—
370C22⋊Q8—671no—
371C22⋊Q8—1341no—
372C22⋊Q8C4×D6711yescharacteristic
373#121C4×D6711yescharacteristic
374C4×Q8—671no—
375C4×Q8—671no—
376C4×Q8—1341no—
377C4×Q8C22×D6711yescharacteristic
378C4×D4—21no—
379C4×D4C22×D6711yescharacteristic
380#117—21no—
381#117—1341no—
382C2×C42C22×D6711yescharacteristic
383#107—671no—
384#107—671no—
385#106—671no—
386#106—671no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—671no—
390C4⋊C8—671no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8C22×D6711yescharacteristic
397C4⋊C8C22×D6711yescharacteristic
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#103—671no—
404#102—671no—
405#102—671no—
406C22⋊C8—21no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409#97—21no—
410#97—671no—
411#97—671no—
412#97—1341no—
413#97—1341no—
414#97—1341no—
415#97C22⋊D6711yescharacteristic
416#97C22⋊D6711yescharacteristic
417C4×C8—21no—
418C4×C8—21no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8C22×D6711yescharacteristic
426C4×C8C22×D6711yescharacteristic
427C2×Q8—671no—
428C2×Q8—671no—
429C2×Q8—1341no—
430C2×Q8C2×C4×D6711yescharacteristic
431C2×D4—21no—
432C2×D4C2×C4×D6711yescharacteristic
433C22×C4—21no—
434C22×C4C2×C4×D6711yescharacteristic
435C22×C4C2×C22⋊D6711yescharacteristic
436C22×C4#1176162811yescharacteristic
437Q16—2681no—
438Q16—1342no—
439SD16—2681no—
440SD16—1342no—
441M16—22no—
442M16—41no—
443M16—42no—
444M16—2682no—
445M16—2681no—
446M16—1342no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—42no—
450C2×C8—2682no—
451C2×C8—671no—
452C2×C8—671no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—1341no—
456C2×C8—1341no—
457C2×C8—1341no—
458C2×C8—1341no—
459C2×C8C2×C22⋊D6711yescharacteristic
460C2×C8C2×C22⋊D6711yescharacteristic
461C2×C8C2×C22⋊D6711yescharacteristic
462C2×C8C2×C22⋊D6711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—671no—
468C4⋊C4—671no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C4⋊C4—1341no—
475C4⋊C4—1341no—
476C4⋊C4C2×C4×D6711yescharacteristic
477C4⋊C4C2×C4×D6711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—1341no—
484C22⋊C4—1341no—
485C42—21no—
486C42—1341no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42C2×C22⋊D6711yescharacteristic
490C42#1176162811yescharacteristic
491C42C23×D6711yescharacteristic
492C23—21no—
493C23C4×C22⋊D6711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—5361no—
497Q8—1341no—
498Q8—1341no—
499Q8—1341no—
500Q8—1341no—
501D4—21no—
502D4—21no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—1341no—
515C2×C4—1341no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4C4×C22⋊D6711yescharacteristic
519C2×C4C4×C22⋊D6711yescharacteristic
520C2×C4C4×C22⋊D6711yescharacteristic
521C2×C4C22×C4×D6711yescharacteristic
522C2×C4C22×C22⋊D6711yescharacteristic
523C2×C4C2×#1176162811yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—41no—
528C8—41no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8—672no—
533C8—1342no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8#1452407312yes—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22C2×C4×C22⋊D6711yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—2681no—
553C4—2681no—
554C4—2681no—
555C4—2681no—
556C4#131×D6711yescharacteristic
557C4#1686416111yescharacteristic
558C4C2×#1452407311yescharacteristic
559C4#1686419411yescharacteristic
560C2—41no—
561C2—81no—
562C2#2256895111yescharacteristic
563C2C4×#1452407311yescharacteristic
564C2#2256911611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2969C2
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
cosocleC42×C67C24
hypercenter#2969D67
hypocenterC67#18077
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #22569156C2semidirect
3. #22559111C2semidirect
4. #22559104C2semidirect
5. #22559085C2semidirect
6. #22559080C2semidirect
7. #22559057C2semidirect
8. #22559045C2semidirect
9. #22559026C2semidirect
10. C8×#11761616C2semidirect
11. #22558962C2semidirect
12. #22558629C2semidirect
14. #22558512C2semidirect
298. C67#18077semidirect

Maximal subgroups

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

subgroupindex
#225708162
#225691562
#225591112
#225591042
#225590852
#225590802
#225590572
#225590452
#225590262
C8×#117616162
#225589622
#225586292
#225586222
#225585122
C67×#29692
#1807767