Finite Group Databasebeta

← #30826472

Subgroups — #30826472

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570816C211yesmaximal, characteristic
2#22569303C211yesmaximal, characteristic, complemented
3#22559116C211yesmaximal, characteristic, complemented
4#22559111C211yesmaximal, characteristic, complemented
5#22559104C211yesmaximal, characteristic, complemented
6#22559097C211yesmaximal, characteristic, complemented
7#22559026C211yesmaximal, characteristic, complemented
8#22559021C211yesmaximal, characteristic, complemented
9C8×#11761618C211yesmaximal, characteristic, complemented
10C8×#11761616C211yesmaximal, characteristic, complemented
11#22558974C211yesmaximal, characteristic, complemented
12#22558669C211yesmaximal, characteristic
13#22558663C211yesmaximal, characteristic, complemented
14#22558490C211yesmaximal, characteristic, complemented
15C67×#2993C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863806C411yescharacteristic
18#16863791C411yescharacteristic
19#16863766C411yescharacteristic
20#16863755C411yescharacteristic
21#16863652C411yescharacteristic
22#16863652C411yescharacteristic
23#16863640C411yescharacteristic
24#16863640C411yescharacteristic
25#16863526C2211yescharacteristic
26#16863526C2211yescharacteristic
27#16863525—21no—
28#16863525—21no—
29C8×Dic134C2211yescharacteristic
30C8×Dic134C2211yescharacteristic
31C4×#11761618C2211yescharacteristic
32C4×#11761618C2211yescharacteristic
33C4×#11761616C2211yescharacteristic
34C4×#11761616C2211yescharacteristic
35#16863454—21no—
36#16863454—21no—
37#16863453C2211yescharacteristic
38C4×#11761612C2211yescharacteristic
39C2×C4×C67⋊C8C2211yescharacteristic
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
54C8×C67⋊C8C2211yescharacteristic
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#424×C67—21no—
66#423×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761616C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71C2×#11761612C2×C411yescharacteristic
72C2×#11761612C2×C411yescharacteristic
73C22×C67⋊C8—21no—
74#14524043C2×C411yescharacteristic
75#14524043C2×C411yescharacteristic
76#14524035—21no—
77#14524022—21no—
78#14524022C2×C411yescharacteristic
79#14524022C2×C411yescharacteristic
80C4×Dic134—21no—
81C4×Dic134C2311yescharacteristic
82C4×Dic134C2311yescharacteristic
83#14524002—21no—
84#14523976C2×C411yescharacteristic
85#14523976C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87#14523974C2×C411yescharacteristic
88C4⋊(C67⋊C8)C2311yescharacteristic
89C4⋊(C67⋊C8)C2311yescharacteristic
90Dic67⋊C8—21no—
91Dic67⋊C8—21no—
92Dic67⋊C8—21no—
93Dic67⋊C8—21no—
94#14523967C2×C411yescharacteristic
95#14523967C2×C411yescharacteristic
96#14523966C2×C411yescharacteristic
97#14523966C2×C411yescharacteristic
98#14523966C2×C411yescharacteristic
99#14523966C2×C411yescharacteristic
100#14523964C2×C411yescharacteristic
101#14523964C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103#14523954—21no—
104#14523954C2×C411yescharacteristic
105#14523954C2×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—
115C2×M16×C67—21no—
116C2×M16×C67—21no—
117C22×C8×C67—21no—
118#126×C67—21no—
119#125×C67C2×C411yescharacteristic
120#124×C67C2×C411yescharacteristic
121#123×C67—21no—
122C22⋊Q8×C67C2×C411yescharacteristic
123#121×C67C2×C411yescharacteristic
124C4×Q8×C67C2311yescharacteristic
125C4×D4×C67—21no—
126C4×D4×C67C2311yescharacteristic
127#117×C67—21no—
128C2×C42×C67C2311yescharacteristic
129C4⋊C8×C67—21no—
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67D411yescharacteristic
133C4⋊C8×C67D411yescharacteristic
134C4⋊C8×C67C2311yescharacteristic
135C4⋊C8×C67C2311yescharacteristic
136C22⋊C8×C67—21no—
137C22⋊C8×C67—21no—
138C22⋊C8×C67D411yescharacteristic
139C22⋊C8×C67D411yescharacteristic
140#97×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—22no—
149#11761616—22no—
150#11761616—22no—
151#11761612—22no—
152#11761612—22no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—42no—
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C4⋊Dic67—21no—
163C4⋊Dic67—21no—
164C4⋊Dic67—21no—
165C4⋊Dic67C22×C411yescharacteristic
166C4⋊Dic67C22×C411yescharacteristic
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171C22⋊Dic67—21no—
172C22⋊Dic67—21no—
173C4×Dic67—21no—
174C4×Dic67—21no—
175C2×Q8×C67C22×C411yescharacteristic
176C2×D4×C67—21no—
177C2×D4×C67C22×C411yescharacteristic
178C22×C4×C67—21no—
179C22×C4×C67C22×C411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67#4111yescharacteristic
182M16×C67—81no—
183M16×C67—81no—
184C2×C8×C67—21no—
185C2×C8×C67—21no—
186C2×C8×C67—81no—
187C2×C8×C67C2×D411yescharacteristic
188C2×C8×C67C2×D411yescharacteristic
189C2×C8×C67C2×D411yescharacteristic
190C2×C8×C67C2×D411yescharacteristic
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67C22×C411yescharacteristic
196C4⋊C4×C67C22×C411yescharacteristic
197C22⋊C4×C67—21no—
198C22⋊C4×C67—21no—
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C42×C67—21no—
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67—21no—
208C23×C67C4×D411yescharacteristic
209Q8×C67—21no—
210Q8×C67—21no—
211D4×C67—21no—
212D4×C67—21no—
213D4×C67—81no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67—41no—
224C2×C4×C67C4×D411yescharacteristic
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C23×C411yescharacteristic
228C2×C4×C67C22×D411yescharacteristic
229C2×C4×C67C2×#4111yescharacteristic
230C2×Dic67—21no—
231C2×Dic67—21no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234Dic134—21no—
235Dic134—21no—
236Dic134—21no—
237Dic134—21no—
238Dic134—81no—
239C8×C67—21no—
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—41no—
244C8×C67—41no—
245C67⋊C8—22no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—41no—
249C67⋊C8—41no—
250C67⋊C8—41no—
251C67⋊C8#13112yes—
252C67⋊C8#13112yes—
253C22×C67—21no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67—41no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67C2×C4×D411yescharacteristic
260C4×C67—21no—
261C4×C67—21no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67C2×#13111yescharacteristic
268C4×C67C2×#13111yescharacteristic
269C4×C67#57711yescharacteristic
270C4×C67#57811yescharacteristic
271Dic67—41no—
272Dic67—41no—
273Dic67—41no—
274Dic67—41no—
275#17969—671nomaximal, Sylow
276C2×C67—41no—
277C2×C67—81no—
278C2×C67#293111yescharacteristic
279C2×C67#294711yescharacteristic
280C2×C67#295611yescharacteristic
281#3172—671no—
282#2993D6711yescharacteristic
283#2985—671no—
284#1593—671no—
285#1592—671no—
286#1591—671no—
287#1590—671no—
288#1579—671no—
289SD16⋊C8—671no—
290C8×Q16—671no—
291C8×SD16—671no—
292#1569—671no—
293#1494—671no—
294#1493—671no—
295C8×M16—671no—
296C67#1796911yescharacteristic, complemented, Sylow
297#533—671no—
298#529C2×D6711yescharacteristic
299#501—671no—
300#497—671no—
301#491—671no—
302#489—671no—
303#470—671no—
304#470—671no—
305#468—671no—
306#468—671no—
307#446—671no—
308#446—671no—
309#446—1341no—
310#446—1341no—
311#446C2×D6711yescharacteristic
312C8×Q8—671no—
313C8×Q8—671no—
314C8×Q8C2×D6711yescharacteristic
315C4×Q16—671no—
316C4×Q16—671no—
317C4×SD16—671no—
318C4×SD16—671no—
319#436—21no—
320#436C2×D6711yescharacteristic
321#435—21no—
322C8×D4C2×D6711yescharacteristic
323#433—21no—
324#433—1341no—
325#433—1341no—
326#432—671no—
327#431C2×D6711yescharacteristic
328#424—21no—
329#423C2×D6711yescharacteristic
330C4×M16—671no—
331C2×C4×C8—671no—
332#335—671no—
333#335—671no—
334#334—671no—
335#334—671no—
336Q8⋊C8—671no—
337Q8⋊C8—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342D4⋊C8—671no—
343D4⋊C8—671no—
344#322—671no—
345#322—671no—
346C82—671no—
347C82—671no—
348C2×Q16—671no—
349C2×Q16—671no—
350C2×SD16—671no—
351C2×SD16—671no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—671no—
355C2×M16—671no—
356C22×C8—21no—
357C22×C8—1341no—
358C4⋊Q8—671no—
359C4⋊Q8—671no—
360#126—21no—
361#125—1341no—
362#125C4×D6711yescharacteristic
363#124C4×D6711yescharacteristic
364#123—21no—
365C22⋊Q8—671no—
366C22⋊Q8—671no—
367C22⋊Q8—1341no—
368C22⋊Q8C4×D6711yescharacteristic
369#121C4×D6711yescharacteristic
370C4×Q8—671no—
371C4×Q8—671no—
372C4×Q8—1341no—
373C4×Q8C22×D6711yescharacteristic
374C4×D4—21no—
375C4×D4C22×D6711yescharacteristic
376#117—21no—
377#117—1341no—
378C2×C42C22×D6711yescharacteristic
379#107—671no—
380#107—671no—
381#106—671no—
382#106—671no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8C22⋊D6711yescharacteristic
393C4⋊C8C22⋊D6711yescharacteristic
394C4⋊C8C22×D6711yescharacteristic
395C4⋊C8C22×D6711yescharacteristic
396#103—671no—
397#103—671no—
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#102—671no—
403#102—671no—
404C22⋊C8—21no—
405C22⋊C8—21no—
406C22⋊C8—1341no—
407C22⋊C8C22⋊D6711yescharacteristic
408C22⋊C8C22⋊D6711yescharacteristic
409#97—21no—
410#97—671no—
411#97—671no—
412#97—1341no—
413#97—1341no—
414#97—1341no—
415C4×C8—21no—
416C4×C8—671no—
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—1341no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8C22×D6711yescharacteristic
424C4×C8C22×D6711yescharacteristic
425C2×Q8—671no—
426C2×Q8—671no—
427C2×Q8—1341no—
428C2×Q8C2×C4×D6711yescharacteristic
429C2×D4—21no—
430C2×D4C2×C4×D6711yescharacteristic
431C22×C4—21no—
432C22×C4C2×C4×D6711yescharacteristic
433C22×C4C2×C22⋊D6711yescharacteristic
434C22×C4#1176162811yescharacteristic
435Q16—1342no—
436Q16—1342no—
437SD16—1342no—
438SD16—1342no—
439M16—81no—
440M16—81no—
441M16—1342no—
442M16—1342no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—81no—
446C2×C8—2682no—
447C2×C8—671no—
448C2×C8—671no—
449C2×C8—671no—
450C2×C8—671no—
451C2×C8—1341no—
452C2×C8—1341no—
453C2×C8—1341no—
454C2×C8—1341no—
455C2×C8C2×C22⋊D6711yescharacteristic
456C2×C8C2×C22⋊D6711yescharacteristic
457C2×C8C2×C22⋊D6711yescharacteristic
458C2×C8C2×C22⋊D6711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—671no—
464C4⋊C4—671no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1341no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4C2×C4×D6711yescharacteristic
473C4⋊C4C2×C4×D6711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—1341no—
480C22⋊C4—1341no—
481C42—21no—
482C42—1341no—
483C42—1341no—
484C42C2×C4×D6711yescharacteristic
485C42C2×C22⋊D6711yescharacteristic
486C42#1176162811yescharacteristic
487C42C23×D6711yescharacteristic
488C23—21no—
489C23C4×C22⋊D6711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—5361no—
493Q8—1341no—
494Q8—1341no—
495Q8—1341no—
496Q8—1341no—
497D4—21no—
498D4—21no—
499D4—81no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—1341no—
511C2×C4—1341no—
512C2×C4—1341no—
513C2×C4—1341no—
514C2×C4C4×C22⋊D6711yescharacteristic
515C2×C4C4×C22⋊D6711yescharacteristic
516C2×C4C4×C22⋊D6711yescharacteristic
517C2×C4C22×C4×D6711yescharacteristic
518C2×C4C22×C22⋊D6711yescharacteristic
519C2×C4C2×#1176162811yescharacteristic
520C8—21no—
521C8—21no—
522C8—21no—
523C8—21no—
524C8—41no—
525C8—41no—
526C8—2681no—
527C8—2681no—
528C8—2681no—
529C8—672no—
530C8—672no—
531C8—1342no—
532C8—1341no—
533C8—1341no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22C2×C4×C22⋊D6711yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—2681no—
549C4—2681no—
550C4—2681no—
551C4—2681no—
552C4#131×D6711yescharacteristic
553C4#131×D6711yescharacteristic
554C4#1686420711yescharacteristic
555C4#1686421111yescharacteristic
556C2—41no—
557C2—81no—
558C2#2256891911yescharacteristic
559C2#2256904811yescharacteristic
560C2#2256911611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2993C2
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
cosocleC42×C67C24
hypercenter#2993D67
hypocenterC67#17969
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
2. #22569303C2semidirect
3. #22559116C2semidirect
4. #22559111C2semidirect
5. #22559104C2semidirect
6. #22559097C2semidirect
7. #22559026C2semidirect
8. #22559021C2semidirect
9. C8×#11761618C2semidirect
10. C8×#11761616C2semidirect
11. #22558974C2semidirect
13. #22558663C2semidirect
14. #22558490C2semidirect
296. C67#17969semidirect

Maximal subgroups

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

subgroupindex
#225708162
#225693032
#225591162
#225591112
#225591042
#225590972
#225590262
#225590212
C8×#117616182
C8×#117616162
#225589742
#225586692
#225586632
#225584902
C67×#29932
#1796967