Finite Group Databasebeta

← #30826968

Subgroups — #30826968

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570274C211yesmaximal, characteristic
2#22569227C211yesmaximal, characteristic
3#22559109C211yesmaximal, characteristic, complemented
4#22559099C211yesmaximal, characteristic, complemented
5#22559087C211yesmaximal, characteristic, complemented
6#22559078C211yesmaximal, characteristic, complemented
7#22559057C211yesmaximal, characteristic, complemented
8#22559052C211yesmaximal, characteristic, complemented
9#22559026C211yesmaximal, characteristic, complemented
10C8×#11761618C211yesmaximal, characteristic, complemented
11#22558992C211yesmaximal, characteristic, complemented
12#22558682C211yesmaximal, characteristic, complemented
13#22558594C211yesmaximal, characteristic
14#22558506C211yesmaximal, characteristic, complemented
15C67×#2962C211yesmaximal, characteristic, complemented
16C4×#11761628C2211yescharacteristic
17#16863734C411yescharacteristic
18#16863734C411yescharacteristic
19#16863678C411yescharacteristic
20#16863673C411yescharacteristic
21#16863603C411yescharacteristic
22#16863598C411yescharacteristic
23#16863556C411yescharacteristic
24#16863556C411yescharacteristic
25C8×Dic134C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C4×#11761618C2211yescharacteristic
28C4×#11761618C2211yescharacteristic
29C4×#11761615C2211yescharacteristic
30C4×#11761615C2211yescharacteristic
31#16863475C2211yescharacteristic
32#16863475C2211yescharacteristic
33#16863472—21no—
34#16863472—21no—
35#16863453C2211yescharacteristic
36#16863451—21no—
37#16863449C2211yescharacteristic
38#16863449C2211yescharacteristic
39#16863348—21no—
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46Q8⋊(C67⋊C8)C2211yescharacteristic
47Q8⋊(C67⋊C8)C2211yescharacteristic
48#16863126C2211yescharacteristic
49#16863126C2211yescharacteristic
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863117C2211yescharacteristic
55C8×C67⋊C8C2211yescharacteristic
56#533×C67C2211yescharacteristic
57#446×C67—21no—
58#446×C67—21no—
59#446×C67C2211yescharacteristic
60#446×C67C2211yescharacteristic
61C8×Q8×C67C2211yescharacteristic
62C8×Q8×C67C2211yescharacteristic
63#424×C67—21no—
64C2×C4⋊C8×C67C2211yescharacteristic
65#405×C67—21no—
66C4×M16×C67C2211yescharacteristic
67C2×#11761628C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761618C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761615C2×C411yescharacteristic
72C2×#11761611—21no—
73C2×#11761611—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#14523967C2×C411yescharacteristic
97#14523967C2×C411yescharacteristic
98#14523966C2×C411yescharacteristic
99#14523966C2×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—
108#14523954—21no—
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C8×Dic67—21no—
114C8×Dic67—21no—
115C2×M16×C67D411yescharacteristic
116C2×M16×C67D411yescharacteristic
117C22×C8×C67—21no—
118C4⋊Q8×C67C2×C411yescharacteristic
119C4⋊Q8×C67C2×C411yescharacteristic
120#125×C67—21no—
121C22⋊Q8×C67—21no—
122C22⋊Q8×C67C2×C411yescharacteristic
123C22⋊Q8×C67C2×C411yescharacteristic
124C4×Q8×C67—21no—
125C4×Q8×C67C2311yescharacteristic
126C4×Q8×C67C2311yescharacteristic
127#117×C67—21no—
128C2×C42×C67C2311yescharacteristic
129C4⋊C8×C67—21no—
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67—21no—
133C4⋊C8×C67—21no—
134C4⋊C8×C67C2311yescharacteristic
135C4⋊C8×C67C2311yescharacteristic
136#97×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
144#11761628—22no—
145#11761628—41no—
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#11761618—22no—
154#11761615—41no—
155#11761615—41no—
156#11761611—42no—
157#11761611—42no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—21no—
160C2×C67⋊C8—21no—
161C2×C67⋊C8—21no—
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C2×C67⋊C8C22×C411yescharacteristic
165C2×C67⋊C8C22×C411yescharacteristic
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×C67—21no—
177C2×Q8×C67C22×C411yescharacteristic
178C2×Q8×C67C22×C411yescharacteristic
179C22×C4×C67C22×C411yescharacteristic
180C22×C4×C67C2×D411yescharacteristic
181C22×C4×C67#4111yescharacteristic
182M16×C67—22no—
183M16×C67—41no—
184C2×C8×C67—21no—
185C2×C8×C67—21no—
186C2×C8×C67—42no—
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×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67C22×C411yescharacteristic
199C4⋊C4×C67C22×C411yescharacteristic
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67C22×C411yescharacteristic
205C42×C67C2×D411yescharacteristic
206C42×C67#4111yescharacteristic
207C42×C67C2411yescharacteristic
208C23×C67C4×D411yescharacteristic
209C22×D67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—81no—
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—21no—
222C2×C4×C67—21no—
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
230C22⋊D67—42no—
231C2×Dic67—21no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234D268—21no—
235D268—21no—
236C4×D67—42no—
237C4×D67—41no—
238C4×D67—41no—
239C4×D67—81no—
240Dic134—21no—
241Dic134#13112yes—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—41no—
246C8×C67—41no—
247C8×C67#13712yes—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—21no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C67⋊C8—41no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—21no—
259C22×C67C2×C4×D411yescharacteristic
260C2×D67—41no—
261C2×D67—41no—
262C2×D67—41no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67C2×#13111yescharacteristic
272C4×C67#56811yescharacteristic
273C4×C67C2×#13711yescharacteristic
274C4×C67#57511yescharacteristic
275Dic67—22no—
276Dic67—41no—
277Dic67—41no—
278#18013—671nomaximal, Sylow
279C2×C67—41no—
280C2×C67#293311yescharacteristic
281C2×C67C4×#13711yescharacteristic
282C2×C67#295711yescharacteristic
283D67—81no—
284#3107—671no—
285#2971—671no—
286#2962D6711yescharacteristic, complemented
287#1592—671no—
288#1590—671no—
289#1588—671no—
290#1587—671no—
291#1584—671no—
292#1583—671no—
293#1579—671no—
294C8×Q16—671no—
295#1573—671no—
296#1497—671no—
297#1477—671no—
298#1453—671no—
299C67#1801311yescharacteristic, complemented, Sylow
300#533C2×D6711yescharacteristic
301C4×#41—671no—
302#485—671no—
303#485—671no—
304#475—671no—
305#474—671no—
306#462—671no—
307#461—671no—
308#452—671no—
309#452—671no—
310#446—21no—
311#446—21no—
312#446C2×D6711yescharacteristic
313#446C2×D6711yescharacteristic
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8C2×D6711yescharacteristic
317C8×Q8C2×D6711yescharacteristic
318C4×Q16—671no—
319C4×Q16—671no—
320C4×SD16—671no—
321C4×SD16—671no—
322#436—671no—
323#436—671no—
324#436—1341no—
325#436—1341no—
326#432—671no—
327#432—1341no—
328#431—671no—
329#431—671no—
330#424—21no—
331C2×C4⋊C8C2×D6711yescharacteristic
332#405—21no—
333C4×M16—1341no—
334C4×M16C2×D6711yescharacteristic
335#335—671no—
336#335—671no—
337#334—671no—
338#334—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344Q8⋊C8—671no—
345D4⋊C8—671no—
346D4⋊C8—671no—
347#322—671no—
348#322—671no—
349#322—671no—
350C82—671no—
351C2×#41—671no—
352C2×Q16—671no—
353C2×Q16—671no—
354C2×SD16—671no—
355C2×SD16—671no—
356C2×M16—1341no—
357C2×M16—1341no—
358C2×M16C22⋊D6711yescharacteristic
359C2×M16C22⋊D6711yescharacteristic
360C22×C8—21no—
361C4⋊Q8C4×D6711yescharacteristic
362C4⋊Q8C4×D6711yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C4×D6711yescharacteristic
366C22⋊Q8C4×D6711yescharacteristic
367C4×Q8—21no—
368C4×Q8—671no—
369C4×Q8—671no—
370C4×Q8C22×D6711yescharacteristic
371C4×Q8C22×D6711yescharacteristic
372C4×D4—671no—
373C4×D4—671no—
374C4×D4—1341no—
375C4×D4—1341no—
376#117—21no—
377#117—671no—
378#117—1341no—
379C2×C42—1341no—
380C2×C42C22×D6711yescharacteristic
381#107—671no—
382#107—671no—
383#106—671no—
384#106—671no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—671no—
391C4⋊C8—671no—
392C4⋊C8—671no—
393C4⋊C8—671no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8—1341no—
397C4⋊C8C22×D6711yescharacteristic
398C4⋊C8C22×D6711yescharacteristic
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#103—671no—
404#103—671no—
405#102—671no—
406#102—671no—
407C22⋊C8—671no—
408C22⋊C8—671no—
409C22⋊C8—1341no—
410C22⋊C8—1341no—
411C22⋊C8—1341no—
412C22⋊C8—1341no—
413#97—21no—
414#97—21no—
415#97—1341no—
416#97C22⋊D6711yescharacteristic
417#97C22⋊D6711yescharacteristic
418C4×C8—21no—
419C4×C8—21no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—671no—
424C4×C8—1341no—
425C4×C8—1341no—
426C4×C8C22×D6711yescharacteristic
427C4×C8C22×D6711yescharacteristic
428#41—2681no—
429#41—1342no—
430C2×Q8—21no—
431C2×Q8—671no—
432C2×Q8C2×C4×D6711yescharacteristic
433C2×Q8C2×C4×D6711yescharacteristic
434C2×D4—671no—
435C2×D4—1341no—
436C22×C4—1341no—
437C22×C4—1341no—
438C22×C4—1341no—
439C22×C4C2×C4×D6711yescharacteristic
440C22×C4C2×C22⋊D6711yescharacteristic
441C22×C4#1176162811yescharacteristic
442Q16—1342no—
443Q16—1342no—
444SD16—2681no—
445SD16—2681no—
446M16—22no—
447M16—41no—
448M16—2682no—
449M16—2682no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—42no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—1341no—
458C2×C8—1341no—
459C2×C8—1341no—
460C2×C8—1341no—
461C2×C8C2×C22⋊D6711yescharacteristic
462C2×C8C2×C22⋊D6711yescharacteristic
463C2×C8C2×C22⋊D6711yescharacteristic
464C2×C8C2×C22⋊D6711yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—671no—
473C4⋊C4—671no—
474C4⋊C4—1341no—
475C4⋊C4—1341no—
476C4⋊C4C2×C4×D6711yescharacteristic
477C4⋊C4C2×C4×D6711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1341no—
481C22⋊C4—1341no—
482C22⋊C4—1341no—
483C42—21no—
484C42—21no—
485C42—1341no—
486C42—1341no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42C2×C22⋊D6711yescharacteristic
490C42#1176162811yescharacteristic
491C42C23×D6711yescharacteristic
492C23—1341no—
493C23C4×C22⋊D6711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—672no—
500Q8—1341no—
501D4—2682no—
502D4—1341no—
503D4—1341no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—2682no—
514C2×C4—2681no—
515C2×C4—2681no—
516C2×C4—5361no—
517C2×C4—1341no—
518C2×C4—1341no—
519C2×C4—1341no—
520C2×C4C4×C22⋊D6711yescharacteristic
521C2×C4C4×C22⋊D6711yescharacteristic
522C2×C4C4×C22⋊D6711yescharacteristic
523C2×C4C22×C4×D6711yescharacteristic
524C2×C4C22×C22⋊D6711yescharacteristic
525C2×C4C2×#1176162811yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—41no—
530C8—41no—
531C8—2681no—
532C8—2681no—
533C8—2681no—
534C8—2681no—
535C8—1341no—
536C8—1341no—
537C8—1341no—
538C8—1341no—
539C8#1452407412yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2681no—
544C22—2681no—
545C22—2681no—
546C22C2×C4×C22⋊D6711yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—2681no—
556C4—2681no—
557C4—1342no—
558C4#131×D6711yescharacteristic
559C4#1686416111yescharacteristic
560C4C2×#1452407411yescharacteristic
561C4#1686419411yescharacteristic
562C2—41no—
563C2—5361no—
564C2#2256893111yescharacteristic
565C2C4×#1452407411yescharacteristic
566C2#2256912411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2962C2
socleC22×C67C2×C4×D4
prime radicalC2×#11761628C2×C4
cosocleC42×C67C24
hypercenter#2962D67
hypocenterC67#18013
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #22559109C2semidirect
4. #22559099C2semidirect
5. #22559087C2semidirect
6. #22559078C2semidirect
7. #22559057C2semidirect
8. #22559052C2semidirect
9. #22559026C2semidirect
10. C8×#11761618C2semidirect
11. #22558992C2semidirect
12. #22558682C2semidirect
14. #22558506C2semidirect
15. C67×#2962C2semidirect
286. #2962D67semidirect
299. C67#18013semidirect

Maximal subgroups

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

subgroupindex
#225702742
#225692272
#225591092
#225590992
#225590872
#225590782
#225590572
#225590522
#225590262
C8×#117616182
#225589922
#225586822
#225585942
#225585062
C67×#29622
#1801367