Finite Group Databasebeta

← #30827694

Subgroups — #30827694

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#22570809C211yesmaximal, characteristic
2#22569207C211yesmaximal, characteristic
3#22569158C211yesmaximal, characteristic, complemented
4#22559114C211yesmaximal, characteristic, complemented
5#22559103C211yesmaximal, characteristic, complemented
6#22559084C211yesmaximal, characteristic, complemented
7#22559083C211yesmaximal, characteristic, complemented
8#22559056C211yesmaximal, characteristic, complemented
9#22559048C211yesmaximal, characteristic, complemented
10#22559025C211yesmaximal, characteristic, complemented
11#22559006C211yesmaximal, characteristic, complemented
12#22558962C211yesmaximal, characteristic, complemented
13#22558623C211yesmaximal, characteristic
14#22558512C211yesmaximal, characteristic, complemented
15C67×#1485C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863802C411yescharacteristic
18#16863788C411yescharacteristic
19#16863764C411yescharacteristic
20#16863758C411yescharacteristic
21#16863652C411yescharacteristic
22#16863652C411yescharacteristic
23#16863644C411yescharacteristic
24#16863644C411yescharacteristic
25#16863527—21no—
26#16863527—21no—
27#16863527C2211yescharacteristic
28#16863527C2211yescharacteristic
29#16863523C2211yescharacteristic
30Q8×C67⋊C8C2211yescharacteristic
31Q8×C67⋊C8C2211yescharacteristic
32#16863520C2211yescharacteristic
33C4×#11761618C2211yescharacteristic
34C4×#11761618C2211yescharacteristic
35C4×#11761615C2211yescharacteristic
36C4×#11761615C2211yescharacteristic
37#16863473C2211yescharacteristic
38#16863471—21no—
39#16863470—21no—
40#16863461C2211yescharacteristic
41#16863455—21no—
42#16863423—21no—
43C2×C4⋊(C67⋊C8)C2211yescharacteristic
44#16863352—21no—
45#16863350—21no—
46C4×#11761612C2211yescharacteristic
47C4×#11761612C2211yescharacteristic
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863118C2211yescharacteristic
57#16863118C2211yescharacteristic
58#16863117C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60#533×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62C4×M16×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65Q8⋊C8×C67C2211yescharacteristic
66Q8⋊C8×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761612C2×C411yescharacteristic
72C2×#11761612C2×C411yescharacteristic
73C2×#11761612D411yescharacteristic
74C2×#11761612D411yescharacteristic
75C2×#11761611—21no—
76C22×C67⋊C8—21no—
77C2×C8×D67—21no—
78#14524037—21no—
79#14524035C2×C411yescharacteristic
80#14524031C2×C411yescharacteristic
81#14524026—21no—
82#14524022C2×C411yescharacteristic
83#14524017C2×C411yescharacteristic
84C4×Dic134C2311yescharacteristic
85C4×D268C2311yescharacteristic
86C4×C22⋊D67—21no—
87#14524000—21no—
88#14523976C2×C411yescharacteristic
89#14523976C2×C411yescharacteristic
90#14523974C2×C411yescharacteristic
91#14523974C2×C411yescharacteristic
92C4⋊(C67⋊C8)—21no—
93C4⋊(C67⋊C8)—21no—
94C4⋊(C67⋊C8)—21no—
95C4⋊(C67⋊C8)—21no—
96C4⋊(C67⋊C8)—21no—
97C4⋊(C67⋊C8)C2311yescharacteristic
98C4⋊(C67⋊C8)C2311yescharacteristic
99Dic67⋊C8—21no—
100Dic67⋊C8—21no—
101#14523967C2×C411yescharacteristic
102#14523967C2×C411yescharacteristic
103#14523967C2×C411yescharacteristic
104#14523967C2×C411yescharacteristic
105#14523966C2×C411yescharacteristic
106#14523966C2×C411yescharacteristic
107#14523963C2×C411yescharacteristic
108#14523963C2×C411yescharacteristic
109(C2×D67)⋊C8—21no—
110(C2×D67)⋊C8—21no—
111#14523954—21no—
112#14523954—21no—
113#14523954C2×C411yescharacteristic
114#14523954C2×C411yescharacteristic
115#14523954D411yescharacteristic
116#14523954D411yescharacteristic
117#14523953—21no—
118C4×C67⋊C8—21no—
119C4×C67⋊C8—21no—
120C4×C67⋊C8C2311yescharacteristic
121C4×C67⋊C8C2311yescharacteristic
122C4×C67⋊C8C2311yescharacteristic
123C4×C67⋊C8C2311yescharacteristic
124C8×Dic67—21no—
125C2×M16×C67—21no—
126C4⋊Q8×C67C2×C411yescharacteristic
127C4⋊Q8×C67C2×C411yescharacteristic
128#125×C67—21no—
129C22⋊Q8×C67—21no—
130C22⋊Q8×C67C2×C411yescharacteristic
131C22⋊Q8×C67C2×C411yescharacteristic
132C4×Q8×C67—21no—
133C4×Q8×C67C2311yescharacteristic
134C4×Q8×C67C2311yescharacteristic
135#117×C67—21no—
136C2×C42×C67C2311yescharacteristic
137C4⋊C8×C67C2311yescharacteristic
138C4⋊C8×C67C2311yescharacteristic
139C22⋊C8×C67—21no—
140#97×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134C22×C411yescharacteristic
145C2×D268C22×C411yescharacteristic
146C2×C22⋊D67—21no—
147C2×C4×D67—21no—
148#11761618—22no—
149#11761618—41no—
150#11761615—22no—
151#11761615—41no—
152#11761612—22no—
153#11761612—22no—
154#11761612—41no—
155#11761612—41no—
156#11761611—42no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8—42no—
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C2×C67⋊C8C22×C411yescharacteristic
164C2×C67⋊C8C2×D411yescharacteristic
165C2×C67⋊C8C2×D411yescharacteristic
166C2×C67⋊C8C2×D411yescharacteristic
167C2×C67⋊C8C2×D411yescharacteristic
168C8×D67—42no—
169C4⋊Dic67C22×C411yescharacteristic
170C4⋊Dic67C22×C411yescharacteristic
171Dic67⋊C4—21no—
172Dic67⋊C4—21no—
173Dic67⋊C4—21no—
174Dic67⋊C4—21no—
175C22⋊Dic67—21no—
176(C2×D67)⋊C4—21no—
177(C2×D67)⋊C4—21no—
178(C2×D67)⋊C4—21no—
179(C2×D67)⋊C4—21no—
180C4×Dic67—21no—
181C2×Q8×C67—21no—
182C2×Q8×C67C22×C411yescharacteristic
183C2×Q8×C67C22×C411yescharacteristic
184C22×C4×C67C22×C411yescharacteristic
185C22×C4×C67C2×D411yescharacteristic
186C22×C4×C67#4111yescharacteristic
187M16×C67—42no—
188C2×C8×C67—21no—
189C2×C8×C67—21no—
190C2×C8×C67—21no—
191C2×C8×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×C67—21no—
199C4⋊C4×C67C22×C411yescharacteristic
200C4⋊C4×C67C22×C411yescharacteristic
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67C4×D411yescharacteristic
210C22×D67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×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—21no—
224C2×C4×C67—41no—
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C23×C411yescharacteristic
229C2×C4×C67C22×D411yescharacteristic
230C2×C4×C67C2×#4111yescharacteristic
231C22⋊D67—81no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235D268—21no—
236D268—21no—
237C4×D67—41no—
238C4×D67—41no—
239Dic134—21no—
240Dic134—21no—
241C8×C67—22no—
242C8×C67—41no—
243C8×C67—41no—
244C8×C67—41no—
245C67⋊C8—21no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8#13112yes—
254C67⋊C8#13712yes—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67C2×C4×D411yescharacteristic
259C2×D67—41no—
260C2×D67—41no—
261C2×D67—41no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67C2×#13111yescharacteristic
271C4×C67#56811yescharacteristic
272C4×C67C2×#13711yescharacteristic
273C4×C67#57511yescharacteristic
274Dic67—41no—
275Dic67—41no—
276Dic67—41no—
277#18077—671nomaximal, Sylow
278C2×C67—41no—
279C2×C67#293511yescharacteristic
280C2×C67C4×#13711yescharacteristic
281C2×C67#295611yescharacteristic
282D67—81no—
283#3172—671no—
284#2969—671no—
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#1485D6711yescharacteristic, complemented
296#1483—671no—
297#1455—671no—
298C67#1807711yescharacteristic, complemented, Sylow
299#533C2×D6711yescharacteristic
300#529—671no—
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—671no—
312#446—1341no—
313#446—1341no—
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8—671no—
317C4×Q16—671no—
318C4×Q16—671no—
319C4×SD16—671no—
320C4×SD16—671no—
321#436—671no—
322#436—1341no—
323#435—1341no—
324C8×D4—671no—
325#433—1341no—
326#431C2×D6711yescharacteristic
327#424—1341no—
328C2×C4⋊C8—671no—
329#405—1341no—
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⋊C8C2×D6711yescharacteristic
341Q8⋊C8C2×D6711yescharacteristic
342Q8⋊C8C2×D6711yescharacteristic
343Q8⋊C8C2×D6711yescharacteristic
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—671no—
358C2×M16—671no—
359C2×M16—1341no—
360C22×C8—1341no—
361C22×C8—1341no—
362C4⋊Q8C4×D6711yescharacteristic
363C4⋊Q8C4×D6711yescharacteristic
364#126—1341no—
365#125—21no—
366#125—671no—
367#124—671no—
368#123—1341no—
369C22⋊Q8—21no—
370C22⋊Q8—671no—
371C22⋊Q8C4×D6711yescharacteristic
372C22⋊Q8C4×D6711yescharacteristic
373#121—671no—
374C4×Q8—21no—
375C4×Q8—671no—
376C4×Q8C22×D6711yescharacteristic
377C4×Q8C22×D6711yescharacteristic
378C4×D4—671no—
379C4×D4—1341no—
380#117—21no—
381#117—1341no—
382C2×C42C22×D6711yescharacteristic
383#107—671no—
384#107—671no—
385#106—671no—
386#106—671no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
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—1341no—
408C22⋊C8—1341no—
409#97—21no—
410#97—671no—
411#97—671no—
412#97—671no—
413#97—671no—
414#97—1341no—
415#97—1341no—
416#97—1341no—
417C4×C8—21no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8C22×D6711yescharacteristic
426C4×C8C22×D6711yescharacteristic
427C2×Q8—21no—
428C2×Q8—671no—
429C2×Q8C2×C4×D6711yescharacteristic
430C2×Q8C2×C4×D6711yescharacteristic
431C2×D4—671no—
432C2×D4—1341no—
433C22×C4—1341no—
434C22×C4C2×C4×D6711yescharacteristic
435C22×C4D4×D6711yescharacteristic
436C22×C4#1176162711yescharacteristic
437Q16—2681no—
438Q16—1342no—
439SD16—2681no—
440SD16—1342no—
441M16—42no—
442M16—2682no—
443M16—2681no—
444M16—2681no—
445M16—1342no—
446M16—1342no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—2682no—
452C2×C8—2682no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—671no—
458C2×C8—671no—
459C2×C8—671no—
460C2×C8—671no—
461C2×C8—1341no—
462C2×C8—1341no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—671no—
471C4⋊C4—671no—
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—1341no—
481C22⋊C4—1341no—
482C22⋊C4—1341no—
483C22⋊C4—1341no—
484C22⋊C4—1341no—
485C42—21no—
486C42—21no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42D4×D6711yescharacteristic
490C42#1176162711yescharacteristic
491C42C23×D6711yescharacteristic
492C23—1341no—
493C23(C22⋊D67)⋊C411yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—1341no—
500Q8—1341no—
501D4—5361no—
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—2681no—
514C2×C4—2681no—
515C2×C4—1341no—
516C2×C4—1341no—
517C2×C4—1341no—
518C2×C4(C22⋊D67)⋊C411yescharacteristic
519C2×C4(C22⋊D67)⋊C411yescharacteristic
520C2×C4(C22⋊D67)⋊C411yescharacteristic
521C2×C4C22×C4×D6711yescharacteristic
522C2×C4C2×D4×D6711yescharacteristic
523C2×C4C2×#1176162711yescharacteristic
524C8—22no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—2681no—
529C8—2681no—
530C8—672no—
531C8—672no—
532C8—1341no—
533C8—1341no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8—1341no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—2681no—
542C22—2681no—
543C22—2681no—
544C22C2×(C22⋊D67)⋊C411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—2681no—
554C4—2681no—
555C4—2681no—
556C4#131×D6711yescharacteristic
557C4#1686416111yescharacteristic
558C4#137×D6711yescharacteristic
559C4#1686419511yescharacteristic
560C2—41no—
561C2—5361no—
562C2#2256894911yescharacteristic
563C2#2256902411yescharacteristic
564C2#2256911511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1485C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenter#1485D67
hypocenterC67#18077
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
3. #22569158C2semidirect
4. #22559114C2semidirect
5. #22559103C2semidirect
6. #22559084C2semidirect
7. #22559083C2semidirect
8. #22559056C2semidirect
9. #22559048C2semidirect
10. #22559025C2semidirect
11. #22559006C2semidirect
12. #22558962C2semidirect
14. #22558512C2semidirect
15. C67×#1485C2semidirect
295. #1485D67semidirect
298. C67#18077semidirect

Maximal subgroups

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

subgroupindex
#225708092
#225692072
#225691582
#225591142
#225591032
#225590842
#225590832
#225590562
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