Finite Group Databasebeta

← #30827018

Subgroups — #30827018

Order 17152.

600 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#22570289C211yesmaximal, characteristic
2#22569365C211yesmaximal, characteristic
3#22569298C211yesmaximal, characteristic, complemented
4#22559103C211yesmaximal, characteristic, complemented
5#22559096C211yesmaximal, characteristic, complemented
6#22559084C211yesmaximal, characteristic, complemented
7#22559083C211yesmaximal, characteristic, complemented
8#22559077C211yesmaximal, characteristic, complemented
9#22559069C211yesmaximal, characteristic, complemented
10#22559013C211yesmaximal, characteristic, complemented
11#22559006C211yesmaximal, characteristic, complemented
12#22558992C211yesmaximal, characteristic, complemented
13#22558670C211yesmaximal, characteristic
14#22558492C211yesmaximal, characteristic, complemented
15C2×Q8⋊C8×C67C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863726C411yescharacteristic
18#16863709C411yescharacteristic
19#16863703C411yescharacteristic
20#16863703C411yescharacteristic
21#16863618C411yescharacteristic
22#16863618C411yescharacteristic
23#16863552C411yescharacteristic
24#16863535C411yescharacteristic
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#16863467—21no—
39#16863461C2211yescharacteristic
40C8×C22⋊D67—21no—
41#16863456—21no—
42#16863453—21no—
43#16863453C2211yescharacteristic
44#16863449—21no—
45#16863449C2211yescharacteristic
46#16863449C2211yescharacteristic
47#16863421—21no—
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863117C2211yescharacteristic
57#16863117C2211yescharacteristic
58C8×C67⋊C8C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60C2×C4×Q8×C67C2211yescharacteristic
61C2×C4⋊C8×C67C2211yescharacteristic
62C2×C4×C8×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×#11761611—21no—
72C2×C8×D67—21no—
73#14524037—21no—
74#14524035C2×C411yescharacteristic
75#14524031C2×C411yescharacteristic
76#14524026—21no—
77#14524022C2×C411yescharacteristic
78#14524017C2×C411yescharacteristic
79C4×Dic134C2311yescharacteristic
80C4×D268C2311yescharacteristic
81C4×C22⋊D67—21no—
82#14524000—21no—
83#14523976C2×C411yescharacteristic
84#14523976C2×C411yescharacteristic
85#14523974C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87C4⋊(C67⋊C8)—21no—
88C4⋊(C67⋊C8)—21no—
89C4⋊(C67⋊C8)—21no—
90C4⋊(C67⋊C8)—21no—
91C4⋊(C67⋊C8)C2×C411yescharacteristic
92C4⋊(C67⋊C8)C2×C411yescharacteristic
93C4⋊(C67⋊C8)D411yescharacteristic
94C4⋊(C67⋊C8)D411yescharacteristic
95C4⋊(C67⋊C8)C2311yescharacteristic
96C4⋊(C67⋊C8)C2311yescharacteristic
97Dic67⋊C8—21no—
98Dic67⋊C8—21no—
99#14523967C2×C411yescharacteristic
100#14523967C2×C411yescharacteristic
101#14523967C2×C411yescharacteristic
102#14523967C2×C411yescharacteristic
103#14523966C2×C411yescharacteristic
104#14523966C2×C411yescharacteristic
105#14523963C2×C411yescharacteristic
106#14523963C2×C411yescharacteristic
107C22⋊(C67⋊C8)—21no—
108C22⋊(C67⋊C8)C2×C411yescharacteristic
109C22⋊(C67⋊C8)C2×C411yescharacteristic
110C22⋊(C67⋊C8)D411yescharacteristic
111C22⋊(C67⋊C8)D411yescharacteristic
112(C2×D67)⋊C8—21no—
113(C2×D67)⋊C8—21no—
114#14523954—21no—
115#14523954—21no—
116#14523954—21no—
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—
125C22×Q8×C67C2×C411yescharacteristic
126C22×C8×C67—21no—
127C22×C8×C67—21no—
128C4×Q8×C67—21no—
129C4×Q8×C67—21no—
130C4×Q8×C67C2×C411yescharacteristic
131C4×Q8×C67C2×C411yescharacteristic
132C4×Q8×C67C2311yescharacteristic
133C4×Q8×C67C2311yescharacteristic
134C2×C4⋊C4×C67—21no—
135C2×C4⋊C4×C67C2×C411yescharacteristic
136C2×C42×C67—21no—
137C2×C42×C67C2311yescharacteristic
138C4⋊C8×C67—21no—
139C4⋊C8×C67C2311yescharacteristic
140C4⋊C8×C67C2311yescharacteristic
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#11761611—81no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8C22×C411yescharacteristic
156C2×C67⋊C8C22×C411yescharacteristic
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C2×D411yescharacteristic
160C2×C67⋊C8C2×D411yescharacteristic
161C2×C67⋊C8C2×D411yescharacteristic
162C2×C67⋊C8C2×D411yescharacteristic
163C8×D67—81no—
164C4⋊Dic67C22×C411yescharacteristic
165C4⋊Dic67C22×C411yescharacteristic
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170C22⋊Dic67—21no—
171(C2×D67)⋊C4—21no—
172(C2×D67)⋊C4—21no—
173(C2×D67)⋊C4—21no—
174(C2×D67)⋊C4—21no—
175C4×Dic67—21no—
176C2×Q8×C67—21no—
177C2×Q8×C67—22no—
178C2×Q8×C67—41no—
179C2×Q8×C67C22×C411yescharacteristic
180C2×Q8×C67C22×C411yescharacteristic
181C22×C4×C67—21no—
182C22×C4×C67—21no—
183C22×C4×C67C22×C411yescharacteristic
184C22×C4×C67C2×D411yescharacteristic
185C22×C4×C67#4111yescharacteristic
186C2×C8×C67—24no—
187C2×C8×C67—21no—
188C2×C8×C67—21no—
189C2×C8×C67—21no—
190C2×C8×C67—21no—
191C2×C8×C67—81no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67C22×C411yescharacteristic
198C4⋊C4×C67C22×C411yescharacteristic
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67C4×D411yescharacteristic
208C22×D67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—42no—
213Q8×C67#13112yes—
214C2×C4×C67—21no—
215C2×C4×C67—22no—
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—42no—
224C2×C4×C67—42no—
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—42no—
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—22no—
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—21no—
252C67⋊C8—21no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67#44312yes—
258C22×C67C2×C4×D411yescharacteristic
259C2×D67—41no—
260C2×D67—41no—
261C2×D67—41no—
262C4×C67—21no—
263C4×C67—22no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67C2×#13111yescharacteristic
271C4×C67#56811yescharacteristic
272C4×C67#57711yescharacteristic
273C4×C67#57811yescharacteristic
274Dic67—41no—
275Dic67—41no—
276Dic67—41no—
277#18017—671nomaximal, Sylow
278C2×C67—22no—
279C2×C67#293311yescharacteristic
280C2×C67#294711yescharacteristic
281C2×C67C2×#44311yescharacteristic
282D67—81no—
283#3108—671no—
284#2994—671no—
285#2984—671no—
286#1591—671no—
287#1590—671no—
288#1588—671no—
289#1587—671no—
290#1587—671no—
291#1585—671no—
292C8×Q16—671no—
293C8×SD16—671no—
294#1573—671no—
295#1494—671no—
296C2×Q8⋊C8D6711yescharacteristic, complemented
297#1450—671no—
298C67#1801711yescharacteristic, complemented, Sylow
299#529—671no—
300C2×C4×Q8C2×D6711yescharacteristic
301#483—671no—
302#480—671no—
303#479—671no—
304#479—671no—
305#464—671no—
306#464—671no—
307#451—671no—
308#448—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#435—1341no—
323C8×D4—671no—
324C8×D4—1341no—
325#433—1341no—
326#432—671no—
327#432—1341no—
328#431—671no—
329#431—671no—
330#431—1341no—
331#424—1341no—
332C2×C4⋊C8C2×D6711yescharacteristic
333C2×C4×C8C2×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—
348C82—671no—
349C82—671no—
350C22×Q8C4×D6711yescharacteristic
351C2×Q16—671no—
352C2×Q16—671no—
353C2×SD16—671no—
354C2×SD16—671no—
355C2×M16—1341no—
356C22×C8—21no—
357C22×C8—21no—
358C22×C8—1341no—
359#126—1341no—
360#125—671no—
361#124—671no—
362#123—1341no—
363C22⋊Q8—671no—
364#121—671no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—671no—
368C4×Q8C4×D6711yescharacteristic
369C4×Q8C4×D6711yescharacteristic
370C4×Q8C22×D6711yescharacteristic
371C4×Q8C22×D6711yescharacteristic
372C4×D4—671no—
373C4×D4—1341no—
374#117—1341no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C4×D6711yescharacteristic
377C2×C42—21no—
378C2×C42C22×D6711yescharacteristic
379#107—671no—
380#107—671no—
381#106—671no—
382#106—671no—
383C4⋊C8—21no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—671no—
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—671no—
407C22⋊C8—671no—
408C22⋊C8—671no—
409C22⋊C8—671no—
410C22⋊C8—1341no—
411C22⋊C8—1341no—
412C22⋊C8—1341no—
413#97—1341no—
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—22no—
429C2×Q8—41no—
430C2×Q8—671no—
431C2×Q8C2×C4×D6711yescharacteristic
432C2×Q8C2×C4×D6711yescharacteristic
433C2×D4—671no—
434C2×D4—1341no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—1341no—
438C22×C4C2×C4×D6711yescharacteristic
439C22×C4D4×D6711yescharacteristic
440C22×C4#1176162711yescharacteristic
441Q16—2681no—
442Q16—1342no—
443SD16—2681no—
444SD16—1342no—
445M16—5361no—
446C2×C8—24no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—81no—
452C2×C8—5361no—
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—671no—
469C4⋊C4—671no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C4⋊C4C2×C4×D6711yescharacteristic
475C4⋊C4C2×C4×D6711yescharacteristic
476C22⋊C4—1341no—
477C22⋊C4—1341no—
478C22⋊C4—1341no—
479C22⋊C4—1341no—
480C22⋊C4—1341no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—1341no—
486C42C2×C4×D6711yescharacteristic
487C42D4×D6711yescharacteristic
488C42#1176162711yescharacteristic
489C42C23×D6711yescharacteristic
490C23—1341no—
491C23(C22⋊D67)⋊C411yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—42no—
496Q8—1341no—
497Q8—1341no—
498Q8#1452404912yes—
499D4—2682no—
500D4—1341no—
501D4—1341no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
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—22no—
526C8—41no—
527C8—41no—
528C8—2681no—
529C8—2681no—
530C8—1341no—
531C8—1341no—
532C8—1341no—
533C8—1341no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8—1341no—
538C22—21no—
539C22—21no—
540C22—2681no—
541C22—2681no—
542C22—2681no—
543C22#1686351012yes—
544C22C2×(C22⋊D67)⋊C411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—2681no—
554C4—2681no—
555C4—2681no—
556C4C2×#1452404911yescharacteristic
557C4#1686416211yescharacteristic
558C4#1686420211yescharacteristic
559C4#1686420911yescharacteristic
560C2—22no—
561C2—5361no—
562C2#2256893011yescharacteristic
563C2#2256904611yescharacteristic
564C2C2×#1686351011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC2×Q8⋊C8×C67C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenterC2×Q8⋊C8D67
hypocenterC67#18017
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #22569298C2semidirect
4. #22559103C2semidirect
5. #22559096C2semidirect
6. #22559084C2semidirect
7. #22559083C2semidirect
8. #22559077C2semidirect
9. #22559069C2semidirect
10. #22559013C2semidirect
11. #22559006C2semidirect
12. #22558992C2semidirect
14. #22558492C2semidirect
15. C2×Q8⋊C8×C67C2semidirect
296. C2×Q8⋊C8D67semidirect
298. C67#18017semidirect

Maximal subgroups

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

subgroupindex
#225702892
#225693652
#225692982
#225591032
#225590962
#225590842
#225590832
#225590772
#225590692
#225590132
#225590062
#225589922
#225586702
#225584922
C2×Q8⋊C8×C672
#1801767