Finite Group Databasebeta

← #30825902

Subgroups — #30825902

Order 17152.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570285C211yesmaximal, characteristic
2#22569289C211yesmaximal, characteristic, complemented
3#22559085C211yesmaximal, characteristic, complemented
4#22559078C211yesmaximal, characteristic, complemented
5#22559057C211yesmaximal, characteristic, complemented
6#22559050C211yesmaximal, characteristic, complemented
7#22559026C211yesmaximal, characteristic, complemented
8#22559021C211yesmaximal, characteristic, complemented
9C8×#11761618C211yesmaximal, characteristic, complemented
10C8×#11761616C211yesmaximal, characteristic, complemented
11#22559000C211yesmaximal, characteristic, complemented
12#22558637C211yesmaximal, characteristic, complemented
13#22558617C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic, complemented
15C67×#2993C211yesmaximal, characteristic
16C2×C4×Dic134C2211yescharacteristic
17#16863724C411yescharacteristic
18#16863707C411yescharacteristic
19#16863700C411yescharacteristic
20#16863700C411yescharacteristic
21#16863615C411yescharacteristic
22#16863615C411yescharacteristic
23#16863551C411yescharacteristic
24#16863534C411yescharacteristic
25C8×Dic134C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C8×Dic134C2211yescharacteristic
28C8×Dic134C2211yescharacteristic
29#16863520—21no—
30#16863520—21no—
31C4×#11761618C2211yescharacteristic
32C4×#11761618C2211yescharacteristic
33C4×#11761616C2211yescharacteristic
34C4×#11761616C2211yescharacteristic
35#16863449C2211yescharacteristic
36#16863449C2211yescharacteristic
37#16863449C2211yescharacteristic
38#16863446—21no—
39#16863446—21no—
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#424×C67—21no—
66#423×C67C2211yescharacteristic
67C22×Dic134C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761618C2×C411yescharacteristic
70C2×#11761616C2×C411yescharacteristic
71C2×#11761616C2×C411yescharacteristic
72C4×Dic134—21no—
73C4×Dic134—21no—
74C4×Dic134C2×C411yescharacteristic
75C4×Dic134C2×C411yescharacteristic
76C4×Dic134C2311yescharacteristic
77C4×Dic134C2311yescharacteristic
78C2×C4⋊Dic67C2×C411yescharacteristic
79C2×Dic67⋊C4—21no—
80C2×C4×Dic67—21no—
81#14523976C2×C411yescharacteristic
82#14523976C2×C411yescharacteristic
83#14523974C2×C411yescharacteristic
84#14523974C2×C411yescharacteristic
85C4⋊(C67⋊C8)—21no—
86C4⋊(C67⋊C8)C2×C411yescharacteristic
87C4⋊(C67⋊C8)C2×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—
103C22⋊(C67⋊C8)—21no—
104C22⋊(C67⋊C8)C2×C411yescharacteristic
105C22⋊(C67⋊C8)C2×C411yescharacteristic
106C4×C67⋊C8—21no—
107C4×C67⋊C8C2311yescharacteristic
108C4×C67⋊C8C2311yescharacteristic
109C4×C67⋊C8C2311yescharacteristic
110C4×C67⋊C8C2311yescharacteristic
111C8×Dic67—21no—
112C8×Dic67—21no—
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×Dic134—22no—
146C2×Dic134—22no—
147C2×Dic134C22×C411yescharacteristic
148C2×Dic134C22×C411yescharacteristic
149C22×Dic67—21no—
150C22×Dic67—21no—
151#11761618—22no—
152#11761618—22no—
153#11761616—22no—
154#11761616—22no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C22×C411yescharacteristic
163C4⋊Dic67—21no—
164C4⋊Dic67C22×C411yescharacteristic
165C4⋊Dic67C22×C411yescharacteristic
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170C4×Dic67—21no—
171C4×Dic67—21no—
172C4×Dic67—21no—
173C4×Dic67—21no—
174C2×Q8×C67C22×C411yescharacteristic
175C2×D4×C67—21no—
176C2×D4×C67C22×C411yescharacteristic
177C22×C4×C67—21no—
178C22×C4×C67C22×C411yescharacteristic
179C22×C4×C67C2×D411yescharacteristic
180C22×C4×C67#4111yescharacteristic
181M16×C67—81no—
182M16×C67—81no—
183C2×C8×C67—21no—
184C2×C8×C67—21no—
185C2×C8×C67—81no—
186C2×C8×C67C2×D411yescharacteristic
187C2×C8×C67C2×D411yescharacteristic
188C2×C8×C67C2×D411yescharacteristic
189C2×C8×C67C2×D411yescharacteristic
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67C22×C411yescharacteristic
195C4⋊C4×C67C22×C411yescharacteristic
196C22⋊C4×C67—21no—
197C22⋊C4×C67—21no—
198C22⋊C4×C67—21no—
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C42×C67—21no—
202C42×C67C22×C411yescharacteristic
203C42×C67C2×D411yescharacteristic
204C42×C67#4111yescharacteristic
205C42×C67C2411yescharacteristic
206C23×C67—21no—
207C23×C67C4×D411yescharacteristic
208Q8×C67—21no—
209Q8×C67—21no—
210D4×C67—21no—
211D4×C67—21no—
212D4×C67—81no—
213C2×C4×C67—21no—
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—41no—
221C2×C4×C67—41no—
222C2×C4×C67—41no—
223C2×C4×C67C4×D411yescharacteristic
224C2×C4×C67C4×D411yescharacteristic
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C23×C411yescharacteristic
227C2×C4×C67C22×D411yescharacteristic
228C2×C4×C67C2×#4111yescharacteristic
229C2×Dic67—24no—
230C2×Dic67—21no—
231C2×Dic67—21no—
232C2×Dic67—21no—
233C2×Dic67—21no—
234C2×Dic67—81no—
235Dic134—24no—
236Dic134—21no—
237Dic134—21no—
238Dic134#13112yes—
239Dic134#13112yes—
240C8×C67—21no—
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—41no—
245C8×C67—41no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—41no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—41no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67C2×C4×D411yescharacteristic
261C4×C67—21no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67C2×#13111yescharacteristic
269C4×C67C2×#13111yescharacteristic
270C4×C67#57711yescharacteristic
271C4×C67#57811yescharacteristic
272Dic67—22no—
273Dic67—22no—
274Dic67—41no—
275Dic67—41no—
276#17919—671nomaximal, Sylow
277C2×C67—41no—
278C2×C67—81no—
279C2×C67#292911yescharacteristic
280C2×C67#294711yescharacteristic
281C2×C67#295711yescharacteristic
282#3108—671no—
283#2993D6711yescharacteristic
284#2982—671no—
285#1588—671no—
286#1587—671no—
287#1584—671no—
288#1583—671no—
289#1579—671no—
290SD16⋊C8—671no—
291C8×Q16—671no—
292C8×SD16—671no—
293#1574—671no—
294#1487—671no—
295#1482—671no—
296#1454—671no—
297C67#1791911yescharacteristic, complemented, Sylow
298#529C2×D6711yescharacteristic
299C2×C4×Q8—671no—
300#483—671no—
301#480—671no—
302#479—671no—
303#479—671no—
304#464—671no—
305#464—671no—
306#451—671no—
307#448—671no—
308#446C2×D6711yescharacteristic
309C8×Q8—671no—
310C8×Q8—671no—
311C8×Q8—671no—
312C8×Q8—671no—
313C8×Q8—1341no—
314C8×Q8—1341no—
315C8×Q8C2×D6711yescharacteristic
316C4×Q16—671no—
317C4×Q16—671no—
318C4×SD16—671no—
319C4×SD16—671no—
320#436—21no—
321#436C2×D6711yescharacteristic
322#435—21no—
323C8×D4C2×D6711yescharacteristic
324#433—21no—
325#431—671no—
326#431—671no—
327#431—671no—
328#431—1341no—
329#431—1341no—
330#431C2×D6711yescharacteristic
331#424—21no—
332#423C2×D6711yescharacteristic
333#335—671no—
334#335—671no—
335#334—671no—
336#334—671no—
337Q8⋊C8—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343D4⋊C8—671no—
344D4⋊C8—671no—
345#322—671no—
346#322—671no—
347#322—671no—
348C82—671no—
349C22×Q8—671no—
350C2×Q16—671no—
351C2×Q16—671no—
352C2×SD16—671no—
353C2×SD16—671no—
354C2×M16—21no—
355C2×M16—21no—
356C22×C8—21no—
357#126—21no—
358#125C4×D6711yescharacteristic
359#124C4×D6711yescharacteristic
360#123—21no—
361C22⋊Q8C4×D6711yescharacteristic
362#121C4×D6711yescharacteristic
363C4×Q8—671no—
364C4×Q8—671no—
365C4×Q8—671no—
366C4×Q8—671no—
367C4×Q8—1341no—
368C4×Q8—1341no—
369C4×Q8C22×D6711yescharacteristic
370C4×D4—21no—
371C4×D4C22×D6711yescharacteristic
372#117—21no—
373C2×C4⋊C4—671no—
374C2×C4⋊C4—1341no—
375C2×C42—1341no—
376C2×C42C22×D6711yescharacteristic
377#107—671no—
378#107—671no—
379#106—671no—
380#106—671no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8C22⋊D6711yescharacteristic
394C4⋊C8C22⋊D6711yescharacteristic
395C4⋊C8C22×D6711yescharacteristic
396C4⋊C8C22×D6711yescharacteristic
397#103—671no—
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#102—671no—
404#102—671no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—671no—
408C22⋊C8—671no—
409C22⋊C8—1341no—
410C22⋊C8—1341no—
411C22⋊C8C22⋊D6711yescharacteristic
412C22⋊C8C22⋊D6711yescharacteristic
413#97—21no—
414C4×C8—21no—
415C4×C8—671no—
416C4×C8—671no—
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—1341no—
420C4×C8—1341no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8C22×D6711yescharacteristic
425C4×C8C22×D6711yescharacteristic
426C2×Q8—671no—
427C2×Q8—671no—
428C2×Q8—1342no—
429C2×Q8—1342no—
430C2×Q8—1341no—
431C2×Q8C2×C4×D6711yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D6711yescharacteristic
434C22×C4—21no—
435C22×C4—1341no—
436C22×C4—1341no—
437C22×C4C2×C4×D6711yescharacteristic
438C22×C4C2×C22⋊D6711yescharacteristic
439C22×C4#1176162811yescharacteristic
440Q16—1342no—
441Q16—1342no—
442SD16—1342no—
443SD16—1342no—
444M16—81no—
445M16—81no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—81no—
449C2×C8—671no—
450C2×C8—671no—
451C2×C8—671no—
452C2×C8—671no—
453C2×C8—1341no—
454C2×C8—1341no—
455C2×C8—1341no—
456C2×C8—1341no—
457C2×C8C2×C22⋊D6711yescharacteristic
458C2×C8C2×C22⋊D6711yescharacteristic
459C2×C8C2×C22⋊D6711yescharacteristic
460C2×C8C2×C22⋊D6711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—671no—
466C4⋊C4—671no—
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—
479C42—21no—
480C42—1341no—
481C42—1341no—
482C42—1341no—
483C42—1341no—
484C42C2×C4×D6711yescharacteristic
485C42C2×C22⋊D6711yescharacteristic
486C42#1176162811yescharacteristic
487C42C23×D6711yescharacteristic
488C23—21no—
489C23C4×C22⋊D6711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—672no—
493Q8—672no—
494Q8—1344no—
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—5361no—
511C2×C4—1341no—
512C2×C4—1341no—
513C2×C4—1344no—
514C2×C4—1341no—
515C2×C4—1341no—
516C2×C4C4×C22⋊D6711yescharacteristic
517C2×C4C4×C22⋊D6711yescharacteristic
518C2×C4C4×C22⋊D6711yescharacteristic
519C2×C4C22×C4×D6711yescharacteristic
520C2×C4C22×C22⋊D6711yescharacteristic
521C2×C4C2×#1176162811yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—2681no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8—1341no—
533C8—1341no—
534C8—1341no—
535C8—1341no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D6711yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—2681no—
551C4—2681no—
552C4—1342no—
553C4—1342no—
554C4#131×D6711yescharacteristic
555C4#131×D6711yescharacteristic
556C4#1686420711yescharacteristic
557C4#1686421111yescharacteristic
558C2—41no—
559C2—81no—
560C2#2256890111yescharacteristic
561C2#2256904811yescharacteristic
562C2#2256912411yescharacteristic
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#17919
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. #22569289C2semidirect
3. #22559085C2semidirect
4. #22559078C2semidirect
5. #22559057C2semidirect
6. #22559050C2semidirect
7. #22559026C2semidirect
8. #22559021C2semidirect
9. C8×#11761618C2semidirect
10. C8×#11761616C2semidirect
11. #22559000C2semidirect
12. #22558637C2semidirect
14. #22558509C2semidirect
297. C67#17919semidirect

Maximal subgroups

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

subgroupindex
#225702852
#225692892
#225590852
#225590782
#225590572
#225590502
#225590262
#225590212
C8×#117616182
C8×#117616162
#225590002
#225586372
#225586172
#225585092
C67×#29932
#1791967