Finite Group Databasebeta

← #30827605

Subgroups — #30827605

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570792C211yesmaximal, characteristic
2#22569157C211yesmaximal, characteristic
3#22569151C211yesmaximal, characteristic
4#22559104C211yesmaximal, characteristic, complemented
5#22559103C211yesmaximal, characteristic, complemented
6#22559061C211yesmaximal, characteristic, complemented
7#22559060C211yesmaximal, characteristic, complemented
8#22559058C211yesmaximal, characteristic, complemented
9C8×#11761618C211yesmaximal, characteristic, complemented
10#22559013C211yesmaximal, characteristic, complemented
11#335×D67C211yesmaximal, characteristic
12#22558610C211yesmaximal, characteristic
13#22558610C211yesmaximal, characteristic
14#22558488C211yesmaximal, characteristic
15C67×#1584C211yesmaximal, characteristic, complemented
16#16863950C2211yescharacteristic
17#16863950C2211yescharacteristic
18#16863800C411yescharacteristic
19#16863774C411yescharacteristic
20#16863745C411yescharacteristic
21#16863745C411yescharacteristic
22#16863638C411yescharacteristic
23#16863636C411yescharacteristic
24#16863636C411yescharacteristic
25#16863634C411yescharacteristic
26#16863527C2211yescharacteristic
27#16863526—21no—
28#16863526C2211yescharacteristic
29#16863525—21no—
30#16863524—21no—
31#16863523—21no—
32#16863523C2211yescharacteristic
33Q8×C67⋊C8C2211yescharacteristic
34C8×Dic134C2211yescharacteristic
35#16863520C2211yescharacteristic
36C4×Dic268C2211yescharacteristic
37C4×#11761618C2211yescharacteristic
38C4×#11761618C2211yescharacteristic
39#16863447C2211yescharacteristic
40#16863421—21no—
41#16863421—21no—
42C4⋊C8×D67C2211yescharacteristic
43C4⋊C8×D67C2211yescharacteristic
44#16863353—21no—
45#16863350—21no—
46C4×#11761611C2211yescharacteristic
47C4×C8×D67C2211yescharacteristic
48#16863149C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863148C2211yescharacteristic
51#16863128C2211yescharacteristic
52#16863128C2211yescharacteristic
53Q8⋊(C67⋊C8)C2211yescharacteristic
54Q8⋊(C67⋊C8)C2211yescharacteristic
55#16863126C2211yescharacteristic
56#16863126C2211yescharacteristic
57#16863118C2211yescharacteristic
58C8×C67⋊C8C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60#446×C67C2211yescharacteristic
61C8×Q8×C67C2211yescharacteristic
62C4×Q16×C67C2211yescharacteristic
63#335×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65Q8⋊C8×C67C2211yescharacteristic
66#322×C67C2211yescharacteristic
67C2×Dic268C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761618C2×C411yescharacteristic
70C2×#11761611D411yescharacteristic
71C2×#11761611D411yescharacteristic
72C2×C8×D67—21no—
73C2×C8×D67—21no—
74C2×C8×D67C2×C411yescharacteristic
75C2×C8×D67C2×C411yescharacteristic
76#14524042C2×C411yescharacteristic
77#14524042C2×C411yescharacteristic
78#14524041C2×C411yescharacteristic
79#14524041C2×C411yescharacteristic
80#14524033—21no—
81#14524033—21no—
82#14524023C2×C411yescharacteristic
83#14524023C2×C411yescharacteristic
84#14524021—21no—
85#14524021—21no—
86#14524019C2×C411yescharacteristic
87#14524019C2×C411yescharacteristic
88C4×Dic134—21no—
89C4×Dic134—21no—
90C4×Dic134C2311yescharacteristic
91C4×Dic134C2311yescharacteristic
92#14524000—21no—
93#14524000—21no—
94C42×D67C2311yescharacteristic
95#14523977C2×C411yescharacteristic
96#14523976C2×C411yescharacteristic
97#14523976C2×C411yescharacteristic
98C4⋊(C67⋊C8)—21no—
99C4⋊(C67⋊C8)—21no—
100C4⋊(C67⋊C8)C2311yescharacteristic
101C4⋊(C67⋊C8)C2311yescharacteristic
102Dic67⋊C8—21no—
103Dic67⋊C8—21no—
104Dic67⋊C8—21no—
105Dic67⋊C8—21no—
106Dic67⋊C8—21no—
107Dic67⋊C8—21no—
108Dic67⋊C8—21no—
109#14523968C2×C411yescharacteristic
110#14523968C2×C411yescharacteristic
111#14523967C2×C411yescharacteristic
112#14523967C2×C411yescharacteristic
113#14523966C2×C411yescharacteristic
114#14523966C2×C411yescharacteristic
115(C2×D67)⋊C8—21no—
116#14523954—21no—
117#14523953—21no—
118#14523953—21no—
119#14523953D411yescharacteristic
120#14523953D411yescharacteristic
121C4×C67⋊C8—21no—
122C4×C67⋊C8C2311yescharacteristic
123C4×C67⋊C8C2311yescharacteristic
124C4×C67⋊C8C2311yescharacteristic
125C8×Dic67—21no—
126C8×Dic67—21no—
127C8×Dic67C2×C411yescharacteristic
128C8×Dic67C2×C411yescharacteristic
129C2×Q16×C67C2×C411yescharacteristic
130C4×Q8×C67C2311yescharacteristic
131C4×Q8×C67C2311yescharacteristic
132#107×C67C2×C411yescharacteristic
133C4⋊C8×C67—21no—
134C4⋊C8×C67—21no—
135C4⋊C8×C67C2311yescharacteristic
136C4⋊C8×C67C2311yescharacteristic
137#103×C67C2×C411yescharacteristic
138#103×C67C2×C411yescharacteristic
139#97×C67—21no—
140C4×C8×C67—21no—
141C4×C8×C67C2311yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134—21no—
145C2×Dic134—21no—
146C2×Dic134C22×C411yescharacteristic
147C2×Dic134C22×C411yescharacteristic
148C2×C4×D67C22×C411yescharacteristic
149C2×C4×D67C2×D411yescharacteristic
150C2×C4×D67#4111yescharacteristic
151Dic268—41no—
152#11761618—22no—
153#11761618—22no—
154#11761611—22no—
155#11761611Q1614yescomplemented
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8—21no—
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C2×D411yescharacteristic
162C2×C67⋊C8C2×D411yescharacteristic
163C8×D67—22no—
164C8×D67—41no—
165C8×D67—42no—
166C8×D67—42no—
167C4⋊Dic67—21no—
168C4⋊Dic67—21no—
169C4⋊Dic67C22×C411yescharacteristic
170C4⋊Dic67C22×C411yescharacteristic
171Dic67⋊C4—21no—
172Dic67⋊C4—21no—
173Dic67⋊C4—21no—
174Dic67⋊C4—21no—
175Dic67⋊C4—21no—
176Dic67⋊C4—21no—
177Dic67⋊C4—21no—
178Dic67⋊C4—21no—
179Dic67⋊C4—21no—
180Dic67⋊C4—21no—
181(C2×D67)⋊C4—21no—
182(C2×D67)⋊C4—21no—
183(C2×D67)⋊C4—21no—
184(C2×D67)⋊C4—21no—
185C4×Dic67—21no—
186C4×Dic67—21no—
187C4×Dic67C22×C411yescharacteristic
188C4×Dic67C2×D411yescharacteristic
189C4×Dic67#4111yescharacteristic
190C2×Q8×C67C22×C411yescharacteristic
191C2×Q8×C67C22×C411yescharacteristic
192Q16×C67—41no—
193C2×C8×C67—21no—
194C2×C8×C67—21no—
195C2×C8×C67—21no—
196C2×C8×C67C22×C411yescharacteristic
197C2×C8×C67C22×C411yescharacteristic
198C2×C8×C67C2×D411yescharacteristic
199C2×C8×C67C2×D411yescharacteristic
200C4⋊C4×C67—21no—
201C4⋊C4×C67—21no—
202C4⋊C4×C67C22×C411yescharacteristic
203C4⋊C4×C67C22×C411yescharacteristic
204C42×C67—21no—
205C42×C67—21no—
206C42×C67C2411yescharacteristic
207C22×D67C4×D411yescharacteristic
208Q8×C67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67C23×C411yescharacteristic
217C2×C4×C67C22×D411yescharacteristic
218C2×C4×C67C2×#4111yescharacteristic
219C2×Dic67—21no—
220C2×Dic67—21no—
221C2×Dic67—21no—
222C2×Dic67—21no—
223C2×Dic67C4×D411yescharacteristic
224C2×Dic67C4×D411yescharacteristic
225C2×Dic67C4×D411yescharacteristic
226C4×D67—21no—
227C4×D67—21no—
228C4×D67—41no—
229C4×D67C2×Q1612yes—
230C4×D67#13512yes—
231Dic134—21no—
232Dic134—21no—
233Dic134—21no—
234Dic134—21no—
235Dic134—81no—
236Dic134—81no—
237C8×C67—21no—
238C8×C67—21no—
239C8×C67—21no—
240C8×C67—41no—
241C8×C67—41no—
242C8×C67—41no—
243C8×C67C2×Q1612yes—
244C67⋊C8—21no—
245C67⋊C8—21no—
246C67⋊C8—41no—
247C67⋊C8—41no—
248C67⋊C8—41no—
249C67⋊C8#13112yes—
250C67⋊C8C2×Q1612yes—
251C22×C67C2×C4×D411yescharacteristic
252C2×D67—21no—
253C2×D67—21no—
254C2×D67C4×Q1612yes—
255C4×C67—21no—
256C4×C67—41no—
257C4×C67—41no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67C2×#13111yescharacteristic
261C4×C67#56811yescharacteristic
262C4×C67C22×Q1611yescharacteristic
263C4×C67C2×#13511yescharacteristic
264Dic67—21no—
265Dic67—21no—
266Dic67—41no—
267Dic67—41no—
268Dic67—41no—
269Dic67—41no—
270Dic67C4×Q1612yes—
271#18068—671nomaximal, Sylow
272C2×C67#293511yescharacteristic
273C2×C67C2×C4×Q1611yescharacteristic
274C2×C67#295711yescharacteristic
275D67—22no—
276#3170—671no—
277#2962—671no—
278#2962—671no—
279#1591—671no—
280#1591—671no—
281#1584—671no—
282#1584—671no—
283#1584—671no—
284#1584D6711yescharacteristic, complemented
285C8×Q16—671no—
286C8×Q16—671no—
287C2×#335—671no—
288#1481—671no—
289#1481—671no—
290C8×M16—671no—
291C67#1806811yescharacteristic, complemented, Sylow
292#533—671no—
293#533—671no—
294#500—671no—
295#494—671no—
296#487—671no—
297#487—671no—
298#467—671no—
299#467—671no—
300#467—671no—
301#467—671no—
302#446—671no—
303#446—671no—
304#446—671no—
305#446—1341no—
306#446—1341no—
307#446—1341no—
308#446—1341no—
309#446C2×D6711yescharacteristic
310C8×Q8—671no—
311C8×Q8—671no—
312C8×Q8—671no—
313C8×Q8C2×D6711yescharacteristic
314C4×Q16—671no—
315C4×Q16—671no—
316C4×Q16—671no—
317C4×Q16C2×D6711yescharacteristic
318#431—671no—
319#424—1341no—
320#424—1341no—
321C2×C4⋊C8—671no—
322C2×C4⋊C8—671no—
323#405—1341no—
324#405—1341no—
325C4×M16—671no—
326C2×C4×C8—671no—
327#335—671no—
328#335—671no—
329#335—671no—
330#335C2×D6711yescharacteristic
331Q8⋊C8—671no—
332Q8⋊C8—671no—
333Q8⋊C8—671no—
334Q8⋊C8—671no—
335Q8⋊C8—671no—
336Q8⋊C8—671no—
337Q8⋊C8C2×D6711yescharacteristic
338Q8⋊C8C2×D6711yescharacteristic
339#322—671no—
340#322C2×D6711yescharacteristic
341C82—671no—
342C82—671no—
343C2×Q16—671no—
344C2×Q16—671no—
345C2×Q16—671no—
346C2×Q16C4×D6711yescharacteristic
347C2×M16—671no—
348C2×M16—671no—
349C22×C8—671no—
350C22×C8—671no—
351C22×C8—1341no—
352C22×C8—1341no—
353C4⋊Q8—671no—
354C4⋊Q8—671no—
355C4⋊Q8—671no—
356C4⋊Q8—671no—
357#125—1341no—
358#125—1341no—
359C22⋊Q8—671no—
360C22⋊Q8—671no—
361C22⋊Q8—671no—
362C22⋊Q8—671no—
363C22⋊Q8—1341no—
364C22⋊Q8—1341no—
365C4×Q8—671no—
366C4×Q8—671no—
367C4×Q8—1341no—
368C4×Q8—1341no—
369C4×Q8C22×D6711yescharacteristic
370C4×Q8C22×D6711yescharacteristic
371#117—1341no—
372#117—1341no—
373C2×C42—671no—
374#107—671no—
375#107—671no—
376#107—671no—
377#107C4×D6711yescharacteristic
378C4⋊C8—21no—
379C4⋊C8—21no—
380C4⋊C8—671no—
381C4⋊C8—671no—
382C4⋊C8—1341no—
383C4⋊C8—1341no—
384C4⋊C8—1341no—
385C4⋊C8—1341no—
386C4⋊C8—1341no—
387C4⋊C8—1341no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8C22×D6711yescharacteristic
392C4⋊C8C22×D6711yescharacteristic
393#103—671no—
394#103—671no—
395#103—671no—
396#103—671no—
397#103—671no—
398#103—671no—
399#103C4×D6711yescharacteristic
400#103C4×D6711yescharacteristic
401C22⋊C8—1341no—
402#97—21no—
403#97—671no—
404#97—671no—
405#97—1341no—
406#97—1341no—
407#97—1341no—
408C4×C8—21no—
409C4×C8—671no—
410C4×C8—671no—
411C4×C8—671no—
412C4×C8—671no—
413C4×C8—671no—
414C4×C8—1341no—
415C4×C8—1341no—
416C4×C8—1341no—
417C4×C8C22×D6711yescharacteristic
418C4×C8C22×D6711yescharacteristic
419C4×C8C22×D6711yescharacteristic
420C2×Q8—671no—
421C2×Q8—671no—
422C2×Q8—1341no—
423C2×Q8—1341no—
424C2×Q8C2×C4×D6711yescharacteristic
425C2×Q8C2×C4×D6711yescharacteristic
426C22×C4—671no—
427C22×C4—671no—
428C22×C4—671no—
429Q16—41no—
430Q16—2681no—
431Q16—1342no—
432Q16—1342no—
433M16—674no—
434M16—1342no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—2681no—
439C2×C8—2682no—
440C2×C8—2682no—
441C2×C8—671no—
442C2×C8—671no—
443C2×C8—671no—
444C2×C8—671no—
445C2×C8—1342no—
446C2×C8—1341no—
447C2×C8—1341no—
448C2×C8—1341no—
449C2×C8C2×C4×D6711yescharacteristic
450C2×C8C2×C4×D6711yescharacteristic
451C2×C8D4×D6711yescharacteristic
452C2×C8D4×D6711yescharacteristic
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—671no—
456C4⋊C4—671no—
457C4⋊C4—1341no—
458C4⋊C4—1341no—
459C4⋊C4—1341no—
460C4⋊C4—1341no—
461C4⋊C4—1341no—
462C4⋊C4—1341no—
463C4⋊C4—1341no—
464C4⋊C4—1341no—
465C4⋊C4—1341no—
466C4⋊C4—1341no—
467C4⋊C4—1341no—
468C4⋊C4—1341no—
469C4⋊C4C2×C4×D6711yescharacteristic
470C4⋊C4C2×C4×D6711yescharacteristic
471C22⋊C4—1341no—
472C22⋊C4—1341no—
473C22⋊C4—1341no—
474C22⋊C4—1341no—
475C42—21no—
476C42—21no—
477C42—671no—
478C42—671no—
479C42—671no—
480C42—1341no—
481C42—1341no—
482C42C23×D6711yescharacteristic
483C23—671no—
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—21no—
488Q8—5361no—
489Q8—5361no—
490Q8—1341no—
491Q8—1341no—
492Q8—1341no—
493Q8—1341no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—2681no—
499C2×C4—671no—
500C2×C4—672no—
501C2×C4—672no—
502C2×C4—671no—
503C2×C4—671no—
504C2×C4—1341no—
505C2×C4—1341no—
506C2×C4—1341no—
507C2×C4—1341no—
508C2×C4—1341no—
509C2×C4—1341no—
510C2×C4C22×C4×D6711yescharacteristic
511C2×C4C2×D4×D6711yescharacteristic
512C2×C4#41×D6711yescharacteristic
513C8—21no—
514C8—21no—
515C8—21no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—2681no—
520C8—2681no—
521C8—2681no—
522C8—672no—
523C8—672no—
524C8—1341no—
525C8—1341no—
526C8Q16×D6712yes—
527C22—672no—
528C22—1341no—
529C22—1341no—
530C22C4×D4×D6711yescharacteristic
531C4—21no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—2681no—
537C4—2681no—
538C4—2681no—
539C4—2681no—
540C4—672no—
541C4—1341no—
542C4—1341no—
543C4#131×D6711yescharacteristic
544C4#1686416211yescharacteristic
545C4C2×Q16×D6711yescharacteristic
546C4#135×D6711yescharacteristic
547C2—1342no—
548C2#2256895011yescharacteristic
549C2C4×Q16×D6711yescharacteristic
550C2#2256912311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1584C2
socleC22×C67C2×C4×D4
prime radicalC22×D67C4×D4
cosocleC42×C67C24
hypercenter#1584D67
hypocenterC67#18068
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #22559104C2semidirect
5. #22559103C2semidirect
6. #22559061C2semidirect
7. #22559060C2semidirect
8. #22559058C2semidirect
9. C8×#11761618C2semidirect
10. #22559013C2semidirect
15. C67×#1584C2semidirect
155. #11761611Q16semidirect
284. #1584D67semidirect
291. C67#18068semidirect

Maximal subgroups

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

subgroupindex
#225707922
#225691572
#225691512
#225591042
#225591032
#225590612
#225590602
#225590582
C8×#117616182
#225590132
#335×D672
#225586102
#225584882
C67×#15842
#1806867