Finite Group Databasebeta

← #30826815

Subgroups — #30826815

Order 17152.

610 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570227C211yesmaximal, characteristic
2#22569213C211yesmaximal, characteristic, complemented
3#22559116C211yesmaximal, characteristic, complemented
4#22559109C211yesmaximal, characteristic, complemented
5#22559078C211yesmaximal, characteristic, complemented
6#22559073C211yesmaximal, characteristic, complemented
7#22559066C211yesmaximal, characteristic, complemented
8#22559064C211yesmaximal, characteristic, complemented
9#22559031C211yesmaximal, characteristic, complemented
10#22559021C211yesmaximal, characteristic, complemented
11#22558990C211yesmaximal, characteristic, complemented
12#22558667C211yesmaximal, characteristic, complemented
13#22558591C211yesmaximal, characteristic, complemented
14#22558496C211yesmaximal, characteristic, complemented
15C67×#2970C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863737C411yescharacteristic
18#16863720C411yescharacteristic
19#16863703C411yescharacteristic
20#16863696C411yescharacteristic
21#16863618C411yescharacteristic
22#16863611C411yescharacteristic
23#16863558C411yescharacteristic
24#16863541C411yescharacteristic
25#16863526C2211yescharacteristic
26C8×Dic134C2211yescharacteristic
27C4×#11761618C2211yescharacteristic
28C4×#11761615C2211yescharacteristic
29C4×#11761616C2211yescharacteristic
30C4×#11761613C2211yescharacteristic
31#16863475C2211yescharacteristic
32#16863472—21no—
33#16863465—21no—
34C8×D268C2211yescharacteristic
35#16863458—21no—
36#16863449C2211yescharacteristic
37#16863449C2211yescharacteristic
38#16863353—21no—
39C2×C4×C67⋊C8C2211yescharacteristic
40#16863148C2211yescharacteristic
41#16863148C2211yescharacteristic
42#16863146C2211yescharacteristic
43#16863146C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46#16863126C2211yescharacteristic
47#16863126C2211yescharacteristic
48D4⋊(C67⋊C8)C2211yescharacteristic
49D4⋊(C67⋊C8)C2211yescharacteristic
50#16863123C2211yescharacteristic
51#16863123C2211yescharacteristic
52#16863118C2211yescharacteristic
53#16863118C2211yescharacteristic
54#16863118C2211yescharacteristic
55#16863118C2211yescharacteristic
56C4×#41×C67C2211yescharacteristic
57#446×C67C2211yescharacteristic
58#446×C67C2211yescharacteristic
59#435×C67—21no—
60#435×C67—21no—
61C8×D4×C67C2211yescharacteristic
62C8×D4×C67C2211yescharacteristic
63#432×C67—21no—
64#432×C67C2211yescharacteristic
65C4×M16×C67C2211yescharacteristic
66C2×#97×C67—21no—
67C2×#11761618C2×C411yescharacteristic
68C2×#11761616C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761613C2×C411yescharacteristic
71C2×#11761611—21no—
72C22×C67⋊C8—21no—
73C2×C8×D67—21no—
74#14524037—21no—
75#14524035C2×C411yescharacteristic
76#14524031C2×C411yescharacteristic
77#14524026—21no—
78#14524022C2×C411yescharacteristic
79#14524017C2×C411yescharacteristic
80C4×Dic134C2311yescharacteristic
81C4×D268C2311yescharacteristic
82C4×C22⋊D67—21no—
83#14524000—21no—
84#14523976C2×C411yescharacteristic
85#14523976C2×C411yescharacteristic
86#14523974C2×C411yescharacteristic
87#14523974C2×C411yescharacteristic
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#14523966C2×C411yescharacteristic
97#14523966C2×C411yescharacteristic
98#14523964C2×C411yescharacteristic
99#14523964C2×C411yescharacteristic
100#14523963C2×C411yescharacteristic
101#14523963C2×C411yescharacteristic
102C22⋊(C67⋊C8)—21no—
103C22⋊(C67⋊C8)C2×C411yescharacteristic
104C22⋊(C67⋊C8)C2×C411yescharacteristic
105(C2×D67)⋊C8—21no—
106(C2×D67)⋊C8—21no—
107#14523953—21no—
108C4×C67⋊C8—21no—
109C4×C67⋊C8—21no—
110C4×C67⋊C8C2311yescharacteristic
111C4×C67⋊C8C2311yescharacteristic
112C4×C67⋊C8C2311yescharacteristic
113C4×C67⋊C8C2311yescharacteristic
114C8×Dic67—21no—
115C2×#41×C67C2×C411yescharacteristic
116C2×M16×C67D411yescharacteristic
117C2×M16×C67D411yescharacteristic
118C22×C8×C67—21no—
119C22×C8×C67—21no—
120C4×Q8×C67C2×C411yescharacteristic
121C4×Q8×C67C2311yescharacteristic
122C4×D4×C67—21no—
123C4×D4×C67—21no—
124C4×D4×C67C2×C411yescharacteristic
125C4×D4×C67C2311yescharacteristic
126#117×C67—21no—
127#117×C67C2×C411yescharacteristic
128C2×C42×C67—21no—
129C2×C42×C67C2311yescharacteristic
130C4⋊C8×C67—21no—
131C4⋊C8×C67—21no—
132C4⋊C8×C67C2311yescharacteristic
133C4⋊C8×C67C2311yescharacteristic
134C22⋊C8×C67—21no—
135C22⋊C8×C67—21no—
136C22⋊C8×C67—21no—
137#97×C67—21no—
138#97×C67—21no—
139#97×C67—21no—
140#97×C67D411yescharacteristic
141#97×C67D411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134C22×C411yescharacteristic
145C2×D268C22×C411yescharacteristic
146C2×C22⋊D67—21no—
147C2×C4×D67—21no—
148#11761618—41no—
149#11761616—22no—
150#11761615—41no—
151#11761613—22no—
152#11761611—42no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—21no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—42no—
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C8×D67—42no—
163C4⋊Dic67C22×C411yescharacteristic
164C4⋊Dic67C22×C411yescharacteristic
165Dic67⋊C4—21no—
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169C22⋊Dic67—21no—
170(C2×D67)⋊C4—21no—
171(C2×D67)⋊C4—21no—
172(C2×D67)⋊C4—21no—
173(C2×D67)⋊C4—21no—
174C4×Dic67—21no—
175#41×C67—22no—
176#41×C67—41no—
177C2×Q8×C67C22×C411yescharacteristic
178C2×D4×C67—21no—
179C2×D4×C67C22×C411yescharacteristic
180C22×C4×C67—21no—
181C22×C4×C67—21no—
182C22×C4×C67—21no—
183C22×C4×C67C22×C411yescharacteristic
184C22×C4×C67C2×D411yescharacteristic
185C22×C4×C67#4111yescharacteristic
186M16×C67—41no—
187M16×C67—41no—
188C2×C8×C67—21no—
189C2×C8×C67—21no—
190C2×C8×C67—42no—
191C2×C8×C67—42no—
192C2×C8×C67C2×D411yescharacteristic
193C2×C8×C67C2×D411yescharacteristic
194C2×C8×C67C2×D411yescharacteristic
195C2×C8×C67C2×D411yescharacteristic
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67C22×C411yescharacteristic
199C4⋊C4×C67C22×C411yescharacteristic
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C22⋊C4×C67—21no—
203C42×C67—21no—
204C42×C67—21no—
205C42×C67—21no—
206C42×C67C22×C411yescharacteristic
207C42×C67C2×D411yescharacteristic
208C42×C67#4111yescharacteristic
209C42×C67C2411yescharacteristic
210C23×C67—21no—
211C23×C67C4×D411yescharacteristic
212C22×D67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215D4×C67—21no—
216D4×C67—42no—
217D4×C67#13112yes—
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—21no—
225C2×C4×C67—41no—
226C2×C4×C67—42no—
227C2×C4×C67—41no—
228C2×C4×C67—41no—
229C2×C4×C67—81no—
230C2×C4×C67C4×D411yescharacteristic
231C2×C4×C67C4×D411yescharacteristic
232C2×C4×C67C4×D411yescharacteristic
233C2×C4×C67C23×C411yescharacteristic
234C2×C4×C67C22×D411yescharacteristic
235C2×C4×C67C2×#4111yescharacteristic
236C22⋊D67—81no—
237C2×Dic67—21no—
238C2×Dic67—21no—
239C2×Dic67—21no—
240D268—21no—
241D268—21no—
242C4×D67—41no—
243C4×D67—41no—
244Dic134—21no—
245Dic134—21no—
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—21no—
250C8×C67—41no—
251C8×C67—41no—
252C67⋊C8—22no—
253C67⋊C8—21no—
254C67⋊C8—21no—
255C67⋊C8—21no—
256C67⋊C8—21no—
257C67⋊C8—41no—
258C67⋊C8—41no—
259C67⋊C8—41no—
260C22×C67—21no—
261C22×C67—22no—
262C22×C67—21no—
263C22×C67—21no—
264C22×C67—41no—
265C22×C67—41no—
266C22×C67C2×C4×D411yescharacteristic
267C2×D67—41no—
268C2×D67—41no—
269C2×D67—41no—
270C4×C67—21no—
271C4×C67—21no—
272C4×C67—21no—
273C4×C67—21no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67C2×#13111yescharacteristic
278C4×C67#56811yescharacteristic
279C4×C67#57511yescharacteristic
280C4×C67#57511yescharacteristic
281Dic67—41no—
282Dic67—41no—
283Dic67—41no—
284#18002—671nomaximal, Sylow
285C2×C67—41no—
286C2×C67—42no—
287C2×C67#293311yescharacteristic
288C2×C67#294311yescharacteristic
289C2×C67#295611yescharacteristic
290D67—81no—
291#3102—671no—
292#2970D6711yescharacteristic, complemented
293#2969—671no—
294#1593—671no—
295#1592—671no—
296#1587—671no—
297#1586—671no—
298#1585—671no—
299#1585—671no—
300#1580—671no—
301SD16⋊C8—671no—
302#1572—671no—
303#1494—671no—
304#1476—671no—
305#1451—671no—
306C67#1800211yescharacteristic, complemented, Sylow
307#529—671no—
308C4×#41C2×D6711yescharacteristic
309#485—671no—
310#482—671no—
311#479—671no—
312#478—671no—
313#464—671no—
314#463—671no—
315#452—671no—
316#449—671no—
317#446—671no—
318#446C2×D6711yescharacteristic
319#446C2×D6711yescharacteristic
320C8×Q8—671no—
321C4×Q16—671no—
322C4×SD16—671no—
323C4×SD16—671no—
324C4×D8—671no—
325#436—671no—
326#436—1341no—
327#435—21no—
328#435—21no—
329#435—1341no—
330C8×D4—671no—
331C8×D4C2×D6711yescharacteristic
332C8×D4C2×D6711yescharacteristic
333#433—1341no—
334#432—21no—
335#432C2×D6711yescharacteristic
336#431—671no—
337#431—671no—
338#405—1341no—
339C4×M16C2×D6711yescharacteristic
340C2×#97—21no—
341C2×C4×C8—671no—
342#335—671no—
343#335—671no—
344#334—671no—
345#334—671no—
346Q8⋊C8—671no—
347Q8⋊C8—671no—
348Q8⋊C8—671no—
349Q8⋊C8—671no—
350D4⋊C8—671no—
351D4⋊C8—671no—
352D4⋊C8—671no—
353D4⋊C8—671no—
354#322—671no—
355#322—671no—
356#322—671no—
357#322—671no—
358C2×#41C4×D6711yescharacteristic
359C2×Q16—671no—
360C2×SD16—671no—
361C2×SD16—671no—
362C2×D8—671no—
363C2×M16—1341no—
364C2×M16C22⋊D6711yescharacteristic
365C2×M16C22⋊D6711yescharacteristic
366C22×C8—21no—
367C22×C8—21no—
368C22×C8—1341no—
369C22×C8—1341no—
370#126—1341no—
371#125—671no—
372#124—671no—
373#123—1341no—
374C22⋊Q8—671no—
375#121—671no—
376C4×Q8—671no—
377C4×Q8C4×D6711yescharacteristic
378C4×Q8C22×D6711yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—671no—
382C4×D4—1341no—
383C4×D4C4×D6711yescharacteristic
384C4×D4C22×D6711yescharacteristic
385#117—21no—
386#117—1341no—
387#117C4×D6711yescharacteristic
388C2×C42—21no—
389C2×C42C22×D6711yescharacteristic
390#107—671no—
391#107—671no—
392#106—671no—
393#106—671no—
394C4⋊C8—21no—
395C4⋊C8—21no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—671no—
399C4⋊C8—671no—
400C4⋊C8—1341no—
401C4⋊C8—1341no—
402C4⋊C8C22×D6711yescharacteristic
403C4⋊C8C22×D6711yescharacteristic
404#103—671no—
405#103—671no—
406#103—671no—
407#103—671no—
408#102—671no—
409#102—671no—
410#102—671no—
411#102—671no—
412C22⋊C8—21no—
413C22⋊C8—21no—
414C22⋊C8—21no—
415C22⋊C8—671no—
416C22⋊C8—671no—
417C22⋊C8—1341no—
418C22⋊C8—1341no—
419C22⋊C8—1341no—
420#97—21no—
421#97—21no—
422#97—21no—
423#97—1341no—
424#97C22⋊D6711yescharacteristic
425#97C22⋊D6711yescharacteristic
426C4×C8—671no—
427C4×C8—671no—
428C4×C8—671no—
429C4×C8—671no—
430C4×C8—1341no—
431C4×C8—1341no—
432C4×C8—1341no—
433C4×C8C22×D6711yescharacteristic
434C4×C8C22×D6711yescharacteristic
435#41—22no—
436#41—41no—
437C2×Q8—671no—
438C2×Q8C2×C4×D6711yescharacteristic
439C2×D4—21no—
440C2×D4—671no—
441C2×D4—1341no—
442C2×D4C2×C4×D6711yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1341no—
447C22×C4C2×C4×D6711yescharacteristic
448C22×C4C2×C22⋊D6711yescharacteristic
449C22×C4#1176162811yescharacteristic
450Q16—2681no—
451SD16—2681no—
452SD16—1342no—
453D8—1342no—
454M16—41no—
455M16—41no—
456M16—2682no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—2682no—
462C2×C8—2682no—
463C2×C8—671no—
464C2×C8—671no—
465C2×C8—671no—
466C2×C8—671no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8—1341no—
470C2×C8—1341no—
471C2×C8C2×C22⋊D6711yescharacteristic
472C2×C8C2×C22⋊D6711yescharacteristic
473C2×C8C2×C22⋊D6711yescharacteristic
474C2×C8C2×C22⋊D6711yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—671no—
478C4⋊C4—671no—
479C4⋊C4—1341no—
480C4⋊C4—1341no—
481C4⋊C4—1341no—
482C4⋊C4—1341no—
483C4⋊C4C2×C4×D6711yescharacteristic
484C4⋊C4C2×C4×D6711yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—1341no—
489C22⋊C4—1341no—
490C22⋊C4—1341no—
491C22⋊C4—1341no—
492C22⋊C4—1341no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—1341no—
497C42C2×C4×D6711yescharacteristic
498C42C2×C22⋊D6711yescharacteristic
499C42#1176162811yescharacteristic
500C42C23×D6711yescharacteristic
501C23—21no—
502C23—1341no—
503C23C4×C22⋊D6711yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—1341no—
507Q8—1341no—
508D4—21no—
509D4—42no—
510D4—5361no—
511D4—1341no—
512D4—1341no—
513D4#1452405112yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—81no—
526C2×C4—2681no—
527C2×C4—2681no—
528C2×C4—1341no—
529C2×C4—1341no—
530C2×C4—1341no—
531C2×C4C4×C22⋊D6711yescharacteristic
532C2×C4C4×C22⋊D6711yescharacteristic
533C2×C4C4×C22⋊D6711yescharacteristic
534C2×C4C22×C4×D6711yescharacteristic
535C2×C4C22×C22⋊D6711yescharacteristic
536C2×C4C2×#1176162811yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—2681no—
544C8—2681no—
545C8—2681no—
546C8—1342no—
547C8—1341no—
548C8—1341no—
549C8—1341no—
550C8—1341no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—41no—
556C22—41no—
557C22—2681no—
558C22—2681no—
559C22—2681no—
560C22C2×C4×C22⋊D6711yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—2681no—
569C4—2681no—
570C4—2681no—
571C4C2×#1452405111yescharacteristic
572C4#1686416211yescharacteristic
573C4#1686419411yescharacteristic
574C4#1686419411yescharacteristic
575C2—41no—
576C2—42no—
577C2—5361no—
578C2#2256893411yescharacteristic
579C2#2256900711yescharacteristic
580C2#2256911611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#2970C2
socleC22×C67C2×C4×D4
prime radical#16863541C4
cosocleC42×C67C24
hypercenter#2970D67
hypocenterC67#18002
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #22569213C2semidirect
3. #22559116C2semidirect
4. #22559109C2semidirect
5. #22559078C2semidirect
6. #22559073C2semidirect
7. #22559066C2semidirect
8. #22559064C2semidirect
9. #22559031C2semidirect
10. #22559021C2semidirect
11. #22558990C2semidirect
12. #22558667C2semidirect
13. #22558591C2semidirect
14. #22558496C2semidirect
15. C67×#2970C2semidirect
292. #2970D67semidirect
306. C67#18002semidirect

Maximal subgroups

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

subgroupindex
#225702272
#225692132
#225591162
#225591092
#225590782
#225590732
#225590662
#225590642
#225590312
#225590212
#225589902
#225586672
#225585912
#225584962
C67×#29702
#1800267