Finite Group Databasebeta

← #30826817

Subgroups — #30826817

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#22569223C211yesmaximal, characteristic, complemented
2#22569214C211yesmaximal, characteristic, complemented
3#22559118C211yesmaximal, characteristic, complemented
4#22559112C211yesmaximal, characteristic, complemented
5#22559081C211yesmaximal, characteristic, complemented
6#22559074C211yesmaximal, characteristic, complemented
7#22559067C211yesmaximal, characteristic, complemented
8#22559067C211yesmaximal, characteristic, complemented
9#22559033C211yesmaximal, characteristic, complemented
10SD16⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
11#22558989C211yesmaximal, characteristic, complemented
12#22558669C211yesmaximal, characteristic, complemented
13#22558591C211yesmaximal, characteristic, complemented
14#22558499C211yesmaximal, characteristic, complemented
15C67×#3102C211yesmaximal, characteristic
16#16863527C2211yescharacteristic
17#16863527C2211yescharacteristic
18#16863527C2211yescharacteristic
19Q8×C67⋊C8C2211yescharacteristic
20#16863474—21no—
21#16863474C2211yescharacteristic
22#16863468—21no—
23#16863468—21no—
24#16863468—21no—
25D4×C67⋊C8C2211yescharacteristic
26D4×C67⋊C8C2211yescharacteristic
27D4×C67⋊C8C2211yescharacteristic
28#16863457—21no—
29#16863453—21no—
30#16863453C2211yescharacteristic
31#16863449C2211yescharacteristic
32#16863352—21no—
33C4×#11761612C2211yescharacteristic
34C2×#14523954—21no—
35C2×C4×C67⋊C8C2211yescharacteristic
36#16863149C2211yescharacteristic
37#16863149C2211yescharacteristic
38#16863147C2211yescharacteristic
39#16863147C2211yescharacteristic
40Q8⋊(C67⋊C8)C2211yescharacteristic
41Q8⋊(C67⋊C8)C2211yescharacteristic
42Q8⋊(C67⋊C8)C2211yescharacteristic
43Q8⋊(C67⋊C8)C2211yescharacteristic
44D4⋊(C67⋊C8)C2211yescharacteristic
45D4⋊(C67⋊C8)C2211yescharacteristic
46D4⋊(C67⋊C8)C2211yescharacteristic
47D4⋊(C67⋊C8)C2211yescharacteristic
48#16863118C2211yescharacteristic
49#16863118C2211yescharacteristic
50#16863117C2211yescharacteristic
51#16863117C2211yescharacteristic
52#529×C67C2211yescharacteristic
53C4×#41×C67C2211yescharacteristic
54#485×C67C411yescharacteristic
55#482×C67C411yescharacteristic
56#479×C67C411yescharacteristic
57#478×C67C411yescharacteristic
58#464×C67C411yescharacteristic
59#463×C67C411yescharacteristic
60#452×C67C411yescharacteristic
61#449×C67C411yescharacteristic
62C4×Q16×C67C2211yescharacteristic
63C4×SD16×C67C2211yescharacteristic
64C4×SD16×C67C2211yescharacteristic
65C4×D8×C67C2211yescharacteristic
66#431×C67C2211yescharacteristic
67C2×#11761612—21no—
68C2×#11761612D411yescharacteristic
69C2×#11761612D411yescharacteristic
70C22×C67⋊C8—21no—
71C22×C67⋊C8—21no—
72C22×C67⋊C8—21no—
73C22×C67⋊C8—21no—
74C4⋊(C67⋊C8)—21no—
75C4⋊(C67⋊C8)—21no—
76C4⋊(C67⋊C8)—21no—
77C4⋊(C67⋊C8)—21no—
78C4⋊(C67⋊C8)C2311yescharacteristic
79C4⋊(C67⋊C8)C2311yescharacteristic
80C4⋊(C67⋊C8)C2311yescharacteristic
81C4⋊(C67⋊C8)C2311yescharacteristic
82C22⋊(C67⋊C8)—21no—
83C22⋊(C67⋊C8)—21no—
84C22⋊(C67⋊C8)—21no—
85C22⋊(C67⋊C8)—21no—
86C22⋊(C67⋊C8)—21no—
87C22⋊(C67⋊C8)—21no—
88#14523954—21no—
89#14523954—21no—
90#14523954—21no—
91#14523954—21no—
92#14523954D411yescharacteristic
93#14523954D411yescharacteristic
94C4×C67⋊C8—21no—
95C4×C67⋊C8—21no—
96C4×C67⋊C8—21no—
97C4×C67⋊C8C2311yescharacteristic
98C4×C67⋊C8C2311yescharacteristic
99C4×C67⋊C8C2311yescharacteristic
100C4×C67⋊C8C2311yescharacteristic
101C2×#41×C67C2×C411yescharacteristic
102C2×Q16×C67C2×C411yescharacteristic
103C2×SD16×C67C2×C411yescharacteristic
104C2×SD16×C67C2×C411yescharacteristic
105C2×D8×C67C2×C411yescharacteristic
106#126×C67—21no—
107#125×C67C2×C411yescharacteristic
108#124×C67C2×C411yescharacteristic
109#123×C67—21no—
110C22⋊Q8×C67C2×C411yescharacteristic
111#121×C67C2×C411yescharacteristic
112C4×Q8×C67C2×C411yescharacteristic
113C4×Q8×C67C2311yescharacteristic
114C4×Q8×C67C2311yescharacteristic
115C4×D4×C67—21no—
116C4×D4×C67—21no—
117C4×D4×C67—21no—
118C4×D4×C67C2×C411yescharacteristic
119C4×D4×C67C2311yescharacteristic
120C4×D4×C67C2311yescharacteristic
121#117×C67—21no—
122#117×C67—21no—
123#117×C67C2×C411yescharacteristic
124C2×C42×C67—21no—
125C2×C42×C67C2311yescharacteristic
126#107×C67C2×C411yescharacteristic
127#107×C67C2×C411yescharacteristic
128#106×C67C2×C411yescharacteristic
129#106×C67C2×C411yescharacteristic
130C4⋊C8×C67C2×C411yescharacteristic
131C4⋊C8×C67C2×C411yescharacteristic
132#103×C67C2×C411yescharacteristic
133#103×C67C2×C411yescharacteristic
134#103×C67C2×C411yescharacteristic
135#103×C67C2×C411yescharacteristic
136#102×C67C2×C411yescharacteristic
137#102×C67C2×C411yescharacteristic
138#102×C67C2×C411yescharacteristic
139#102×C67C2×C411yescharacteristic
140C22⋊C8×C67C2×C411yescharacteristic
141C22⋊C8×C67C2×C411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#11761612—41no—
145#11761612—41no—
146#11761612—42no—
147C2×C67⋊C8—21no—
148C2×C67⋊C8—21no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8—42no—
154C2×C67⋊C8—42no—
155C2×C67⋊C8—42no—
156C2×C67⋊C8—42no—
157C2×C67⋊C8C2×D411yescharacteristic
158C2×C67⋊C8C2×D411yescharacteristic
159C2×C67⋊C8C2×D411yescharacteristic
160C2×C67⋊C8C2×D411yescharacteristic
161#41×C67—22no—
162#41×C67—41no—
163C2×Q8×C67C22×C411yescharacteristic
164C2×Q8×C67C22×C411yescharacteristic
165C2×D4×C67—21no—
166C2×D4×C67—21no—
167C2×D4×C67C22×C411yescharacteristic
168C2×D4×C67C22×C411yescharacteristic
169C22×C4×C67—21no—
170C22×C4×C67—21no—
171C22×C4×C67—21no—
172C22×C4×C67—21no—
173C22×C4×C67C22×C411yescharacteristic
174C22×C4×C67C2×D411yescharacteristic
175C22×C4×C67#4111yescharacteristic
176Q16×C67—41no—
177SD16×C67—22no—
178SD16×C67—41no—
179D8×C67—22no—
180C2×C8×C67C22×C411yescharacteristic
181C2×C8×C67C22×C411yescharacteristic
182C2×C8×C67C22×C411yescharacteristic
183C2×C8×C67C22×C411yescharacteristic
184C4⋊C4×C67—21no—
185C4⋊C4×C67—21no—
186C4⋊C4×C67—21no—
187C4⋊C4×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67C22×C411yescharacteristic
191C4⋊C4×C67C22×C411yescharacteristic
192C4⋊C4×C67C22×C411yescharacteristic
193C4⋊C4×C67C22×C411yescharacteristic
194C22⋊C4×C67—21no—
195C22⋊C4×C67—21no—
196C22⋊C4×C67—21no—
197C22⋊C4×C67—21no—
198C22⋊C4×C67—21no—
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C22⋊C4×C67—21no—
202C42×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×C67—21no—
212C23×C67C4×D411yescharacteristic
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217D4×C67—21no—
218D4×C67—21no—
219D4×C67—21no—
220D4×C67—42no—
221D4×C67—81no—
222D4×C67#13112yes—
223C2×C4×C67—21no—
224C2×C4×C67—21no—
225C2×C4×C67—21no—
226C2×C4×C67—21no—
227C2×C4×C67—21no—
228C2×C4×C67—21no—
229C2×C4×C67—21no—
230C2×C4×C67—21no—
231C2×C4×C67—21no—
232C2×C4×C67—21no—
233C2×C4×C67—41no—
234C2×C4×C67—42no—
235C2×C4×C67—41no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—41no—
239C2×C4×C67—81no—
240C2×C4×C67C4×D411yescharacteristic
241C2×C4×C67C4×D411yescharacteristic
242C2×C4×C67C4×D411yescharacteristic
243C2×C4×C67C23×C411yescharacteristic
244C2×C4×C67C22×D411yescharacteristic
245C2×C4×C67C2×#4111yescharacteristic
246C8×C67—21no—
247C8×C67—21no—
248C8×C67—21no—
249C8×C67—21no—
250C67⋊C8—22no—
251C67⋊C8—21no—
252C67⋊C8—21no—
253C67⋊C8—21no—
254C67⋊C8—21no—
255C67⋊C8—41no—
256C67⋊C8—41no—
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×C67—41no—
267C22×C67—41no—
268C22×C67—41no—
269C22×C67C2×C4×D411yescharacteristic
270C4×C67—21no—
271C4×C67—21no—
272C4×C67—21no—
273C4×C67—21no—
274C4×C67—41no—
275C4×C67—41no—
276C4×C67—41no—
277C4×C67—41no—
278C4×C67—41no—
279C4×C67—41no—
280C4×C67C2×#13111yescharacteristic
281C4×C67#56811yescharacteristic
282C4×C67#57511yescharacteristic
283C4×C67#57511yescharacteristic
284#18002—671nomaximal, Sylow
285C2×C67—41no—
286C2×C67—42no—
287C2×C67—81no—
288C2×C67#293311yescharacteristic
289C2×C67#294311yescharacteristic
290C2×C67#295611yescharacteristic
291#3102D6711yescharacteristic
292#2970—671no—
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#529C2×D6711yescharacteristic
308C4×#41C2×D6711yescharacteristic
309#485Dic6711yescharacteristic
310#482Dic6711yescharacteristic
311#479Dic6711yescharacteristic
312#478Dic6711yescharacteristic
313#464Dic6711yescharacteristic
314#463Dic6711yescharacteristic
315#452Dic6711yescharacteristic
316#449Dic6711yescharacteristic
317#446—671no—
318#446—671no—
319#446—671no—
320C8×Q8—671no—
321C4×Q16C2×D6711yescharacteristic
322C4×SD16C2×D6711yescharacteristic
323C4×SD16C2×D6711yescharacteristic
324C4×D8C2×D6711yescharacteristic
325#436—671no—
326#436—1341no—
327#435—1341no—
328#435—1341no—
329#435—1341no—
330C8×D4—671no—
331C8×D4—671no—
332C8×D4—671no—
333#433—1341no—
334#432—671no—
335#432—1341no—
336#431—671no—
337#431C2×D6711yescharacteristic
338#405—1341no—
339C4×M16—671no—
340C2×#97—1341no—
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×#41C2×Dic6711yescharacteristic
359C2×Q16C2×Dic6711yescharacteristic
360C2×SD16C2×Dic6711yescharacteristic
361C2×SD16C2×Dic6711yescharacteristic
362C2×D8C2×Dic6711yescharacteristic
363C2×M16—671no—
364C2×M16—671no—
365C2×M16—1341no—
366C22×C8—1341no—
367C22×C8—1341no—
368C22×C8—1341no—
369C22×C8—1341no—
370#126—21no—
371#125C2×Dic6711yescharacteristic
372#124C2×Dic6711yescharacteristic
373#123—21no—
374C22⋊Q8C2×Dic6711yescharacteristic
375#121C2×Dic6711yescharacteristic
376C4×Q8C2×Dic6711yescharacteristic
377C4×Q8C22×D6711yescharacteristic
378C4×Q8C22×D6711yescharacteristic
379C4×D4—21no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4C2×Dic6711yescharacteristic
383C4×D4C22×D6711yescharacteristic
384C4×D4C22×D6711yescharacteristic
385#117—21no—
386#117—21no—
387#117C2×Dic6711yescharacteristic
388C2×C42—21no—
389C2×C42C22×D6711yescharacteristic
390#107C2×Dic6711yescharacteristic
391#107C2×Dic6711yescharacteristic
392#106C2×Dic6711yescharacteristic
393#106C2×Dic6711yescharacteristic
394C4⋊C8—671no—
395C4⋊C8—671no—
396C4⋊C8—671no—
397C4⋊C8—671no—
398C4⋊C8—1341no—
399C4⋊C8—1341no—
400C4⋊C8—1341no—
401C4⋊C8—1341no—
402C4⋊C8C2×Dic6711yescharacteristic
403C4⋊C8C2×Dic6711yescharacteristic
404#103C2×Dic6711yescharacteristic
405#103C2×Dic6711yescharacteristic
406#103C2×Dic6711yescharacteristic
407#103C2×Dic6711yescharacteristic
408#102C2×Dic6711yescharacteristic
409#102C2×Dic6711yescharacteristic
410#102C2×Dic6711yescharacteristic
411#102C2×Dic6711yescharacteristic
412C22⋊C8—1341no—
413C22⋊C8—1341no—
414C22⋊C8—1341no—
415C22⋊C8—1341no—
416C22⋊C8—1341no—
417C22⋊C8—1341no—
418C22⋊C8C2×Dic6711yescharacteristic
419C22⋊C8C2×Dic6711yescharacteristic
420#97—671no—
421#97—671no—
422#97—1341no—
423#97—1341no—
424#97—1341no—
425#97—1341no—
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×Q8C22×Dic6711yescharacteristic
438C2×Q8C22×Dic6711yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C22×Dic6711yescharacteristic
442C2×D4C22×Dic6711yescharacteristic
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4C22×Dic6711yescharacteristic
448C22×C4D4×D6711yescharacteristic
449C22×C4#1176162711yescharacteristic
450Q16—41no—
451SD16—22no—
452SD16—41no—
453D8—22no—
454M16—2682no—
455M16—2681no—
456M16—2681no—
457C2×C8—2682no—
458C2×C8—2682no—
459C2×C8—2682no—
460C2×C8—2682no—
461C2×C8—671no—
462C2×C8—671no—
463C2×C8—671no—
464C2×C8—671no—
465C2×C8—1341no—
466C2×C8—1341no—
467C2×C8—1341no—
468C2×C8—1341no—
469C2×C8—1341no—
470C2×C8—1341no—
471C2×C8C22×Dic6711yescharacteristic
472C2×C8C22×Dic6711yescharacteristic
473C2×C8C22×Dic6711yescharacteristic
474C2×C8C22×Dic6711yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4C22×Dic6711yescharacteristic
482C4⋊C4C22×Dic6711yescharacteristic
483C4⋊C4C22×Dic6711yescharacteristic
484C4⋊C4C22×Dic6711yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42C22×Dic6711yescharacteristic
498C42D4×D6711yescharacteristic
499C42#1176162711yescharacteristic
500C42C23×D6711yescharacteristic
501C23—21no—
502C23—21no—
503C23D4×Dic6711yescharacteristic
504Q8—21no—
505Q8—21no—
506Q8—21no—
507Q8—21no—
508D4—21no—
509D4—21no—
510D4—21no—
511D4—42no—
512D4—81no—
513D4#1452405012yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4D4×Dic6711yescharacteristic
532C2×C4D4×Dic6711yescharacteristic
533C2×C4D4×Dic6711yescharacteristic
534C2×C4C23×Dic6711yescharacteristic
535C2×C4C2×D4×D6711yescharacteristic
536C2×C4C2×#1176162711yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—2681no—
542C8—2681no—
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—41no—
558C22—41no—
559C22—41no—
560C22C2×D4×Dic6711yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4C2×#1452405011yescharacteristic
572C4#1686416311yescharacteristic
573C4#1686419311yescharacteristic
574C4#1686419311yescharacteristic
575C2—41no—
576C2—42no—
577C2—81no—
578C2#2256893611yescharacteristic
579C2#2256901011yescharacteristic
580C2#2256911911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#3102C2
socleC22×C67C2×C4×D4
prime radical#449×C67C4
cosocleC42×C67C24
hypercenter#3102D67
hypocenterC67#18002
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3102

Complements

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

NH ≅ G/Ntype
1. #22569223C2semidirect
2. #22569214C2semidirect
3. #22559118C2semidirect
4. #22559112C2semidirect
5. #22559081C2semidirect
6. #22559074C2semidirect
7. #22559067C2semidirect
8. #22559067C2semidirect
9. #22559033C2semidirect
10. SD16⋊(C67⋊C8)C2semidirect
11. #22558989C2semidirect
12. #22558669C2semidirect
13. #22558591C2semidirect
14. #22558499C2semidirect
306. C67#18002semidirect

Maximal subgroups

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

subgroupindex
#225692232
#225692142
#225591182
#225591122
#225590812
#225590742
#225590672
#225590332
SD16⋊(C67⋊C8)2
#225589892
#225586692
#225585912
#225584992
C67×#31022
#1800267