Finite Group Databasebeta

← #30827702

Subgroups — #30827702

Order 17152.

588 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#22569214C211yesmaximal, characteristic
2#22569158C211yesmaximal, characteristic, complemented
3#22559112C211yesmaximal, characteristic, complemented
4#22559106C211yesmaximal, characteristic, complemented
5#22559088C211yesmaximal, characteristic, complemented
6#22559081C211yesmaximal, characteristic, complemented
7#22559060C211yesmaximal, characteristic, complemented
8#22559046C211yesmaximal, characteristic, complemented
9#22559028C211yesmaximal, characteristic, complemented
10SD16×C67⋊C8C211yesmaximal, characteristic, complemented
11#22558961C211yesmaximal, characteristic, complemented
12#22558630C211yesmaximal, characteristic, complemented
13#22558622C211yesmaximal, characteristic
14#22558512C211yesmaximal, characteristic, complemented
15C67×#3172C211yesmaximal, characteristic
16#16863527—21no—
17#16863527—21no—
18#16863527C2211yescharacteristic
19#16863527C2211yescharacteristic
20#16863527C2211yescharacteristic
21Q8×C67⋊C8C2211yescharacteristic
22Q8×C67⋊C8C2211yescharacteristic
23Q8×C67⋊C8C2211yescharacteristic
24#16863474—21no—
25#16863474C2211yescharacteristic
26#16863468—21no—
27D4×C67⋊C8C2211yescharacteristic
28#16863457—21no—
29#16863449C2211yescharacteristic
30#16863423—21no—
31C2×C4⋊(C67⋊C8)C2211yescharacteristic
32#16863352—21no—
33#16863352—21no—
34C4×#11761612C2211yescharacteristic
35C4×#11761612C2211yescharacteristic
36#16863149C2211yescharacteristic
37#16863149C2211yescharacteristic
38#16863147C2211yescharacteristic
39#16863147C2211yescharacteristic
40Q8⋊(C67⋊C8)C2211yescharacteristic
41Q8⋊(C67⋊C8)C2211yescharacteristic
42Q8⋊(C67⋊C8)C2211yescharacteristic
43Q8⋊(C67⋊C8)C2211yescharacteristic
44Q8⋊(C67⋊C8)C2211yescharacteristic
45Q8⋊(C67⋊C8)C2211yescharacteristic
46D4⋊(C67⋊C8)C2211yescharacteristic
47D4⋊(C67⋊C8)C2211yescharacteristic
48#16863118C2211yescharacteristic
49#16863118C2211yescharacteristic
50#16863117C2211yescharacteristic
51C8×C67⋊C8C2211yescharacteristic
52#533×C67C2211yescharacteristic
53#529×C67C2211yescharacteristic
54#501×C67C411yescharacteristic
55#497×C67C411yescharacteristic
56#491×C67C411yescharacteristic
57#489×C67C411yescharacteristic
58#470×C67C411yescharacteristic
59#470×C67C411yescharacteristic
60#468×C67C411yescharacteristic
61#468×C67C411yescharacteristic
62C4×Q16×C67C2211yescharacteristic
63C4×Q16×C67C2211yescharacteristic
64C4×SD16×C67C2211yescharacteristic
65C4×SD16×C67C2211yescharacteristic
66C4×M16×C67C2211yescharacteristic
67C2×#11761612—21no—
68C2×#11761612—21no—
69C2×#11761612D411yescharacteristic
70C2×#11761612D411yescharacteristic
71C22×C67⋊C8—21no—
72C22×C67⋊C8—21no—
73C4⋊(C67⋊C8)—21no—
74C4⋊(C67⋊C8)—21no—
75C4⋊(C67⋊C8)—21no—
76C4⋊(C67⋊C8)—21no—
77C4⋊(C67⋊C8)—21no—
78C4⋊(C67⋊C8)—21no—
79C4⋊(C67⋊C8)—21no—
80C4⋊(C67⋊C8)C2311yescharacteristic
81C4⋊(C67⋊C8)C2311yescharacteristic
82C4⋊(C67⋊C8)C2311yescharacteristic
83C4⋊(C67⋊C8)C2311yescharacteristic
84C22⋊(C67⋊C8)—21no—
85C22⋊(C67⋊C8)—21no—
86C22⋊(C67⋊C8)—21no—
87#14523954—21no—
88#14523954—21no—
89#14523954—21no—
90#14523954—21no—
91#14523954D411yescharacteristic
92#14523954D411yescharacteristic
93C4×C67⋊C8—21no—
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×Q16×C67C2×C411yescharacteristic
102C2×Q16×C67C2×C411yescharacteristic
103C2×SD16×C67C2×C411yescharacteristic
104C2×SD16×C67C2×C411yescharacteristic
105C2×M16×C67C2×C411yescharacteristic
106C2×M16×C67C2×C411yescharacteristic
107C4⋊Q8×C67C2×C411yescharacteristic
108C4⋊Q8×C67C2×C411yescharacteristic
109#126×C67—21no—
110#125×C67—21no—
111#125×C67C2×C411yescharacteristic
112#124×C67C2×C411yescharacteristic
113#123×C67—21no—
114C22⋊Q8×C67—21no—
115C22⋊Q8×C67C2×C411yescharacteristic
116C22⋊Q8×C67C2×C411yescharacteristic
117C22⋊Q8×C67C2×C411yescharacteristic
118#121×C67C2×C411yescharacteristic
119C4×Q8×C67—21no—
120C4×Q8×C67C2311yescharacteristic
121C4×Q8×C67C2311yescharacteristic
122C4×Q8×C67C2311yescharacteristic
123C4×D4×C67—21no—
124C4×D4×C67C2311yescharacteristic
125#117×C67—21no—
126#117×C67—21no—
127C2×C42×C67C2311yescharacteristic
128#107×C67C2×C411yescharacteristic
129#107×C67C2×C411yescharacteristic
130#106×C67C2×C411yescharacteristic
131#106×C67C2×C411yescharacteristic
132#103×C67C2×C411yescharacteristic
133#103×C67C2×C411yescharacteristic
134#103×C67C2×C411yescharacteristic
135#103×C67C2×C411yescharacteristic
136#103×C67C2×C411yescharacteristic
137#103×C67C2×C411yescharacteristic
138#102×C67C2×C411yescharacteristic
139#102×C67C2×C411yescharacteristic
140#97×C67C2×C411yescharacteristic
141#97×C67C2×C411yescharacteristic
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144#11761612—22no—
145#11761612—41no—
146#11761612—42no—
147#11761612—42no—
148C2×C67⋊C8—21no—
149C2×C67⋊C8—21no—
150C2×C67⋊C8—21no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8—21no—
154C2×C67⋊C8—42no—
155C2×C67⋊C8—42no—
156C2×C67⋊C8C2×D411yescharacteristic
157C2×C67⋊C8C2×D411yescharacteristic
158C2×C67⋊C8C2×D411yescharacteristic
159C2×C67⋊C8C2×D411yescharacteristic
160C2×Q8×C67—21no—
161C2×Q8×C67C22×C411yescharacteristic
162C2×Q8×C67C22×C411yescharacteristic
163C2×Q8×C67C22×C411yescharacteristic
164C2×D4×C67—21no—
165C2×D4×C67C22×C411yescharacteristic
166C22×C4×C67—21no—
167C22×C4×C67C22×C411yescharacteristic
168C22×C4×C67C2×D411yescharacteristic
169C22×C4×C67#4111yescharacteristic
170Q16×C67—22no—
171Q16×C67—41no—
172SD16×C67—22no—
173SD16×C67—41no—
174M16×C67—22no—
175M16×C67—41no—
176C2×C8×C67C22×C411yescharacteristic
177C2×C8×C67C22×C411yescharacteristic
178C2×C8×C67C22×C411yescharacteristic
179C2×C8×C67C22×C411yescharacteristic
180C4⋊C4×C67—21no—
181C4⋊C4×C67—21no—
182C4⋊C4×C67—21no—
183C4⋊C4×C67—21no—
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×C67—21no—
191C4⋊C4×C67C22×C411yescharacteristic
192C4⋊C4×C67C22×C411yescharacteristic
193C4⋊C4×C67C22×C411yescharacteristic
194C4⋊C4×C67C22×C411yescharacteristic
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×C67C22×C411yescharacteristic
206C42×C67C2×D411yescharacteristic
207C42×C67#4111yescharacteristic
208C42×C67C2411yescharacteristic
209C23×C67—21no—
210C23×C67C4×D411yescharacteristic
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—21no—
216Q8×C67—21no—
217Q8×C67—81no—
218D4×C67—21no—
219D4×C67—21no—
220D4×C67—81no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
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—41no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67C4×D411yescharacteristic
236C2×C4×C67C4×D411yescharacteristic
237C2×C4×C67C4×D411yescharacteristic
238C2×C4×C67C23×C411yescharacteristic
239C2×C4×C67C22×D411yescharacteristic
240C2×C4×C67C2×#4111yescharacteristic
241C8×C67—21no—
242C8×C67—21no—
243C8×C67—21no—
244C8×C67#13112yes—
245C67⋊C8—22no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—41no—
250C67⋊C8—41no—
251C67⋊C8—41no—
252C67⋊C8—41no—
253C67⋊C8—41no—
254C67⋊C8#13712yes—
255C22×C67—21no—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—41no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67C2×C4×D411yescharacteristic
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—21no—
265C4×C67—21no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67C2×#13111yescharacteristic
274C4×C67#56811yescharacteristic
275C4×C67C2×#13711yescharacteristic
276C4×C67#57511yescharacteristic
277#18077—671nomaximal, Sylow
278C2×C67—41no—
279C2×C67—81no—
280C2×C67#293511yescharacteristic
281C2×C67C4×#13711yescharacteristic
282C2×C67#295611yescharacteristic
283#3172D6711yescharacteristic
284#2969—671no—
285#2962—671no—
286#1592—671no—
287#1591—671no—
288#1588—671no—
289#1587—671no—
290#1584—671no—
291#1582—671no—
292#1579—671no—
293C8×SD16—671no—
294#1565—671no—
295#1485—671no—
296#1483—671no—
297#1455—671no—
298C67#1807711yescharacteristic, complemented, Sylow
299#533C2×D6711yescharacteristic
300#529C2×D6711yescharacteristic
301#501Dic6711yescharacteristic
302#497Dic6711yescharacteristic
303#491Dic6711yescharacteristic
304#489Dic6711yescharacteristic
305#470Dic6711yescharacteristic
306#470Dic6711yescharacteristic
307#468Dic6711yescharacteristic
308#468Dic6711yescharacteristic
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×Q16C2×D6711yescharacteristic
318C4×Q16C2×D6711yescharacteristic
319C4×SD16C2×D6711yescharacteristic
320C4×SD16C2×D6711yescharacteristic
321#436—671no—
322#436—1341no—
323#435—1341no—
324C8×D4—671no—
325#433—1341no—
326#431—671no—
327#424—1341no—
328C2×C4⋊C8—671no—
329#405—1341no—
330#405—1341no—
331C4×M16—671no—
332C4×M16—671no—
333C4×M16C2×D6711yescharacteristic
334#335—671no—
335#335—671no—
336#334—671no—
337#334—671no—
338Q8⋊C8—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344D4⋊C8—671no—
345D4⋊C8—671no—
346#322—671no—
347#322—671no—
348#322—671no—
349C82—671no—
350C2×Q16C2×Dic6711yescharacteristic
351C2×Q16C2×Dic6711yescharacteristic
352C2×SD16C2×Dic6711yescharacteristic
353C2×SD16C2×Dic6711yescharacteristic
354C2×M16—671no—
355C2×M16—671no—
356C2×M16—1341no—
357C2×M16—1341no—
358C2×M16C2×Dic6711yescharacteristic
359C2×M16C2×Dic6711yescharacteristic
360C22×C8—1341no—
361C22×C8—1341no—
362C4⋊Q8C2×Dic6711yescharacteristic
363C4⋊Q8C2×Dic6711yescharacteristic
364#126—21no—
365#125—21no—
366#125C2×Dic6711yescharacteristic
367#124C2×Dic6711yescharacteristic
368#123—21no—
369C22⋊Q8—21no—
370C22⋊Q8C2×Dic6711yescharacteristic
371C22⋊Q8C2×Dic6711yescharacteristic
372C22⋊Q8C2×Dic6711yescharacteristic
373#121C2×Dic6711yescharacteristic
374C4×Q8—21no—
375C4×Q8C22×D6711yescharacteristic
376C4×Q8C22×D6711yescharacteristic
377C4×Q8C22×D6711yescharacteristic
378C4×D4—21no—
379C4×D4C22×D6711yescharacteristic
380#117—21no—
381#117—21no—
382C2×C42C22×D6711yescharacteristic
383#107C2×Dic6711yescharacteristic
384#107C2×Dic6711yescharacteristic
385#106C2×Dic6711yescharacteristic
386#106C2×Dic6711yescharacteristic
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—671no—
390C4⋊C8—671no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8—1341no—
397C4⋊C8—1341no—
398#103C2×Dic6711yescharacteristic
399#103C2×Dic6711yescharacteristic
400#103C2×Dic6711yescharacteristic
401#103C2×Dic6711yescharacteristic
402#103C2×Dic6711yescharacteristic
403#103C2×Dic6711yescharacteristic
404#102C2×Dic6711yescharacteristic
405#102C2×Dic6711yescharacteristic
406C22⋊C8—1341no—
407C22⋊C8—1341no—
408C22⋊C8—1341no—
409#97—671no—
410#97—671no—
411#97—1341no—
412#97—1341no—
413#97—1341no—
414#97—1341no—
415#97C2×Dic6711yescharacteristic
416#97C2×Dic6711yescharacteristic
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8C22×D6711yescharacteristic
426C4×C8C22×D6711yescharacteristic
427C2×Q8—21no—
428C2×Q8C22×Dic6711yescharacteristic
429C2×Q8C22×Dic6711yescharacteristic
430C2×Q8C22×Dic6711yescharacteristic
431C2×D4—21no—
432C2×D4C22×Dic6711yescharacteristic
433C22×C4—21no—
434C22×C4C22×Dic6711yescharacteristic
435C22×C4D4×D6711yescharacteristic
436C22×C4#1176162711yescharacteristic
437Q16—22no—
438Q16—41no—
439SD16—22no—
440SD16—41no—
441M16—22no—
442M16—41no—
443M16—2682no—
444M16—2682no—
445M16—2681no—
446M16—1342no—
447C2×C8—2682no—
448C2×C8—2682no—
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×C8—1341no—
458C2×C8—1341no—
459C2×C8C22×Dic6711yescharacteristic
460C2×C8C22×Dic6711yescharacteristic
461C2×C8C22×Dic6711yescharacteristic
462C2×C8C22×Dic6711yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4C22×Dic6711yescharacteristic
475C4⋊C4C22×Dic6711yescharacteristic
476C4⋊C4C22×Dic6711yescharacteristic
477C4⋊C4C22×Dic6711yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42C22×Dic6711yescharacteristic
489C42D4×D6711yescharacteristic
490C42#1176162711yescharacteristic
491C42C23×D6711yescharacteristic
492C23—21no—
493C23D4×Dic6711yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—21no—
500Q8—81no—
501D4—21no—
502D4—21no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4D4×Dic6711yescharacteristic
519C2×C4D4×Dic6711yescharacteristic
520C2×C4D4×Dic6711yescharacteristic
521C2×C4C23×Dic6711yescharacteristic
522C2×C4C2×D4×D6711yescharacteristic
523C2×C4C2×#1176162711yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—2681no—
528C8—2681no—
529C8—2681no—
530C8—2681no—
531C8—2681no—
532C8—672no—
533C8—1342no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8#1452405012yes—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22C2×D4×Dic6711yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4C2×#1452405011yescharacteristic
557C4#1686416311yescharacteristic
558C4#137×D6711yescharacteristic
559C4#1686419511yescharacteristic
560C2—41no—
561C2—81no—
562C2#2256895311yescharacteristic
563C2#137×Dic6711yescharacteristic
564C2#2256911911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#3172C2
socleC22×C67C2×C4×D4
prime radical#121×C67C2×C4
cosocleC42×C67C24
hypercenter#3172D67
hypocenterC67#18077
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #22569158C2semidirect
3. #22559112C2semidirect
4. #22559106C2semidirect
5. #22559088C2semidirect
6. #22559081C2semidirect
7. #22559060C2semidirect
8. #22559046C2semidirect
9. #22559028C2semidirect
10. SD16×C67⋊C8C2semidirect
11. #22558961C2semidirect
12. #22558630C2semidirect
14. #22558512C2semidirect
298. C67#18077semidirect

Maximal subgroups

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

subgroupindex
#225692142
#225691582
#225591122
#225591062
#225590882
#225590812
#225590602
#225590462
#225590282
SD16×C67⋊C82
#225589612
#225586302
#225586222
#225585122
C67×#31722
#1807767