Finite Group Databasebeta

← #30826974

Subgroups — #30826974

Order 17152.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22569229C211yesmaximal, characteristic
2#22569158C211yesmaximal, characteristic, complemented
3#22559112C211yesmaximal, characteristic, complemented
4#22559100C211yesmaximal, characteristic, complemented
5#22559088C211yesmaximal, characteristic, complemented
6#22559081C211yesmaximal, characteristic, complemented
7#22559060C211yesmaximal, characteristic, complemented
8#22559053C211yesmaximal, characteristic, complemented
9#22559028C211yesmaximal, characteristic, complemented
10Q16×C67⋊C8C211yesmaximal, characteristic, complemented
11#22558994C211yesmaximal, characteristic, complemented
12#22558682C211yesmaximal, characteristic, complemented
13#22558596C211yesmaximal, characteristic
14#22558504C211yesmaximal, characteristic, complemented
15C67×#3107C211yesmaximal, characteristic
16#16863527—21no—
17#16863527—21no—
18#16863527C2211yescharacteristic
19#16863527C2211yescharacteristic
20Q8×C67⋊C8C2211yescharacteristic
21Q8×C67⋊C8C2211yescharacteristic
22Q8×C67⋊C8C2211yescharacteristic
23Q8×C67⋊C8C2211yescharacteristic
24#16863474—21no—
25#16863474—21no—
26#16863474C2211yescharacteristic
27#16863474C2211yescharacteristic
28#16863453—21no—
29#16863453C2211yescharacteristic
30#16863449C2211yescharacteristic
31#16863423—21no—
32C2×C4⋊(C67⋊C8)C2211yescharacteristic
33#16863352—21no—
34C4×#11761612—21no—
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
53C4×#41×C67C2211yescharacteristic
54#485×C67C411yescharacteristic
55#485×C67C411yescharacteristic
56#475×C67C411yescharacteristic
57#474×C67C411yescharacteristic
58#462×C67C411yescharacteristic
59#461×C67C411yescharacteristic
60#452×C67C411yescharacteristic
61#452×C67C411yescharacteristic
62C4×Q16×C67C2211yescharacteristic
63C4×Q16×C67C2211yescharacteristic
64C4×SD16×C67C2211yescharacteristic
65C4×SD16×C67C2211yescharacteristic
66#431×C67C2211yescharacteristic
67C2×#11761612—21no—
68C2×#11761612—21no—
69C2×#11761612D411yescharacteristic
70C2×#11761612D411yescharacteristic
71C22×C67⋊C8—21no—
72C4⋊(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—
87C22⋊(C67⋊C8)—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×#41×C67C2×C411yescharacteristic
102C2×Q16×C67C2×C411yescharacteristic
103C2×Q16×C67C2×C411yescharacteristic
104C2×SD16×C67C2×C411yescharacteristic
105C2×SD16×C67C2×C411yescharacteristic
106C4⋊Q8×C67C2×C411yescharacteristic
107C4⋊Q8×C67C2×C411yescharacteristic
108#125×C67—21no—
109C22⋊Q8×C67—21no—
110C22⋊Q8×C67C2×C411yescharacteristic
111C22⋊Q8×C67C2×C411yescharacteristic
112C4×Q8×C67—21no—
113C4×Q8×C67C2×C411yescharacteristic
114C4×Q8×C67C2311yescharacteristic
115C4×Q8×C67C2311yescharacteristic
116C4×Q8×C67C2311yescharacteristic
117C4×D4×C67—21no—
118C4×D4×C67—21no—
119C4×D4×C67C2×C411yescharacteristic
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#103×C67C2×C411yescharacteristic
137#103×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—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⋊C8C2×D411yescharacteristic
156C2×C67⋊C8C2×D411yescharacteristic
157C2×C67⋊C8C2×D411yescharacteristic
158C2×C67⋊C8C2×D411yescharacteristic
159#41×C67—22no—
160#41×C67—41no—
161C2×Q8×C67—21no—
162C2×Q8×C67C22×C411yescharacteristic
163C2×Q8×C67C22×C411yescharacteristic
164C2×Q8×C67C22×C411yescharacteristic
165C2×D4×C67—21no—
166C2×D4×C67C22×C411yescharacteristic
167C22×C4×C67—21no—
168C22×C4×C67—21no—
169C22×C4×C67—21no—
170C22×C4×C67C22×C411yescharacteristic
171C22×C4×C67C2×D411yescharacteristic
172C22×C4×C67#4111yescharacteristic
173Q16×C67—22no—
174Q16×C67—22no—
175SD16×C67—41no—
176SD16×C67—41no—
177C2×C8×C67C22×C411yescharacteristic
178C2×C8×C67C22×C411yescharacteristic
179C2×C8×C67C22×C411yescharacteristic
180C2×C8×C67C22×C411yescharacteristic
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×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—
199C42×C67—21no—
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67C22×C411yescharacteristic
205C42×C67C2×D411yescharacteristic
206C42×C67#4111yescharacteristic
207C42×C67C2411yescharacteristic
208C23×C67—21no—
209C23×C67C4×D411yescharacteristic
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—21no—
214Q8×C67—21no—
215Q8×C67—81no—
216Q8×C67#13112yes—
217D4×C67—21no—
218D4×C67—21no—
219D4×C67—42no—
220C2×C4×C67—21no—
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—41no—
232C2×C4×C67—42no—
233C2×C4×C67—41no—
234C2×C4×C67—41no—
235C2×C4×C67—81no—
236C2×C4×C67C4×D411yescharacteristic
237C2×C4×C67C4×D411yescharacteristic
238C2×C4×C67C4×D411yescharacteristic
239C2×C4×C67C23×C411yescharacteristic
240C2×C4×C67C22×D411yescharacteristic
241C2×C4×C67C2×#4111yescharacteristic
242C8×C67—21no—
243C8×C67—21no—
244C8×C67—21no—
245C8×C67—21no—
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—41no—
255C67⋊C8#13712yes—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—21no—
259C22×C67—41no—
260C22×C67—41no—
261C22×C67—41no—
262C22×C67C2×C4×D411yescharacteristic
263C4×C67—21no—
264C4×C67—22no—
265C4×C67—21no—
266C4×C67—21no—
267C4×C67—21no—
268C4×C67—41no—
269C4×C67—41no—
270C4×C67—41no—
271C4×C67—41no—
272C4×C67—41no—
273C4×C67—41no—
274C4×C67C2×#13111yescharacteristic
275C4×C67#56811yescharacteristic
276C4×C67C2×#13711yescharacteristic
277C4×C67#57511yescharacteristic
278#18013—671nomaximal, Sylow
279C2×C67—41no—
280C2×C67—81no—
281C2×C67#293311yescharacteristic
282C2×C67C4×#13711yescharacteristic
283C2×C67#295711yescharacteristic
284#3107D6711yescharacteristic
285#2971—671no—
286#2962—671no—
287#1592—671no—
288#1590—671no—
289#1588—671no—
290#1587—671no—
291#1584—671no—
292#1583—671no—
293#1579—671no—
294C8×Q16—671no—
295#1573—671no—
296#1497—671no—
297#1477—671no—
298#1453—671no—
299C67#1801311yescharacteristic, complemented, Sylow
300#533C2×D6711yescharacteristic
301C4×#41C2×D6711yescharacteristic
302#485Dic6711yescharacteristic
303#485Dic6711yescharacteristic
304#475Dic6711yescharacteristic
305#474Dic6711yescharacteristic
306#462Dic6711yescharacteristic
307#461Dic6711yescharacteristic
308#452Dic6711yescharacteristic
309#452Dic6711yescharacteristic
310#446—671no—
311#446—671no—
312#446—1341no—
313#446—1341no—
314C8×Q8—671no—
315C8×Q8—671no—
316C8×Q8—671no—
317C8×Q8—671no—
318C4×Q16C2×D6711yescharacteristic
319C4×Q16C2×D6711yescharacteristic
320C4×SD16C2×D6711yescharacteristic
321C4×SD16C2×D6711yescharacteristic
322#436—671no—
323#436—671no—
324#436—1341no—
325#436—1341no—
326#432—671no—
327#432—1341no—
328#431—671no—
329#431C2×D6711yescharacteristic
330#424—1341no—
331C2×C4⋊C8—671no—
332#405—1341no—
333C4×M16—671no—
334C4×M16—1341no—
335#335—671no—
336#335—671no—
337#334—671no—
338#334—671no—
339Q8⋊C8—671no—
340Q8⋊C8—671no—
341Q8⋊C8—671no—
342Q8⋊C8—671no—
343Q8⋊C8—671no—
344Q8⋊C8—671no—
345D4⋊C8—671no—
346D4⋊C8—671no—
347#322—671no—
348#322—671no—
349#322—671no—
350C82—671no—
351C2×#41C2×Dic6711yescharacteristic
352C2×Q16C2×Dic6711yescharacteristic
353C2×Q16C2×Dic6711yescharacteristic
354C2×SD16C2×Dic6711yescharacteristic
355C2×SD16C2×Dic6711yescharacteristic
356C2×M16—671no—
357C2×M16—671no—
358C2×M16—1341no—
359C2×M16—1341no—
360C22×C8—1341no—
361C4⋊Q8C2×Dic6711yescharacteristic
362C4⋊Q8C2×Dic6711yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C2×Dic6711yescharacteristic
366C22⋊Q8C2×Dic6711yescharacteristic
367C4×Q8—21no—
368C4×Q8C2×Dic6711yescharacteristic
369C4×Q8C22×D6711yescharacteristic
370C4×Q8C22×D6711yescharacteristic
371C4×Q8C22×D6711yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C2×Dic6711yescharacteristic
375C4×D4C22×D6711yescharacteristic
376#117—21no—
377#117—21no—
378#117C2×Dic6711yescharacteristic
379C2×C42—21no—
380C2×C42C22×D6711yescharacteristic
381#107C2×Dic6711yescharacteristic
382#107C2×Dic6711yescharacteristic
383#106C2×Dic6711yescharacteristic
384#106C2×Dic6711yescharacteristic
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8—1341no—
397C4⋊C8C2×Dic6711yescharacteristic
398C4⋊C8C2×Dic6711yescharacteristic
399#103C2×Dic6711yescharacteristic
400#103C2×Dic6711yescharacteristic
401#103C2×Dic6711yescharacteristic
402#103C2×Dic6711yescharacteristic
403#103C2×Dic6711yescharacteristic
404#103C2×Dic6711yescharacteristic
405#102C2×Dic6711yescharacteristic
406#102C2×Dic6711yescharacteristic
407C22⋊C8—1341no—
408C22⋊C8—1341no—
409C22⋊C8—1341no—
410C22⋊C8—1341no—
411C22⋊C8C2×Dic6711yescharacteristic
412C22⋊C8C2×Dic6711yescharacteristic
413#97—671no—
414#97—671no—
415#97—1341no—
416#97—1341no—
417#97—1341no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—1341no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8—1341no—
426C4×C8C22×D6711yescharacteristic
427C4×C8C22×D6711yescharacteristic
428#41—22no—
429#41—41no—
430C2×Q8—21no—
431C2×Q8C22×Dic6711yescharacteristic
432C2×Q8C22×Dic6711yescharacteristic
433C2×Q8C22×Dic6711yescharacteristic
434C2×D4—21no—
435C2×D4C22×Dic6711yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4C22×Dic6711yescharacteristic
440C22×C4D4×D6711yescharacteristic
441C22×C4#1176162711yescharacteristic
442Q16—22no—
443Q16—22no—
444SD16—41no—
445SD16—41no—
446M16—2682no—
447M16—2682no—
448M16—2681no—
449M16—1342no—
450C2×C8—2682no—
451C2×C8—671no—
452C2×C8—671no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—1341no—
456C2×C8—1341no—
457C2×C8—1341no—
458C2×C8—1341no—
459C2×C8—1341no—
460C2×C8—1341no—
461C2×C8C22×Dic6711yescharacteristic
462C2×C8C22×Dic6711yescharacteristic
463C2×C8C22×Dic6711yescharacteristic
464C2×C8C22×Dic6711yescharacteristic
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—
483C42—21no—
484C42—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—81no—
500Q8#1452405012yes—
501D4—21no—
502D4—21no—
503D4—42no—
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—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4D4×Dic6711yescharacteristic
521C2×C4D4×Dic6711yescharacteristic
522C2×C4D4×Dic6711yescharacteristic
523C2×C4C23×Dic6711yescharacteristic
524C2×C4C2×D4×D6711yescharacteristic
525C2×C4C2×#1176162711yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—2681no—
531C8—2681no—
532C8—2681no—
533C8—2681no—
534C8—2681no—
535C8—2681no—
536C8—672no—
537C8—1341no—
538C8—1341no—
539C8—1341no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×D4×Dic6711yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4C2×#1452405011yescharacteristic
559C4#1686416311yescharacteristic
560C4#137×D6711yescharacteristic
561C4#1686419511yescharacteristic
562C2—41no—
563C2—81no—
564C2#2256893611yescharacteristic
565C2#137×Dic6711yescharacteristic
566C2#2256912611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#3107C2
socleC22×C67C2×C4×D4
prime radicalC2×#41×C67C2×C4
cosocleC42×C67C24
hypercenter#3107D67
hypocenterC67#18013
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #22569158C2semidirect
3. #22559112C2semidirect
4. #22559100C2semidirect
5. #22559088C2semidirect
6. #22559081C2semidirect
7. #22559060C2semidirect
8. #22559053C2semidirect
9. #22559028C2semidirect
10. Q16×C67⋊C8C2semidirect
11. #22558994C2semidirect
12. #22558682C2semidirect
14. #22558504C2semidirect
299. C67#18013semidirect

Maximal subgroups

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

subgroupindex
#225692292
#225691582
#225591122
#225591002
#225590882
#225590812
#225590602
#225590532
#225590282
Q16×C67⋊C82
#225589942
#225586822
#225585962
#225585042
C67×#31072
#1801367