Finite Group Databasebeta

← #30826951

Subgroups — #30826951

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#22570289C211yesmaximal, characteristic
2#22569207C211yesmaximal, characteristic
3Q8×#11761612C211yesmaximal, characteristic, complemented
4#22559115C211yesmaximal, characteristic, complemented
5#22559096C211yesmaximal, characteristic, complemented
6#22559077C211yesmaximal, characteristic, complemented
7#22559069C211yesmaximal, characteristic, complemented
8#22559062C211yesmaximal, characteristic, complemented
9#22559049C211yesmaximal, characteristic, complemented
10#22559024C211yesmaximal, characteristic, complemented
11#22559013C211yesmaximal, characteristic, complemented
12#22558997C211yesmaximal, characteristic, complemented
13#22558675C211yesmaximal, characteristic
14#22558504C211yesmaximal, characteristic, complemented
15C67×#1479C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863726C411yescharacteristic
18#16863709C411yescharacteristic
19#16863703C411yescharacteristic
20#16863703C411yescharacteristic
21#16863618C411yescharacteristic
22#16863618C411yescharacteristic
23#16863552C411yescharacteristic
24#16863535C411yescharacteristic
25#16863527—21no—
26#16863527—21no—
27#16863527C2211yescharacteristic
28#16863527C2211yescharacteristic
29#16863523C2211yescharacteristic
30Q8×C67⋊C8C2211yescharacteristic
31Q8×C67⋊C8C2211yescharacteristic
32#16863520C2211yescharacteristic
33C4×#11761618C2211yescharacteristic
34C4×#11761618C2211yescharacteristic
35C4×#11761615C2211yescharacteristic
36C4×#11761615C2211yescharacteristic
37#16863473C2211yescharacteristic
38#16863471—21no—
39#16863470—21no—
40#16863461C2211yescharacteristic
41#16863455—21no—
42#16863449C2211yescharacteristic
43#16863420—21no—
44#16863420C2211yescharacteristic
45#16863350—21no—
46C4×#11761612—21no—
47C4×#11761612C2211yescharacteristic
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863118C2211yescharacteristic
57#16863118C2211yescharacteristic
58#16863117C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60C2×C4×Q8×C67C2211yescharacteristic
61#431×C67C2211yescharacteristic
62#431×C67C2211yescharacteristic
63Q8⋊C8×C67C2211yescharacteristic
64Q8⋊C8×C67C2211yescharacteristic
65Q8⋊C8×C67C2211yescharacteristic
66Q8⋊C8×C67C2211yescharacteristic
67C2×#11761618C2×C411yescharacteristic
68C2×#11761618C2×C411yescharacteristic
69C2×#11761615C2×C411yescharacteristic
70C2×#11761615C2×C411yescharacteristic
71C2×#11761612—21no—
72C2×#11761612D411yescharacteristic
73C2×#11761612D411yescharacteristic
74C2×#11761611—21no—
75C2×C8×D67—21no—
76#14524037—21no—
77#14524035C2×C411yescharacteristic
78#14524031C2×C411yescharacteristic
79#14524026—21no—
80#14524022C2×C411yescharacteristic
81#14524017C2×C411yescharacteristic
82C4×Dic134C2311yescharacteristic
83C4×D268C2311yescharacteristic
84C4×C22⋊D67—21no—
85#14524000—21no—
86#14523976C2×C411yescharacteristic
87#14523976C2×C411yescharacteristic
88#14523974C2×C411yescharacteristic
89#14523974C2×C411yescharacteristic
90C4⋊(C67⋊C8)—21no—
91C4⋊(C67⋊C8)—21no—
92C4⋊(C67⋊C8)—21no—
93C4⋊(C67⋊C8)—21no—
94C4⋊(C67⋊C8)—21no—
95C4⋊(C67⋊C8)C2×C411yescharacteristic
96C4⋊(C67⋊C8)C2×C411yescharacteristic
97C4⋊(C67⋊C8)C2311yescharacteristic
98C4⋊(C67⋊C8)C2311yescharacteristic
99Dic67⋊C8—21no—
100Dic67⋊C8—21no—
101#14523967C2×C411yescharacteristic
102#14523967C2×C411yescharacteristic
103#14523967C2×C411yescharacteristic
104#14523967C2×C411yescharacteristic
105#14523966C2×C411yescharacteristic
106#14523966C2×C411yescharacteristic
107#14523963C2×C411yescharacteristic
108#14523963C2×C411yescharacteristic
109C22⋊(C67⋊C8)C2×C411yescharacteristic
110C22⋊(C67⋊C8)C2×C411yescharacteristic
111(C2×D67)⋊C8—21no—
112(C2×D67)⋊C8—21no—
113#14523954—21no—
114#14523954—21no—
115#14523954D411yescharacteristic
116#14523954D411yescharacteristic
117#14523953—21no—
118C4×C67⋊C8—21no—
119C4×C67⋊C8—21no—
120C4×C67⋊C8C2311yescharacteristic
121C4×C67⋊C8C2311yescharacteristic
122C4×C67⋊C8C2311yescharacteristic
123C4×C67⋊C8C2311yescharacteristic
124C8×Dic67—21no—
125C22×Q8×C67C2×C411yescharacteristic
126C4×Q8×C67—21no—
127C4×Q8×C67—21no—
128C4×Q8×C67C2×C411yescharacteristic
129C4×Q8×C67C2×C411yescharacteristic
130C4×Q8×C67C2311yescharacteristic
131C4×Q8×C67C2311yescharacteristic
132C2×C4⋊C4×C67—21no—
133C2×C4⋊C4×C67C2×C411yescharacteristic
134C2×C42×C67—21no—
135C2×C42×C67C2311yescharacteristic
136C4⋊C8×C67—21no—
137C4⋊C8×C67C2311yescharacteristic
138C4⋊C8×C67C2311yescharacteristic
139C22⋊C8×C67—21no—
140C22⋊C8×C67—21no—
141C4×C8×C67—21no—
142C4×C8×C67C2311yescharacteristic
143C4×C8×C67C2311yescharacteristic
144C2×Dic134C22×C411yescharacteristic
145C2×D268C22×C411yescharacteristic
146C2×C22⋊D67—21no—
147C2×C4×D67—21no—
148#11761618—22no—
149#11761618—41no—
150#11761615—22no—
151#11761615—41no—
152#11761612—22no—
153#11761612—41no—
154#11761612—42no—
155#11761611—42no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8—21no—
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C22×C411yescharacteristic
162C2×C67⋊C8C2×D411yescharacteristic
163C2×C67⋊C8C2×D411yescharacteristic
164C2×C67⋊C8C2×D411yescharacteristic
165C2×C67⋊C8C2×D411yescharacteristic
166C8×D67—42no—
167C4⋊Dic67C22×C411yescharacteristic
168C4⋊Dic67C22×C411yescharacteristic
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171Dic67⋊C4—21no—
172Dic67⋊C4—21no—
173C22⋊Dic67—21no—
174(C2×D67)⋊C4—21no—
175(C2×D67)⋊C4—21no—
176(C2×D67)⋊C4—21no—
177(C2×D67)⋊C4—21no—
178C4×Dic67—21no—
179C2×Q8×C67—21no—
180C2×Q8×C67—22no—
181C2×Q8×C67—41no—
182C2×Q8×C67C22×C411yescharacteristic
183C2×Q8×C67C22×C411yescharacteristic
184C22×C4×C67—21no—
185C22×C4×C67—21no—
186C22×C4×C67C22×C411yescharacteristic
187C22×C4×C67C2×D411yescharacteristic
188C22×C4×C67#4111yescharacteristic
189C2×C8×C67—21no—
190C2×C8×C67—21no—
191C2×C8×C67—21no—
192C2×C8×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67—21no—
198C4⋊C4×C67C22×C411yescharacteristic
199C4⋊C4×C67C22×C411yescharacteristic
200C42×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67—21no—
204C42×C67C22×C411yescharacteristic
205C42×C67C2×D411yescharacteristic
206C42×C67#4111yescharacteristic
207C42×C67C2411yescharacteristic
208C23×C67C4×D411yescharacteristic
209C22×D67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—42no—
214Q8×C67#13112yes—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—21no—
218C2×C4×C67—21no—
219C2×C4×C67—21no—
220C2×C4×C67—21no—
221C2×C4×C67—21no—
222C2×C4×C67—21no—
223C2×C4×C67—41no—
224C2×C4×C67—42no—
225C2×C4×C67—81no—
226C2×C4×C67C4×D411yescharacteristic
227C2×C4×C67C4×D411yescharacteristic
228C2×C4×C67C4×D411yescharacteristic
229C2×C4×C67C23×C411yescharacteristic
230C2×C4×C67C22×D411yescharacteristic
231C2×C4×C67C2×#4111yescharacteristic
232C22⋊D67—81no—
233C2×Dic67—21no—
234C2×Dic67—21no—
235C2×Dic67—21no—
236D268—21no—
237D268—21no—
238C4×D67—41no—
239C4×D67—41no—
240Dic134—21no—
241Dic134—21no—
242C8×C67—41no—
243C8×C67—41no—
244C8×C67—41no—
245C8×C67—41no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—21no—
250C67⋊C8—21no—
251C67⋊C8—21no—
252C67⋊C8—21no—
253C67⋊C8—41no—
254C67⋊C8—41no—
255C67⋊C8#13712yes—
256C22×C67—21no—
257C22×C67—21no—
258C22×C67—21no—
259C22×C67C2×C4×D411yescharacteristic
260C2×D67—41no—
261C2×D67—41no—
262C2×D67—41no—
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×C67C2×#13111yescharacteristic
272C4×C67#56811yescharacteristic
273C4×C67C2×#13711yescharacteristic
274C4×C67#57511yescharacteristic
275Dic67—41no—
276Dic67—41no—
277Dic67—41no—
278#18012—671nomaximal, Sylow
279C2×C67—41no—
280C2×C67#293311yescharacteristic
281C2×C67C4×#13711yescharacteristic
282C2×C67#295711yescharacteristic
283D67—81no—
284#3108—671no—
285#2969—671no—
286Q8×M16—671no—
287#1593—671no—
288#1590—671no—
289#1587—671no—
290#1585—671no—
291#1584—671no—
292#1583—671no—
293SD16⋊C8—671no—
294C8×Q16—671no—
295#1574—671no—
296#1495—671no—
297#1479D6711yescharacteristic, complemented
298#1453—671no—
299C67#1801211yescharacteristic, complemented, Sylow
300#529—671no—
301C2×C4×Q8C2×D6711yescharacteristic
302#483—671no—
303#480—671no—
304#479—671no—
305#479—671no—
306#464—671no—
307#464—671no—
308#451—671no—
309#448—671no—
310#446—671no—
311#446—671no—
312#446—671no—
313#446—1341no—
314#446—1341no—
315C8×Q8—671no—
316C8×Q8—671no—
317C8×Q8—671no—
318C4×Q16—671no—
319C4×Q16—671no—
320C4×SD16—671no—
321C4×SD16—671no—
322#436—671no—
323#436—1341no—
324#435—1341no—
325C8×D4—671no—
326#433—1341no—
327#431—671no—
328#431C2×D6711yescharacteristic
329#431C2×D6711yescharacteristic
330#423—671no—
331#423—1341no—
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⋊C8C2×D6711yescharacteristic
342Q8⋊C8C2×D6711yescharacteristic
343Q8⋊C8C2×D6711yescharacteristic
344Q8⋊C8C2×D6711yescharacteristic
345D4⋊C8—671no—
346D4⋊C8—671no—
347#322—671no—
348#322—671no—
349#322—671no—
350C82—671no—
351C22×Q8C4×D6711yescharacteristic
352C2×Q16—671no—
353C2×Q16—671no—
354C2×SD16—671no—
355C2×SD16—671no—
356C2×M16—671no—
357C2×M16—671no—
358C2×M16—1341no—
359C2×M16—1341no—
360C22×C8—1341no—
361#126—1341no—
362#125—671no—
363#124—671no—
364#123—1341no—
365C22⋊Q8—671no—
366#121—671no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—671no—
370C4×Q8C4×D6711yescharacteristic
371C4×Q8C4×D6711yescharacteristic
372C4×Q8C22×D6711yescharacteristic
373C4×Q8C22×D6711yescharacteristic
374C4×D4—671no—
375C4×D4—1341no—
376#117—1341no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D6711yescharacteristic
379C2×C42—21no—
380C2×C42C22×D6711yescharacteristic
381#107—671no—
382#107—671no—
383#106—671no—
384#106—671no—
385C4⋊C8—21no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—671no—
389C4⋊C8—671no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8—1341no—
395C4⋊C8—1341no—
396C4⋊C8—1341no—
397C4⋊C8C22×D6711yescharacteristic
398C4⋊C8C22×D6711yescharacteristic
399#103—671no—
400#103—671no—
401#103—671no—
402#103—671no—
403#103—671no—
404#103—671no—
405#102—671no—
406#102—671no—
407C22⋊C8—21no—
408C22⋊C8—21no—
409C22⋊C8—671no—
410C22⋊C8—671no—
411C22⋊C8—1341no—
412C22⋊C8—1341no—
413#97—671no—
414#97—671no—
415#97—1341no—
416#97—1341no—
417#97—1341no—
418C4×C8—21no—
419C4×C8—671no—
420C4×C8—671no—
421C4×C8—671no—
422C4×C8—671no—
423C4×C8—1341no—
424C4×C8—1341no—
425C4×C8—1341no—
426C4×C8C22×D6711yescharacteristic
427C4×C8C22×D6711yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—671no—
432C2×Q8C2×C4×D6711yescharacteristic
433C2×Q8C2×C4×D6711yescharacteristic
434C2×D4—671no—
435C2×D4—1341no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1341no—
439C22×C4C2×C4×D6711yescharacteristic
440C22×C4D4×D6711yescharacteristic
441C22×C4#1176162711yescharacteristic
442Q16—2681no—
443Q16—1342no—
444SD16—2681no—
445SD16—1342no—
446M16—2682no—
447M16—2682no—
448M16—2681no—
449M16—1342no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—2682no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—671no—
458C2×C8—671no—
459C2×C8—671no—
460C2×C8—671no—
461C2×C8—671no—
462C2×C8—671no—
463C2×C8—1341no—
464C2×C8—1341no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—671no—
471C4⋊C4—671no—
472C4⋊C4—1341no—
473C4⋊C4—1341no—
474C4⋊C4—1341no—
475C4⋊C4—1341no—
476C4⋊C4C2×C4×D6711yescharacteristic
477C4⋊C4C2×C4×D6711yescharacteristic
478C22⋊C4—1341no—
479C22⋊C4—1341no—
480C22⋊C4—1341no—
481C22⋊C4—1341no—
482C22⋊C4—1341no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—1341no—
488C42C2×C4×D6711yescharacteristic
489C42D4×D6711yescharacteristic
490C42#1176162711yescharacteristic
491C42C23×D6711yescharacteristic
492C23—1341no—
493C23(C22⋊D67)⋊C411yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8—1341no—
499Q8—1341no—
500Q8#1452404912yes—
501D4—5361no—
502D4—1341no—
503D4—1341no—
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—41no—
513C2×C4—42no—
514C2×C4—81no—
515C2×C4—2681no—
516C2×C4—2681no—
517C2×C4—1341no—
518C2×C4—1341no—
519C2×C4—1341no—
520C2×C4(C22⋊D67)⋊C411yescharacteristic
521C2×C4(C22⋊D67)⋊C411yescharacteristic
522C2×C4(C22⋊D67)⋊C411yescharacteristic
523C2×C4C22×C4×D6711yescharacteristic
524C2×C4C2×D4×D6711yescharacteristic
525C2×C4C2×#1176162711yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—2681no—
531C8—2681no—
532C8—672no—
533C8—1341no—
534C8—1341no—
535C8—1341no—
536C8—1341no—
537C8—1341no—
538C8—1341no—
539C8—1341no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—2681no—
544C22—2681no—
545C22—2681no—
546C22C2×(C22⋊D67)⋊C411yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—2681no—
556C4—2681no—
557C4—2681no—
558C4C2×#1452404911yescharacteristic
559C4#1686416211yescharacteristic
560C4#137×D6711yescharacteristic
561C4#1686419511yescharacteristic
562C2—41no—
563C2—5361no—
564C2#2256893011yescharacteristic
565C2#2256902411yescharacteristic
566C2#2256912211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1479C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenter#1479D67
hypocenterC67#18012
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. Q8×#11761612C2semidirect
4. #22559115C2semidirect
5. #22559096C2semidirect
6. #22559077C2semidirect
7. #22559069C2semidirect
8. #22559062C2semidirect
9. #22559049C2semidirect
10. #22559024C2semidirect
11. #22559013C2semidirect
12. #22558997C2semidirect
14. #22558504C2semidirect
15. C67×#1479C2semidirect
297. #1479D67semidirect
299. C67#18012semidirect

Maximal subgroups

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

subgroupindex
#225702892
#225692072
Q8×#117616122
#225591152
#225590962
#225590772
#225590692
#225590622
#225590492
#225590242
#225590132
#225589972
#225586752
#225585042
C67×#14792
#1801267