Finite Group Databasebeta

← #30826470

Subgroups — #30826470

Order 17152.

582 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570809C211yesmaximal, characteristic
2#22569354C211yesmaximal, characteristic
3#22569304C211yesmaximal, characteristic, complemented
4#22559115C211yesmaximal, characteristic, complemented
5#22559114C211yesmaximal, characteristic, complemented
6#22559103C211yesmaximal, characteristic, complemented
7#22559096C211yesmaximal, characteristic, complemented
8#22559025C211yesmaximal, characteristic, complemented
9#22559024C211yesmaximal, characteristic, complemented
10#22559013C211yesmaximal, characteristic, complemented
11#22559006C211yesmaximal, characteristic, complemented
12#22558974C211yesmaximal, characteristic, complemented
13#22558670C211yesmaximal, characteristic
14C8×#11761612C211yesmaximal, characteristic, complemented
15C67×#1493C211yesmaximal, characteristic, complemented
16#16863930C2211yescharacteristic
17#16863802C411yescharacteristic
18#16863788C411yescharacteristic
19#16863764C411yescharacteristic
20#16863758C411yescharacteristic
21#16863652C411yescharacteristic
22#16863652C411yescharacteristic
23#16863644C411yescharacteristic
24#16863644C411yescharacteristic
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#16863472—21no—
39#16863465—21no—
40#16863461C2211yescharacteristic
41#16863457—21no—
42#16863457—21no—
43#16863456—21no—
44#16863453C2211yescharacteristic
45#16863449C2211yescharacteristic
46#16863421—21no—
47C4×#11761612C2211yescharacteristic
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863125C2211yescharacteristic
55#16863125C2211yescharacteristic
56#16863118C2211yescharacteristic
57#16863118C2211yescharacteristic
58C8×C67⋊C8C2211yescharacteristic
59C8×C67⋊C8C2211yescharacteristic
60#533×C67C2211yescharacteristic
61#423×C67C2211yescharacteristic
62C2×C4×C8×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×#11761612C2×C411yescharacteristic
72C2×#11761612C2×C411yescharacteristic
73C2×#11761611—21no—
74C2×C8×D67—21no—
75#14524037—21no—
76#14524035C2×C411yescharacteristic
77#14524031C2×C411yescharacteristic
78#14524026—21no—
79#14524022C2×C411yescharacteristic
80#14524017C2×C411yescharacteristic
81C4×Dic134C2311yescharacteristic
82C4×D268C2311yescharacteristic
83C4×C22⋊D67—21no—
84#14524000—21no—
85#14523976C2×C411yescharacteristic
86#14523976C2×C411yescharacteristic
87#14523974C2×C411yescharacteristic
88#14523974C2×C411yescharacteristic
89C4⋊(C67⋊C8)—21no—
90C4⋊(C67⋊C8)—21no—
91C4⋊(C67⋊C8)—21no—
92C4⋊(C67⋊C8)—21no—
93C4⋊(C67⋊C8)D411yescharacteristic
94C4⋊(C67⋊C8)D411yescharacteristic
95C4⋊(C67⋊C8)C2311yescharacteristic
96C4⋊(C67⋊C8)C2311yescharacteristic
97Dic67⋊C8—21no—
98Dic67⋊C8—21no—
99#14523967C2×C411yescharacteristic
100#14523967C2×C411yescharacteristic
101#14523967C2×C411yescharacteristic
102#14523967C2×C411yescharacteristic
103#14523966C2×C411yescharacteristic
104#14523966C2×C411yescharacteristic
105#14523963C2×C411yescharacteristic
106#14523963C2×C411yescharacteristic
107C22⋊(C67⋊C8)—21no—
108C22⋊(C67⋊C8)D411yescharacteristic
109C22⋊(C67⋊C8)D411yescharacteristic
110(C2×D67)⋊C8—21no—
111(C2×D67)⋊C8—21no—
112#14523954—21no—
113#14523954—21no—
114#14523954—21no—
115#14523954C2×C411yescharacteristic
116#14523954C2×C411yescharacteristic
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—
125C2×M16×C67—21no—
126C22×C8×C67—21no—
127C4⋊Q8×C67C2×C411yescharacteristic
128C4⋊Q8×C67C2×C411yescharacteristic
129#125×C67—21no—
130C22⋊Q8×C67—21no—
131C22⋊Q8×C67C2×C411yescharacteristic
132C22⋊Q8×C67C2×C411yescharacteristic
133C4×Q8×C67—21no—
134C4×Q8×C67C2311yescharacteristic
135C4×Q8×C67C2311yescharacteristic
136#117×C67—21no—
137C2×C42×C67C2311yescharacteristic
138C4⋊C8×C67—21no—
139C4⋊C8×C67C2311yescharacteristic
140C4⋊C8×C67C2311yescharacteristic
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—22no—
150#11761615—22no—
151#11761615—22no—
152#11761612—22no—
153#11761612—22no—
154#11761611—81no—
155C2×C67⋊C8—21no—
156C2×C67⋊C8—21no—
157C2×C67⋊C8C22×C411yescharacteristic
158C2×C67⋊C8C22×C411yescharacteristic
159C2×C67⋊C8C22×C411yescharacteristic
160C2×C67⋊C8C22×C411yescharacteristic
161C2×C67⋊C8C2×D411yescharacteristic
162C2×C67⋊C8C2×D411yescharacteristic
163C2×C67⋊C8C2×D411yescharacteristic
164C2×C67⋊C8C2×D411yescharacteristic
165C8×D67—81no—
166C4⋊Dic67C22×C411yescharacteristic
167C4⋊Dic67C22×C411yescharacteristic
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170Dic67⋊C4—21no—
171Dic67⋊C4—21no—
172C22⋊Dic67—21no—
173(C2×D67)⋊C4—21no—
174(C2×D67)⋊C4—21no—
175(C2×D67)⋊C4—21no—
176(C2×D67)⋊C4—21no—
177C4×Dic67—21no—
178C2×Q8×C67—21no—
179C2×Q8×C67C22×C411yescharacteristic
180C2×Q8×C67C22×C411yescharacteristic
181C22×C4×C67C22×C411yescharacteristic
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67#4111yescharacteristic
184M16×C67—81no—
185C2×C8×C67—21no—
186C2×C8×C67—21no—
187C2×C8×C67—21no—
188C2×C8×C67—21no—
189C2×C8×C67—42no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67—21no—
194C4⋊C4×C67—21no—
195C4⋊C4×C67—21no—
196C4⋊C4×C67—21no—
197C4⋊C4×C67C22×C411yescharacteristic
198C4⋊C4×C67C22×C411yescharacteristic
199C22⋊C4×C67—21no—
200C22⋊C4×C67—21no—
201C42×C67—21no—
202C42×C67—21no—
203C42×C67C22×C411yescharacteristic
204C42×C67C2×D411yescharacteristic
205C42×C67#4111yescharacteristic
206C42×C67C2411yescharacteristic
207C23×C67C4×D411yescharacteristic
208C22×D67—21no—
209Q8×C67—21no—
210Q8×C67—21no—
211Q8×C67—21no—
212Q8×C67—21no—
213Q8×C67—81no—
214C2×C4×C67—21no—
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—41no—
223C2×C4×C67C4×D411yescharacteristic
224C2×C4×C67C4×D411yescharacteristic
225C2×C4×C67C4×D411yescharacteristic
226C2×C4×C67C23×C411yescharacteristic
227C2×C4×C67C22×D411yescharacteristic
228C2×C4×C67C2×#4111yescharacteristic
229C22⋊D67—81no—
230C2×Dic67—21no—
231C2×Dic67—21no—
232C2×Dic67—21no—
233D268—21no—
234D268—21no—
235C4×D67—41no—
236C4×D67—41no—
237Dic134—21no—
238Dic134—21no—
239C8×C67—22no—
240C8×C67—41no—
241C8×C67—41no—
242C8×C67—41no—
243C67⋊C8—21no—
244C67⋊C8—21no—
245C67⋊C8—21no—
246C67⋊C8—21no—
247C67⋊C8—21no—
248C67⋊C8—21no—
249C67⋊C8—41no—
250C67⋊C8—41no—
251C67⋊C8#13112yes—
252C67⋊C8#13112yes—
253C22×C67—21no—
254C22×C67—21no—
255C22×C67—21no—
256C22×C67C2×C4×D411yescharacteristic
257C2×D67—41no—
258C2×D67—41no—
259C2×D67—41no—
260C4×C67—21no—
261C4×C67—21no—
262C4×C67—21no—
263C4×C67—21no—
264C4×C67—41no—
265C4×C67—41no—
266C4×C67—41no—
267C4×C67—41no—
268C4×C67C2×#13111yescharacteristic
269C4×C67C2×#13111yescharacteristic
270C4×C67#57711yescharacteristic
271C4×C67#57811yescharacteristic
272Dic67—41no—
273Dic67—41no—
274Dic67—41no—
275#17969—671nomaximal, Sylow
276C2×C67—41no—
277C2×C67#293111yescharacteristic
278C2×C67#294711yescharacteristic
279C2×C67#295611yescharacteristic
280D67—81no—
281#3172—671no—
282#2993—671no—
283#2985—671no—
284#1593—671no—
285#1592—671no—
286#1591—671no—
287#1590—671no—
288#1579—671no—
289SD16⋊C8—671no—
290C8×Q16—671no—
291C8×SD16—671no—
292#1569—671no—
293#1494—671no—
294#1493D6711yescharacteristic, complemented
295C8×M16—671no—
296C67#1796911yescharacteristic, complemented, Sylow
297#533C2×D6711yescharacteristic
298#529—671no—
299#501—671no—
300#497—671no—
301#491—671no—
302#489—671no—
303#470—671no—
304#470—671no—
305#468—671no—
306#468—671no—
307#446—671no—
308#446—671no—
309#446—671no—
310#446—1341no—
311#446—1341no—
312C8×Q8—671no—
313C8×Q8—671no—
314C8×Q8—671no—
315C4×Q16—671no—
316C4×Q16—671no—
317C4×SD16—671no—
318C4×SD16—671no—
319#436—671no—
320#436—1341no—
321#435—1341no—
322C8×D4—671no—
323#433—1341no—
324#433—1341no—
325#433—1341no—
326#432—671no—
327#431—671no—
328#424—1341no—
329#423C2×D6711yescharacteristic
330C4×M16—671no—
331C2×C4×C8C2×D6711yescharacteristic
332#335—671no—
333#335—671no—
334#334—671no—
335#334—671no—
336Q8⋊C8—671no—
337Q8⋊C8—671no—
338Q8⋊C8C2×D6711yescharacteristic
339Q8⋊C8C2×D6711yescharacteristic
340Q8⋊C8C2×D6711yescharacteristic
341Q8⋊C8C2×D6711yescharacteristic
342D4⋊C8—671no—
343D4⋊C8—671no—
344#322—671no—
345#322—671no—
346C82—671no—
347C82—671no—
348C2×Q16—671no—
349C2×Q16—671no—
350C2×SD16—671no—
351C2×SD16—671no—
352C2×M16—21no—
353C2×M16—671no—
354C2×M16—671no—
355C2×M16—1341no—
356C22×C8—21no—
357C22×C8—1341no—
358C4⋊Q8C4×D6711yescharacteristic
359C4⋊Q8C4×D6711yescharacteristic
360#126—1341no—
361#125—21no—
362#125—671no—
363#124—671no—
364#123—1341no—
365C22⋊Q8—21no—
366C22⋊Q8—671no—
367C22⋊Q8C4×D6711yescharacteristic
368C22⋊Q8C4×D6711yescharacteristic
369#121—671no—
370C4×Q8—21no—
371C4×Q8—671no—
372C4×Q8C22×D6711yescharacteristic
373C4×Q8C22×D6711yescharacteristic
374C4×D4—671no—
375C4×D4—1341no—
376#117—21no—
377#117—1341no—
378C2×C42C22×D6711yescharacteristic
379#107—671no—
380#107—671no—
381#106—671no—
382#106—671no—
383C4⋊C8—21no—
384C4⋊C8—671no—
385C4⋊C8—671no—
386C4⋊C8—671no—
387C4⋊C8—671no—
388C4⋊C8—1341no—
389C4⋊C8—1341no—
390C4⋊C8—1341no—
391C4⋊C8—1341no—
392C4⋊C8—1341no—
393C4⋊C8—1341no—
394C4⋊C8C22×D6711yescharacteristic
395C4⋊C8C22×D6711yescharacteristic
396#103—671no—
397#103—671no—
398#103—671no—
399#103—671no—
400#103—671no—
401#103—671no—
402#102—671no—
403#102—671no—
404C22⋊C8—671no—
405C22⋊C8—671no—
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—
415C4×C8—21no—
416C4×C8—671no—
417C4×C8—671no—
418C4×C8—671no—
419C4×C8—671no—
420C4×C8—1341no—
421C4×C8—1341no—
422C4×C8—1341no—
423C4×C8C22×D6711yescharacteristic
424C4×C8C22×D6711yescharacteristic
425C2×Q8—21no—
426C2×Q8—671no—
427C2×Q8C2×C4×D6711yescharacteristic
428C2×Q8C2×C4×D6711yescharacteristic
429C2×D4—671no—
430C2×D4—1341no—
431C22×C4—1341no—
432C22×C4C2×C4×D6711yescharacteristic
433C22×C4D4×D6711yescharacteristic
434C22×C4#1176162711yescharacteristic
435Q16—1342no—
436Q16—1342no—
437SD16—1342no—
438SD16—1342no—
439M16—81no—
440M16—5361no—
441M16—1342no—
442M16—1342no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—42no—
448C2×C8—5361no—
449C2×C8—671no—
450C2×C8—671no—
451C2×C8—671no—
452C2×C8—671no—
453C2×C8—671no—
454C2×C8—671no—
455C2×C8—671no—
456C2×C8—671no—
457C2×C8—1341no—
458C2×C8—1341no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—671no—
467C4⋊C4—671no—
468C4⋊C4—1341no—
469C4⋊C4—1341no—
470C4⋊C4—1341no—
471C4⋊C4—1341no—
472C4⋊C4C2×C4×D6711yescharacteristic
473C4⋊C4C2×C4×D6711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—1341no—
477C22⋊C4—1341no—
478C22⋊C4—1341no—
479C22⋊C4—1341no—
480C22⋊C4—1341no—
481C42—21no—
482C42—21no—
483C42—1341no—
484C42C2×C4×D6711yescharacteristic
485C42D4×D6711yescharacteristic
486C42#1176162711yescharacteristic
487C42C23×D6711yescharacteristic
488C23—1341no—
489C23(C22⋊D67)⋊C411yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—21no—
494Q8—81no—
495Q8—1341no—
496Q8—1341no—
497D4—5361no—
498D4—1341no—
499D4—1341no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—2681no—
510C2×C4—2681no—
511C2×C4—1341no—
512C2×C4—1341no—
513C2×C4—1341no—
514C2×C4(C22⋊D67)⋊C411yescharacteristic
515C2×C4(C22⋊D67)⋊C411yescharacteristic
516C2×C4(C22⋊D67)⋊C411yescharacteristic
517C2×C4C22×C4×D6711yescharacteristic
518C2×C4C2×D4×D6711yescharacteristic
519C2×C4C2×#1176162711yescharacteristic
520C8—22no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—2681no—
525C8—2681no—
526C8—672no—
527C8—672no—
528C8—1341no—
529C8—1341no—
530C8—1341no—
531C8—1341no—
532C8—1341no—
533C8—1341no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—2681no—
538C22—2681no—
539C22—2681no—
540C22C2×(C22⋊D67)⋊C411yescharacteristic
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—2681no—
550C4—2681no—
551C4—2681no—
552C4#131×D6711yescharacteristic
553C4#131×D6711yescharacteristic
554C4#1686420211yescharacteristic
555C4#1686420911yescharacteristic
556C2—41no—
557C2—5361no—
558C2#2256891711yescharacteristic
559C2#2256904611yescharacteristic
560C2#2256911511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1493C2
socleC22×C67C2×C4×D4
prime radical#14524017C2×C4
cosocleC42×C67C24
hypercenter#1493D67
hypocenterC67#17969
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
3. #22569304C2semidirect
4. #22559115C2semidirect
5. #22559114C2semidirect
6. #22559103C2semidirect
7. #22559096C2semidirect
8. #22559025C2semidirect
9. #22559024C2semidirect
10. #22559013C2semidirect
11. #22559006C2semidirect
12. #22558974C2semidirect
14. C8×#11761612C2semidirect
15. C67×#1493C2semidirect
294. #1493D67semidirect
296. C67#17969semidirect

Maximal subgroups

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

subgroupindex
#225708092
#225693542
#225693042
#225591152
#225591142
#225591032
#225590962
#225590252
#225590242
#225590132
#225590062
#225589742
#225586702
C8×#117616122
C67×#14932
#1796967