Finite Group Databasebeta

← #30827219

Subgroups — #30827219

Order 17152.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570248C211yesmaximal, characteristic
2#22569310C211yesmaximal, characteristic
3#22569298C211yesmaximal, characteristic
4#22559115C211yesmaximal, characteristic, complemented
5#22559103C211yesmaximal, characteristic, complemented
6#22559083C211yesmaximal, characteristic, complemented
7#22559077C211yesmaximal, characteristic, complemented
8#22559062C211yesmaximal, characteristic, complemented
9#22559056C211yesmaximal, characteristic, complemented
10#22559025C211yesmaximal, characteristic, complemented
11#22559013C211yesmaximal, characteristic, complemented
12#22558987C211yesmaximal, characteristic
13#22558683C211yesmaximal, characteristic
14#22558509C211yesmaximal, characteristic
15C67×#1479C211yesmaximal, characteristic
16#16863953C2211yescharacteristic
17#16863731C411yescharacteristic
18#16863731C411yescharacteristic
19#16863691C411yescharacteristic
20#16863691C411yescharacteristic
21#16863606C411yescharacteristic
22#16863606C411yescharacteristic
23#16863552C411yescharacteristic
24#16863552C411yescharacteristic
25#16863527—21no—
26#16863527—21no—
27#16863527C2211yescharacteristic
28#16863527C2211yescharacteristic
29#16863526—21no—
30#16863523C2211yescharacteristic
31#16863523C2211yescharacteristic
32Q8×C67⋊C8C2211yescharacteristic
33Q8×C67⋊C8C2211yescharacteristic
34C8×Dic134—21no—
35#16863520C2211yescharacteristic
36#16863520C2211yescharacteristic
37C4×#11761618C2211yescharacteristic
38C4×#11761618C2211yescharacteristic
39C4×#11761618C2211yescharacteristic
40C4×#11761618C2211yescharacteristic
41#16863454—21no—
42#16863454—21no—
43#16863453—21no—
44#16863453C2211yescharacteristic
45#16863449—21no—
46#16863449C2211yescharacteristic
47#16863449C2211yescharacteristic
48#16863148C2211yescharacteristic
49#16863148C2211yescharacteristic
50#16863146C2211yescharacteristic
51#16863146C2211yescharacteristic
52#16863128C2211yescharacteristic
53#16863128C2211yescharacteristic
54#16863128C2211yescharacteristic
55#16863128C2211yescharacteristic
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×#11761618C2×C411yescharacteristic
70C2×#11761618C2×C411yescharacteristic
71#14524043C2×C411yescharacteristic
72#14524043C2×C411yescharacteristic
73#14524035—21no—
74#14524022—21no—
75#14524022C2×C411yescharacteristic
76#14524022C2×C411yescharacteristic
77C4×Dic134—21no—
78C4×Dic134C2311yescharacteristic
79C4×Dic134C2311yescharacteristic
80#14524002—21no—
81#14523976C2×C411yescharacteristic
82#14523976C2×C411yescharacteristic
83#14523976C2×C411yescharacteristic
84#14523976C2×C411yescharacteristic
85C4⋊(C67⋊C8)—21no—
86C4⋊(C67⋊C8)—21no—
87C4⋊(C67⋊C8)—21no—
88C4⋊(C67⋊C8)—21no—
89C4⋊(C67⋊C8)C2×C411yescharacteristic
90C4⋊(C67⋊C8)C2×C411yescharacteristic
91C4⋊(C67⋊C8)D411yescharacteristic
92C4⋊(C67⋊C8)D411yescharacteristic
93C4⋊(C67⋊C8)C2311yescharacteristic
94C4⋊(C67⋊C8)C2311yescharacteristic
95Dic67⋊C8—21no—
96Dic67⋊C8—21no—
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#14523966C2×C411yescharacteristic
106#14523966C2×C411yescharacteristic
107C22⋊(C67⋊C8)—21no—
108C22⋊(C67⋊C8)C2×C411yescharacteristic
109C22⋊(C67⋊C8)C2×C411yescharacteristic
110C22⋊(C67⋊C8)D411yescharacteristic
111C22⋊(C67⋊C8)D411yescharacteristic
112#14523954—21no—
113#14523954—21no—
114#14523954—21no—
115#14523953—21no—
116#14523953—21no—
117C4×C67⋊C8—21no—
118C4×C67⋊C8—21no—
119C4×C67⋊C8C2311yescharacteristic
120C4×C67⋊C8C2311yescharacteristic
121C4×C67⋊C8C2311yescharacteristic
122C4×C67⋊C8C2311yescharacteristic
123C8×Dic67—21no—
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×Dic134—21no—
145C2×Dic134C22×C411yescharacteristic
146C2×Dic134C22×C411yescharacteristic
147#11761618—22no—
148#11761618—22no—
149#11761618—41no—
150#11761618—41no—
151C2×C67⋊C8—21no—
152C2×C67⋊C8—21no—
153C2×C67⋊C8C22×C411yescharacteristic
154C2×C67⋊C8C22×C411yescharacteristic
155C2×C67⋊C8C22×C411yescharacteristic
156C2×C67⋊C8C22×C411yescharacteristic
157C2×C67⋊C8C2×D411yescharacteristic
158C2×C67⋊C8C2×D411yescharacteristic
159C2×C67⋊C8C2×D411yescharacteristic
160C2×C67⋊C8C2×D411yescharacteristic
161C4⋊Dic67—21no—
162C4⋊Dic67—21no—
163C4⋊Dic67—21no—
164C4⋊Dic67C22×C411yescharacteristic
165C4⋊Dic67C22×C411yescharacteristic
166Dic67⋊C4—21no—
167Dic67⋊C4—21no—
168Dic67⋊C4—21no—
169Dic67⋊C4—21no—
170C22⋊Dic67—21no—
171C22⋊Dic67—21no—
172C4×Dic67—21no—
173C4×Dic67—21no—
174C2×Q8×C67—21no—
175C2×Q8×C67—22no—
176C2×Q8×C67—41no—
177C2×Q8×C67C22×C411yescharacteristic
178C2×Q8×C67C22×C411yescharacteristic
179C22×C4×C67—21no—
180C22×C4×C67—21no—
181C22×C4×C67C22×C411yescharacteristic
182C22×C4×C67C2×D411yescharacteristic
183C22×C4×C67#4111yescharacteristic
184C2×C8×C67—21no—
185C2×C8×C67—21no—
186C2×C8×C67—21no—
187C2×C8×C67—21no—
188C4⋊C4×C67—21no—
189C4⋊C4×C67—21no—
190C4⋊C4×C67—21no—
191C4⋊C4×C67—21no—
192C4⋊C4×C67—21no—
193C4⋊C4×C67C22×C411yescharacteristic
194C4⋊C4×C67C22×C411yescharacteristic
195C42×C67—21no—
196C42×C67—21no—
197C42×C67—21no—
198C42×C67—21no—
199C42×C67C22×C411yescharacteristic
200C42×C67C2×D411yescharacteristic
201C42×C67#4111yescharacteristic
202C42×C67C2411yescharacteristic
203C23×C67C4×D411yescharacteristic
204Q8×C67—21no—
205Q8×C67—21no—
206Q8×C67—21no—
207Q8×C67—42no—
208Q8×C67#13112yes—
209C2×C4×C67—21no—
210C2×C4×C67—21no—
211C2×C4×C67—21no—
212C2×C4×C67—21no—
213C2×C4×C67—21no—
214C2×C4×C67—21no—
215C2×C4×C67—21no—
216C2×C4×C67—21no—
217C2×C4×C67—41no—
218C2×C4×C67—42no—
219C2×C4×C67—42no—
220C2×C4×C67C4×D411yescharacteristic
221C2×C4×C67C4×D411yescharacteristic
222C2×C4×C67C4×D411yescharacteristic
223C2×C4×C67C23×C411yescharacteristic
224C2×C4×C67C22×D411yescharacteristic
225C2×C4×C67C2×#4111yescharacteristic
226C2×Dic67—21no—
227C2×Dic67—21no—
228C2×Dic67—21no—
229C2×Dic67—21no—
230Dic134—21no—
231Dic134—21no—
232Dic134—21no—
233Dic134—21no—
234Dic134—42no—
235C8×C67—41no—
236C8×C67—41no—
237C8×C67—41no—
238C8×C67—41no—
239C67⋊C8—21no—
240C67⋊C8—21no—
241C67⋊C8—21no—
242C67⋊C8—21no—
243C67⋊C8—21no—
244C67⋊C8—21no—
245C67⋊C8—21no—
246C67⋊C8—21no—
247C67⋊C8—41no—
248C67⋊C8—41no—
249C22×C67—21no—
250C22×C67—21no—
251C22×C67—21no—
252C22×C67C2×C4×D411yescharacteristic
253C4×C67—21no—
254C4×C67—22no—
255C4×C67—21no—
256C4×C67—21no—
257C4×C67—21no—
258C4×C67—41no—
259C4×C67—41no—
260C4×C67—41no—
261C4×C67C2×#13111yescharacteristic
262C4×C67#56811yescharacteristic
263C4×C67#57811yescharacteristic
264C4×C67#57811yescharacteristic
265Dic67—41no—
266Dic67—41no—
267Dic67—41no—
268Dic67—41no—
269#18032—671nomaximal, Sylow
270C2×C67—41no—
271C2×C67#293311yescharacteristic
272C2×C67#295011yescharacteristic
273C2×C67#295511yescharacteristic
274#3104—671no—
275#2986—671no—
276#2984—671no—
277#1593—671no—
278#1591—671no—
279#1587—671no—
280#1587—671no—
281#1584—671no—
282#1584—671no—
283#1579—671no—
284C8×Q16—671no—
285#1572—671no—
286#1497—671no—
287#1479D6711yescharacteristic
288#1454—671no—
289C67#1803211yescharacteristic, complemented, Sylow
290#533—671no—
291C2×C4×Q8C2×D6711yescharacteristic
292#484—671no—
293#484—671no—
294#477—671no—
295#477—671no—
296#462—671no—
297#462—671no—
298#451—671no—
299#451—671no—
300#446—671no—
301#446—671no—
302#446—671no—
303#446—671no—
304#446—1341no—
305#446—1341no—
306#446—1341no—
307C8×Q8—671no—
308C8×Q8—671no—
309C8×Q8—671no—
310C8×Q8—671no—
311C8×Q8—1341no—
312C4×Q16—671no—
313C4×Q16—671no—
314C4×Q16—671no—
315C4×Q16—671no—
316#433—1341no—
317#433—1341no—
318#432—671no—
319#432—1341no—
320#431—671no—
321#431—671no—
322#431—1341no—
323#431C2×D6711yescharacteristic
324#431C2×D6711yescharacteristic
325#335—671no—
326#335—671no—
327#334—671no—
328#334—671no—
329Q8⋊C8—671no—
330Q8⋊C8—671no—
331Q8⋊C8—671no—
332Q8⋊C8—671no—
333Q8⋊C8C2×D6711yescharacteristic
334Q8⋊C8C2×D6711yescharacteristic
335Q8⋊C8C2×D6711yescharacteristic
336Q8⋊C8C2×D6711yescharacteristic
337#322—671no—
338#322—671no—
339#322—671no—
340C82—671no—
341C22×Q8C4×D6711yescharacteristic
342C2×Q16—671no—
343C2×Q16—671no—
344C2×Q16—671no—
345C2×Q16—671no—
346C4⋊Q8—671no—
347C4⋊Q8—671no—
348#125—1341no—
349C22⋊Q8—671no—
350C22⋊Q8—671no—
351C22⋊Q8—1341no—
352C4×Q8—21no—
353C4×Q8—21no—
354C4×Q8—671no—
355C4×Q8—671no—
356C4×Q8—1341no—
357C4×Q8C4×D6711yescharacteristic
358C4×Q8C4×D6711yescharacteristic
359C4×Q8C22×D6711yescharacteristic
360C4×Q8C22×D6711yescharacteristic
361#117—1341no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4C4×D6711yescharacteristic
364C2×C42—21no—
365C2×C42C22×D6711yescharacteristic
366#107—671no—
367#107—671no—
368#107—671no—
369#107—671no—
370C4⋊C8—21no—
371C4⋊C8—671no—
372C4⋊C8—671no—
373C4⋊C8—671no—
374C4⋊C8—671no—
375C4⋊C8—671no—
376C4⋊C8—671no—
377C4⋊C8—1341no—
378C4⋊C8—1341no—
379C4⋊C8—1341no—
380C4⋊C8—1341no—
381C4⋊C8—1341no—
382C4⋊C8—1341no—
383C4⋊C8—1341no—
384C4⋊C8—1341no—
385C4⋊C8C22×D6711yescharacteristic
386C4⋊C8C22×D6711yescharacteristic
387#103—671no—
388#103—671no—
389#103—671no—
390#103—671no—
391#103—671no—
392#103—671no—
393#103—671no—
394#103—671no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397C22⋊C8—671no—
398C22⋊C8—671no—
399C22⋊C8—671no—
400C22⋊C8—671no—
401C22⋊C8—1341no—
402#97—1341no—
403#97—1341no—
404#97—1341no—
405#97—1341no—
406#97—1341no—
407C4×C8—21no—
408C4×C8—671no—
409C4×C8—671no—
410C4×C8—671no—
411C4×C8—671no—
412C4×C8—1341no—
413C4×C8—1341no—
414C4×C8—1341no—
415C4×C8—1341no—
416C4×C8C22×D6711yescharacteristic
417C4×C8C22×D6711yescharacteristic
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—671no—
422C2×Q8—671no—
423C2×Q8—1341no—
424C2×Q8C2×C4×D6711yescharacteristic
425C2×Q8C2×C4×D6711yescharacteristic
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C2×C4×D6711yescharacteristic
429C22×C4D4×D6711yescharacteristic
430C22×C4#1176162711yescharacteristic
431Q16—2681no—
432Q16—2681no—
433Q16—1342no—
434Q16—1342no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—671no—
440C2×C8—671no—
441C2×C8—671no—
442C2×C8—671no—
443C2×C8—671no—
444C2×C8—671no—
445C2×C8—671no—
446C2×C8—671no—
447C2×C8—1341no—
448C2×C8—1341no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—671no—
455C4⋊C4—671no—
456C4⋊C4—1341no—
457C4⋊C4—1341no—
458C4⋊C4—1341no—
459C4⋊C4—1341no—
460C4⋊C4—1341no—
461C4⋊C4—1341no—
462C4⋊C4—1341no—
463C4⋊C4C2×C4×D6711yescharacteristic
464C4⋊C4C2×C4×D6711yescharacteristic
465C22⋊C4—1341no—
466C22⋊C4—1341no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—1341no—
472C42—1341no—
473C42C2×C4×D6711yescharacteristic
474C42D4×D6711yescharacteristic
475C42#1176162711yescharacteristic
476C42C23×D6711yescharacteristic
477C23(C22⋊D67)⋊C411yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—42no—
482Q8—2682no—
483Q8—1341no—
484Q8—1341no—
485Q8—1341no—
486Q8—1341no—
487Q8#1452404912yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1341no—
500C2×C4—1341no—
501C2×C4—1341no—
502C2×C4—1341no—
503C2×C4(C22⋊D67)⋊C411yescharacteristic
504C2×C4(C22⋊D67)⋊C411yescharacteristic
505C2×C4(C22⋊D67)⋊C411yescharacteristic
506C2×C4C22×C4×D6711yescharacteristic
507C2×C4C2×D4×D6711yescharacteristic
508C2×C4C2×#1176162711yescharacteristic
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—2681no—
514C8—2681no—
515C8—1341no—
516C8—1341no—
517C8—1341no—
518C8—1341no—
519C8—1341no—
520C8—1341no—
521C8—1341no—
522C8—1341no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×(C22⋊D67)⋊C411yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—2681no—
536C4—2681no—
537C4—2681no—
538C4—2681no—
539C4C2×#1452404911yescharacteristic
540C4#1686416211yescharacteristic
541C4#1686421211yescharacteristic
542C4#1686421211yescharacteristic
543C2—41no—
544C2#2256893011yescharacteristic
545C2#2256907311yescharacteristic
546C2#2256910811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC42C23×D67
FittingC67×#1479C2
socleC22×C67C2×C4×D4
prime radicalC23×C67C4×D4
cosocleC42×C67C24
hypercenter#1479D67
hypocenterC67#18032
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
4. #22559115C2semidirect
5. #22559103C2semidirect
6. #22559083C2semidirect
7. #22559077C2semidirect
8. #22559062C2semidirect
9. #22559056C2semidirect
10. #22559025C2semidirect
11. #22559013C2semidirect
289. C67#18032semidirect

Maximal subgroups

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

subgroupindex
#225702482
#225693102
#225692982
#225591152
#225591032
#225590832
#225590772
#225590622
#225590562
#225590252
#225590132
#225589872
#225586832
#225585092
C67×#14792
#1803267