Finite Group Databasebeta

← #49170324

Subgroups — #49170324

Order 37504.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627648C211yesmaximal, characteristic, complemented
2C2×#22627645C211yesmaximal, characteristic, complemented
3#34036799C211yesmaximal, characteristic, complemented
4#34036772C211yesmaximal, characteristic, complemented
5#34036372C211yesmaximal, characteristic, complemented
6#34036362C211yesmaximal, characteristic, complemented
7#34036348C211yesmaximal, characteristic, complemented
8#34036321C211yesmaximal, characteristic, complemented
9#34036288C211yesmaximal, characteristic, complemented
10#34036273C211yesmaximal, characteristic, complemented
11#34036271C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13#34036231C211yesmaximal, characteristic, complemented
14#34036202C211yesmaximal, characteristic, complemented
15#496×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×#17989801C2211yescharacteristic, complemented
19C2×D4×D293C2211yescharacteristic
20#22627648—24no—
21#22627645—24no—
22C2×#17989792C2211yescharacteristic, complemented
23C2×#17989791C2211yescharacteristic
24C2×#17989790C2211yescharacteristic
25C2×#17989787C2211yescharacteristic
26C2×#17989787C2211yescharacteristic
27C2×#17989785C2211yescharacteristic
28C2×C8×D293C2211yescharacteristic, complemented
29#22627620—21no—
30#22627606—21no—
31#22627600C2211yescharacteristic
32#22627597—21no—
33#22627596—21no—
34#22627596—21no—
35#22627595C2211yescharacteristic
36#22627595C2211yescharacteristic
37#22627595C2211yescharacteristic, complemented
38#22627594—21no—
39C4×Dic586C2211yescharacteristic, complemented
40C4×D1172C2211yescharacteristic
41C4×D1172C2211yescharacteristic
42D4×Dic293—21no—
43C22⋊C4×D293—21no—
44#22627558C2211yescharacteristic, complemented
45C4⋊#14556873C2211yescharacteristic, complemented
46#22627549C2211yescharacteristic, complemented
47#22627548C2211yescharacteristic, complemented
48#22627546C2211yescharacteristic
49#22627545C2211yescharacteristic
50#22627545C2211yescharacteristic, complemented
51#22627545C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic
53#22627536C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×SD16×C293C2211yescharacteristic, complemented
56C2×D8×C293C2211yescharacteristic
57C2×D8×C293C2211yescharacteristic
58#127×C293C2211yescharacteristic, complemented
59#124×C293C2211yescharacteristic
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989803—22no—
63#17989803—41no—
64#17989801—22no—
65#17989801—22no—
66#17989801—41no—
67#17989801—41no—
68C2×Dic586C2311yescharacteristic
69C2×D1172—21no—
70C2×D1172D411yescharacteristic
71C2×D1172C2311yescharacteristic
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78D4×D293—22no—
79D4×D293—41no—
80C22×Dic293—21no—
81C22×Dic293—21no—
82C2×C4×D293—21no—
83C2×C4×D293—21no—
84C2×C4×D293C2311yescharacteristic
85C2×C4×D293C2311yescharacteristic
86#17989792—22no—
87#17989791D414yescomplemented
88#17989790—22no—
89#17989787—22no—
90#17989787—22no—
91#17989785—22no—
92C2×#14556873C2311yescharacteristic
93C2×#14556873C2311yescharacteristic
94C8×D293—22no—
95C4⋊Dic293D411yescharacteristic
96C4⋊Dic293C2311yescharacteristic
97C4⋊Dic293C2311yescharacteristic
98#17989781—21no—
99#17989781—21no—
100C22⋊Dic293—21no—
101C22⋊Dic293—21no—
102C22⋊Dic293—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779D411yescharacteristic
106#17989779D411yescharacteristic
107C4×Dic293—21no—
108C2×Q8×C293C2311yescharacteristic
109C2×D4×C293—21no—
110C2×D4×C293—21no—
111C2×D4×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C2×D4×C293C2311yescharacteristic
114SD16×C293—22no—
115SD16×C293—41no—
116D8×C293—22no—
117D8×C293—41no—
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C22⋊C4×C293—21no—
121C22⋊C4×C293—21no—
122C42×C293C2311yescharacteristic
123C22×D293—21no—
124C22×D293—42no—
125C22×D293—42no—
126C22×D293—81no—
127C22×D293C2×D411yescharacteristic
128C22×D293C2×D411yescharacteristic
129Dic586—21no—
130Dic586C2×D412yes—
131D1172—21no—
132D1172—41no—
133D1172—42no—
134D1172C2×D412yes—
135C22⋊D293—42no—
136C22⋊D293—42no—
137C22⋊D293—42no—
138C22⋊D293—81no—
139C22⋊D293—81no—
140C22⋊D293—81no—
141C2×Dic293—21no—
142C2×Dic293—21no—
143C2×Dic293—21no—
144C2×Dic293—42no—
145C2×Dic293—42no—
146C2×Dic293C2×D411yescharacteristic
147C2×Dic293C2×D411yescharacteristic
148C4×D293—21no—
149C4×D293—21no—
150C4×D293—21no—
151C4×D293—21no—
152C4×D293—42no—
153C4×D293—81no—
154#14556873—21no—
155#14556873—21no—
156C23×C293—21no—
157C23×C293—21no—
158C23×C293—21no—
159Q8×C293—21no—
160Q8×C293—21no—
161D4×C293—21no—
162D4×C293—21no—
163D4×C293—21no—
164D4×C293—21no—
165D4×C293—21no—
166D4×C293—21no—
167D4×C293—42no—
168D4×C293—81no—
169C2×C4×C293—21no—
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2411yescharacteristic
173C8×C293—21no—
174C8×C293C2×D412yes—
175C22×C293—41no—
176C22×C293—41no—
177C22×C293—41no—
178C22×C293—41no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293C22×D411yescharacteristic
185C2×D293—21no—
186C2×D293—21no—
187C2×D293—22no—
188C2×D293—21no—
189C2×D293—21no—
190C2×D293—21no—
191C2×D293—21no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—84no—
195C4×C293—21no—
196C4×C293—21no—
197C4×C293—41no—
198C4×C293C22×D411yescharacteristic
199C4×C293H(5,𝔽2)11yescharacteristic
200Dic293—21no—
201Dic293—21no—
202Dic293—22no—
203Dic293—41no—
204Dic293—41no—
205C2×C293—81no—
206C2×C293—81no—
207C2×C293—81no—
208C2×C293D4211yescharacteristic
209C2×C293#57511yescharacteristic
210C2×C293#57611yescharacteristic
211D293—41no—
212D293—41no—
213D293—42no—
214C293#326811yescharacteristic, complemented, Sylow
215#3268—2931nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2931no—
217C2×#135—2931no—
218#550—2931no—
219#546—2931no—
220#496D29311yescharacteristic, complemented
221#468—2931no—
222#466—2931no—
223#464—2931no—
224#459—2931no—
225#452—2931no—
226#449—2931no—
227#449—2931no—
228#447—2931no—
229#440—2931no—
230#435—2931no—
231C2×#41—2931no—
232C2×#41—2931no—
233C2×#41—2931no—
234C22×D4—2931no—
235AGL(1,ℤ/8)—5864no—
236#135—5864no—
237C2×Q16—2931no—
238C2×SD16—2931no—
239C2×SD16—2931no—
240C2×SD16C2×D29311yescharacteristic, complemented
241C2×SD16C2×D29311yescharacteristic, complemented
242C2×D8—2931no—
243C2×D8—2931no—
244C2×D8C2×D29311yescharacteristic
245C2×D8C2×D29311yescharacteristic
246C2×M16—2931no—
247C22×C8—2931no—
248#127—5861no—
249#127C2×D29311yescharacteristic, complemented
250#124C2×D29311yescharacteristic
251#123—5861no—
252C22⋊Q8—2931no—
253#121—2931no—
254#121—2931no—
255#121—2931no—
256#121—5861no—
257#121—5861no—
258#121—5861no—
259#120—5861no—
260C4×Q8—2931no—
261C4×D4—2931no—
262C4×D4—2931no—
263C4×D4—5861no—
264C2×C22⋊C4—5861no—
265#107—2931no—
266C4⋊C8—2931no—
267#103—2931no—
268#103—2931no—
269#102—2931no—
270#102—2931no—
271#102—2931no—
272#102—2931no—
273C22⋊C8—2931no—
274C22⋊C8—2931no—
275#97C2×D29311yescharacteristic, complemented
276C24—5861no—
277#41—5862no—
278#41—5862no—
279#41—5862no—
280#41—11721no—
281#41—11721no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8C22×D29311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2931no—
288C2×D4—2931no—
289C2×D4—5861no—
290C2×D4—5861no—
291C2×D4—5862no—
292C2×D4—5861no—
293C2×D4—5861no—
294C2×D4—5861no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—11721no—
298C2×D4C22×D29311yescharacteristic
299C2×D4C22×D29311yescharacteristic
300C2×D4C22×D29311yescharacteristic
301C22×C4—2931no—
302C22×C4—2931no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4—5861no—
306C22×C4—5861no—
307Q16—5862no—
308SD16—22no—
309SD16—41no—
310SD16—2934no—
311SD16—5862no—
312D8—22no—
313D8—41no—
314D8—5862no—
315D8—5862no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8—5862no—
320C2×C8C22×D29311yescharacteristic
321C2×C8C22×D29311yescharacteristic
322C4⋊C4—2931no—
323C4⋊C4—2931no—
324C4⋊C4—2931no—
325C4⋊C4—5861no—
326C4⋊C4—5861no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2931no—
330C22⋊C4—2931no—
331C22⋊C4—5861no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C42—5861no—
337C42C22×D29311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2931no—
342C23—2931no—
343C23—23441no—
344C23—5861no—
345C23—11722no—
346C23—11722no—
347Q8—21no—
348Q8—21no—
349Q8—2932no—
350Q8—5861no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—2932no—
360D4—23441no—
361D4—23441no—
362D4—23441no—
363D4—5861no—
364D4—11721no—
365D4—11722no—
366D4—11722no—
367D4—11722no—
368D4—11722no—
369C2×C4—21no—
370C2×C4—2931no—
371C2×C4—2931no—
372C2×C4—23441no—
373C2×C4—5861no—
374C2×C4—5861no—
375C2×C4—5861no—
376C2×C4—5861no—
377C2×C4—5861no—
378C2×C4—5861no—
379C2×C4—5861no—
380C2×C4—11722no—
381C2×C4—11722no—
382C2×C4—11722no—
383C2×C4D4×D29311yescharacteristic
384C2×C4D4×D29311yescharacteristic
385C2×C4C23×D29311yescharacteristic
386C8—21no—
387C8—5861no—
388C8—5861no—
389C8D4×D29312yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—23444no—
400C22—5861no—
401C22—5861no—
402C22—5862no—
403C22—5861no—
404C22—5861no—
405C22—5861no—
406C22—5861no—
407C22—11721no—
408C22—11721no—
409C22C2×D4×D29311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5861no—
414C4—5861no—
415C4—5862no—
416C4—11721no—
417C4—11721no—
418C4C2×D4×D29311yescharacteristic
419C4#2262766911yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—11721no—
424C2—11722no—
425C2—11721no—
426C2#3403676411yescharacteristic
427C2#3403692511yescharacteristic
428C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293#3268
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. C2×#22627645C2semidirect
3. #34036799C2semidirect
4. #34036772C2semidirect
5. #34036372C2semidirect
6. #34036362C2semidirect
7. #34036348C2semidirect
8. #34036321C2semidirect
9. #34036288C2semidirect
10. #34036273C2semidirect
11. #34036271C2semidirect
12. #34036261C2semidirect
13. #34036231C2semidirect
14. #34036202C2semidirect
15. #496×C293C2semidirect
18. C2×#17989801C22semidirect
22. C2×#17989792C22semidirect
28. C2×C8×D293C22semidirect
37. #22627595C22semidirect
39. C4×Dic586C22semidirect
44. #22627558C22semidirect
45. C4⋊#14556873C22semidirect
46. #22627549C22semidirect
47. #22627548C22semidirect
50. #22627545C22semidirect
51. #22627545C22semidirect
53. #22627536C22semidirect
54. C2×SD16×C293C22semidirect
55. C2×SD16×C293C22semidirect
58. #127×C293C22semidirect
60. #97×C293C22semidirect
87. #17989791D4semidirect
214. C293#3268semidirect
220. #496D293semidirect
240. C2×SD16C2×D293semidirect
241. C2×SD16C2×D293semidirect
249. #127C2×D293semidirect
275. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
C2×#226276452
#340367992
#340367722
#340363722
#340363622
#340363482
#340363212
#340362882
#340362732
#340362712
#340362612
#340362312
#340362022
#496×C2932
#3268293