Finite Group Databasebeta

← #49132807

Subgroups — #49132807

Order 36224.

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×#22583634C211yesmaximal, characteristic, complemented
2C2×#22583629C211yesmaximal, characteristic, complemented
3#31827135C211yesmaximal, characteristic, complemented
4#31827110C211yesmaximal, characteristic, complemented
5#31826852C211yesmaximal, characteristic, complemented
6#31826710C211yesmaximal, characteristic, complemented
7#31826700C211yesmaximal, characteristic, complemented
8#31826688C211yesmaximal, characteristic, complemented
9#31826628C211yesmaximal, characteristic, complemented
10#31826611C211yesmaximal, characteristic, complemented
11#31826607C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826565C211yesmaximal, characteristic, complemented
14#31826535C211yesmaximal, characteristic, complemented
15#459×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×#17985391C2211yescharacteristic, complemented
19C2×D4×D283C2211yescharacteristic
20#22583634—24no—
21#22583629—24no—
22C2×Dic1132C2211yescharacteristic, complemented
23C2×#17985381C2211yescharacteristic
24C2×#17985380C2211yescharacteristic
25C2×#17985380C2211yescharacteristic, complemented
26C2×#17985380C2211yescharacteristic, complemented
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic
29C2×#17985375C2211yescharacteristic
30C2×C8×D283C2211yescharacteristic, complemented
31#22583606—21no—
32#22583598C2211yescharacteristic
33#22583594—21no—
34#22583590C2211yescharacteristic
35#22583585C2211yescharacteristic
36#22583584—21no—
37#22583582—21no—
38#22583582—21no—
39#22583581C2211yescharacteristic
40#22583581C2211yescharacteristic, complemented
41#22583580—21no—
42C4×Dic566C2211yescharacteristic, complemented
43C4×D1132C2211yescharacteristic
44D4×Dic283—21no—
45C2×(C2×D283)⋊C4—21no—
46#22583543C2211yescharacteristic, complemented
47#22583535C2211yescharacteristic, complemented
48#22583533C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic
50#22583531C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583521C2211yescharacteristic, complemented
54C2×D8×C283C2211yescharacteristic
55C2×D8×C283C2211yescharacteristic
56#127×C283C2211yescharacteristic, complemented
57C4×D4×C283C2211yescharacteristic
58C4⋊C8×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60#102×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985391—22no—
63#17985391—22no—
64#17985391—22no—
65#17985391—41no—
66#17985391—41no—
67#17985391—41no—
68C2×Dic566C2311yescharacteristic
69C2×Dic566C2311yescharacteristic
70C2×D1132—21no—
71C2×D1132D411yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78C2×C22⋊D283—21no—
79D4×D283—22no—
80D4×D283—41no—
81C22×Dic283—21no—
82C22×Dic283—21no—
83C22×Dic283—21no—
84C2×C4×D283C2311yescharacteristic
85C2×C4×D283C2311yescharacteristic
86Dic1132—22no—
87#17985381—22no—
88#17985380—22no—
89#17985380—41no—
90#17985380D414yescomplemented
91#17985377—22no—
92#17985377—41no—
93#17985375—22no—
94C2×C283⋊C8C2311yescharacteristic
95C2×C283⋊C8C2311yescharacteristic
96C8×D283—22no—
97C4⋊Dic283—21no—
98C4⋊Dic283D411yescharacteristic
99C4⋊Dic283C2311yescharacteristic
100Dic283⋊C4—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4D411yescharacteristic
108(C2×D283)⋊C4D411yescharacteristic
109C4×Dic283—21no—
110C2×D4×C283—21no—
111C2×D4×C283—21no—
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115C22×C4×C283—21no—
116D8×C283—22no—
117D8×C283—22no—
118C2×C8×C283C2311yescharacteristic
119C2×C8×C283C2311yescharacteristic
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—42no—
125C22×D283—42no—
126C22×D283—81no—
127C22×D283C2×D411yescharacteristic
128C22×D283C2×D411yescharacteristic
129Dic566—21no—
130Dic566—21no—
131Dic566—21no—
132Dic566C2×D412yes—
133D1132—21no—
134D1132—21no—
135D1132—41no—
136D1132—81no—
137C22⋊D283—42no—
138C22⋊D283—42no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C22⋊D283—81no—
142C22⋊D283—81no—
143C2×Dic283—21no—
144C2×Dic283—21no—
145C2×Dic283—21no—
146C2×Dic283—42no—
147C2×Dic283—42no—
148C2×Dic283—42no—
149C2×Dic283C2×D411yescharacteristic
150C2×Dic283C2×D411yescharacteristic
151C4×D283—21no—
152C4×D283—21no—
153C4×D283—21no—
154C4×D283—21no—
155C283⋊C8—21no—
156C283⋊C8C2×D412yes—
157C23×C283—21no—
158C23×C283—21no—
159C23×C283—21no—
160D4×C283—21no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—21no—
165D4×C283—42no—
166D4×C283—81no—
167D4×C283C2×D412yes—
168C2×C4×C283—21no—
169C2×C4×C283—81no—
170C2×C4×C283C2×D411yescharacteristic
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2411yescharacteristic
173C8×C283—21no—
174C8×C283—21no—
175C22×C283—22no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283—41no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283C22×D411yescharacteristic
185C2×D283—21no—
186C2×D283—21no—
187C2×D283—21no—
188C2×D283—21no—
189C2×D283—21no—
190C2×D283—21no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—84no—
195C4×C283—21no—
196C4×C283—21no—
197C4×C283—41no—
198C4×C283C22×D411yescharacteristic
199C4×C283H(5,𝔽2)11yescharacteristic
200Dic283—21no—
201Dic283—21no—
202Dic283—22no—
203Dic283—41no—
204Dic283—41no—
205C2×C283—42no—
206C2×C283—81no—
207C2×C283—81no—
208C2×C283D4211yescharacteristic
209C2×C283#57511yescharacteristic
210C2×C283#57611yescharacteristic
211D283—41no—
212D283—41no—
213D283—81no—
214C283#326811yescharacteristic, complemented, Sylow
215#3268—2831nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2831no—
217C2×#135—2831no—
218#550—2831no—
219#546—2831no—
220#496—2831no—
221#468—2831no—
222#466—2831no—
223#464—2831no—
224#459D28311yescharacteristic, complemented
225#452—2831no—
226#449—2831no—
227#449—2831no—
228#447—2831no—
229#440—2831no—
230#435—2831no—
231C2×#41—2831no—
232C2×#41—2831no—
233C2×#41—2831no—
234C22×D4—2831no—
235AGL(1,ℤ/8)—5664no—
236#135—5664no—
237C2×Q16—2831no—
238C2×SD16—2831no—
239C2×SD16—2831no—
240C2×SD16—2831no—
241C2×SD16—2831no—
242C2×D8—2831no—
243C2×D8—2831no—
244C2×D8C2×D28311yescharacteristic
245C2×D8C2×D28311yescharacteristic
246C2×M16—2831no—
247C22×C8—2831no—
248#127—5661no—
249#127C2×D28311yescharacteristic, complemented
250#124—2831no—
251#123—5661no—
252C22⋊Q8—2831no—
253#121—2831no—
254#121—2831no—
255#121—2831no—
256#121—5661no—
257#121—5661no—
258#121—5661no—
259#120—5661no—
260C4×Q8—2831no—
261C4×D4—2831no—
262C4×D4—5661no—
263C4×D4C2×D28311yescharacteristic
264C2×C22⋊C4—5661no—
265#107—2831no—
266C4⋊C8C2×D28311yescharacteristic, complemented
267#103—2831no—
268#103—2831no—
269#102—2831no—
270#102—2831no—
271#102C2×D28311yescharacteristic, complemented
272#102C2×D28311yescharacteristic, complemented
273C22⋊C8—2831no—
274C22⋊C8—2831no—
275#97—2831no—
276C24—5661no—
277#41—5662no—
278#41—5662no—
279#41—5662no—
280#41—11321no—
281#41—11321no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8—2831no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2831no—
288C2×D4—2831no—
289C2×D4—5661no—
290C2×D4—5661no—
291C2×D4—5662no—
292C2×D4—5661no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—11321no—
298C2×D4C22×D28311yescharacteristic
299C2×D4C22×D28311yescharacteristic
300C2×D4C22×D28311yescharacteristic
301C22×C4—21no—
302C22×C4—2831no—
303C22×C4—2831no—
304C22×C4—5661no—
305C22×C4—5661no—
306C22×C4—5661no—
307Q16—5662no—
308SD16—2834no—
309SD16—5662no—
310SD16—5662no—
311SD16—11321no—
312D8—22no—
313D8—22no—
314D8—5662no—
315D8—11321no—
316M16—5662no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8—5662no—
320C2×C8C22×D28311yescharacteristic
321C2×C8C22×D28311yescharacteristic
322C4⋊C4—2831no—
323C4⋊C4—2831no—
324C4⋊C4—5661no—
325C4⋊C4—5661no—
326C4⋊C4C22×D28311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2831no—
329C22⋊C4—2831no—
330C22⋊C4—5661no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4—5661no—
335C22⋊C4—5661no—
336C42—5661no—
337C42C22×D28311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2831no—
342C23—2831no—
343C23—5661no—
344C23—11322no—
345C23—11322no—
346C23—22641no—
347Q8—2832no—
348Q8—5661no—
349Q8—5661no—
350Q8—5661no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—5661no—
359D4—5661no—
360D4—11321no—
361D4—11322no—
362D4—11322no—
363D4—11322no—
364D4—11322no—
365D4—22641no—
366D4—22641no—
367D4—22641no—
368D4D4×D28312yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—2831no—
372C2×C4—2831no—
373C2×C4—5661no—
374C2×C4—5661no—
375C2×C4—5661no—
376C2×C4—5661no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—5661no—
380C2×C4—11322no—
381C2×C4—11322no—
382C2×C4—11322no—
383C2×C4D4×D28311yescharacteristic
384C2×C4D4×D28311yescharacteristic
385C2×C4C23×D28311yescharacteristic
386C8—21no—
387C8—21no—
388C8—2832no—
389C8—5661no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5661no—
400C22—5661no—
401C22—5661no—
402C22—5661no—
403C22—5661no—
404C22—5661no—
405C22—11321no—
406C22—11321no—
407C22—11321no—
408C22—22644no—
409C22C2×D4×D28311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5661no—
414C4—5661no—
415C4—5662no—
416C4—11321no—
417C4—11321no—
418C4C2×D4×D28311yescharacteristic
419C4#2258365611yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—11321no—
424C2—11321no—
425C2—22641no—
426C2#3182709911yescharacteristic
427C2#3182726111yescharacteristic
428C2#3182726611yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#459×C283C2
socleC22×C283C22×D4
hypercenter#459D283
hypocenterC283#3268
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. C2×#22583629C2semidirect
3. #31827135C2semidirect
4. #31827110C2semidirect
5. #31826852C2semidirect
6. #31826710C2semidirect
7. #31826700C2semidirect
8. #31826688C2semidirect
9. #31826628C2semidirect
10. #31826611C2semidirect
11. #31826607C2semidirect
12. #31826597C2semidirect
13. #31826565C2semidirect
14. #31826535C2semidirect
15. #459×C283C2semidirect
18. C2×#17985391C22semidirect
22. C2×Dic1132C22semidirect
25. C2×#17985380C22semidirect
26. C2×#17985380C22semidirect
30. C2×C8×D283C22semidirect
40. #22583581C22semidirect
42. C4×Dic566C22semidirect
46. #22583543C22semidirect
47. #22583535C22semidirect
48. #22583533C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583521C22semidirect
56. #127×C283C22semidirect
58. C4⋊C8×C283C22semidirect
59. #102×C283C22semidirect
60. #102×C283C22semidirect
90. #17985380D4semidirect
214. C283#3268semidirect
224. #459D283semidirect
249. #127C2×D283semidirect
266. C4⋊C8C2×D283semidirect
271. #102C2×D283semidirect
272. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
C2×#225836292
#318271352
#318271102
#318268522
#318267102
#318267002
#318266882
#318266282
#318266112
#318266072
#318265972
#318265652
#318265352
#459×C2832
#3268283