Finite Group Databasebeta

← #49129807

Subgroups — #49129807

Order 36224.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827301C211yesmaximal, characteristic
2#31827231C211yesmaximal, characteristic, complemented
3#31826933C211yesmaximal, characteristic, complemented
4#31826483C211yesmaximal, characteristic, complemented
5#31826480C214yesmaximal, complemented
6C2×#22583537C211yesmaximal, characteristic, complemented
7C2×#22583536C211yesmaximal, characteristic, complemented
8#104×D283C214yesmaximal, complemented
9#421×C283C211yesmaximal, characteristic, complemented
10#22583658C412yescomplemented
11#22583657—21no—
12#22583657C412yescomplemented
13#22583656—21no—
14#22583656C412yescomplemented
15#22583655—22no—
16#22583655C412yescomplemented
17C2×#17985392—22no—
18C2×#17985392C2211yescharacteristic
19C2×#17985392C2211yescharacteristic
20#41×D283—22no—
21#41×D283C2214yes—
22#22583618—22no—
23#22583616—22no—
24C2×#17985376C2211yescharacteristic, complemented
25M16×D283C2214yescomplemented
26#22583575—21no—
27#22583572—21no—
28(C22⋊D283)⋊C4—22no—
29#22583567C2212yescomplemented
30C2×C4×Dic283C2211yescharacteristic, complemented
31C42×D283C2212yescomplemented
32#22583538C2214yescomplemented
33#22583537C2212yescomplemented
34#22583537C2212yescomplemented
35#22583536C2212yescomplemented
36#22583536C2212yescomplemented
37C2×#41×C283C2211yescharacteristic
38C2×M16×C283C2211yescharacteristic, complemented
39#117×C283C2211yescharacteristic, complemented
40#104×C283C2214yescomplemented
41#17985393—22no—
42#17985393C2×C414yescomplemented
43#17985392—22no—
44#17985392—22no—
45#17985392—24no—
46#17985392—24no—
47#17985392—21no—
48#17985392—21no—
49#17985392C2×C412yes—
50#17985392C2×C412yes—
51#17985392C2×C412yescomplemented
52#17985392D412yes—
53#17985392D412yes—
54#17985392C2312yes—
55#17985391—22no—
56#17985391—24no—
57#17985391—24no—
58#17985391C2×C414yescomplemented
59C2×Dic566—22no—
60C2×Dic566C2×C411yescharacteristic, complemented
61C2×Dic566D411yescharacteristic
62Q8×D283—22no—
63Q8×D283C2×C414yescomplemented
64C2×D1132—22no—
65C2×D1132C2×C411yescharacteristic, complemented
66C2×D1132D411yescharacteristic
67C2×C22⋊D283—24no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283D412yes—
70D4×D283—24no—
71D4×D283—22no—
72D4×D283—24no—
73D4×D283C2×C414yescomplemented
74C22×Dic283—21no—
75C2×C4×D283—24no—
76C2×C4×D283—22no—
77C2×C4×D283—21no—
78C2×C4×D283C2312yes—
79#17985376—21no—
80#17985376C2312yes—
81#17985375—24no—
82C2×C283⋊C8—21no—
83C8×D283—24no—
84Dic283⋊C4—22no—
85(C2×D283)⋊C4—22no—
86C4×Dic283—22no—
87C4×Dic283—21no—
88C4×Dic283C2311yescharacteristic
89C4×Dic283C2311yescharacteristic
90#41×C283—21no—
91#41×C283—21no—
92#41×C283C2×C412yescomplemented
93#41×C283C2312yes—
94C2×Q8×C283C2×C411yescharacteristic, complemented
95C2×D4×C283—21no—
96C2×D4×C283C2×C411yescharacteristic, complemented
97C22×C4×C283—21no—
98C22×C4×C283C2311yescharacteristic
99M16×C283—21no—
100M16×C283C2312yes—
101C2×C8×C283—21no—
102C4⋊C4×C283—21no—
103C22⋊C4×C283—21no—
104C42×C283C2312yes—
105C22×D283—21no—
106C22×D283—24no—
107C22×D283C2×D412yes—
108Dic566—21no—
109Dic566—24no—
110Dic566C22⋊C412yes—
111Dic566C22⋊C412yes—
112Dic566C22×C412yes—
113D1132—24no—
114D1132—21no—
115D1132C22⋊C412yes—
116D1132C22⋊C412yes—
117D1132C22×C412yes—
118C22⋊D283—22no—
119C22⋊D283—22no—
120C22⋊D283—24no—
121C22⋊D283—24no—
122C22⋊D283C22⋊C414yes—
123C22⋊D283C22⋊C414yes—
124C2×Dic283—21no—
125C2×Dic283—21no—
126C2×Dic283—24no—
127C2×Dic283—21no—
128C2×Dic283—42no—
129C2×Dic283C2×D412yes—
130C4×D283—24no—
131C4×D283—22no—
132C4×D283—22no—
133C4×D283—24no—
134C4×D283—44no—
135C4×D283C22×C414yes—
136C4×D283C2×D414yes—
137C283⋊C8—22no—
138C23×C283—21no—
139C23×C283C2×D411yescharacteristic
140Q8×C283—21no—
141Q8×C283C22×C412yes—
142D4×C283—22no—
143D4×C283—22no—
144D4×C283—21no—
145D4×C283C22×C412yes—
146C2×C4×C283—21no—
147C2×C4×C283—22no—
148C2×C4×C283—22no—
149C2×C4×C283—22no—
150C2×C4×C283C22×C411yescharacteristic
151C2×C4×C283C22×C411yescharacteristic
152C2×C4×C283C2×D411yescharacteristic
153C2×C4×C283C2×D411yescharacteristic
154C2×C4×C283C2×D411yescharacteristic
155C2×C4×C283C2411yescharacteristic
156C8×C283—22no—
157C22×C283—22no—
158C22×C283—41no—
159C22×C283—81no—
160C22×C283C2×C22⋊C411yescharacteristic
161C22×C283C2×C22⋊C411yescharacteristic
162C22×C283C22×D411yescharacteristic
163C2×D283—22no—
164C2×D283—42no—
165C2×D283—84no—
166C2×D283—81no—
167C2×D283C2×C22⋊C414yes—
168C4×C283—22no—
169C4×C283—42no—
170C4×C283C2×C22⋊C411yescharacteristic
171C4×C283C2×C22⋊C411yescharacteristic
172C4×C283C23×C411yescharacteristic
173C4×C283C22×D411yescharacteristic
174Dic283—22no—
175Dic283—22no—
176Dic283—41no—
177Dic283C2×C22⋊C414yes—
178C2×C283—21no—
179C2×C283—21no—
180C2×C283—21no—
181C2×C283—42no—
182C2×C283C22×C22⋊C411yescharacteristic
183D283—24no—
184D283—42no—
185C283#290111yescharacteristic, complemented, Sylow
186#2901—2831nomaximal, Sylow
187#585—2831no—
188#568—2831no—
189C4×#41—2831no—
190#421—2831no—
191#421—2834no—
192#421D28311yescharacteristic, complemented
193C2×#104—2831no—
194C2×#104—2831no—
195C2×#104—2834no—
1962−1+4—2832no—
1972−1+4—2832no—
1982−1+4—5661no—
199H(5,𝔽2)—2832no—
200H(5,𝔽2)—2832no—
201H(5,𝔽2)—5662no—
202H(5,𝔽2)—5661no—
203C2×#41—2831no—
204C2×#41—2834no—
205C2×#41—2831no—
206C2×#41—5662no—
207C2×#41—5662no—
208C2×#41C2×D28311yescharacteristic
209#131—5662no—
210#131—5662no—
211C2×M16—2831no—
212C2×M16—2834no—
213C2×M16C2×D28311yescharacteristic, complemented
214C4×Q8—5661no—
215C4×D4—5662no—
216C4×D4—5661no—
217#117—2832no—
218#117C2×D28311yescharacteristic, complemented
219C2×C42—2831no—
220C2×C42—2832no—
221#104—2832no—
222#104—2832no—
223#104—2832no—
224#104—2832no—
225#104—2834no—
226#104C2×D28314yescomplemented
227#41—21no—
228#41—21no—
229#41—2832no—
230#41—2832no—
231#41—2832no—
232#41—2832no—
233#41—2834no—
234#41—2834no—
235#41—2832no—
236#41—2832no—
237#41—5664no—
238#41—5664no—
239#41—5662no—
240#41—5662no—
241#41—5664no—
242#41—5664no—
243#41—5662no—
244#41—5662no—
245#41—5661no—
246#41—5661no—
247#41C4×D28312yescomplemented
248#41C22×D28312yes—
249C2×Q8—2831no—
250C2×Q8—2834no—
251C2×Q8—2831no—
252C2×Q8—5662no—
253C2×Q8—5662no—
254C2×Q8C4×D28311yescharacteristic, complemented
255C2×D4—21no—
256C2×D4—2832no—
257C2×D4—2834no—
258C2×D4—2831no—
259C2×D4—2831no—
260C2×D4—5662no—
261C2×D4—5664no—
262C2×D4—5662no—
263C2×D4—5664no—
264C2×D4—5664no—
265C2×D4—5661no—
266C2×D4C4×D28311yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—2832no—
269C22×C4—5664no—
270C22×C4—5662no—
271C22×C4—5661no—
272C22×C4—5661no—
273C22×C4C22×D28311yescharacteristic
274M16—21no—
275M16—2832no—
276M16—5664no—
277M16—5661no—
278M16C22×D28312yes—
279C2×C8—21no—
280C2×C8—5664no—
281C2×C8—5661no—
282C4⋊C4—21no—
283C4⋊C4—5662no—
284C22⋊C4—21no—
285C22⋊C4—5662no—
286C42—2831no—
287C42—2831no—
288C42—5662no—
289C42—5661no—
290C42C22×D28312yes—
291C23—21no—
292C23—2832no—
293C23—5664no—
294C23—5661no—
295C23D4×D28311yescharacteristic
296Q8—21no—
297Q8—2832no—
298Q8—2832no—
299Q8—2832no—
300Q8—5664no—
301Q8—5661no—
302Q8C2×C4×D28312yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—2834no—
307D4—2834no—
308D4—2832no—
309D4—2832no—
310D4—2832no—
311D4—5664no—
312D4—5664no—
313D4—5664no—
314D4—5662no—
315D4—5662no—
316D4—5661no—
317D4C2×C4×D28312yes—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—2832no—
323C2×C4—2834no—
324C2×C4—2834no—
325C2×C4—5664no—
326C2×C4—5664no—
327C2×C4—5664no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5662no—
331C2×C4—5662no—
332C2×C4—5661no—
333C2×C4—11324no—
334C2×C4—11322no—
335C2×C4C2×C4×D28311yescharacteristic
336C2×C4C2×C4×D28311yescharacteristic
337C2×C4D4×D28311yescharacteristic
338C2×C4D4×D28311yescharacteristic
339C2×C4D4×D28311yescharacteristic
340C2×C4C23×D28311yescharacteristic
341C8—22no—
342C8—5662no—
343C22—22no—
344C22—41no—
345C22—81no—
346C22—2834no—
347C22—5662no—
348C22—11322no—
349C22—22641no—
350C22—22644no—
351C22C22⋊C4×D28311yescharacteristic
352C22C22⋊C4×D28311yescharacteristic
353C22C2×D4×D28311yescharacteristic
354C4—22no—
355C4—42no—
356C4—2834no—
357C4—5662no—
358C4—5662no—
359C4—11321no—
360C4C22⋊C4×D28311yescharacteristic
361C4C22⋊C4×D28311yescharacteristic
362C4C22×C4×D28311yescharacteristic
363C4C2×D4×D28311yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—5664no—
369C2—11322no—
370C2C2×C22⋊C4×D28311yescharacteristic
subgroupG/H
centerC4C2×D4×D283
commutatorC4×C283C23×C4
FrattiniC2×C4C23×D283
Fitting#421×C283C2
socleC2×C283C22×C22⋊C4
prime radical#31827301C2
cosocleC2×C4×C283C24
hypercenter#421D283
hypocenterC283#2901
LCSG ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #31827231C2semidirect
3. #31826933C2semidirect
4. #31826483C2semidirect
5. #31826480C2semidirect
6. C2×#22583537C2semidirect
7. C2×#22583536C2semidirect
8. #104×D283C2semidirect
9. #421×C283C2semidirect
10. #22583658C4semidirect
12. #22583657C4semidirect
14. #22583656C4semidirect
16. #22583655C4semidirect
24. C2×#17985376C22semidirect
25. M16×D283C22semidirect
29. #22583567C22semidirect
30. C2×C4×Dic283C22semidirect
31. C42×D283C22semidirect
32. #22583538C22semidirect
33. #22583537C22semidirect
34. #22583537C22semidirect
35. #22583536C22semidirect
36. #22583536C22semidirect
38. C2×M16×C283C22semidirect
39. #117×C283C22semidirect
40. #104×C283C22semidirect
42. #17985393C2×C4semidirect
51. #17985392C2×C4semidirect
58. #17985391C2×C4semidirect
60. C2×Dic566C2×C4semidirect
63. Q8×D283C2×C4semidirect
65. C2×D1132C2×C4semidirect
73. D4×D283C2×C4semidirect
92. #41×C283C2×C4semidirect
94. C2×Q8×C283C2×C4semidirect
96. C2×D4×C283C2×C4semidirect
185. C283#2901semidirect
192. #421D283semidirect
213. C2×M16C2×D283semidirect
218. #117C2×D283semidirect
226. #104C2×D283semidirect
247. #41C4×D283semidirect
254. C2×Q8C4×D283semidirect
266. C2×D4C4×D283semidirect

Maximal subgroups

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

subgroupindex
#318273012
#318272312
#318269332
#318264832
#318264802
C2×#225835372
C2×#225835362
#104×D2832
#421×C2832
#2901283