Finite Group Databasebeta

← #49133307

Subgroups — #49133307

Order 36224.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827149C211yesmaximal, characteristic
2#31827128C211yesmaximal, characteristic
3#31826832C211yesmaximal, characteristic, complemented
4#31826803C211yesmaximal, characteristic, complemented
5#31826797C211yesmaximal, characteristic, complemented
6#31826775C211yesmaximal, characteristic, complemented
7#31826705C211yesmaximal, characteristic, complemented
8#31826673C211yesmaximal, characteristic, complemented
9#31826620C211yesmaximal, characteristic, complemented
10#31826619C211yesmaximal, characteristic, complemented
11#31826569C211yesmaximal, characteristic, complemented
12#31826534C211yesmaximal, characteristic
13#31826475C211yesmaximal, characteristic, complemented
14#103×D283C211yesmaximal, characteristic, complemented
15#470×C283C211yesmaximal, characteristic, complemented
16C22×Dic566C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985382C2211yescharacteristic
20C2×C8×D283C2211yescharacteristic
21#22583609—21no—
22#22583608C2211yescharacteristic
23#22583603—21no—
24#22583601C2211yescharacteristic
25#22583600—21no—
26#22583592—21no—
27#22583590C2211yescharacteristic
28#22583588—21no—
29#22583587—21no—
30#22583586C2211yescharacteristic
31Q8×Dic283C2211yescharacteristic
32#22583575C2211yescharacteristic
33C4×C22⋊D283C2211yescharacteristic
34(C22⋊D283)⋊C4—21no—
35#22583569C2211yescharacteristic
36#22583568—21no—
37C4⋊C4×D283C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583541C2211yescharacteristic
40Dic283⋊C8C2211yescharacteristic
41#22583535C2211yescharacteristic
42#22583535C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583533C2211yescharacteristic
48#22583533C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51#22583520C2211yescharacteristic
52C2×Q16×C283C2211yescharacteristic
53C2×M16×C283C2211yescharacteristic
54C22⋊Q8×C283C2211yescharacteristic
55C22⋊Q8×C283C2211yescharacteristic
56#106×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic
59C2×Dic566—21no—
60C2×Dic566—22no—
61C2×Dic566—21no—
62C2×Dic566—41no—
63C2×Dic566C2311yescharacteristic
64C2×Dic566C2311yescharacteristic
65Q8×D283—22no—
66Q8×D283—41no—
67C2×C22⋊D283D411yescharacteristic
68C22×Dic283—21no—
69C2×C4×D283—21no—
70C2×C4×D283—21no—
71C2×C4×D283C2311yescharacteristic
72#17985382—22no—
73#17985382—41no—
74C2×C283⋊C8C2311yescharacteristic
75C2×C283⋊C8C2311yescharacteristic
76C8×D283—41no—
77C4⋊Dic283—21no—
78C4⋊Dic283—21no—
79C4⋊Dic283—21no—
80C4⋊Dic283C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4—21no—
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4D411yescharacteristic
89C22⋊Dic283—21no—
90C22⋊Dic283D411yescharacteristic
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4D411yescharacteristic
94C4×Dic283—21no—
95C4×Dic283—21no—
96C4×Dic283—21no—
97C4×Dic283C2311yescharacteristic
98C2×Q8×C283C2311yescharacteristic
99C2×Q8×C283C2311yescharacteristic
100C22×C4×C283C2311yescharacteristic
101Q16×C283—41no—
102M16×C283—41no—
103C2×C8×C283C2311yescharacteristic
104C2×C8×C283C2311yescharacteristic
105C4⋊C4×C283—21no—
106C4⋊C4×C283—21no—
107C4⋊C4×C283C2311yescharacteristic
108C4⋊C4×C283C2311yescharacteristic
109C22⋊C4×C283—21no—
110C22⋊C4×C283—21no—
111C22×D283C2×D411yescharacteristic
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—42no—
116Dic566—42no—
117Dic566#4112yes—
118C22⋊D283—22no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—42no—
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C4×D283—21no—
128C4×D283—21no—
129C4×D283—42no—
130C4×D283—42no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283C2×D411yescharacteristic
134Q8×C283—21no—
135Q8×C283—21no—
136Q8×C283—21no—
137Q8×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283C2×D411yescharacteristic
145C2×C4×C283C2411yescharacteristic
146C8×C283—21no—
147C8×C283—21no—
148C22×C283—21no—
149C22×C283—21no—
150C22×C283—21no—
151C22×C283C22×D411yescharacteristic
152C2×D283—21no—
153C2×D283—21no—
154C2×D283#13712yes—
155C4×C283—21no—
156C4×C283—41no—
157C4×C283—41no—
158C4×C283—41no—
159C4×C283—41no—
160C4×C283C2×#4111yescharacteristic
161C4×C283H(5,𝔽2)11yescharacteristic
162Dic283—21no—
163Dic283—22no—
164Dic283—21no—
165Dic283—21no—
166Dic283—41no—
167Dic283—41no—
168Dic283—41no—
169C2×C283—41no—
170C2×C283#54611yescharacteristic
171C2×C283C2×#13711yescharacteristic
172C2×C283#57811yescharacteristic
173D283—22no—
174C283#331211yescharacteristic, complemented, Sylow
175#3312—2831nomaximal, Sylow
176#552—2831no—
177D4×Q8—2831no—
178#491—2831no—
179#485—2831no—
180#483—2831no—
181#479—2831no—
182#470D28311yescharacteristic, complemented
183#467—2831no—
184#462—2831no—
185#451—2831no—
186#451—2831no—
187#441—2831no—
188#435—2831no—
189#419—2831no—
190C2×#103—2831no—
191C22×Q8—2831no—
192C22×Q8—2831no—
193C2×Q16—2831no—
194C2×Q16—2831no—
195C2×Q16C2×D28311yescharacteristic
196C2×M16C2×D28311yescharacteristic
197C22×C8—2831no—
198C4⋊Q8—2831no—
199C4⋊Q8—5661no—
200#126—5661no—
201#125—2831no—
202#125—5661no—
203#123—5661no—
204C22⋊Q8—2831no—
205C22⋊Q8—2831no—
206C22⋊Q8—5661no—
207C22⋊Q8—5661no—
208C22⋊Q8C2×D28311yescharacteristic
209C22⋊Q8C2×D28311yescharacteristic
210C4×Q8—2831no—
211C4×Q8—2831no—
212C4×D4—2831no—
213C4×D4—5661no—
214#117—2831no—
215#117—5661no—
216C2×C4⋊C4—2831no—
217#107—2831no—
218#106—2831no—
219#106C2×D28311yescharacteristic
220C4⋊C8—2831no—
221#103—2831no—
222#103—2831no—
223#103—2831no—
224#103—2831no—
225#103—2831no—
226#103—2831no—
227#103—2831no—
228#103—2831no—
229#103C2×D28311yescharacteristic
230#103C2×D28311yescharacteristic
231C22⋊C8—2831no—
232C22⋊C8—2831no—
233#97—2831no—
234C2×Q8—2831no—
235C2×Q8—2831no—
236C2×Q8—5661no—
237C2×Q8—5662no—
238C2×Q8—5661no—
239C2×Q8—5662no—
240C2×Q8—11321no—
241C2×Q8—11321no—
242C2×Q8C22×D28311yescharacteristic
243C2×Q8C22×D28311yescharacteristic
244C2×D4—2831no—
245C22×C4—2831no—
246C22×C4—5661no—
247C22×C4—5661no—
248C22×C4—5661no—
249C22×C4C22×D28311yescharacteristic
250Q16—41no—
251Q16—5662no—
252Q16—11321no—
253M16—41no—
254C2×C8—2831no—
255C2×C8—2831no—
256C2×C8—11321no—
257C2×C8C22×D28311yescharacteristic
258C2×C8C22×D28311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—2831no—
262C4⋊C4—2831no—
263C4⋊C4—2831no—
264C4⋊C4—5661no—
265C4⋊C4—5661no—
266C4⋊C4—5661no—
267C4⋊C4—5661no—
268C4⋊C4—5661no—
269C4⋊C4—5661no—
270C4⋊C4—5661no—
271C4⋊C4—5661no—
272C4⋊C4—5661no—
273C4⋊C4C22×D28311yescharacteristic
274C4⋊C4C22×D28311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2831no—
278C22⋊C4—2831no—
279C22⋊C4—5661no—
280C22⋊C4—5661no—
281C22⋊C4—5661no—
282C42—2831no—
283C42—5661no—
284C42—5661no—
285C42—5661no—
286C23—2831no—
287C23D4×D28311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—2832no—
293Q8—5661no—
294Q8—5661no—
295Q8—5661no—
296Q8—11322no—
297Q8—11322no—
298D4—5662no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—2831no—
306C2×C4—2831no—
307C2×C4—2831no—
308C2×C4—5661no—
309C2×C4—5661no—
310C2×C4—5661no—
311C2×C4—5661no—
312C2×C4—5661no—
313C2×C4—5661no—
314C2×C4—11322no—
315C2×C4—11322no—
316C2×C4—11322no—
317C2×C4D4×D28311yescharacteristic
318C2×C4C23×D28311yescharacteristic
319C8—21no—
320C8—21no—
321C8—5661no—
322C8—5661no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—2832no—
327C22—5661no—
328C22—5661no—
329C22C2×D4×D28311yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—5661no—
336C4—5661no—
337C4—5661no—
338C4—5662no—
339C4—11321no—
340C4—11321no—
341C4—11321no—
342C4#41×D28311yescharacteristic
343C4#2258365511yescharacteristic
344C2—41no—
345C2—5662no—
346C2#3182710411yescharacteristic
347C2#137×D28311yescharacteristic
348C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#470×C283C2
socleC22×C283C22×D4
prime radicalC2×C22⋊D283D4
hypercenter#470D283
hypocenterC283#3312
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. #31826832C2semidirect
4. #31826803C2semidirect
5. #31826797C2semidirect
6. #31826775C2semidirect
7. #31826705C2semidirect
8. #31826673C2semidirect
9. #31826620C2semidirect
10. #31826619C2semidirect
11. #31826569C2semidirect
13. #31826475C2semidirect
14. #103×D283C2semidirect
15. #470×C283C2semidirect
174. C283#3312semidirect
182. #470D283semidirect

Maximal subgroups

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

subgroupindex
#318271492
#318271282
#318268322
#318268032
#318267972
#318267752
#318267052
#318266732
#318266202
#318266192
#318265692
#318265342
#318264752
#103×D2832
#470×C2832
#3312283