Finite Group Databasebeta

← #49133358

Subgroups — #49133358

Order 36224.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827150C211yesmaximal, characteristic, complemented
2#31827114C211yesmaximal, characteristic
3#31826832C211yesmaximal, characteristic, complemented
4#31826804C211yesmaximal, characteristic, complemented
5#31826800C211yesmaximal, characteristic, complemented
6#31826767C211yesmaximal, characteristic, complemented
7#31826723C211yesmaximal, characteristic, complemented
8#31826705C211yesmaximal, characteristic, complemented
9#31826674C211yesmaximal, characteristic, complemented
10#31826628C211yesmaximal, characteristic, complemented
11#31826627C211yesmaximal, characteristic, complemented
12#31826562C211yesmaximal, characteristic, complemented
13#31826534C211yesmaximal, characteristic, complemented
14#31826458C211yesmaximal, characteristic, complemented
15#417×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×Dic1132C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985380C2211yescharacteristic
20C2×C8×D283C2211yescharacteristic
21#22583608C2211yescharacteristic
22#22583605—21no—
23#22583601C2211yescharacteristic
24#22583600—21no—
25#22583593—21no—
26#22583593—21no—
27#22583591—21no—
28#22583590C2211yescharacteristic
29#22583589C2211yescharacteristic
30#22583589C2211yescharacteristic
31#22583586C2211yescharacteristic
32#22583582—21no—
33#22583575C2211yescharacteristic
34D4×Dic283C2211yescharacteristic
35C4×C22⋊D283C2211yescharacteristic
36#22583569—21no—
37#22583568C2211yescharacteristic
38C2×Dic283⋊C4—21no—
39#22583543C2211yescharacteristic
40#22583543C2211yescharacteristic
41#22583542C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583533C2211yescharacteristic
47#22583533C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51#22583520C2211yescharacteristic
52C2×#41×C283C2211yescharacteristic
53C2×M16×C283C2211yescharacteristic
54C2×C4⋊C4×C283C2211yescharacteristic
55#103×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985391—22no—
60#17985391—41no—
61C2×Dic566C2311yescharacteristic
62C2×Dic566C2311yescharacteristic
63C2×C22⋊D283—21no—
64C2×C22⋊D283D411yescharacteristic
65C22×Dic283—21no—
66C22×Dic283—21no—
67C2×C4×D283C2311yescharacteristic
68Dic1132—41no—
69#17985382—41no—
70#17985380—22no—
71C2×C283⋊C8C2311yescharacteristic
72C2×C283⋊C8C2311yescharacteristic
73C8×D283—41no—
74C4⋊Dic283—21no—
75C4⋊Dic283—21no—
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4D411yescharacteristic
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283D411yescharacteristic
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4D411yescharacteristic
94C4×Dic283—21no—
95C4×Dic283—21no—
96C4×Dic283C2311yescharacteristic
97#41×C283—22no—
98#41×C283—41no—
99C2×Q8×C283C2311yescharacteristic
100C2×D4×C283—21no—
101C2×D4×C283C2311yescharacteristic
102C22×C4×C283—21no—
103C22×C4×C283—21no—
104C22×C4×C283C2311yescharacteristic
105M16×C283—41no—
106C2×C8×C283C2311yescharacteristic
107C2×C8×C283C2311yescharacteristic
108C4⋊C4×C283—21no—
109C4⋊C4×C283C2311yescharacteristic
110C4⋊C4×C283C2311yescharacteristic
111C22×D283C2×D411yescharacteristic
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—21no—
116C22⋊D283—41no—
117C22⋊D283—42no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—42no—
123C2×Dic283—42no—
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C4×D283—21no—
128C4×D283—21no—
129C283⋊C8—21no—
130C283⋊C8—21no—
131C23×C283—21no—
132C23×C283C2×D411yescharacteristic
133Q8×C283—21no—
134Q8×C283—21no—
135D4×C283—21no—
136D4×C283—42no—
137D4×C283#4112yes—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—42no—
144C2×C4×C283—42no—
145C2×C4×C283C2×D411yescharacteristic
146C2×C4×C283C2411yescharacteristic
147C8×C283—21no—
148C8×C283—21no—
149C22×C283—21no—
150C22×C283—22no—
151C22×C283—21no—
152C22×C283—21no—
153C22×C283—41no—
154C22×C283—41no—
155C22×C283C22×D411yescharacteristic
156C2×D283—21no—
157C2×D283—21no—
158C2×D283—21no—
159C4×C283—21no—
160C4×C283—41no—
161C4×C283—41no—
162C4×C283—41no—
163C4×C283C2×#4111yescharacteristic
164C4×C283H(5,𝔽2)11yescharacteristic
165Dic283—21no—
166Dic283—21no—
167Dic283—21no—
168Dic283—41no—
169Dic283—41no—
170Dic283—41no—
171Dic283—41no—
172C2×C283—41no—
173C2×C283—42no—
174C2×C283#54611yescharacteristic
175C2×C283#57511yescharacteristic
176C2×C283#57811yescharacteristic
177D283—41no—
178C283#331511yescharacteristic, complemented, Sylow
179#3315—2831nomaximal, Sylow
180#552—2831no—
181#547—2831no—
182#491—2831no—
183#485—2831no—
184#484—2831no—
185#478—2831no—
186#470—2831no—
187#467—2831no—
188#462—2831no—
189#452—2831no—
190#452—2831no—
191#440—2831no—
192#435—2831no—
193#417D28311yescharacteristic, complemented
194#416—2831no—
195C2×#41—2831no—
196C2×#41C2×D28311yescharacteristic
197C2×Q16—2831no—
198C2×Q16—2831no—
199C2×SD16—2831no—
200C2×M16C2×D28311yescharacteristic
201C22×C8—2831no—
202C4⋊Q8—2831no—
203#126—5661no—
204#125—2831no—
205#125—5661no—
206#123—5661no—
207#123—5661no—
208#123—5661no—
209C22⋊Q8—2831no—
210C22⋊Q8—2831no—
211C22⋊Q8—2831no—
212C22⋊Q8—2831no—
213#121—5661no—
214C4×Q8—2831no—
215C4×D4—2831no—
216C4×D4—2831no—
217#117—2831no—
218#117—5661no—
219C2×C4⋊C4—5661no—
220C2×C4⋊C4C2×D28311yescharacteristic
221#107—2831no—
222#107—2831no—
223#106—2831no—
224C4⋊C8—2831no—
225#103—2831no—
226#103—2831no—
227#103—2831no—
228#103—2831no—
229#103—2831no—
230#103C2×D28311yescharacteristic
231#103C2×D28311yescharacteristic
232#102—2831no—
233#102C2×D28311yescharacteristic
234#102C2×D28311yescharacteristic
235C22⋊C8—2831no—
236C22⋊C8—2831no—
237#97—2831no—
238#41—22no—
239#41—41no—
240#41—5662no—
241#41—11321no—
242C2×Q8—2831no—
243C2×Q8—2831no—
244C2×Q8C22×D28311yescharacteristic
245C2×D4—21no—
246C2×D4—2831no—
247C2×D4—5661no—
248C2×D4C22×D28311yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—2831no—
252C22×C4—5661no—
253C22×C4—5661no—
254C22×C4C22×D28311yescharacteristic
255Q16—11321no—
256Q16—11321no—
257SD16—5662no—
258M16—41no—
259C2×C8—2831no—
260C2×C8—2831no—
261C2×C8—11321no—
262C2×C8C22×D28311yescharacteristic
263C2×C8C22×D28311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—2831no—
266C4⋊C4—2831no—
267C4⋊C4—2831no—
268C4⋊C4—5661no—
269C4⋊C4—5661no—
270C4⋊C4—5661no—
271C4⋊C4—5661no—
272C4⋊C4—5661no—
273C4⋊C4—5661no—
274C4⋊C4—5661no—
275C4⋊C4—5661no—
276C4⋊C4—5661no—
277C4⋊C4C22×D28311yescharacteristic
278C4⋊C4C22×D28311yescharacteristic
279C22⋊C4—2831no—
280C22⋊C4—2831no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C22⋊C4—5661no—
284C22⋊C4—5661no—
285C22⋊C4—5661no—
286C22⋊C4—5661no—
287C42—2831no—
288C42—5661no—
289C42—5661no—
290C23—21no—
291C23—2831no—
292C23D4×D28311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—5661no—
296Q8—5661no—
297Q8—5661no—
298Q8—5661no—
299D4—21no—
300D4—42no—
301D4—11321no—
302D4—11322no—
303D4#1798539212yes—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—2831no—
312C2×C4—2831no—
313C2×C4—2831no—
314C2×C4—5661no—
315C2×C4—5661no—
316C2×C4—5661no—
317C2×C4—5661no—
318C2×C4—5661no—
319C2×C4—5661no—
320C2×C4—11322no—
321C2×C4—11322no—
322C2×C4D4×D28311yescharacteristic
323C2×C4C23×D28311yescharacteristic
324C8—21no—
325C8—21no—
326C8—5661no—
327C8—5661no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—5661no—
335C22—5661no—
336C22—5661no—
337C22C2×D4×D28311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—5661no—
343C4—5661no—
344C4—5661no—
345C4—11321no—
346C4—11321no—
347C4—11321no—
348C4—11321no—
349C4C2×#1798539211yescharacteristic
350C4#2258365511yescharacteristic
351C2—41no—
352C2—42no—
353C2—11321no—
354C2#3182710511yescharacteristic
355C2#3182726111yescharacteristic
356C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radical#31827114C2
hypercenter#417D283
hypocenterC283#3315
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #31827150C2semidirect
3. #31826832C2semidirect
4. #31826804C2semidirect
5. #31826800C2semidirect
6. #31826767C2semidirect
7. #31826723C2semidirect
8. #31826705C2semidirect
9. #31826674C2semidirect
10. #31826628C2semidirect
11. #31826627C2semidirect
12. #31826562C2semidirect
13. #31826534C2semidirect
14. #31826458C2semidirect
15. #417×C283C2semidirect
178. C283#3315semidirect
193. #417D283semidirect

Maximal subgroups

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

subgroupindex
#318271502
#318271142
#318268322
#318268042
#318268002
#318267672
#318267232
#318267052
#318266742
#318266282
#318266272
#318265622
#318265342
#318264582
#417×C2832
#3315283