Finite Group Databasebeta

← #49133154

Subgroups — #49133154

Order 36224.

386 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#31827157C211yesmaximal, characteristic, complemented
2#31827121C211yesmaximal, characteristic
3#31826818C211yesmaximal, characteristic, complemented
4#31826809C211yesmaximal, characteristic, complemented
5#31826776C211yesmaximal, characteristic, complemented
6#31826775C211yesmaximal, characteristic, complemented
7#31826705C211yesmaximal, characteristic, complemented
8#31826685C211yesmaximal, characteristic, complemented
9#31826624C211yesmaximal, characteristic, complemented
10#31826620C211yesmaximal, characteristic, complemented
11#31826557C211yesmaximal, characteristic, complemented
12C8×C22⋊D283C211yesmaximal, characteristic, complemented
13#31826458C211yesmaximal, characteristic, complemented
14C2×#22583535C211yesmaximal, characteristic, complemented
15#465×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C22×Dic566C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985382C2211yescharacteristic
20C2×C8×D283C2211yescharacteristic
21#22583605—21no—
22#22583601—21no—
23#22583601C2211yescharacteristic
24#22583597C2211yescharacteristic
25#22583596—21no—
26#22583594—21no—
27#22583590C2211yescharacteristic
28#22583587—21no—
29#22583586C2211yescharacteristic
30#22583582—21no—
31Q8×Dic283C2211yescharacteristic
32#22583575C2211yescharacteristic
33D4×Dic283—21no—
34C4×C22⋊D283C2211yescharacteristic
35#22583568C2211yescharacteristic
36#22583566—21no—
37C2×C4⋊Dic283C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583543C2211yescharacteristic
40Dic283⋊C8C2211yescharacteristic
41#22583535C2211yescharacteristic
42#22583535C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583533C2211yescharacteristic
46#22583533C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C2×SD16×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic
54C22⋊Q8×C283C2211yescharacteristic
55#121×C283C2211yescharacteristic
56#106×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985391—22no—
60#17985391—41no—
61C2×Dic566—22no—
62C2×Dic566—21no—
63C2×Dic566—41no—
64C2×Dic566C2311yescharacteristic
65C2×Dic566C2311yescharacteristic
66C2×C22⋊D283—21no—
67C2×C22⋊D283D411yescharacteristic
68C22×Dic283—21no—
69C22×Dic283—21no—
70C22×Dic283—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⋊Dic283—21no—
81C4⋊Dic283C2311yescharacteristic
82C4⋊Dic283C2311yescharacteristic
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4—21no—
86Dic283⋊C4D411yescharacteristic
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×Dic283—21no—
97C4×Dic283C2311yescharacteristic
98C2×Q8×C283C2311yescharacteristic
99C2×D4×C283—21no—
100C2×D4×C283C2311yescharacteristic
101C22×C4×C283C2311yescharacteristic
102SD16×C283—41no—
103C2×C8×C283—41no—
104C2×C8×C283C2311yescharacteristic
105C2×C8×C283C2311yescharacteristic
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#4112yes—
117C22⋊D283—22no—
118C22⋊D283—42no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—42no—
124C2×Dic283—42no—
125C2×Dic283—42no—
126C2×Dic283C2×D411yescharacteristic
127C2×Dic283C2×D411yescharacteristic
128C2×Dic283C2×D411yescharacteristic
129C4×D283—21no—
130C4×D283—21no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283—21no—
134C23×C283C2×D411yescharacteristic
135Q8×C283—21no—
136Q8×C283—21no—
137D4×C283—21no—
138D4×C283—21no—
139D4×C283—42no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283C2×D411yescharacteristic
146C2×C4×C283C2411yescharacteristic
147C8×C283—21no—
148C8×C283—21no—
149C22×C283—21no—
150C22×C283—21no—
151C22×C283—41no—
152C22×C283—41no—
153C22×C283—41no—
154C22×C283#13512yes—
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—22no—
167Dic283—21no—
168Dic283—21no—
169Dic283—41no—
170Dic283—41no—
171Dic283—41no—
172C2×C283—22no—
173C2×C283—81no—
174C2×C283#54611yescharacteristic
175C2×C283C2×#13511yescharacteristic
176C2×C283#57811yescharacteristic
177D283—41no—
178C283#330011yescharacteristic, complemented, Sylow
179#3300—2831nomaximal, Sylow
180#552—2831no—
181#548—2831no—
182#488—2831no—
183#485—2831no—
184#480—2831no—
185#479—2831no—
186#467—2831no—
187#465D28311yescharacteristic, complemented
188#464—2831no—
189#452—2831no—
190#451—2831no—
191C4×Q16—2831no—
192C8×D4—2831no—
193#416—2831no—
194C2×#103—2831no—
195C2×#41—2831no—
196C22×Q8—2831no—
197C2×Q16—2831no—
198C2×Q16—2831no—
199C2×SD16C2×D28311yescharacteristic
200C22×C8—2831no—
201C22×C8C2×D28311yescharacteristic
202#126—5661no—
203#125—2831no—
204#125—5661no—
205#124—2831no—
206#124—5661no—
207#123—5661no—
208C22⋊Q8—2831no—
209C22⋊Q8—2831no—
210C22⋊Q8—5661no—
211C22⋊Q8C2×D28311yescharacteristic
212#121—5661no—
213#121C2×D28311yescharacteristic
214C4×Q8—2831no—
215C4×Q8—2831no—
216C4×D4—2831no—
217C4×D4—5661no—
218#117—2831no—
219#117—5661no—
220C2×C4⋊C4—2831no—
221#107—2831no—
222#107—2831no—
223#106C2×D28311yescharacteristic
224C4⋊C8—2831no—
225#103—2831no—
226#103—2831no—
227#103—2831no—
228#103—2831no—
229#103—2831no—
230#103—2831no—
231#103C2×D28311yescharacteristic
232#102—2831no—
233#102—2831no—
234#102C2×D28311yescharacteristic
235C22⋊C8—2831no—
236C22⋊C8—2831no—
237C4×C8—2831no—
238#41—5662no—
239#41—11321no—
240C2×Q8—2831no—
241C2×Q8—2831no—
242C2×Q8—5662no—
243C2×Q8—5661no—
244C2×Q8—11321no—
245C2×Q8C22×D28311yescharacteristic
246C2×D4—21no—
247C2×D4—2831no—
248C2×D4—5661no—
249C2×D4C22×D28311yescharacteristic
250C22×C4—2831no—
251C22×C4—5661no—
252C22×C4—5661no—
253C22×C4—5661no—
254C22×C4C22×D28311yescharacteristic
255Q16—5662no—
256Q16—11321no—
257SD16—41no—
258C2×C8—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⋊C4C22×D28311yescharacteristic
276C4⋊C4C22×D28311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2831no—
280C22⋊C4—2831no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C22⋊C4—5661no—
284C22⋊C4—5661no—
285C22⋊C4—5661no—
286C42—2831no—
287C42—5661no—
288C42—5661no—
289C42—5661no—
290C23—21no—
291C23—2831no—
292C23D4×D28311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2832no—
296Q8—5661no—
297Q8—5661no—
298Q8—5661no—
299Q8—11322no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—5662no—
304D4—11322no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—2831no—
311C2×C4—2831no—
312C2×C4—2831no—
313C2×C4—5661no—
314C2×C4—5661no—
315C2×C4—5661no—
316C2×C4—5661no—
317C2×C4—5661no—
318C2×C4—5661no—
319C2×C4—11322no—
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—41no—
331C22—41no—
332C22—41no—
333C22—5661no—
334C22—5661no—
335C22—5661no—
336C22#2258363112yes—
337C22C2×D4×D28311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—5661no—
343C4—5661no—
344C4—5661no—
345C4—5662no—
346C4—11321no—
347C4—11321no—
348C4—11321no—
349C4#41×D28311yescharacteristic
350C4#2258365511yescharacteristic
351C2—22no—
352C2—81no—
353C2—11321no—
354C2#3182710411yescharacteristic
355C2C2×#2258363111yescharacteristic
356C2#3182727911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#465×C283C2
socleC22×C283C22×D4
prime radical#31827121C2
hypercenter#465D283
hypocenterC283#3300
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31827157C2semidirect
3. #31826818C2semidirect
4. #31826809C2semidirect
5. #31826776C2semidirect
6. #31826775C2semidirect
7. #31826705C2semidirect
8. #31826685C2semidirect
9. #31826624C2semidirect
10. #31826620C2semidirect
11. #31826557C2semidirect
12. C8×C22⋊D283C2semidirect
13. #31826458C2semidirect
14. C2×#22583535C2semidirect
15. #465×C283C2semidirect
178. C283#3300semidirect
187. #465D283semidirect

Maximal subgroups

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

subgroupindex
#318271572
#318271212
#318268182
#318268092
#318267762
#318267752
#318267052
#318266852
#318266242
#318266202
#318265572
C8×C22⋊D2832
#318264582
C2×#225835352
#465×C2832
#3300283