Finite Group Databasebeta

← #214641

Subgroups — #214641

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37424C211yesmaximal, characteristic
2#21755C211yesmaximal, characteristic
3#21750C211yesmaximal, characteristic
4#9459C214yesmaximal, complemented
5#9443C214yesmaximal, complemented
6#8741C211yesmaximal, characteristic, complemented
7#8740C211yesmaximal, characteristic, complemented
8#8521C211yesmaximal, characteristic, complemented
9#8521C211yesmaximal, characteristic, complemented
10C2×#541C2211yescharacteristic
11C2×#535C2211yescharacteristic
12#2879C2212yes—
13#2861C411yescharacteristic, complemented
14#2858C411yescharacteristic, complemented
15#2711—24no—
16#2685—24no—
17#2658C411yescharacteristic, complemented
18#2641C411yescharacteristic, complemented
19#2548C412yescomplemented
20#2545C412yescomplemented
21#2435C2212yes—
22#2432C2212yes—
23#2308C2212yes—
24#2298C2211yescharacteristic
25#1900—24no—
26#1898—24no—
27#1822C2211yescharacteristic
28#1822C2211yescharacteristic
29#1820C2211yescharacteristic
30#1820C2211yescharacteristic
31#1783C2212yes—
32#1782C2212yes—
33#1511C2214yes—
34#1511C2214yes—
35#1284C2214yes—
36#1284C2214yes—
37C22×#41C2311yescharacteristic
38C23×D4C2311yescharacteristic
39C22×M16—22no—
40#541—44no—
41#535—44no—
42C2×C4⋊Q8C2×C411yescharacteristic, complemented
43C2×C4⋊Q8C2×C412yescomplemented
44C2×#127C2×C411yescharacteristic
45C2×#127C2×C411yescharacteristic
46C2×#126—22no—
47C2×#125—22no—
48C2×#125C2×C411yescharacteristic, complemented
49C2×#124C2×C412yes—
50C2×#124C2×C412yes—
51C2×#124C2×C411yescharacteristic
52C2×#124C2×C411yescharacteristic
53C2×#124C2×C412yescomplemented
54C2×C22⋊Q8C2312yes—
55C2×#121D414yes—
56C2×#120D414yes—
57C2×C4×Q8C2×C412yescomplemented
58C2×#117C2311yescharacteristic
59C22×C22⋊C4—22no—
60C2×#100—24no—
61C2×#99—24no—
62#408—44no—
63#407—22no—
64#407C2312yes—
65#407C2312yes—
66#399—24no—
67#396—24no—
68#394—24no—
69#394—24no—
70#394—24no—
71#394—24no—
72#391C2×C412yescomplemented
73#390—24no—
74#390—24no—
75#390C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yescomplemented
78#388—24no—
79#388—24no—
80#388—22no—
81#387—22no—
82#386—44no—
83#385—24no—
84#381C2312yes—
85#381C2312yes—
86#379—24no—
87#379—22no—
88C4×C4⋊C4C2×C412yescomplemented
89C4×C22⋊C4C2312yes—
90#376C2×C411yescharacteristic, complemented
91C43C2×C411yescharacteristic, complemented
92#343—22no—
93#343—22no—
94#342—22no—
95#342—22no—
96#328—44no—
97#328—44no—
98C25C2×D412yes—
99C2×#41—24no—
100C2×#41—82no—
101C22×Q8C22×C412yes—
102C22×Q8C22×C411yescharacteristic
103C22×D4—24no—
104C22×D4—82no—
105C22×D4C22⋊C414yes—
106C22×D4C22⋊C414yes—
107C22×D4C22×C411yescharacteristic
108C22×D4C22×C411yescharacteristic
109C22×D4C22×C411yescharacteristic
110C22×D4C2×D412yes—
111C22×D4C2×D412yes—
112C23×C4—42no—
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—42no—
118C2×M16—84no—
119C22×C8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—82no—
122#127—42no—
123#127—42no—
124#126—48no—
125#125—42no—
126#125—48no—
127#124—42no—
128#124—42no—
129#124—82no—
130#124—82no—
131#124—82no—
132C22⋊Q8—24no—
133C22⋊Q8—24no—
134#121—44no—
135#121—44no—
136#120—44no—
137#120—44no—
138C4×Q8—42no—
139C4×Q8—42no—
140#117—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—84no—
162C2×C22⋊C4C22⋊C414yes—
163C2×C22⋊C4C22⋊C414yes—
164C2×C22⋊C4C2×D412yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171#100—84no—
172#99—84no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175C22⋊C8—44no—
176#95—22no—
177#95—22no—
178#95—24no—
179#95—22no—
180#95—24no—
181#95—24no—
182C24—22no—
183C24—22no—
184C24—44no—
185C24—84no—
186C24C2×C22⋊C412yes—
187C24C2×C22⋊C412yes—
188C24C2×C22⋊C412yes—
189C24C22×D411yescharacteristic
190#41—84no—
191#41—84no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—24no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—84no—
211C2×D4—84no—
212C2×D4—84no—
213C2×D4—84no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4—164no—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4H(5,𝔽2)14yes—
232M16—44no—
233M16—44no—
234M16—84no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—84no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—84no—
246C4⋊C4—82no—
247C22⋊C4—24no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C22⋊C4—84no—
256C22⋊C4—84no—
257C22⋊C4—82no—
258C22⋊C4—84no—
259C22⋊C4—84no—
260C22⋊C4—84no—
261C22⋊C4—164no—
262C42—22no—
263C42—22no—
264C42—42no—
265C42—42no—
266C42—44no—
267C42—82no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—44no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—84no—
283C23—84no—
284C23—168no—
285C23C22×C22⋊C411yescharacteristic
286C23#53412yes—
287Q8—82no—
288Q8—82no—
289Q8—82no—
290D4—84no—
291D4—82no—
292D4—82no—
293D4—82no—
294D4—84no—
295D4—164no—
296D4—164no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—168no—
325C2×C4#51812yes—
326C2×C4#53412yes—
327C8—82no—
328C8—82no—
329C22—22no—
330C22—24no—
331C22—22no—
332C22—44no—
333C22—44no—
334C22—44no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—82no—
339C22—84no—
340C22—84no—
341C22—84no—
342C22—84no—
343C22—84no—
344C22—164no—
345C22#230811yescharacteristic
346C22#288312yes—
347C4—22no—
348C4—22no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4—82no—
356C4—162no—
357C4—162no—
358C2—22no—
359C2—42no—
360C2—82no—
361C2—82no—
362C2—82no—
363C2#2175011yescharacteristic
364C2#2178412yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9459C2semidirect
5. #9443C2semidirect
6. #8741C2semidirect
7. #8740C2semidirect
8. #8521C2semidirect
9. #8521C2semidirect
13. #2861C4semidirect
14. #2858C4semidirect
17. #2658C4semidirect
18. #2641C4semidirect
19. #2548C4semidirect
20. #2545C4semidirect
42. C2×C4⋊Q8C2×C4semidirect
43. C2×C4⋊Q8C2×C4semidirect
48. C2×#125C2×C4semidirect
53. C2×#124C2×C4semidirect
57. C2×C4×Q8C2×C4semidirect
72. #391C2×C4semidirect
77. #390C2×C4semidirect
88. C4×C4⋊C4C2×C4semidirect
90. #376C2×C4semidirect
91. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#374242
#217552
#217502
#94592
#94432
#87412
#87402
#85212