Finite Group Databasebeta

← #7622739

Subgroups — #7622739

Order 512.

1263 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60204C211yesmaximal, characteristic
2#33507C218yesmaximal, complemented
3#33500C212yesmaximal, complemented
4#33493C212yesmaximal, complemented
5#33492C214yesmaximal, complemented
6#33485C214yesmaximal, complemented
7#33484C214yesmaximal, complemented
8#33421C214yesmaximal, complemented
9#33418C212yesmaximal, complemented
10#3444C411yescharacteristic
11#2966C414yes—
12#2966C2214yes—
13#2965C414yes—
14C8×#41C2214yes—
15#2960C411yescharacteristic
16#2960C2211yescharacteristic
17#2931C411yescharacteristic
18#2931C2211yescharacteristic
19#2929C412yes—
20#2929C2212yes—
21#2922C412yes—
22#2920C411yescharacteristic
23C4×#131C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2183C2218yescomplemented
26Q8×C16C2218yescomplemented
27#2169C22116yes—
28#2169C2218yescomplemented
29#2168C22116yes—
30D4×C16C2218yescomplemented
31#2164C2218yes—
32#2164C2218yescomplemented
33#2163C2214yes—
34#2163C2212yes—
35#2163C2212yes—
36#2163C2218yescomplemented
37#2162C2214yes—
38#2162C2214yes—
39#2151C2218yescomplemented
40#2150C2214yes—
41#2113C2218yes—
42#2112C2214yes—
43C4×M32C2214yes—
44C4×M32C2214yescomplemented
45C2×#346C2214yescomplemented
46C2×#131C2×C412yes—
47C2×#131C2312yes—
48C23×C8C2×C411yescharacteristic
49C23×C8C2311yescharacteristic
50#533C812yes—
51#533C2×C411yescharacteristic
52#532C812yes—
53#532C2×C411yescharacteristic
54#529C814yes—
55#529C814yes—
56#529C2×C414yes—
57#525C814yes—
58#525C2×C412yes—
59C4×#41C2×C414yes—
60C4×#41C2×C412yes—
61C22×C42C2×C411yescharacteristic
62C2×M32C2318yes—
63C2×M32C2318yes—
64C22×C16C2318yes—
65#446C2×C414yes—
66#446C2×C414yes—
67C8×Q8C2×C414yes—
68C8×Q8C2314yes—
69#436C2×C414yes—
70#436C2×C414yes—
71#435C2×C418yes—
72#435C2×C418yes—
73C8×D4C2×C418yes—
74C8×D4C2×C414yes—
75C8×D4C2318yes—
76C8×D4C2314yes—
77#433C2×C418yes—
78#432C2×C414yes—
79#431C2×C414yes—
80#431C2×C418yes—
81#424C2×C412yes—
82#424C2×C412yes—
83#423C2×C411yescharacteristic
84#423C2×C411yescharacteristic
85#408C2×C412yes—
86#408C2×C412yes—
87#407C2×C412yes—
88#405C2×C412yes—
89#405C2×C414yes—
90#405C2314yes—
91#405C2312yes—
92C4×M16C2×C412yes—
93C4×M16C2×C414yes—
94C2×C4×C8C2×C411yescharacteristic
95C2×C4×C8C2×C411yescharacteristic
96C2×C4×C8C2×C412yes—
97C2×C4×C8C2×C412yes—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2312yes—
100C2×C4×C8C2312yes—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2314yes—
103C4⋊C16C2318yes—
104C4⋊C16C2314yes—
105C4⋊C16C2314yes—
106C4⋊C16C23116yes—
107C22⋊C16C2318yes—
108C22⋊C16C2314yes—
109C22⋊C16C2314yes—
110C22⋊C16C23116yes—
111#346C2314yes—
112#346C2318yes—
113#346C2314yes—
114C4×C16C2318yes—
115C4×C16C2318yes—
116C2×#41C2×C814yes—
117C2×#41C22×C412yes—
118C23×C4C2×C812yes—
119C23×C4C22×C411yescharacteristic
120#131—44no—
121#131—44no—
122C2×M16C22×C412yes—
123C2×M16C22×C414yes—
124C2×M16C22×C412yes—
125C2×M16C22×C414yes—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8C22×C411yescharacteristic
129C22×C8C22×C412yes—
130C22×C8C22×C414yes—
131C22×C8C22×C411yescharacteristic
132C22×C8C22×C412yes—
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C412yes—
135C22×C8C2411yescharacteristic
136C22×C8C2412yes—
137C22×C8C2414yes—
138C4⋊Q8C2×C812yes—
139C4⋊Q8C2×C811yescharacteristic
140#127C2×C811yescharacteristic
141#126C2×C818yes—
142#125C2×C812yes—
143#125C2×C814yes—
144#124C2×C812yes—
145#124C2×C814yes—
146#123C2×C818yes—
147#123C2×C814yes—
148C22⋊Q8C2×C814yes—
149C22⋊Q8C2×C814yes—
150C22⋊Q8C2×C814yes—
151#121C2×C814yes—
152#121C2×C814yes—
153#121C2×C814yes—
154#120C2×C814yes—
155C4×Q8C2×C814yes—
156C4×Q8C2×C814yes—
157C4×Q8C22×C414yes—
158C4×D4C2×C818yes—
159C4×D4C2×C818yes—
160C4×D4C2×C814yes—
161C4×D4C2×C814yes—
162C4×D4C22×C418yes—
163C4×D4C22×C414yes—
164#117C2×C814yes—
165#117C2×C818yes—
166#117C22×C412yes—
167#117C22×C414yes—
168C2×C42C2×C812yes—
169C2×C42C2×C812yes—
170C2×C42C2×C812yes—
171C2×C42C22×C414yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C412yes—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176M32—48no—
177M32—48no—
178C2×C16—48no—
179C2×C16C2418yes—
180C2×C16C2418yes—
181C4⋊C8C22×C414yes—
182C4⋊C8C22×C414yes—
183C4⋊C8C22×C418yes—
184C4⋊C8C22×C418yes—
185C22⋊C8C22×C418yes—
186C22⋊C8C22×C418yes—
187C22⋊C8C22×C418yes—
188#97C22×C414yes—
189#97C22×C414yes—
190#97C22×C414yes—
191C4×C8C22×C411yescharacteristic
192C4×C8C22×C411yescharacteristic
193C4×C8C22×C412yes—
194C4×C8C22×C412yes—
195C4×C8C22×C412yes—
196C4×C8C22×C412yes—
197C4×C8C22×C412yes—
198C4×C8C22×C414yes—
199C4×C8C22×C414yes—
200C4×C8C2412yes—
201C4×C8C2414yes—
202C4×C8C2411yescharacteristic
203C4×C8C2411yescharacteristic
204C24C22×C811yescharacteristic
205#41—42no—
206#41—42no—
207#41—48no—
208C2×Q8C22×C814yes—
209C2×Q8C22×C812yes—
210C2×D4C22×C814yes—
211C2×D4C22×C814yes—
212C2×D4C22×C812yes—
213C2×D4C22×C818yes—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—44no—
217C22×C4C22×C812yes—
218C22×C4C22×C812yes—
219C22×C4C22×C812yes—
220C22×C4C22×C812yes—
221C22×C4C22×C811yescharacteristic
222C22×C4C22×C812yes—
223C22×C4C22×C812yes—
224C22×C4C22×C812yes—
225C22×C4C22×C818yes—
226C22×C4C23×C412yes—
227C22×C4C23×C414yes—
228C22×C4C23×C411yescharacteristic
229M16—44no—
230M16—42no—
231M16—44no—
232M16—42no—
233C2×C8—24no—
234C2×C8—22no—
235C2×C8—28no—
236C2×C8—28no—
237C2×C8—24no—
238C2×C8—22no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8C23×C411yescharacteristic
244C2×C8C23×C412yes—
245C2×C8C23×C411yescharacteristic
246C2×C8C23×C412yes—
247C2×C8C23×C414yes—
248C2×C8C23×C414yes—
249C2×C8C23×C411yescharacteristic
250C2×C8C2511yescharacteristic
251C4⋊C4C22×C814yes—
252C4⋊C4C22×C818yes—
253C4⋊C4C22×C814yes—
254C4⋊C4C22×C818yes—
255C4⋊C4C22×C814yes—
256C4⋊C4C22×C818yes—
257C22⋊C4C22×C818yes—
258C22⋊C4C22×C818yes—
259C22⋊C4C22×C818yes—
260C22⋊C4C22×C814yes—
261C22⋊C4C22×C818yes—
262C42C22×C812yes—
263C42C22×C812yes—
264C42C22×C812yes—
265C42C22×C812yes—
266C42C22×C811yescharacteristic
267C42C22×C812yes—
268C42C22×C811yescharacteristic
269C42C22×C818yes—
270C42C23×C411yescharacteristic
271C42C23×C411yescharacteristic
272C42C23×C412yes—
273C42C23×C414yes—
274C16—28no—
275C16—28no—
276C23—41no—
277C23—42no—
278C23C23×C814yes—
279C23C23×C811yescharacteristic
280C23C23×C812yes—
281Q8—28no—
282Q8—42no—
283D4—28no—
284D4—42no—
285D4—44no—
286D4—48no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—48no—
302C2×C4C23×C812yes—
303C2×C4C23×C812yes—
304C2×C4C23×C811yescharacteristic
305C2×C4C23×C812yes—
306C2×C4C23×C812yes—
307C2×C4C23×C814yes—
308C2×C4C23×C818yes—
309C2×C4C23×C812yes—
310C2×C4C24×C411yescharacteristic
311C8—22no—
312C8—21no—
313C8—24no—
314C8—22no—
315C8—21no—
316C8—24no—
317C8#56812yes—
318C8#58512yes—
319C22—21no—
320C22—22no—
321C22—24no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—28no—
326C22—44no—
327C22C24×C811yescharacteristic
328C4—22no—
329C4—21no—
330C4—24no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—28no—
335C4#340714yes—
336C4#342811yescharacteristic
337C4C2×#56811yescharacteristic
338C2—44no—
339C2—41no—
340C2—42no—
341C2#6027611yescharacteristic
342C2C2×#340712yes—
subgroupG/H
center, Frattini, cosocleC2×C8C25
commutator, socleC22C24×C8
prime radical#3444C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33507C2semidirect
3. #33500C2semidirect
4. #33493C2semidirect
5. #33492C2semidirect
6. #33485C2semidirect
7. #33484C2semidirect
8. #33421C2semidirect
9. #33418C2semidirect
25. #2183C22semidirect
26. Q8×C16C22semidirect
28. #2169C22semidirect
30. D4×C16C22semidirect
32. #2164C22semidirect
36. #2163C22semidirect
39. #2151C22semidirect
44. C4×M32C22semidirect
45. C2×#346C22semidirect

Maximal subgroups

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

subgroupindex
#602042
#335072
#335002
#334932
#334922
#334852
#334842
#334212
#334182