Finite Group Databasebeta

← #7624282

Subgroups — #7624282

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62246C211yesmaximal, characteristic
2C2×#3342C211yesmaximal, characteristic
3#33923C218yesmaximal, complemented
4#33895C218yesmaximal, complemented
5#33795C212yesmaximal, complemented
6#33761C214yesmaximal, complemented
7#33664C212yesmaximal, complemented
8#33649C214yesmaximal, complemented
9#33489C211yesmaximal, characteristic
10#3501—21no—
11C2×#556—21no—
12#3342—24no—
13#3342C2218yes—
14#3338C2214yes—
15#3299—24no—
16D4×Q16C2214yes—
17#3249—21no—
18#3189—22no—
19#3170C2212yes—
20C2×#500C2211yescharacteristic
21C2×#485—22no—
22C2×#475—21no—
23C2×#467C2214yes—
24C2×#467C2212yes—
25C2×C4×Q16C2211yescharacteristic
26C2×C8×D4C2211yescharacteristic
27C2×#427C2212yes—
28#2254C2214yescomplemented
29#2241C2214yescomplemented
30#2237C22116yescomplemented
31#2236C2214yes—
32#2235C2218yes—
33#2235C2218yescomplemented
34#2227C2214yescomplemented
35#2225C2218yescomplemented
36#2219C22116yescomplemented
37#2209C2214yescomplemented
38#2188C2214yes—
39#2188C2218yescomplemented
40C4×Q32C2218yescomplemented
41#2169C2218yes—
42#2162C2212yes—
43#2159C2218yescomplemented
44#2141C2218yescomplemented
45#2112C2214yes—
46C22×Q16C2314yes—
47C22×Q16C2311yescharacteristic
48C23×C8C2312yes—
49#556—28no—
50#556—44no—
51#554—24no—
52#554—44no—
53D4×Q8—24no—
54#544—41no—
55#543—42no—
56#533—22no—
57C2×C4⋊Q8—21no—
58C2×#126—42no—
59C2×#124—41no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—24no—
62#518—41no—
63C2×C4×Q8—21no—
64C2×C4×Q8—41no—
65C2×C4×D4D411yescharacteristic
66C2×#117—22no—
67C2×Q32C2318yes—
68C2×M32C2318yes—
69#500—22no—
70#500C2312yes—
71#500C2314yes—
72#494C2312yes—
73#487—22no—
74#485—216no—
75#484—28no—
76#477—24no—
77#475—28no—
78#467—24no—
79#467—22no—
80#467—24no—
81#467—24no—
82#467C2318yes—
83#467C2318yes—
84#467C2314yes—
85#462—24no—
86#451—28no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×Q16C2314yes—
90C4×Q16C2314yes—
91C8×D4D414yes—
92C8×D4C2314yes—
93#427—24no—
94#427C2318yes—
95C2×#107—21no—
96C2×#107C2314yes—
97C2×#107C2312yes—
98C2×#107C2311yescharacteristic
99C2×#106—22no—
100C2×C4⋊C8D411yescharacteristic
101C2×#103—22no—
102C2×#103—21no—
103C2×#103—24no—
104C2×C22⋊C8D412yes—
105C2×C4×C8C2311yescharacteristic
106#367C23116yes—
107#366C2318yes—
108C4⋊C16C2314yes—
109#358C2318yes—
110#358C23116yes—
111#358C2318yes—
112C22⋊C16C2314yes—
113C22⋊C16C2318yes—
114C4×C16C2314yes—
115C22×Q8—21no—
116C22×Q8—24no—
117C22×Q8—41no—
118C22×D4D811yescharacteristic
119C23×C4C2×D412yes—
120C2×Q16—21no—
121C2×Q16—24no—
122C2×Q16—22no—
123C2×Q16—42no—
124C2×Q16—48no—
125C2×Q16C2414yes—
126C2×Q16C2412yes—
127C22×C8—44no—
128C22×C8C2×D412yes—
129C22×C8C2×D414yes—
130C22×C8C2×D411yescharacteristic
131C22×C8C2411yescharacteristic
132C22×C8C2414yes—
133C4⋊Q8—24no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138#126—416no—
139#125—44no—
140#125—42no—
141#124—48no—
142C22⋊Q8—24no—
143C22⋊Q8—28no—
144C22⋊Q8—28no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148C22⋊Q8—42no—
149C22⋊Q8—48no—
150C22⋊Q8—48no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—48no—
154C4×Q8—42no—
155C4×Q8—42no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4D814yes—
159C4×D4C2×D414yes—
160#117—28no—
161#117—44no—
162#117—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4D811yescharacteristic
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4D812yes—
175C2×C42—42no—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178Q32—48no—
179M32—48no—
180C2×C16C2418yes—
181#107—24no—
182#107—21no—
183#107—22no—
184#107—24no—
185#107—22no—
186#107C2412yes—
187#107C2414yes—
188#107C2414yes—
189#106—28no—
190C4⋊C8C2×D414yes—
191#103—24no—
192#103—24no—
193#103—28no—
194#103—28no—
195#103—24no—
196C22⋊C8C2×D418yes—
197C4×C8C2×D412yes—
198C4×C8C2412yes—
199C24C2×D812yes—
200C2×Q8—22no—
201C2×Q8—24no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×Q8—44no—
206C2×Q8—42no—
207C2×Q8—44no—
208C2×Q8—84no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×D4—24no—
212C2×D4C2×D814yes—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4C2×D812yes—
221C22×C4C2×D811yescharacteristic
222C22×C4C2×D814yes—
223C22×C4C2×D811yescharacteristic
224C22×C4C22×D414yes—
225C22×C4C22×D411yescharacteristic
226Q16—24no—
227Q16—28no—
228Q16—42no—
229Q16—82no—
230Q16—84no—
231C2×C8—28no—
232C2×C8—22no—
233C2×C8—44no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8C22×D411yescharacteristic
237C2×C8C22×D412yes—
238C2×C8C22×D414yes—
239C2×C8C2511yescharacteristic
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4C2×D814yes—
256C22⋊C4—44no—
257C22⋊C4—48no—
258C22⋊C4—48no—
259C22⋊C4—48no—
260C22⋊C4C2×D818yes—
261C42—48no—
262C42—44no—
263C42—44no—
264C42C2×D812yes—
265C42C22×D412yes—
266C16—28no—
267C23—24no—
268C23—42no—
269C23—42no—
270C23C22×D814yes—
271C23C22×D811yescharacteristic
272Q8—44no—
273Q8—42no—
274Q8—42no—
275Q8—44no—
276Q8—82no—
277Q8—88no—
278Q8—162no—
279Q8—164no—
280D4—28no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—84no—
296C2×C4—164no—
297C2×C4—164no—
298C2×C4#51014yes—
299C2×C4C22×D814yes—
300C2×C4C22×D811yescharacteristic
301C2×C4C22×D812yes—
302C2×C4C22×D812yes—
303C2×C4C23×D411yescharacteristic
304C8—22no—
305C8—21no—
306C8—24no—
307C8#58512yes—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—24no—
312C22—24no—
313C22—84no—
314C22C2×#51012yes—
315C22C23×D811yescharacteristic
316C4—22no—
317C4—21no—
318C4—24no—
319C4—84no—
320C4—82no—
321C4—84no—
322C4—82no—
323C4—84no—
324C4C2×#51014yes—
325C4#348511yescharacteristic
326C4#358611yescharacteristic
327C2—22no—
328C2—44no—
329C2#6169411yescharacteristic
330C2C22×#51011yescharacteristic
331C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC22×D4D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33923C2semidirect
4. #33895C2semidirect
5. #33795C2semidirect
6. #33761C2semidirect
7. #33664C2semidirect
8. #33649C2semidirect
28. #2254C22semidirect
29. #2241C22semidirect
30. #2237C22semidirect
33. #2235C22semidirect
34. #2227C22semidirect
35. #2225C22semidirect
36. #2219C22semidirect
37. #2209C22semidirect
39. #2188C22semidirect
40. C4×Q32C22semidirect
43. #2159C22semidirect
44. #2141C22semidirect

Maximal subgroups

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

subgroupindex
#622462
C2×#33422
#339232
#338952
#337952
#337612
#336642
#336492
#334892