Finite Group Databasebeta

← #7623219

Subgroups — #7623219

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60660C211yesmaximal, characteristic
2C2×#3164C211yesmaximal, characteristic
3#33732C214yesmaximal, complemented
4#33731C214yesmaximal, complemented
5#33730C218yesmaximal, complemented
6#33729C218yesmaximal, complemented
7#33409C212yesmaximal
8C2×#2159C212yesmaximal
9#33324C211yesmaximal, characteristic
10#3458—21no—
11C2×#544—21no—
12C22×M32C2212yes—
13#3164—24no—
14#3164C2214yes—
15#3164C2218yes—
16#3164C2214yes—
17#3163—24no—
18#3162—22no—
19C2×#501—22no—
20#3160C2214yes—
21C2×#500C2212yes—
22C2×#499C2212yes—
23#2911—21no—
24C2×#428—21no—
25C2×#427C2212yes—
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2255C2218yescomplemented
30#2255C22116yescomplemented
31#2255C2218yescomplemented
32#2254C2218yescomplemented
33#2253C22116yescomplemented
34#2252C2218yescomplemented
35#2159C2218yes—
36#2159C2218yes—
37#2159C2214yes—
38#2159C2214yes—
39#2159C2214yes—
40#2159C2214yes—
41#2158C2218yes—
42C2×#367C2214yes—
43C2×#366C2214yes—
44#2108C2218yes—
45C4×M32C2214yes—
46C22×M16D411yescharacteristic
47C23×C8C2311yescharacteristic
48#544—24no—
49#544—24no—
50#544—28no—
51#544—44no—
52#544—41no—
53#544—41no—
54#543—42no—
55#533—24no—
56#533—44no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—41no—
59C2×#125—22no—
60C2×#125—41no—
61C2×C22⋊Q8—42no—
62C2×#117—21no—
63C2×#117—22no—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C2×M32Q8116yescomplemented
67C2×M32C2314yes—
68C2×M32C2314yes—
69C22×C16C2314yes—
70#501—216no—
71#501—28no—
72#501—28no—
73#500—24no—
74#500C2318yes—
75#500C2314yes—
76#500C2314yes—
77#499—24no—
78#499—24no—
79#499C2318yes—
80#499C2318yes—
81#498—24no—
82#428—24no—
83#428—24no—
84#428—28no—
85#427—24no—
86#427—22no—
87#427—22no—
88#427—22no—
89#427C2314yes—
90#427C2314yes—
91#427C2314yes—
92C2×#107—22no—
93C2×#107—21no—
94C2×#107—22no—
95C2×#107C2311yescharacteristic
96C2×#107C2312yes—
97C2×#107C2311yescharacteristic
98C2×#107C2314yes—
99C2×#107C2312yes—
100C2×#107C2312yes—
101C2×#106—22no—
102C2×#106—21no—
103C2×#106—22no—
104#405D414yes—
105#405C2314yes—
106C2×#97D412yes—
107C2×C4×C8C2312yes—
108#367C23116yes—
109#367C2318yes—
110#367C2318yes—
111#366C23116yes—
112#366C2318yes—
113#366C2318yes—
114#346C2318yes—
115C4×C16C2318yes—
116C22×Q8—42no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C2×M16Q1618yes—
120C2×M16C2×D414yes—
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D412yes—
123C22×C8C2×D412yes—
124C22×C8C2×Q818yes—
125C22×C8C2411yescharacteristic
126C22×C8C2412yes—
127C4⋊Q8—28no—
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8—48no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134#125—28no—
135#125—28no—
136#125—48no—
137#125—44no—
138#125—44no—
139#125—44no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—416no—
143C22⋊Q8—48no—
144C22⋊Q8—48no—
145C4×Q8—44no—
146C4×Q8—44no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—42no—
151#117—42no—
152#117—42no—
153#117D814yes—
154#117C2×D414yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4D811yescharacteristic
167C2×C4⋊C4D811yescharacteristic
168C2×C22⋊C4—41no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4D812yes—
171C2×C42—41no—
172C2×C42—42no—
173C2×C42C2×D412yes—
174M32C2×Q8116yes—
175M32C2×Q8116yescomplemented
176C2×C16C2×Q8116yescomplemented
177C2×C16C2418yes—
178#107—22no—
179#107—24no—
180#107—28no—
181#107—21no—
182#107—21no—
183#107—22no—
184#107—22no—
185#107—24no—
186#107—24no—
187#107C2414yes—
188#107C2414yes—
189#107C2412yes—
190#107C2412yes—
191#107C2414yes—
192#106—24no—
193#106—24no—
194#106—24no—
195#106—28no—
196#106—28no—
197#97C2×D418yes—
198C4×C8C2×D414yes—
199C4×C8C2414yes—
200C24C2×D811yescharacteristic
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—48no—
204C2×Q8—164no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—162no—
212C22×C4C4⋊Q814yes—
213C22×C4C2×D812yes—
214C22×C4C2×D812yes—
215C22×C4C2×D811yescharacteristic
216C22×C4C2×D812yes—
217C22×C4C2×Q1614yes—
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D412yes—
220M16C2×Q16116yes—
221C2×C8C4⋊Q814yes—
222C2×C8C4⋊Q818yes—
223C2×C8C2×Q1618yes—
224C2×C8C22×D411yescharacteristic
225C2×C8C22×D412yes—
226C2×C8C22×D414yes—
227C2×C8C22×Q814yes—
228C2×C8C22×Q814yes—
229C2×C8C22×Q814yes—
230C2×C8C2511yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—48no—
243C4⋊C4—48no—
244C4⋊C4—48no—
245C4⋊C4—48no—
246C4⋊C4—48no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4C2×D814yes—
250C4⋊C4C2×D814yes—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4C2×D818yes—
255C42—44no—
256C42—44no—
257C42—44no—
258C42C2×D814yes—
259C42C22×D414yes—
260C16C22×Q8116yes—
261C23—41no—
262C23—41no—
263C23#50014yes—
264C23C22×D812yes—
265C23C22×D811yescharacteristic
266Q8—164no—
267Q8—164no—
268Q8—168no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—82no—
279C2×C4—162no—
280C2×C4—162no—
281C2×C4—164no—
282C2×C4#50014yes—
283C2×C4#50018yes—
284C2×C4C2×C4⋊Q814yes—
285C2×C4C2×C4⋊Q812yes—
286C2×C4C22×D814yes—
287C2×C4C22×D814yes—
288C2×C4C22×D811yescharacteristic
289C2×C4C22×D812yes—
290C2×C4C22×Q1612yes—
291C2×C4C22×Q1614yes—
292C2×C4C23×D411yescharacteristic
293C8C2×C4⋊Q812yes—
294C8C2×C4⋊Q814yes—
295C8C22×Q1618yes—
296C8C23×Q812yes—
297C22—21no—
298C22—22no—
299C22—21no—
300C22—22no—
301C22—44no—
302C22C2×#50012yes—
303C22C2×#50014yes—
304C22C23×D811yescharacteristic
305C4—24no—
306C4—24no—
307C4—84no—
308C4—82no—
309C4—84no—
310C4—82no—
311C4—84no—
312C4C2×#50012yes—
313C4C2×#50014yes—
314C4C22×C4⋊Q811yescharacteristic
315C4C23×Q1611yescharacteristic
316C2—22no—
317C2—24no—
318C2C22×#50011yescharacteristic
319C2#6263611yescharacteristic
320C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33732C2semidirect
4. #33731C2semidirect
5. #33730C2semidirect
6. #33729C2semidirect
29. #2255C22semidirect
30. #2255C22semidirect
31. #2255C22semidirect
32. #2254C22semidirect
33. #2253C22semidirect
34. #2252C22semidirect
66. C2×M32Q8semidirect
175. M32C2×Q8semidirect
176. C2×C16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#606602
C2×#31642
#337322
#337312
#337302
#337292
#334092
C2×#21592
#333242