Finite Group Databasebeta

← #224161

Subgroups — #224161

Order 512.

779 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36567C211yesmaximal, characteristic
2#20520C212yesmaximal
3#9892C212yesmaximal
4#9892C212yesmaximal
5#9891C212yesmaximal
6#8584C211yesmaximal, characteristic
7#8582C212yesmaximal
8C4×#330C212yesmaximal
9C4×#330C211yesmaximal, characteristic
10C2×#446—22no—
11#2864C2212yes—
12#2859C2211yescharacteristic
13#2590—22no—
14#2585—22no—
15#2584C412yes—
16#2507C412yes—
17#2506C412yes—
18#2462—22no—
19#2370—22no—
20#2370C412yes—
21C4×#125C2212yes—
22C4×#125C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1771—22no—
25#1771—22no—
26C8×C4⋊C4—22no—
27#1767C2212yes—
28#1767C2212yes—
29#1767C2212yes—
30C4×#97C2212yes—
31C4×#97C2212yes—
32C4×#97C2212yes—
33C2×#330C2211yescharacteristic
34C2×#330C2212yes—
35C2×#330C2211yescharacteristic
36#1295C2212yes—
37#1295C2212yes—
38#1295C2212yes—
39#1295C2212yes—
40#1295C2212yes—
41#1295C2211yescharacteristic
42#1295C2211yescharacteristic
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×#125C2311yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D412yes—
48#446—216no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×#97—22no—
53C2×#97—22no—
54C2×#97—22no—
55C2×#97C2312yes—
56C2×#97C2312yes—
57C2×#97C2312yes—
58C2×#97C2312yes—
59C2×C4×C8—22no—
60C2×C4×C8—22no—
61C2×C4×C8—22no—
62#400—22no—
63#398—22no—
64#398—22no—
65#398—22no—
66#398—22no—
67#395—22no—
68#389—22no—
69#389—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#384C2×C411yescharacteristic
93#384C2×C411yescharacteristic
94#382C2312yes—
95#382C2311yescharacteristic
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C43C2311yescharacteristic
109C43C2312yes—
110#336—22no—
111#336—22no—
112#336—22no—
113#330—22no—
114#330—24no—
115#330Q814yescomplemented
116C22×Q8C22⋊C411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121#125—22no—
122#125—44no—
123#125—48no—
124#125C2×Q812yes—
125#125#4112yes—
126C4×Q8—28no—
127C4×Q8—42no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C412yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C412yes—
154C2×C42C22×C412yes—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—28no—
164C4⋊C8—28no—
165C4⋊C8—28no—
166#97—28no—
167#97—28no—
168#97—22no—
169#97—22no—
170#97—22no—
171#97—22no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—44no—
176#97C2×Q814yes—
177C4×C8—28no—
178C4×C8—44no—
179C4×C8—44no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95C22⋊C411yescharacteristic
194#95C22⋊C411yescharacteristic
195#95C22⋊C412yes—
196#95C22⋊C412yes—
197C2×Q8—24no—
198C2×Q8—41no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C23×C411yescharacteristic
211C22×C4C22×D412yes—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4#10418yes—
246C4⋊C4C4×Q814yes—
247C42—22no—
248C42—24no—
249C42—24no—
250C42—24no—
251C42—42no—
252C42—42no—
253C42—42no—
254C42—41no—
255C42—42no—
256C42—42no—
257C42—44no—
258C42—82no—
259C42—82no—
260C42C22×Q811yescharacteristic
261C42C2×#4111yescharacteristic
262C422−1+412yes—
263C23C22×C22⋊C411yescharacteristic
264Q8—44no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4#41312yes—
297C2×C4C2×#10414yes—
298C2×C4C2×#10414yes—
299C2×C4C2×#10414yes—
300C2×C4C2×C4×Q811yescharacteristic
301C2×C4#51911yescharacteristic
302C2×C4#54812yes—
303C2×C4D4×Q812yes—
304C8—28no—
305C8—44no—
306C8—44no—
307C8—44no—
308C22Q8×C22⋊C411yescharacteristic
309C22C2×#41311yescharacteristic
310C22#288911yescharacteristic
311C22C22×#10412yes—
312C22#290112yes—
313C4—22no—
314C4—22no—
315C4—22no—
316C4—24no—
317C4—24no—
318C4—24no—
319C4—21no—
320C4—21no—
321C4—44no—
322C4—42no—
323C4—41no—
324C4—42no—
325C4—82no—
326C4—82no—
327C4—82no—
328C2#1763811yescharacteristic
329C2#1765312yes—
330C2#1769011yescharacteristic
331C2#2251711yescharacteristic
332C2Q8×#10412yes—
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
115. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#365672
#205202
#98922
#98912
#85842
#85822
C4×#3302