Finite Group Databasebeta

← #214634

Subgroups — #214634

Order 512.

864 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35670C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic
3#20691C211yesmaximal, characteristic
4#9462C214yesmaximal
5#9444C214yesmaximal
6#8741C212yesmaximal
7#8521C212yesmaximal
8C2×#541C2211yescharacteristic
9C2×#519C2211yescharacteristic
10#2921C2212yes—
11#2565C412yescomplemented
12#2546C412yescomplemented
13#2499C2212yes—
14#2478—22no—
15#2478—22no—
16#2462—22no—
17#2462—22no—
18#2434C2212yes—
19#2327C412yescomplemented
20#2310C2212yes—
21C4×C4⋊Q8C412yescomplemented
22#2298C2211yescharacteristic
23#1822C2212yes—
24#1820—22no—
25#1820C2212yes—
26#1820C2212yes—
27#1801—22no—
28#1784C2212yes—
29C4×#101—22no—
30#1740—22no—
31#1511C2218yes—
32#1284C2218yes—
33C23×Q8C2311yescharacteristic
34#541—44no—
35C2×C4⋊Q8C2×C412yes—
36C2×C4⋊Q8C2×C412yes—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×#126—22no—
39C2×#125—22no—
40C2×#125C2×C411yescharacteristic
41C2×C22⋊Q8C2312yes—
42#519—44no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8D412yes—
47C2×C4×Q8D412yes—
48C2×#117—22no—
49C2×#117D412yes—
50C2×#117D412yes—
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53#433—44no—
54#423—22no—
55C2×#101—24no—
56#407C2314yes—
57C4×M16—22no—
58#398—22no—
59#398—22no—
60#398—22no—
61#398—22no—
62#397—22no—
63#397—22no—
64#395—22no—
65#395—22no—
66#391—44no—
67#391C2×C412yes—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#388—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#385—22no—
86#385—22no—
87#384C2×C412yes—
88#382C2×C412yes—
89#381—22no—
90#381C2314yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C412yes—
94C4×C22⋊C4C2312yes—
95#376C2×C411yescharacteristic
96C43C2×C411yescharacteristic
97#343—24no—
98#342—24no—
99#342—24no—
100#328—48no—
101C22×Q8—24no—
102C22×Q8—44no—
103C22×Q8C22×C412yes—
104C22×Q8C22×C412yes—
105C22×Q8C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C2×M16—24no—
110C4⋊Q8—42no—
111C4⋊Q8—44no—
112C4⋊Q8—44no—
113#126—48no—
114#125—42no—
115#125—48no—
116C22⋊Q8—28no—
117C4×Q8—44no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×Q8—44no—
122#117—24no—
123#117—24no—
124#117—44no—
125#117—44no—
126#117—48no—
127C2×C4⋊C4—22no—
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—42no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4C2×D412yes—
150C2×C22⋊C4C2×D412yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—42no—
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C412yes—
156C2×C42C22⋊C412yes—
157C2×C42C22⋊C412yes—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C412yes—
161C2×C42C22×C412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C4⋊C8—44no—
165#101—48no—
166C22⋊C8—28no—
167#97—42no—
168C4×C8—42no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—28no—
183#95—44no—
184C24C22×D411yescharacteristic
185C2×Q8—24no—
186C2×Q8—24no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—48no—
193C2×Q8—48no—
194C2×Q8—42no—
195C2×Q8—42no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C23×C411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4H(5,𝔽2)14yes—
215M16—48no—
216C2×C8—44no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—84no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—42no—
249C42—42no—
250C42—42no—
251C42—44no—
252C42—44no—
253C42—84no—
254C23—22no—
255C23—42no—
256C23C22×C22⋊C411yescharacteristic
257C23#53412yes—
258Q8—48no—
259Q8—48no—
260Q8—84no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—84no—
293C2×C4—88no—
294C2×C4#41014yes—
295C2×C4#51812yes—
296C2×C4#53412yes—
297C8—84no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—82no—
302C22#230811yescharacteristic
303C22C2×#41012yes—
304C4—22no—
305C4—24no—
306C4—22no—
307C4—44no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4—42no—
313C4—82no—
314C4—82no—
315C4—82no—
316C4—164no—
317C2—22no—
318C2—42no—
319C2#2174811yescharacteristic
320C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
11. #2565C4semidirect
12. #2546C4semidirect
19. #2327C4semidirect
21. C4×C4⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
#356702
#207302
#206912
#94622
#94442
#87412
#85212