Finite Group Databasebeta

← #214615

Subgroups — #214615

Order 512.

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

normal subgroups only

This group has 904 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#35678C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic, complemented
3#20691C211yesmaximal, characteristic, complemented
4#9466C214yesmaximal, complemented
5#9446C214yesmaximal, complemented
6#8740C212yesmaximal, complemented
7#8518C212yesmaximal
8C2×#540C2211yescharacteristic
9C2×#519C2211yescharacteristic
10#2921C2212yes—
11#2565C412yescomplemented
12#2548C412yescomplemented
13#2500C2212yes—
14#2476—22no—
15#2476—22no—
16#2458—22no—
17#2458—22no—
18#2436C2212yes—
19#2326C412yescomplemented
20#2310C2212yes—
21C4×#124C412yescomplemented
22#2298C2211yescharacteristic
23#1822C2212yes—
24#1820—22no—
25#1820C2212yes—
26#1820C2212yes—
27#1801—22no—
28#1784C2212yes—
29C4×#101—22no—
30#1740—22no—
31#1510C2218yes—
32#1284C2218yes—
33C23×Q8C2311yescharacteristic
34#540—44no—
35C2×#127C2×C411yescharacteristic
36C2×#125C2×C411yescharacteristic
37C2×#124C2×C412yes—
38C2×#124C2×C412yes—
39C2×#123—22no—
40C2×#121—22no—
41C2×#121C2312yes—
42#519—44no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C2×C4×D4C2×C412yes—
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#394—22no—
61#394—22no—
62#394—22no—
63#394—22no—
64#393—22no—
65#393—22no—
66#391—44no—
67#391C2×C412yes—
68#391C2×C412yes—
69#390—22no—
70#390—22no—
71#390C2×C412yes—
72#390C2×C412yes—
73#388—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#386—22no—
84#386—22no—
85#385—22no—
86#385—22no—
87#384C2×C412yes—
88#382C2×C412yes—
89#381—22no—
90#381C2314yes—
91C4×C4⋊C4C2×C412yes—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2312yes—
95#376C2×C411yescharacteristic
96C43C2×C411yescharacteristic
97#343—24no—
98#342—24no—
99#342—24no—
100#327—48no—
101C22×Q8—24no—
102C22×Q8—44no—
103C22×Q8C22×C412yes—
104C22×Q8C2×D412yes—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4C22×C412yes—
108C23×C4—22no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C2×M16—24no—
113#127—42no—
114#125—42no—
115#124—44no—
116#124—44no—
117#123—48no—
118#121—28no—
119#121—48no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×D4—44no—
125#117—24no—
126#117—24no—
127#117—44no—
128#117—44no—
129#117—48no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C22⋊C412yes—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C2×D412yes—
155C2×C22⋊C4C2×D412yes—
156C2×C42—22no—
157C2×C42—42no—
158C2×C42C22⋊C412yes—
159C2×C42C22⋊C412yes—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C412yes—
165C2×C42C22×C412yes—
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C4⋊C8—44no—
169#101—48no—
170C22⋊C8—28no—
171#97—42no—
172C4×C8—42no—
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—
184C24—22no—
185C24C22×D411yescharacteristic
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×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—82no—
200C2×D4—82no—
201C2×D4—82no—
202C2×D4—82no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4H(5,𝔽2)14yes—
224M16—48no—
225C2×C8—44no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—84no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—48no—
247C22⋊C4—82no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—42no—
260C42—42no—
261C42—42no—
262C42—84no—
263C42—82no—
264C23—22no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23C22×C22⋊C411yescharacteristic
274C23#53412yes—
275Q8—48no—
276Q8—48no—
277D4—84no—
278D4—164no—
279D4—164no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—84no—
309C2×C4—88no—
310C2×C4—164no—
311C2×C4#41014yes—
312C2×C4#51812yes—
313C2×C4#53412yes—
314C8—84no—
315C22—22no—
316C22—24no—
317C22—22no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22#230811yescharacteristic
327C22C2×#41012yes—
328C4—22no—
329C4—24no—
330C4—22no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—82no—
337C4—82no—
338C4—82no—
339C4—164no—
340C2—22no—
341C2—42no—
342C2—162no—
343C2#2174811yescharacteristic
344C2#2178512yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#540C22
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
2. #20730C2semidirect
3. #20691C2semidirect
4. #9466C2semidirect
5. #9446C2semidirect
6. #8740C2semidirect
11. #2565C4semidirect
12. #2548C4semidirect
19. #2326C4semidirect
21. C4×#124C4semidirect

Maximal subgroups

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

subgroupindex
#356782
#207302
#206912
#94662
#94462
#87402
#85182