Finite Group Databasebeta

← #214631

Subgroups — #214631

Order 512.

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

normal subgroups only

This group has 956 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#35624C211yesmaximal, characteristic
2#20729C211yesmaximal, characteristic
3#20689C211yesmaximal, characteristic
4#9459C214yesmaximal
5#9444C214yesmaximal, complemented
6#8741C212yesmaximal, complemented
7#8521C212yesmaximal, complemented
8C2×#541C2211yescharacteristic
9C2×#518C2211yescharacteristic
10#2921C2212yes—
11#2566C412yescomplemented
12#2547C412yescomplemented
13#2499C2212yes—
14#2480—22no—
15#2480—22no—
16#2460—22no—
17#2460—22no—
18#2435C2212yes—
19#2326C412yescomplemented
20#2308C2212yes—
21C4×#124C412yescomplemented
22#2298C2211yescharacteristic
23#1822C2212yes—
24#1820C2212yes—
25#1820C2212yes—
26#1819—22no—
27#1800—22no—
28#1783C2212yes—
29C4×#100—22no—
30C2×#342—22no—
31#1511C2218yes—
32#1284C2218yes—
33C23×D4C2311yescharacteristic
34#541—44no—
35C2×C4⋊Q8C2×C411yescharacteristic, complemented
36C2×#126—22no—
37C2×#125—22no—
38C2×#125C2×C411yescharacteristic, complemented
39C2×#124C2×C412yes—
40C2×#124C2×C412yes—
41C2×C22⋊Q8C2312yes—
42#518—44no—
43C2×C4×Q8C2×C412yescomplemented
44C2×C4×D4C2×C412yes—
45C2×C4×D4C2×C412yes—
46C2×C4×D4D412yes—
47C2×C4×D4D412yes—
48C2×#117C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C22⋊C4D412yes—
53#433—44no—
54#423—22no—
55C2×#100—24no—
56#407C2314yes—
57C4×M16—22no—
58#397—22no—
59#397—22no—
60#397—22no—
61#397—22no—
62#396—22no—
63#396—22no—
64#394—22no—
65#394—22no—
66#391C2×C412yescomplemented
67#391C2×C412yescomplemented
68#390C2×C412yes—
69#390C2×C412yes—
70#388—22no—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#386—44no—
79#385—22no—
80#385—22no—
81#385—22no—
82#385—22no—
83#384C2×C412yescomplemented
84#382C2×C412yescomplemented
85#381C2314yes—
86#380—22no—
87#380—22no—
88#379—22no—
89#379—22no—
90#379—22no—
91C4×C4⋊C4C2×C412yescomplemented
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2312yes—
95#376C2×C411yescharacteristic, complemented
96C43C2×C411yescharacteristic, complemented
97#343—24no—
98#342—24no—
99#342—24no—
100#328—48no—
101C25C2×D411yescharacteristic
102C25C2×D411yescharacteristic
103C22×Q8C22×C412yes—
104C22×D4—24no—
105C22×D4—44no—
106C22×D4C22×C412yes—
107C22×D4C2×D412yes—
108C23×C4—42no—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C412yes—
113C23×C4C2411yescharacteristic
114C2×M16—24no—
115C4⋊Q8—42no—
116#126—48no—
117#125—42no—
118#125—48no—
119#124—44no—
120#124—44no—
121C22⋊Q8—28no—
122C4×Q8—44no—
123C4×D4—44no—
124C4×D4—44no—
125C4×D4—44no—
126C4×D4—44no—
127#117—24no—
128#117—24no—
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⋊C4C22×C412yes—
137C2×C4⋊C4C22×C412yes—
138C2×C4⋊C4C2×D412yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—48no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4C22⋊C412yes—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C2×D412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C412yes—
166C2×C42C22×C412yes—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C4⋊C8—44no—
170#100—48no—
171C22⋊C8—28no—
172#97—42no—
173C4×C8—42no—
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—22no—
183#95—28no—
184C24—22no—
185C24—24no—
186C24—22no—
187C24—42no—
188C24—82no—
189C24—82no—
190C24C2×C22⋊C412yes—
191C24C2×C22⋊C412yes—
192C24C22×D411yescharacteristic
193C2×Q8—24no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—48no—
203C2×D4—48no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—84no—
219C22×C4—164no—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4H(5,𝔽2)14yes—
227M16—48no—
228C2×C8—44no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C22⋊C4—48no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C22⋊C4—84no—
257C22⋊C4—164no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42—42no—
264C42—44no—
265C42—82no—
266C23—24no—
267C23—24no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—84no—
281C23—84no—
282C23—164no—
283C23—164no—
284C23#41014yes—
285C23C22×C22⋊C411yescharacteristic
286C23#53412yes—
287Q8—84no—
288D4—48no—
289D4—48no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
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—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—84no—
314C2×C4—88no—
315C2×C4—162no—
316C2×C4—162no—
317C2×C4—162no—
318C2×C4—162no—
319C2×C4—168no—
320C2×C4#51812yes—
321C2×C4#53412yes—
322C8—84no—
323C22—22no—
324C22—24no—
325C22—22no—
326C22—24no—
327C22—44no—
328C22—44no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—84no—
338C22—84no—
339C22—164no—
340C22#230811yescharacteristic
341C22C2×#41012yes—
342C4—22no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—82no—
349C4—82no—
350C4—82no—
351C4—82no—
352C4—164no—
353C2—22no—
354C2—42no—
355C2—44no—
356C2—82no—
357C2#2174811yescharacteristic
358C2#2178412yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
5. #9444C2semidirect
6. #8741C2semidirect
7. #8521C2semidirect
11. #2566C4semidirect
12. #2547C4semidirect
19. #2326C4semidirect
21. C4×#124C4semidirect
35. C2×C4⋊Q8C2×C4semidirect
38. C2×#125C2×C4semidirect
43. C2×C4×Q8C2×C4semidirect
66. #391C2×C4semidirect
67. #391C2×C4semidirect
83. #384C2×C4semidirect
84. #382C2×C4semidirect
91. C4×C4⋊C4C2×C4semidirect
95. #376C2×C4semidirect
96. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#356242
#207292
#206892
#94592
#94442
#87412
#85212