Finite Group Databasebeta

← #214042

Subgroups — #214042

Order 512.

876 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 186 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#37422C211yesmaximal, characteristic
2#17408C212yesmaximal, complemented
3#9451C214yesmaximal, complemented
4#9447C214yesmaximal, complemented
5#8745C211yesmaximal, characteristic, complemented
6#8744C211yesmaximal, characteristic, complemented
7#8524C212yesmaximal, complemented
8C2×#540C2211yescharacteristic
9C2×#536C2211yescharacteristic
10#2973C2214yes—
11#2861C411yescharacteristic
12#2858C411yescharacteristic
13#2703—24no—
14#2696—24no—
15#2657C411yescharacteristic
16#2642C411yescharacteristic
17#2547C2212yes—
18#2544C2212yes—
19#2436C412yes—
20#2431C412yes—
21#2298C2211yescharacteristic
22#1558C2214yes—
23#1533C2214yes—
24#1471—24no—
25#1468—24no—
26#1466C2214yes—
27#1464C2212yes—
28#1464C2212yes—
29#1285C2214yes—
30#1285C2214yes—
31C22×#41C2×C411yescharacteristic
32C23×Q8C2×C411yescharacteristic
33#540—44no—
34#536—44no—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4⋊Q8C2311yescharacteristic
37C2×#127C2×C411yescharacteristic
38C2×#127C2×C412yes—
39C2×#125C2×C411yescharacteristic
40C2×#124C2×C411yescharacteristic
41C2×#124C2×C412yes—
42C2×#124C2×C412yes—
43C2×#124C2×C412yes—
44C2×#124C2311yescharacteristic
45C2×#123—22no—
46C2×#123D414yes—
47C2×C22⋊Q8—22no—
48C2×#121C2×C412yes—
49C2×#121D414yes—
50C2×C4×D4C2312yes—
51C2×#117C2311yescharacteristic
52#435—44no—
53#435—44no—
54#432C2314yes—
55#432C2314yes—
56#407—24no—
57C2×C22⋊C8—24no—
58#400—24no—
59#397—24no—
60#394—24no—
61#393—24no—
62#392—24no—
63#392—24no—
64#391C2×C412yes—
65#391C2×C412yes—
66#391C2×C412yes—
67#390—24no—
68#390C2×C412yes—
69#388—24no—
70#388—22no—
71#387—22no—
72#386—24no—
73#386—24no—
74#386—24no—
75#380—24no—
76C4×C4⋊C4C2312yes—
77C4×C22⋊C4C2×C412yes—
78#376C2×C411yescharacteristic
79C43C2×C411yescharacteristic
80#333—44no—
81#329—44no—
82#324—22no—
83#324—22no—
84#323—22no—
85#323—22no—
86#323—28no—
87C2×#41—42no—
88C2×#41—82no—
89C22×Q8—21no—
90C22×Q8—42no—
91C22×Q8—82no—
92C22×Q8C22×C411yescharacteristic
93C22×Q8C22×C411yescharacteristic
94C22×Q8C22×C411yescharacteristic
95C22×D4—21no—
96C22×D4—24no—
97C22×D4C22⋊C414yes—
98C22×D4C22⋊C414yes—
99C22×D4C22×C411yescharacteristic
100C22×D4C22×C412yes—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D414yes—
105C2×M16—44no—
106C22×C8—44no—
107C4⋊Q8—24no—
108C4⋊Q8—42no—
109#127—42no—
110#127—82no—
111#125—42no—
112#124—24no—
113#124—42no—
114#124—82no—
115#124—82no—
116#124—82no—
117#123—44no—
118#123—44no—
119#123—48no—
120C22⋊Q8—48no—
121#121—44no—
122#121—44no—
123#121—42no—
124#121—42no—
125C4×D4—24no—
126C4×D4—24no—
127#117—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4C22⋊C414yes—
132C2×C4⋊C4C22×C412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4C22⋊C414yes—
142C2×C22⋊C4C2×D412yes—
143C2×C22⋊C4C2×D414yes—
144C2×C42—22no—
145C2×C42C22×C411yescharacteristic
146C2×C42C22×C412yes—
147C2×C42C22×C412yes—
148C2×C42C2411yescharacteristic
149C4⋊C8—24no—
150C4⋊C8—24no—
151C22⋊C8—24no—
152C22⋊C8—24no—
153C22⋊C8—44no—
154C22⋊C8—44no—
155#97—24no—
156#97—24no—
157#95—22no—
158#95—22no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—22no—
163C24—21no—
164C24C2×C22⋊C414yes—
165C24C2×C22⋊C411yescharacteristic
166#41—88no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—82no—
175C2×Q8—88no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—42no—
185C2×D4—82no—
186C2×D4—84no—
187C2×D4—84no—
188C2×D4—84no—
189C2×D4—84no—
190C2×D4—84no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—84no—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C22⋊C414yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C23×C411yescharacteristic
209C22×C4C22×D412yes—
210M16—164no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—164no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—24no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C22⋊C4—84no—
231C22⋊C4—82no—
232C42—22no—
233C42—42no—
234C42—42no—
235C42—82no—
236C42H(5,𝔽2)14yes—
237C23—21no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—22no—
242C23—42no—
243C23—44no—
244C23—44no—
245C23—44no—
246C23—41no—
247C23—42no—
248C23C22×C22⋊C411yescharacteristic
249Q8—82no—
250Q8—82no—
251Q8—82no—
252Q8—84no—
253D4—82no—
254D4—82no—
255D4—82no—
256D4—84no—
257D4—164no—
258D4—164no—
259D4—168no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—84no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4#51812yes—
286C2×C4#53414yes—
287C8—84no—
288C8—84no—
289C22—22no—
290C22—44no—
291C22—42no—
292C22—44no—
293C22—41no—
294C22—44no—
295C22—84no—
296C22—84no—
297C22—82no—
298C22—84no—
299C22—82no—
300C22—81no—
301C22#230811yescharacteristic
302C22#288812yes—
303C4—24no—
304C4—22no—
305C4—44no—
306C4—42no—
307C4—41no—
308C4—82no—
309C4—82no—
310C4—81no—
311C4—81no—
312C4—81no—
313C2—22no—
314C2—84no—
315C2—81no—
316C2—161no—
317C2#2175511yescharacteristic
318C2#2178412yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#37422C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17408C2semidirect
3. #9451C2semidirect
4. #9447C2semidirect
5. #8745C2semidirect
6. #8744C2semidirect
7. #8524C2semidirect

Maximal subgroups

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

subgroupindex
#374222
#174082
#94512
#94472
#87452
#87442
#85242