Finite Group Databasebeta

← #122162

Subgroups — #122162

Order 512.

948 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 142 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#34958C211yesmaximal, characteristic
2#10732C212yesmaximal
3#10729C211yesmaximal, characteristic
4#8100C211yesmaximal, characteristic
5#7942C212yesmaximal
6C23×M16C2211yescharacteristic
7C2×#407C2211yescharacteristic
8#2608C412yescomplemented
9#2607C412yescomplemented
10#2569—22no—
11#2549—22no—
12#2437—22no—
13#2396—22no—
14C4×C22⋊Q8—24no—
15#2280C2212yes—
16#2280C2211yescharacteristic
17C2×C4×C22⋊C4C2212yes—
18C2×#344—22no—
19C2×#339—24no—
20#1304—22no—
21#1284—22no—
22#1284—22no—
23C24×C4C2311yescharacteristic
24C22×M16—21no—
25C22×M16—22no—
26C22×M16—21no—
27C22×M16—22no—
28C22×M16—44no—
29C22×M16Q812yescomplemented
30C23×C8—21no—
31C2×C22⋊Q8—24no—
32C2×C4×Q8—22no—
33C2×#117—22no—
34C2×#117D412yes—
35C22×C22⋊C4—22no—
36C22×C42D414yes—
37#407—22no—
38#407—21no—
39#407Q812yescomplemented
40C2×C22⋊C8—22no—
41#400—22no—
42#399—22no—
43#398—24no—
44#396—24no—
45#395—22no—
46#393—22no—
47#391—24no—
48#389—24no—
49#387—24no—
50#387—24no—
51#385—24no—
52#385—24no—
53#381—22no—
54#381C2×C411yescharacteristic
55#381C2×C411yescharacteristic
56#380—24no—
57#380—22no—
58#380—22no—
59#380Q814yescomplemented
60#379—22no—
61#379D412yes—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—24no—
64C4×C22⋊C4—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4C2×C414yes—
70C4×C22⋊C4D412yes—
71C4×C22⋊C4D412yes—
72C2×#95—22no—
73#344—44no—
74#339—48no—
75C25C2×D411yescharacteristic
76C22×Q8—22no—
77C23×C4—24no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D412yes—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4C2×Q811yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4C2×Q812yes—
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4112yes—
90C2×M16—24no—
91C2×M16—28no—
92C2×M16—24no—
93C2×M16—22no—
94C2×M16—44no—
95C2×M16—48no—
96C2×M16—42no—
97C2×M16—44no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—42no—
102C2×M16—42no—
103C22×C8—22no—
104C22×C8—44no—
105C22×C8—42no—
106C22×C8—41no—
107C22⋊Q8—416no—
108C4×Q8—82no—
109C4×Q8—82no—
110#117—42no—
111#117—42no—
112#117—44no—
113#117—44no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—48no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4C22⋊C412yes—
130C2×C22⋊C4C22⋊C412yes—
131C2×C42—22no—
132C2×C42—24no—
133C2×C42—44no—
134C2×C42—44no—
135C2×C42—42no—
136C2×C42C22⋊C414yes—
137C2×C42C22⋊C414yes—
138C22⋊C8—24no—
139C22⋊C8—42no—
140#95—22no—
141#95—24no—
142#95—24no—
143#95—22no—
144#95—24no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95—24no—
150#95—42no—
151C24—44no—
152C24—42no—
153C24C2×C22⋊C411yescharacteristic
154C24#12012yes—
155C24C22⋊Q812yes—
156C24#12411yescharacteristic
157C24C4⋊Q811yescharacteristic
158C2×Q8—48no—
159C2×Q8—82no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—48no—
181C22×C4—44no—
182C22×C4—84no—
183C22×C4C2×C22⋊C412yes—
184C22×C4C4×Q811yescharacteristic
185C22×C4C4×Q811yescharacteristic
186C22×C4C4×Q812yes—
187C22×C4#12011yescharacteristic
188C22×C4#12011yescharacteristic
189C22×C4C22⋊Q812yes—
190C22×C4C22⋊Q812yes—
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q812yes—
194C22×C4C22⋊Q812yes—
195C22×C4#12412yes—
196C22×C4#12411yescharacteristic
197C22×C4#12412yes—
198C22×C4C4⋊Q812yes—
199C22×C4C4⋊Q811yescharacteristic
200C22×C4C4⋊Q812yes—
201M16—28no—
202M16—44no—
203M16—44no—
204M16—44no—
205M16—44no—
206M16—82no—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—48no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—81no—
213C2×C8—81no—
214C4⋊C4—48no—
215C4⋊C4—48no—
216C4⋊C4—48no—
217C4⋊C4—48no—
218C4⋊C4—82no—
219C4⋊C4—84no—
220C4⋊C4—82no—
221C22⋊C4—28no—
222C22⋊C4—48no—
223C22⋊C4—44no—
224C22⋊C4—48no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C22⋊C4—84no—
230C42—44no—
231C42—44no—
232C42—44no—
233C42—42no—
234C42—44no—
235C42—42no—
236C42—82no—
237C42—82no—
238C23—24no—
239C23—24no—
240C23—24no—
241C23—24no—
242C23—22no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—88no—
251C23#37911yescharacteristic
252C23#39111yescharacteristic
253C23#39111yescharacteristic
254C23#39312yes—
255C23#39512yes—
256Q8—88no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—28no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—816no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4#39112yes—
293C2×C4#39112yes—
294C2×C4#39311yescharacteristic
295C2×C4#39311yescharacteristic
296C2×C4#39511yescharacteristic
297C2×C4#39511yescharacteristic
298C2×C4B(4,𝔽2)12yes—
299C2×C4#45812yes—
300C8—44no—
301C8—82no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—48no—
313C22—44no—
314C22—82no—
315C22—84no—
316C22—82no—
317C22#144611yescharacteristic
318C22#179012yes—
319C22#203114yes—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—21no—
324C4—48no—
325C4—82no—
326C4—84no—
327C4—82no—
328C4—164no—
329C4#202911yescharacteristic
330C4#203011yescharacteristic
331C4#206212yes—
332C2—22no—
333C2—24no—
334C2—41no—
335C2—42no—
336C2—41no—
337C2#1138712yes—
338C2#1139011yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1446
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #2608C4semidirect
9. #2607C4semidirect
29. C22×M16Q8semidirect
39. #407Q8semidirect
59. #380Q8semidirect

Maximal subgroups

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

subgroupindex
#349582
#107322
#107292
#81002
#79422