Finite Group Databasebeta

← #392033

Subgroups — #392033

Order 512.

1536 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 1536 conjugacy classes of subgroups — too many to draw. Only its 232 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#20591C211yesmaximal, characteristic
2#20584C212yesmaximal
3#11138C218yesmaximal, complemented
4#11100C214yesmaximal, complemented
5C23×#41C2211yescharacteristic
6#2911C2214yescomplemented
7C2×#413C2214yes—
8C2×#410C2214yes—
9C2×#410C2214yes—
10C2×#407C2212yes—
11#2280C2214yes—
12#1785C414yescomplemented
13#1784C412yescomplemented
14#1783C412yescomplemented
15#1782—24no—
16#1781—24no—
17#1733C2218yescomplemented
18C2×H(3,ℤ/4)C2214yescomplemented
19C22×#41—44no—
20C22×#41—44no—
21C22×#41—42no—
22C22×#41—42no—
23C22×#41C2×C412yes—
24C22×#41D412yes—
25C22×#41Q812yes—
26C22×#41C2312yes—
27C23×Q8C2×C411yescharacteristic
28C23×D4C2×C411yescharacteristic
29C23×D4D412yes—
30C24×C4Q812yes—
31C24×C4C2311yescharacteristic
32C22×M16C2314yes—
33C2×#117C2318yes—
34#428—48no—
35#427—44no—
36#413—44no—
37#413—44no—
38C2×#101C2×C418yes—
39C2×#100C2×C418yes—
40#410—44no—
41#410—44no—
42#410—44no—
43#410—44no—
44C2×#99—28no—
45C2×#99—28no—
46#407—42no—
47#407C2×C414yes—
48#407C2×C414yes—
49C2×C22⋊C8—42no—
50#381—24no—
51#380—44no—
52#379—24no—
53C4×C22⋊C4—44no—
54#339—48no—
55H(3,ℤ/4)—44no—
56C25C4⋊C412yes—
57C25C2×D411yescharacteristic
58C2×#41—24no—
59C2×#41—24no—
60C2×#41—24no—
61C2×#41—24no—
62C2×#41—24no—
63C2×#41—24no—
64C2×#41—44no—
65C2×#41—42no—
66C2×#41—42no—
67C2×#41—44no—
68C2×#41—48no—
69C2×#41—44no—
70C2×#41—44no—
71C2×#41—44no—
72C2×#41—44no—
73C2×#41—432no—
74C2×#41—416no—
75C2×#41—416no—
76C22×Q8—42no—
77C22×Q8—44no—
78C22×Q8C4⋊C412yes—
79C22×Q8C22×C412yes—
80C22×D4—42no—
81C22×D4—44no—
82C22×D4—44no—
83C22×D4—48no—
84C22×D4C4⋊C412yes—
85C22×D4C4⋊C414yes—
86C22×D4C22×C412yes—
87C22×D4C2×D414yes—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—44no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—44no—
96C23×C4C4⋊C412yes—
97C23×C4C4⋊C414yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×Q811yescharacteristic
102C23×C4C2×Q814yes—
103C23×C4C2×Q811yescharacteristic
104C23×C4C2411yescharacteristic
105C2×M16—24no—
106C2×M16—28no—
107C2×M16—28no—
108C2×M16C22×C418yes—
109C22×C8—24no—
110#117—28no—
111#117—28no—
112#117—28no—
113#117—28no—
114C2×C4⋊C4—28no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—48no—
117C2×C42—28no—
118#107—48no—
119#106—48no—
120#101—48no—
121#100—48no—
122#99—416no—
123#99—416no—
124C22⋊C8—44no—
125C22⋊C8—44no—
126#95—48no—
127C24—24no—
128C24—22no—
129C24—42no—
130C24—44no—
131C24—82no—
132C24—84no—
133C24C2×C4⋊C411yescharacteristic
134C24C2×C4⋊C414yes—
135C24C2×C4⋊C411yescharacteristic
136C24C22×D411yescharacteristic
137#41—432no—
138#41—432no—
139#41—48no—
140#41—416no—
141#41—48no—
142#41—48no—
143#41—48no—
144#41—48no—
145#41—48no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—24no—
149C2×Q8—44no—
150C2×Q8—42no—
151C2×Q8—416no—
152C2×D4—28no—
153C2×D4—28no—
154C2×D4—28no—
155C2×D4—24no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—44no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—432no—
164C2×D4—416no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—48no—
191C22×C4—416no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4C2×C4⋊C411yescharacteristic
198C22×C4C2×C4⋊C414yes—
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4C23×C411yescharacteristic
201C22×C4C22×Q811yescharacteristic
202C22×C4H(5,𝔽2)14yes—
203C22×C4H(5,𝔽2)12yes—
204C22×C4H(5,𝔽2)14yes—
205C22×C4H(5,𝔽2)12yes—
206M16—216no—
207M16—48no—
208C2×C8—28no—
209C2×C8—44no—
210C2×C8—44no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C22⋊C4—48no—
214C22⋊C4—48no—
215C22⋊C4—816no—
216C42—48no—
217C42—48no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—24no—
223C23—24no—
224C23—24no—
225C23—42no—
226C23—48no—
227C23—44no—
228C23—48no—
229C23—44no—
230C23—88no—
231C23—84no—
232C23—81no—
233C23—81no—
234C23—84no—
235C23—82no—
236C23—84no—
237C23—88no—
238C23—84no—
239C23C22×C4⋊C411yescharacteristic
240C23#53412yes—
241C23#53414yes—
242Q8—48no—
243Q8—48no—
244D4—48no—
245D4—416no—
246D4—416no—
247D4—48no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—48no—
256C2×C4—416no—
257C2×C4—44no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—816no—
271C2×C4—88no—
272C2×C4#51812yes—
273C2×C4#51814yes—
274C2×C4#54312yes—
275C2×C4#54311yescharacteristic
276C2×C4#54311yescharacteristic
277C2×C4#54312yes—
278C2×C4#56018yes—
279C2×C4#56014yes—
280C2×C4H(3,𝔽4)14yes—
281C8—48no—
282C22—24no—
283C22—22no—
284C22—44no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—44no—
294C22—88no—
295C22—84no—
296C22—84no—
297C22—84no—
298C22—816no—
299C22#231511yescharacteristic
300C22#231512yes—
301C22#281214yes—
302C4—41no—
303C4—41no—
304C4—44no—
305C4—168no—
306C4#237514yes—
307C4#284011yescharacteristic
308C4#284212yes—
309C4#284211yescharacteristic
310C2—24no—
311C2—22no—
312C2—21no—
313C2—42no—
314C2—81no—
315C2—84no—
316C2#1467311yescharacteristic
subgroupG/H
centerC4#2842
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC2#14673
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11138C2semidirect
4. #11100C2semidirect
6. #2911C22semidirect
12. #1785C4semidirect
13. #1784C4semidirect
14. #1783C4semidirect
17. #1733C22semidirect
18. C2×H(3,ℤ/4)C22semidirect

Maximal subgroups

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

subgroupindex
#205912
#205842
#111382
#111002