Finite Group Databasebeta

← #124136

Subgroups — #124136

Order 512.

437 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 102 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#21375C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic, complemented
3C4×C22⋊C16C211yesmaximal, characteristic, complemented
4#7128C212yesmaximal, complemented
5#7102C212yesmaximal, complemented
6C2×#529C411yescharacteristic
7C2×#432C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×C22⋊C16—21no—
10C2×C22⋊C16—21no—
11C2×#346—21no—
12C2×C4×C16—21no—
13#1984C411yescharacteristic
14#1984C2211yescharacteristic
15#1954C411yescharacteristic
16#1954C2211yescharacteristic
17#1869—21no—
18#1869—21no—
19#1750—21no—
20#1750—21no—
21#1374C2212yes—
22#1374C2212yes—
23C23×C8D411yescharacteristic
24C23×C8D411yescharacteristic
25#529—42no—
26#529—42no—
27C2×#126—21no—
28C2×#125C811yescharacteristic
29C2×#124C811yescharacteristic
30C2×#123—21no—
31C2×C22⋊Q8C811yescharacteristic
32C2×#121C811yescharacteristic
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2×C411yescharacteristic
36C2×#117—21no—
37C22×C42C2×C411yescharacteristic
38C22×C16—22no—
39C22×C16—22no—
40#432—42no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×C4×C8—42no—
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53C2×C4×C8C2311yescharacteristic
54C22⋊C16—24no—
55C22⋊C16—24no—
56#346—42no—
57C4×C16—42no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×Q8C2×C811yescharacteristic
63C22×D4—21no—
64C22×D4C2×C811yescharacteristic
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C2×C811yescharacteristic
68C23×C4M1611yescharacteristic
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#126—44no—
82#125—42no—
83#124—42no—
84#123—44no—
85C22⋊Q8—42no—
86#121—42no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—42no—
92C4×D4—42no—
93#117—42no—
94#117—42no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C2×C811yescharacteristic
100C2×C4⋊C4C2×C811yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—21no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C2×C811yescharacteristic
111C2×C42M1611yescharacteristic
112C2×C42C22×C411yescharacteristic
113C2×C16—24no—
114C2×C16—24no—
115C2×C16—42no—
116C2×C16—42no—
117C2×C16—42no—
118C2×C16—42no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126#97—42no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—41no—
130C4×C8—41no—
131C24—21no—
132C24C22⋊C811yescharacteristic
133C2×Q8—22no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×D4—22no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4C22⋊C811yescharacteristic
164C22×C4C22⋊C811yescharacteristic
165C22×C4C22⋊C811yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C22×C811yescharacteristic
168C22×C4C2×M1611yescharacteristic
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—44no—
178C2×C8—44no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8#10412yes—
186C2×C8#10412yes—
187C2×C8#13512yes—
188C2×C8#13512yes—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C42—22no—
201C42—22no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42#9912yes—
207C42#10012yes—
208C16—44no—
209C16—44no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23#35012yes—
224C23C2×C22⋊C811yescharacteristic
225Q8—82no—
226D4—82no—
227D4—84no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4#32312yes—
260C2×C4#32312yes—
261C2×C4C2×#9911yescharacteristic
262C2×C4C2×#10011yescharacteristic
263C2×C4#41611yescharacteristic
264C2×C4#41611yescharacteristic
265C2×C4C2×#10411yescharacteristic
266C2×C4C2×#10411yescharacteristic
267C2×C4#50412yes—
268C8—22no—
269C8—22no—
270C8—22no—
271C8—22no—
272C8—81no—
273C8—81no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22C2×#32311yescharacteristic
289C22#147611yescharacteristic
290C22#147611yescharacteristic
291C22#211311yescharacteristic
292C22#211511yescharacteristic
293C22C2×#35011yescharacteristic
294C22#211711yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#150111yescharacteristic
304C4#150111yescharacteristic
305C4#152511yescharacteristic
306C4#154311yescharacteristic
307C2—42no—
308C2—161no—
309C2#756311yescharacteristic
310C2#1141911yescharacteristic
311C2#1142211yescharacteristic
312C2#1161411yescharacteristic
313C2#1161611yescharacteristic
314C2#1161811yescharacteristic
315C2#1162011yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12599C2semidirect
3. C4×C22⋊C16C2semidirect
4. #7128C2semidirect
5. #7102C2semidirect

Maximal subgroups

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

subgroupindex
#213752
#125992
C4×C22⋊C162
#71282
#71022