Finite Group Databasebeta

← #122434

Subgroups — #122434

Order 512.

347 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 72 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#22261C211yesmaximal, characteristic
2#11466C211yesmaximal, characteristic, complemented
3#11465C211yesmaximal, characteristic, complemented
4#7065C212yesmaximal, complemented
5#7060C212yesmaximal, complemented
6C2×C8×D4C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2397C411yescharacteristic
9#2112—21no—
10C2×C22⋊C16—21no—
11#1955C411yescharacteristic
12#1955C2211yescharacteristic
13#1869—21no—
14#1869—21no—
15#1869—21no—
16#1869—21no—
17#1832C2211yescharacteristic
18#1746C411yescharacteristic
19#1322C2212yes—
20#1322C2212yes—
21#1314—21no—
22#1314—21no—
23C23×C8—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4—21no—
28C2×C4×D4C2×C411yescharacteristic
29C2×C4×D4C2×C411yescharacteristic
30C2×M32—41no—
31C22×C16—41no—
32C8×D4—41no—
33C8×D4—41no—
34C8×D4—41no—
35C8×D4—41no—
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2311yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8D411yescharacteristic
44C2×C22⋊C8D411yescharacteristic
45C2×#97C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47#399—21no—
48#399C811yescharacteristic
49#399C811yescharacteristic
50#392—21no—
51#392C811yescharacteristic
52#392C811yescharacteristic
53#376C2×C411yescharacteristic
54C22⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16—41no—
57C22⋊C16—41no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2×C811yescharacteristic
65C22×D4M1611yescharacteristic
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—42no—
76C22×C8—81no—
77C22×C8—81no—
78C22×C8—81no—
79C22×C8—81no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—81no—
89C4×D4—81no—
90C4×D4—81no—
91C4×D4—81no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4C2×C811yescharacteristic
95C2×C4⋊C4M1611yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4C2×C811yescharacteristic
102C2×C22⋊C4C2×C811yescharacteristic
103C2×C42—21no—
104C2×C42C22×C411yescharacteristic
105M32—82no—
106C2×C16—42no—
107C2×C16—42no—
108C2×C16—82no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C22⋊C8—41no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115C22⋊C8—81no—
116C22⋊C8—81no—
117C22⋊C8—81no—
118C22⋊C8—81no—
119#97—41no—
120C4×C8—21no—
121C4×C8—21no—
122#95—21no—
123#95C2×C811yescharacteristic
124#95C2×C811yescharacteristic
125C24—21no—
126C24C22⋊C811yescharacteristic
127C24C22⋊C811yescharacteristic
128C2×D4—22no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—81no—
138C2×D4—81no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4C22⋊C811yescharacteristic
157C22×C4C22⋊C811yescharacteristic
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C22×C811yescharacteristic
160C22×C4C2×M1611yescharacteristic
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8—84no—
179C2×C8#9911yescharacteristic
180C2×C8#9911yescharacteristic
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—42no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C16—82no—
197C16—82no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23#35012yes—
215C23C2×C22⋊C811yescharacteristic
216D4—44no—
217D4—42no—
218D4—42no—
219D4—84no—
220D4—84no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4C2×#9911yescharacteristic
252C2×C4#41311yescharacteristic
253C8—21no—
254C8—21no—
255C8—41no—
256C8—41no—
257C8—41no—
258C8—81no—
259C8—81no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22#145911yescharacteristic
280C22C2×#35011yescharacteristic
281C22#211711yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4#212711yescharacteristic
291C4#213411yescharacteristic
292C2—21no—
293C2—21no—
294C2—42no—
295C2—81no—
296C2—82no—
297C2#1147011yescharacteristic
298C2#1147911yescharacteristic
299C2#1148311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22#1459
prime radical#2397C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11466C2semidirect
3. #11465C2semidirect
4. #7065C2semidirect
5. #7060C2semidirect

Maximal subgroups

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

subgroupindex
#222612
#114662
#114652
#70652
#70602