Finite Group Databasebeta

← #122684

Subgroups — #122684

Order 512.

382 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 84 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#29770C211yesmaximal, characteristic
2C2×#1322C211yesmaximal, characteristic, complemented
3C2×#1292C211yesmaximal, characteristic, complemented
4#7091C212yesmaximal, complemented
5#7088C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×#423—21no—
8C22×C4⋊C8C2211yescharacteristic
9#2854—21no—
10C2×C22⋊C16—21no—
11#2086C411yescharacteristic
12#2086C2211yescharacteristic
13#2067—21no—
14#2067—21no—
15#2054C411yescharacteristic
16#2054C2211yescharacteristic
17#1893—21no—
18#1808C411yescharacteristic
19#1322—21no—
20#1322C2212yes—
21#1292—21no—
22#1292C2212yes—
23C22×D8C2×C411yescharacteristic
24C22×M16—41no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#123—41no—
28C2×#121—21no—
29C22×C42D411yescharacteristic
30C22×C16—41no—
31#466—41no—
32#466—41no—
33C2×#107—21no—
34C2×#107C2×C411yescharacteristic
35#423—22no—
36#423—22no—
37#423—41no—
38#423—41no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—41no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8—41no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8C2311yescharacteristic
47C2×#102—21no—
48C2×#102—21no—
49C2×#102—21no—
50#390—21no—
51#390—21no—
52#389—21no—
53#389—21no—
54#388—41no—
55#388—41no—
56#386—41no—
57#380—21no—
58C22⋊C16—22no—
59C22⋊C16—41no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C22×D4—21no—
65C22×D4—41no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4D811yescharacteristic
68C23×C4SD1611yescharacteristic
69C2×D8—41no—
70C2×D8—81no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—82no—
74C2×M16—82no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—42no—
80C22×C8—81no—
81C22×C8—81no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22×C411yescharacteristic
85C22×C8C2×D411yescharacteristic
86#123—82no—
87#123—82no—
88#121—41no—
89#121—41no—
90#121—41no—
91#121—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42C22⋊C411yescharacteristic
108C2×C42D811yescharacteristic
109C2×C42SD1611yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C16—42no—
112C2×C16—82no—
113#107—41no—
114#107—81no—
115C4⋊C8—24no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123#102—42no—
124#102—81no—
125#102—81no—
126#95—41no—
127#95—41no—
128#95—41no—
129C24—41no—
130C24#10211yescharacteristic
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—81no—
138C2×D4—81no—
139C2×D4—82no—
140C2×D4—82no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—81no—
152C22×C4#9911yescharacteristic
153C22×C4#9911yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4#10211yescharacteristic
156C22×C4#10211yescharacteristic
157C22×C4#10411yescharacteristic
158C22×C4#10411yescharacteristic
159C22×C4D1612yes—
160C22×C4SD3212yes—
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C2×D811yescharacteristic
163C22×C4C2×SD1611yescharacteristic
164D8—82no—
165M16—84no—
166M16—82no—
167M16—82no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—44no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8—84no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—82no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C22⋊C4—82no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C16—82no—
208C23—41no—
209C23—41no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23#32711yescharacteristic
218C23#32711yescharacteristic
219C23#35712yes—
220C23#36012yes—
221C23C2×#10211yescharacteristic
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—162no—
227D4—162no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
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—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4#32711yescharacteristic
261C2×C4#32711yescharacteristic
262C2×C4#35712yes—
263C2×C4C2×#9911yescharacteristic
264C2×C4C2×#10411yescharacteristic
265C2×C4C2×D1611yescharacteristic
266C2×C4C2×SD3211yescharacteristic
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—84no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22—161no—
284C22—161no—
285C22—161no—
286C22—161no—
287C22#134712yes—
288C22#134712yes—
289C22C2×#32711yescharacteristic
290C22C2×#35711yescharacteristic
291C22C2×#36011yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—22no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—161no—
301C4—161no—
302C4—161no—
303C4#212111yescharacteristic
304C4#213311yescharacteristic
305C2—21no—
306C2—21no—
307C2—22no—
308C2—22no—
309C2—321no—
310C2#1168811yescharacteristic
311C2#1169811yescharacteristic
312C2C2×#134711yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1322C2semidirect
3. C2×#1292C2semidirect
4. #7091C2semidirect
5. #7088C2semidirect

Maximal subgroups

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

subgroupindex
#297702
C2×#13222
C2×#12922
#70912
#70882