Finite Group Databasebeta

← #122667

Subgroups — #122667

Order 512.

347 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 74 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#29773C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3C2×#1292C211yesmaximal, characteristic
4#7091C212yesmaximal, complemented
5#7090C212yesmaximal, complemented
6C2×#467C411yescharacteristic
7C2×#423—21no—
8C22×C4⋊C8C2211yescharacteristic
9#2855—21no—
10#2112—21no—
11#2086C411yescharacteristic
12#2086C2211yescharacteristic
13#2066—21no—
14#2066—21no—
15#2056C411yescharacteristic
16#2056C2211yescharacteristic
17#1892—21no—
18#1807C411yescharacteristic
19#1322—21no—
20#1322C2212yes—
21#1292—21no—
22#1292C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×M16—41no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C22⋊Q8—21no—
28C22×C42D411yescharacteristic
29C2×M32—41no—
30#467—41no—
31#467—41no—
32C2×#107C2×C411yescharacteristic
33C2×#106—21no—
34#423—22no—
35#423—22no—
36#423—41no—
37#423—41no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—41no—
40C2×C4⋊C8—41no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8C2311yescharacteristic
46C2×#103—21no—
47C2×#103—21no—
48C2×#103—21no—
49#391—21no—
50#391—21no—
51#389—21no—
52#389—21no—
53#387—41no—
54#387—41no—
55#385—41no—
56#381—41no—
57#380—21no—
58C22⋊C16—22no—
59C22⋊C16—41no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C22×Q8—21no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×Q16—41no—
69C2×Q16—81no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—82no—
73C2×M16—82no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—42no—
79C22×C8—81no—
80C22×C8—81no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C22⋊C411yescharacteristic
102C2×C42D811yescharacteristic
103C2×C42SD1611yescharacteristic
104C2×C42C2×D411yescharacteristic
105M32—82no—
106C2×C16—42no—
107#107—41no—
108#106—81no—
109C4⋊C8—24no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C4⋊C8—41no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#103—42no—
118#103—81no—
119#103—81no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—41no—
126#95—41no—
127C24#10211yescharacteristic
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4#9911yescharacteristic
149C22×C4#9911yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10411yescharacteristic
154C22×C4#10411yescharacteristic
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C2×SD1611yescharacteristic
158Q16—82no—
159M16—84no—
160M16—82no—
161M16—82no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—44no—
172C2×C8—81no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—84no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—82no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C22⋊C4—82no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C16—82no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—41no—
203C23#32711yescharacteristic
204C23#32711yescharacteristic
205C23C2×#10211yescharacteristic
206Q8—81no—
207Q8—81no—
208Q8—81no—
209Q8—81no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4#32711yescharacteristic
246C2×C4#32711yescharacteristic
247C2×C4#36012yes—
248C2×C4C2×#9911yescharacteristic
249C2×C4C2×#10411yescharacteristic
250C2×C4#50911yescharacteristic
251C2×C4#51011yescharacteristic
252C8—41no—
253C8—41no—
254C8—41no—
255C8—41no—
256C8—84no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22#134212yes—
267C22C2×#32711yescharacteristic
268C22#213911yescharacteristic
269C22C2×#36011yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—21no—
274C4—22no—
275C4—21no—
276C4—41no—
277C4—41no—
278C4—161no—
279C4—161no—
280C4—161no—
281C4—161no—
282C4#212111yescharacteristic
283C4#213311yescharacteristic
284C2—21no—
285C2—21no—
286C2—22no—
287C2—41no—
288C2C2×#134211yescharacteristic
289C2#1169811yescharacteristic
290C2#1170411yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
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
4. #7091C2semidirect
5. #7090C2semidirect

Maximal subgroups

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

subgroupindex
#297732
#114402
C2×#12922
#70912
#70902