Finite Group Databasebeta

← #140661

Subgroups — #140661

Order 512.

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

normal subgroups only

This group has 432 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#21053C211yesmaximal, characteristic
2#12789C211yesmaximal, characteristic, complemented
3#12606C211yesmaximal, characteristic, complemented
4#11443C212yesmaximal, complemented
5#11418C212yesmaximal
6C2×#424C2211yescharacteristic
7C2×#407C411yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9C2×#405C2211yescharacteristic
10#2280C411yescharacteristic
11C2×#364—21no—
12#2112C2212yes—
13#2112C2212yes—
14C2×#347—21no—
15#1924—21no—
16#1924—21no—
17#1923—21no—
18#1923—21no—
19#1729C411yescharacteristic
20#1729C411yescharacteristic
21#1376—21no—
22#1375—21no—
23#1323—24no—
24#1314—24no—
25C24×C4C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×#117C2×C411yescharacteristic
32C2×M32—22no—
33C2×M32—22no—
34#424—22no—
35#424—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38#407—42no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—22no—
45#405—42no—
46C2×#97D411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#381—21no—
51#380—21no—
52#380—21no—
53#379—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4—21no—
56#364—42no—
57C22⋊C16—22no—
58C22⋊C16—22no—
59C22⋊C16—22no—
60C22⋊C16D414yes—
61#347—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66C25M1611yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4M1611yescharacteristic
76C23×C4C22×C411yescharacteristic
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C2×D411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D412yes—
95C22×C8#4112yes—
96#117—22no—
97#117—22no—
98#117—41no—
99#117—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4M1611yescharacteristic
102C2×C4⋊C4M1611yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C42—21no—
110C2×C42C22⋊C411yescharacteristic
111C2×C42C22⋊C411yescharacteristic
112M32—42no—
113M32—42no—
114M32—42no—
115M32—42no—
116C2×C16—24no—
117C2×C16—42no—
118C4⋊C8—42no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126#97—22no—
127#97—41no—
128C4×C8—22no—
129C4×C8—41no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134C24—21no—
135C24—22no—
136C24—42no—
137C24—41no—
138C24—41no—
139C24—81no—
140C24C2×M1611yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—82no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4C4×D412yes—
168C22×C4C2×M1611yescharacteristic
169C22×C4#13112yes—
170M16—44no—
171M16—82no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—82no—
190C2×C8#9912yes—
191C2×C8#12112yes—
192C2×C8#12411yescharacteristic
193C2×C8#12711yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—41no—
208C42—41no—
209C16—44no—
210C16—82no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—24no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—82no—
225C23—82no—
226C23—84no—
227C23—84no—
228C23#40711yescharacteristic
229C23#43512yes—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
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—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4#39011yescharacteristic
258C2×C4#40811yescharacteristic
259C2×C4C2×#9911yescharacteristic
260C2×C4#41011yescharacteristic
261C2×C4#41311yescharacteristic
262C2×C4#41311yescharacteristic
263C2×C4#43512yes—
264C8—22no—
265C8—21no—
266C8—21no—
267C8—22no—
268C8—41no—
269C8—81no—
270C8—81no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—84no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22#146411yescharacteristic
288C22#146511yescharacteristic
289C22#195511yescharacteristic
290C4—22no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4#146811yescharacteristic
297C4#147311yescharacteristic
298C4#196411yescharacteristic
299C4#196711yescharacteristic
300C2—21no—
301C2—21no—
302C2—42no—
303C2—82no—
304C2#965211yescharacteristic
305C2#1320011yescharacteristic
306C2#1320111yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC23×C4C22×C4
Frattini, cosocleC23×C8C23
socleC22#1955
prime radicalC25M16
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12789C2semidirect
3. #12606C2semidirect
4. #11443C2semidirect

Maximal subgroups

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

subgroupindex
#210532
#127892
#126062
#114432
#114182