Finite Group Databasebeta

← #140604

Subgroups — #140604

Order 512.

690 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×#1954C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3#12599C211yesmaximal, characteristic, complemented
4C2×#1314C214yesmaximal, complemented
5C22×C4×D4C411yescharacteristic
6C22×C4⋊C8C411yescharacteristic
7C22×C22⋊C8C412yes—
8C22×C22⋊C8C2212yes—
9C22×C4×C8C2211yescharacteristic
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×C22⋊C16C2212yes—
13C2×#346—21no—
14#1954—21no—
15#1954—21no—
16#1954—21no—
17#1954—21no—
18#1374—21no—
19#1374—21no—
20#1314—28no—
21C23×D4C811yescharacteristic
22C24×C4C2×C412yes—
23C23×C8C2×C411yescharacteristic
24C23×C8C2×C412yes—
25C23×C8C2311yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4—21no—
28C2×C4×D4—21no—
29C2×C4×D4—21no—
30C2×C4×D4—42no—
31C2×C4×D4—42no—
32C22×C4⋊C4C811yescharacteristic
33C22×C22⋊C4C812yes—
34C22×C42C2×C411yescharacteristic
35C22×C16—22no—
36C22×C16—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—42no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—24no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—44no—
47C2×C4×C8—42no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C4⋊C16—42no—
53C22⋊C16—22no—
54C22⋊C16—24no—
55C22⋊C16D414yescomplemented
56#346—42no—
57C25C2×C812yes—
58C22×D4—24no—
59C22×D4—21no—
60C22×D4—21no—
61C22×D4—42no—
62C22×D4—42no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—24no—
66C23×C4—24no—
67C23×C4—42no—
68C23×C4—42no—
69C23×C4—42no—
70C23×C4—42no—
71C23×C4C2×C811yescharacteristic
72C23×C4C2×C811yescharacteristic
73C23×C4C2×C812yes—
74C23×C4C22×C411yescharacteristic
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D412yes—
89C22×C8#4112yes—
90C4×D4—44no—
91C4×D4—44no—
92C4×D4—42no—
93C4×D4—42no—
94C4×D4—42no—
95C4×D4—42no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—42no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—44no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22⋊C411yescharacteristic
106C2×C42M1611yescharacteristic
107C2×C42M1611yescharacteristic
108C2×C16—24no—
109C2×C16—42no—
110C2×C16—42no—
111C2×C16—42no—
112C2×C16—42no—
113C2×C16—42no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C22⋊C8—44no—
117C22⋊C8—44no—
118C22⋊C8—44no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C4×C8—22no—
122C4×C8—22no—
123C4×C8—41no—
124C4×C8—41no—
125C24—22no—
126C24—22no—
127C24—24no—
128C24—24no—
129C24—42no—
130C24—42no—
131C24—42no—
132C24—42no—
133C24M3214yes—
134C24C22×C811yescharacteristic
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—42no—
138C2×D4—416no—
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—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—41no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—84no—
166C22×C4—84no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4C22⋊C811yescharacteristic
170C22×C4C22⋊C811yescharacteristic
171C22×C4C22⋊C811yescharacteristic
172C22×C4C22⋊C811yescharacteristic
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C4×D412yes—
175C22×C4C2×M1611yescharacteristic
176C22×C4#13112yes—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—84no—
193C2×C8#10012yes—
194C2×C8#12112yes—
195C2×C8#12411yescharacteristic
196C2×C8#12711yescharacteristic
197C4⋊C4—42no—
198C4⋊C4—42no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C42—22no—
202C42—22no—
203C42—21no—
204C42—21no—
205C42#9912yes—
206C42#10012yes—
207C16—44no—
208C16—82no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—24no—
214C23—28no—
215C23—44no—
216C23—44no—
217C23—44no—
218C23—44no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—44no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—84no—
227C23—84no—
228C23—84no—
229C23—84no—
230C23C2×C22⋊C811yescharacteristic
231C23C8×D412yes—
232C23C2×M3212yes—
233C23#50412yes—
234D4—48no—
235D4—48no—
236C2×C4—28no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—88no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—84no—
262C2×C4#32312yes—
263C2×C4#32312yes—
264C2×C4#34912yes—
265C2×C4#39011yescharacteristic
266C2×C4#40811yescharacteristic
267C2×C4C2×#9911yescharacteristic
268C2×C4#41011yescharacteristic
269C2×C4C2×#10011yescharacteristic
270C2×C4C2×#10011yescharacteristic
271C2×C4#43612yes—
272C8—22no—
273C8—21no—
274C8—21no—
275C8—22no—
276C8—41no—
277C8—82no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—44no—
285C22—44no—
286C22—44no—
287C22—88no—
288C22—88no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22C2×#32311yescharacteristic
294C22#146011yescharacteristic
295C22#195411yescharacteristic
296C22#211211yescharacteristic
297C22#211311yescharacteristic
298C22C2×#34911yescharacteristic
299C22#211511yescharacteristic
300C22#216814yes—
301C4—22no—
302C4—22no—
303C4—22no—
304C4—22no—
305C4—41no—
306C4—82no—
307C4#146911yescharacteristic
308C4#147111yescharacteristic
309C4#196411yescharacteristic
310C4#196511yescharacteristic
311C2—24no—
312C2—84no—
313C2#964911yescharacteristic
314C2#1141811yescharacteristic
315C2#1141911yescharacteristic
316C2#1148811yescharacteristic
317C2#1148911yescharacteristic
318C2#1319011yescharacteristic
319C2#1319411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. #12599C2semidirect
4. C2×#1314C2semidirect
55. C22⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
#127832
#125992
C2×#13142