Finite Group Databasebeta

← #128188

Subgroups — #128188

Order 512.

828 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 138 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×#1463C212yesmaximal
2#13904C211yesmaximal, characteristic
3#7326C214yesmaximal, complemented
4C23×C4⋊C4C2211yescharacteristic
5C2×#407—22no—
6C2×#381—22no—
7C2×#380C411yescharacteristic
8C2×C4×C22⋊C4C411yescharacteristic
9#1463—22no—
10#1463—22no—
11#1463C414yescomplemented
12#1463C2214yes—
13C2×#324—22no—
14C2×#324C414yescomplemented
15#1445C412yes—
16#1445C2212yes—
17C4⋊#95—22no—
18C24×C4C2×C411yescharacteristic
19C24×C4D412yes—
20C22×M16—42no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—41no—
23C22×C4⋊C4—42no—
24C22×C4⋊C4—41no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4D412yes—
30C22×C4⋊C4Q812yes—
31C22×C4⋊C4C2311yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C42—21no—
35#407—24no—
36#407—24no—
37#407—42no—
38#407—42no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8—42no—
41#381—48no—
42#380—41no—
43#380—41no—
44C4×C22⋊C4—42no—
45C2×#95—24no—
46C2×#95—24no—
47C2×#95C2×C412yes—
48C2×#95C2×C412yes—
49#324—24no—
50#324—24no—
51#324—24no—
52#324C2×C418yes—
53C25C22⋊C411yescharacteristic
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—44no—
64C23×C4—44no—
65C23×C4—44no—
66C23×C4C4212yes—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C412yes—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C412yes—
71C23×C4C4⋊C412yes—
72C23×C4C4⋊C412yes—
73C23×C4C4⋊C412yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D412yes—
76C2×M16—44no—
77C2×M16—44no—
78C2×M16—84no—
79C2×M16—84no—
80C22×C8—44no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4C4211yescharacteristic
98C2×C4⋊C4C4211yescharacteristic
99C2×C4⋊C4C22⋊C412yes—
100C2×C4⋊C4C22⋊C412yes—
101C2×C4⋊C4C4⋊C412yes—
102C2×C4⋊C4C4⋊C412yes—
103C2×C4⋊C4C4⋊C412yes—
104C2×C4⋊C4C22×C412yes—
105C2×C4⋊C4C2×D411yescharacteristic
106C2×C4⋊C4C2×Q811yescharacteristic
107C2×C22⋊C4—24no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—44no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C42—42no—
114C2×C42—44no—
115C22⋊C8—28no—
116C22⋊C8—44no—
117C22⋊C8—44no—
118C22⋊C8—44no—
119#95—42no—
120#95—42no—
121#95—48no—
122#95—48no—
123C24—22no—
124C24—22no—
125C24—42no—
126C24—42no—
127C24#9512yes—
128C24#9912yes—
129C24#9911yescharacteristic
130C24#9911yescharacteristic
131C24C2×C22⋊C411yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—84no—
183C22×C4—84no—
184C22×C4#9512yes—
185C22×C4#9512yes—
186C22×C4#9512yes—
187C22×C4C2×C4211yescharacteristic
188C22×C4C2×C22⋊C412yes—
189C22×C4C2×C4⋊C411yescharacteristic
190C22×C4C2×C4⋊C412yes—
191M16—84no—
192M16—84no—
193M16—88no—
194C2×C8—48no—
195C2×C8—88no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—84no—
207C22⋊C4—48no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C42—84no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—21no—
217C23—24no—
218C23—24no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—41no—
223C23—42no—
224C23—41no—
225C23—42no—
226C23—41no—
227C23—41no—
228C23—48no—
229C23—48no—
230C23#34212yes—
231C23#34212yes—
232C23#35312yes—
233C23#35412yes—
234C23C2×#9511yescharacteristic
235C23C2×#9911yescharacteristic
236C23C2×#9911yescharacteristic
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—88no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#34414yes—
283C8—88no—
284C22—24no—
285C22—24no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—48no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22#139314yes—
305C22C2×#34211yescharacteristic
306C22C2×#34412yes—
307C22C2×#35311yescharacteristic
308C22C2×#35411yescharacteristic
309C22#213312yes—
310C4—24no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—84no—
318C4—162no—
319C2—22no—
320C2—21no—
321C2—21no—
322C2—22no—
323C2—24no—
324C2—42no—
325C2#811411yescharacteristic
326C2C2×#139311yescharacteristic
327C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect
11. #1463C4semidirect
14. C2×#324C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14632
#139042
#73262