Finite Group Databasebeta

← #134255

Subgroups — #134255

Order 512.

666 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#384C211yesmaximal, characteristic
2C2×#1325C212yesmaximal
3#8558C212yesmaximal
4#7189C212yesmaximal
5C2×#384C411yescharacteristic
6C2×#384C2211yescharacteristic
7C2×C4×C4⋊C4—22no—
8C2×C4×C4⋊C4C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C43C411yescharacteristic
11#1797—22no—
12C2×#324C2214yes—
13C4⋊#95—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16#1325—24no—
17#1280—22no—
18C22×C4⋊C4—22no—
19C22×C4⋊C4—22no—
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4C2×C411yescharacteristic
22C22×C4⋊C4C2×C411yescharacteristic
23C22×C4⋊C4C2311yescharacteristic
24C22×C42—21no—
25C22×C42—21no—
26C22×C42—21no—
27C22×C42C2×C411yescharacteristic
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31C2×C4⋊C8—42no—
32C2×C22⋊C8—24no—
33#384—22no—
34#384—22no—
35#384—22no—
36#384D412yes—
37#384Q812yes—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—48no—
41C4×C4⋊C4—41no—
42C4×C4⋊C4—41no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95D412yes—
47C43—22no—
48C43C814yescomplemented
49#336—42no—
50#324—24no—
51#324—24no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C22⋊C412yes—
57C23×C4C22⋊C411yescharacteristic
58C23×C4C22⋊C411yescharacteristic
59C23×C4C22×C411yescharacteristic
60C23×C4C2×D412yes—
61C22×C8—44no—
62C2×C4⋊C4—28no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—24no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—44no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—44no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4C4⋊C412yes—
82C2×C4⋊C4M1612yes—
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4C2×Q811yescharacteristic
85C2×C4⋊C4#4112yes—
86C2×C42—24no—
87C2×C42—24no—
88C2×C42—24no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—24no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—22no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42—42no—
102C2×C42—42no—
103C2×C42—44no—
104C2×C42—44no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42C4⋊C412yes—
108C2×C42C2×C812yes—
109C2×C42C2×C812yes—
110C2×C42C2×C812yes—
111C2×C42M1612yes—
112C2×C42M1612yes—
113C4⋊C8—84no—
114C22⋊C8—44no—
115C22⋊C8—44no—
116#95—42no—
117#95—44no—
118#95—44no—
119#95—42no—
120C24C2×C22⋊C411yescharacteristic
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C22⋊C812yes—
165C22×C4C22⋊C812yes—
166C22×C4#9911yescharacteristic
167C22×C4#9911yescharacteristic
168C22×C4#10112yes—
169C22×C4C2×C4⋊C411yescharacteristic
170C22×C4#11711yescharacteristic
171C22×C4#12112yes—
172C22×C4C22⋊Q812yes—
173C22×C4C22×C811yescharacteristic
174C22×C4C2×M1611yescharacteristic
175C22×C4C2×M1611yescharacteristic
176C22×C4C2×M1611yescharacteristic
177C2×C8—84no—
178C2×C8—84no—
179C2×C8—84no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—48no—
186C4⋊C4—48no—
187C4⋊C4—416no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C42—216no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—42no—
201C42—48no—
202C42—42no—
203C42—44no—
204C42—82no—
205C42—82no—
206C42—82no—
207C42C4⋊C814yescomplemented
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23#38011yescharacteristic
213C23C2×C22⋊C811yescharacteristic
214C23#40711yescharacteristic
215C23C2×#9911yescharacteristic
216C23#41011yescharacteristic
217C23C2×#10111yescharacteristic
218C23#41311yescharacteristic
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
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—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4#32412yes—
277C2×C4#32412yes—
278C2×C4#35312yes—
279C2×C4#35612yes—
280C2×C4C2×C4⋊C811yescharacteristic
281C2×C4#42311yescharacteristic
282C2×C4#43111yescharacteristic
283C2×C4#43211yescharacteristic
284C8—164no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22C2×#32411yescharacteristic
298C22#145811yescharacteristic
299C22#146511yescharacteristic
300C22#146711yescharacteristic
301C22#179711yescharacteristic
302C22#179911yescharacteristic
303C22#180111yescharacteristic
304C22C2×#35311yescharacteristic
305C22#212611yescharacteristic
306C22C2×#35611yescharacteristic
307C22#213211yescharacteristic
308C4—28no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—22no—
314C4—44no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—82no—
320C4—82no—
321C4#132514yescomplemented
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2#855811yescharacteristic
327C2C2×#132511yescharacteristic
328C2#1144711yescharacteristic
329C2#1152911yescharacteristic
330C2#1153211yescharacteristic
331C2#1279711yescharacteristic
332C2#1280211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
48. C43C8semidirect
207. C42C4⋊C8semidirect
321. C4#1325semidirect

Maximal subgroups

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

subgroupindex
C4×#3842
C2×#13252
#85582
#71892