Finite Group Databasebeta

← #124152

Subgroups — #124152

Order 512.

466 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 98 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#14298C211yesmaximal, characteristic
2C4×#324C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7123C212yesmaximal
5#7122C212yesmaximal
6C2×#383C411yescharacteristic
7C2×#382C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C4×C22⋊C8—21no—
10C4×C22⋊C8—21no—
11C2×#324—21no—
12C2×#324—21no—
13#1444—21no—
14#1444C411yescharacteristic
15#1444C2211yescharacteristic
16#1442—21no—
17#1442C411yescharacteristic
18#1442C2211yescharacteristic
19#1436—21no—
20#1436—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C4×C8—41no—
35C2×C4×C8—41no—
36#383—22no—
37#383C814yescomplemented
38#382—41no—
39#382—41no—
40C4×C4⋊C4—41no—
41C4×C4⋊C4—41no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2311yescharacteristic
56#336—41no—
57#336—41no—
58#324—24no—
59#324—24no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C22×C8—42no—
72C22×C8—42no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—81no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—81no—
97C2×C42—81no—
98C2×C42C2×C812yes—
99C2×C42M1612yes—
100C22⋊C8—24no—
101C22⋊C8—24no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104C4×C8—82no—
105C4×C8—82no—
106#95—24no—
107#95—24no—
108#95—21no—
109#95—21no—
110#95—41no—
111#95—41no—
112#95—42no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—42no—
117#95—81no—
118#95—81no—
119#95—81no—
120#95—81no—
121#95C2×C812yes—
122#95C2×C812yes—
123C24C2×C22⋊C411yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4C22⋊C812yes—
181C22×C4C22⋊C812yes—
182C22×C4#9911yescharacteristic
183C22×C4#9911yescharacteristic
184C22×C4#10112yes—
185C22×C4#10412yes—
186C22×C4#10412yes—
187C22×C4C22×C811yescharacteristic
188C22×C4C2×M1611yescharacteristic
189C22×C4#13512yes—
190C22×C4#13512yes—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—84no—
202C4⋊C4—84no—
203C42—22no—
204C42—22no—
205C42—22no—
206C42—22no—
207C42—42no—
208C42—42no—
209C42—81no—
210C42—81no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23C2×C22⋊C811yescharacteristic
216C23C2×#9911yescharacteristic
217C23C2×#10111yescharacteristic
218C23#41611yescharacteristic
219C23#41611yescharacteristic
220C23C2×#10411yescharacteristic
221C23C2×#10411yescharacteristic
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
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—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#32412yes—
291C2×C4#32412yes—
292C8—84no—
293C8—84no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22C2×#32411yescharacteristic
307C22#147611yescharacteristic
308C22#147611yescharacteristic
309C22#150111yescharacteristic
310C22#150111yescharacteristic
311C22#152611yescharacteristic
312C22#154411yescharacteristic
313C22#212211yescharacteristic
314C22#212311yescharacteristic
315C22#212711yescharacteristic
316C22#213311yescharacteristic
317C4—22no—
318C4—22no—
319C4—42no—
320C4—41no—
321C4—41no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—42no—
326C4—82no—
327C4—82no—
328C4—161no—
329C4—161no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#756411yescharacteristic
335C2#1144811yescharacteristic
336C2#1145411yescharacteristic
337C2#1173011yescharacteristic
338C2#1173211yescharacteristic
339C2#1173811yescharacteristic
340C2#1174011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #383C8semidirect

Maximal subgroups

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

subgroupindex
#142982
C4×#3242
#71232
#71222