Finite Group Databasebeta

← #125389

Subgroups — #125389

Order 512.

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

normal subgroups only

This group has 476 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#15098C211yesmaximal, characteristic
2#8153C211yesmaximal, characteristic
3#7908C211yesmaximal, characteristic
4#7123C212yesmaximal
5#7123C212yesmaximal
6C2×#431—21no—
7C2×#389C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9#1749—21no—
10C2×#324—21no—
11#1444—21no—
12#1444C2211yescharacteristic
13#1444C2211yescharacteristic
14#1441—21no—
15#1441C411yescharacteristic
16#1441C411yescharacteristic
17#1436—21no—
18#1436—21no—
19#1434C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42C2×C411yescharacteristic
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33#431—24no—
34#431—42no—
35C2×C4⋊C8—41no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×#97—41no—
39C2×C4×C8—41no—
40#389—41no—
41#389—41no—
42C4×C4⋊C4—41no—
43C4×C4⋊C4—41no—
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×#95—21no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95C2311yescharacteristic
57#336—41no—
58#324—24no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C22×C8—42no—
71C22×C8—42no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—81no—
102C2×C42—81no—
103C4⋊C8—44no—
104C22⋊C8—24no—
105C22⋊C8—24no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#97—82no—
109C4×C8—44no—
110#95—24no—
111#95—22no—
112#95—22no—
113#95—41no—
114#95—41no—
115#95—42no—
116#95—41no—
117#95—41no—
118#95—42no—
119#95—42no—
120#95—81no—
121#95—81no—
122#95—81no—
123#95—81no—
124C24C2×C22⋊C411yescharacteristic
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—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
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×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4#9911yescharacteristic
184C22×C4#9911yescharacteristic
185C22×C4#10112yes—
186C22×C4#10412yes—
187C22×C4#10412yes—
188C22×C4#13112yes—
189C22×C4#13512yes—
190C22×C4AGL(1,ℤ/8)11yescharacteristic
191C22×C4#13711yescharacteristic
192C2×C8—44no—
193C2×C8—44no—
194C2×C8—82no—
195C2×C8—82no—
196C2×C8—82no—
197C2×C8—82no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—22no—
211C42—42no—
212C42—81no—
213C42—81no—
214C42—81no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#40811yescharacteristic
220C23C2×#9911yescharacteristic
221C23C2×#10111yescharacteristic
222C23#41611yescharacteristic
223C23#41711yescharacteristic
224C23C2×#10411yescharacteristic
225C23C2×#10411yescharacteristic
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
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—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4#33014yes—
295C2×C4#35612yes—
296C8—84no—
297C8—84no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22#147211yescharacteristic
311C22#148311yescharacteristic
312C22#149411yescharacteristic
313C22#150111yescharacteristic
314C22#150411yescharacteristic
315C22C2×#33011yescharacteristic
316C22#155211yescharacteristic
317C22#212011yescharacteristic
318C22#212111yescharacteristic
319C22#212811yescharacteristic
320C22C2×#35611yescharacteristic
321C4—24no—
322C4—22no—
323C4—42no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—82no—
333C4—161no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—21no—
338C2#769111yescharacteristic
339C2#1157411yescharacteristic
340C2#1157711yescharacteristic
341C2#1172911yescharacteristic
342C2#1173611yescharacteristic
343C2#1175411yescharacteristic
344C2#1176811yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#150982
#81532
#79082
#71232