Finite Group Databasebeta

← #125268

Subgroups — #125268

Order 512.

474 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 78 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#15104C211yesmaximal, characteristic
2#8153C211yesmaximal, characteristic
3C4×#324C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7123C212yesmaximal
6C2×#391C411yescharacteristic
7C2×C4×C4⋊C4C2211yescharacteristic
8#1749—21no—
9C4×C22⋊C8—21no—
10C2×#324—21no—
11C2×#324—21no—
12#1446C411yescharacteristic
13#1446C2211yescharacteristic
14#1445—21no—
15#1444—21no—
16#1444C411yescharacteristic
17#1444C2211yescharacteristic
18#1436—21no—
19#1436—21no—
20#1434C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
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×#97—41no—
35C2×C4×C8—41no—
36#391—41no—
37#391—41no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C2×#95—21no—
41C2×#95—21no—
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×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2311yescharacteristic
54#336—41no—
55#336—41no—
56#324—24no—
57#324—24no—
58C22×Q8—22no—
59C22×Q8—42no—
60C22×Q8—41no—
61C22×Q8—41no—
62C22×Q8—81no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—41no—
98C2×C42—81no—
99C2×C42—81no—
100C22⋊C8—24no—
101C22⋊C8—24no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#97—82no—
105C4×C8—82no—
106#95—24no—
107#95—22no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—42no—
112#95—42no—
113#95—42no—
114#95—41no—
115#95—41no—
116#95—42no—
117#95—81no—
118#95—81no—
119#95—81no—
120#95—81no—
121C24C2×C22⋊C411yescharacteristic
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—82no—
125C2×Q8—84no—
126C2×Q8—84no—
127C2×Q8—82no—
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—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
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—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4#9911yescharacteristic
188C22×C4#9911yescharacteristic
189C22×C4#10412yes—
190C22×C4#10412yes—
191C22×C4#13112yes—
192C22×C4#13512yes—
193C22×C4#13712yes—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C42—42no—
206C42—42no—
207C42—42no—
208C42—42no—
209C42—81no—
210C42—81no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23#40811yescharacteristic
216C23C2×#9911yescharacteristic
217C23#41311yescharacteristic
218C23#41611yescharacteristic
219C23#41911yescharacteristic
220C23C2×#10411yescharacteristic
221C23C2×#10411yescharacteristic
222Q8—84no—
223Q8—84no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
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—22no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
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—42no—
252C2×C4—42no—
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—42no—
266C2×C4—41no—
267C2×C4—41no—
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—81no—
291C2×C4#35612yes—
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—
306C22#146811yescharacteristic
307C22#148511yescharacteristic
308C22#149411yescharacteristic
309C22#150111yescharacteristic
310C22#150811yescharacteristic
311C22#152411yescharacteristic
312C22#155411yescharacteristic
313C22#212011yescharacteristic
314C22#212211yescharacteristic
315C22#212711yescharacteristic
316C22C2×#35611yescharacteristic
317C4—22no—
318C4—42no—
319C4—42no—
320C4—41no—
321C4—42no—
322C4—42no—
323C4—42no—
324C4—82no—
325C4—82no—
326C4—81no—
327C4—82no—
328C4—161no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2#767511yescharacteristic
334C2#1154211yescharacteristic
335C2#1155511yescharacteristic
336C2#1173011yescharacteristic
337C2#1173611yescharacteristic
338C2#1178411yescharacteristic
339C2#1179511yescharacteristic
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
#151042
#81532
C4×#3242
#71312
#71232