Finite Group Databasebeta

← #125613

Subgroups — #125613

Order 512.

458 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 76 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#15097C211yesmaximal, characteristic
2#8701C211yesmaximal, characteristic
3#8558C211yesmaximal, characteristic
4#7131C212yesmaximal
5#7104C212yesmaximal
6C2×#398C411yescharacteristic
7C2×#395C411yescharacteristic
8C2×#391C411yescharacteristic
9C2×C4×C4⋊C4C2211yescharacteristic
10#1817—21no—
11#1797—21no—
12C2×#324—21no—
13#1446C2211yescharacteristic
14#1445C2211yescharacteristic
15#1442—21no—
16#1441—21no—
17#1436—21no—
18#1436—21no—
19#1434C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D411yescharacteristic
31C22×C42D411yescharacteristic
32C2×C4⋊C8—41no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35#398—42no—
36#395—42no—
37#391—41no—
38#391—41no—
39C4×C4⋊C4—41no—
40C4×C4⋊C4—41no—
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×#95—21no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95C2311yescharacteristic
55#336—41no—
56#336—41no—
57#336—41no—
58#324—42no—
59C22×Q8—22no—
60C22×Q8—42no—
61C22×Q8—41no—
62C22×Q8—41no—
63C22×Q8—81no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C22×C8—42no—
76C22×C8—42no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42—22no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—41no—
101C2×C42—81no—
102C2×C42—81no—
103C4⋊C8—82no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108#95—24no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—41no—
117#95—41no—
118#95—42no—
119#95—42no—
120#95—41no—
121#95—41no—
122#95—81no—
123#95—81no—
124C24C2×C22⋊C411yescharacteristic
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—82no—
128C2×Q8—84no—
129C2×Q8—84no—
130C2×Q8—82no—
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—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
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—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4#9911yescharacteristic
192C22×C4#9911yescharacteristic
193C22×C4#10012yes—
194C22×C4#13112yes—
195C22×C4#13512yes—
196C22×C4AGL(1,ℤ/8)11yescharacteristic
197C22×C4#13711yescharacteristic
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C4⋊C4—44no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C4⋊C4—84no—
210C42—42no—
211C42—42no—
212C42—42no—
213C42—81no—
214C42—81no—
215C42—81no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23#40811yescharacteristic
221C23C2×#9911yescharacteristic
222C23C2×#10011yescharacteristic
223C23#41611yescharacteristic
224C23#41711yescharacteristic
225C23#42111yescharacteristic
226C23#42111yescharacteristic
227Q8—84no—
228Q8—84no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
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—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
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—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#35612yes—
299C8—162no—
300C8—162no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22#147111yescharacteristic
314C22#148211yescharacteristic
315C22#149511yescharacteristic
316C22#151311yescharacteristic
317C22#151611yescharacteristic
318C22#153311yescharacteristic
319C22#155111yescharacteristic
320C22#212011yescharacteristic
321C22#212111yescharacteristic
322C22#212911yescharacteristic
323C22C2×#35611yescharacteristic
324C4—22no—
325C4—42no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—42no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—81no—
334C4—82no—
335C4—81no—
336C4—161no—
337C2—21no—
338C2—21no—
339C2—21no—
340C2—21no—
341C2#773011yescharacteristic
342C2#1155911yescharacteristic
343C2#1156811yescharacteristic
344C2#1182611yescharacteristic
345C2#1183111yescharacteristic
346C2#1184211yescharacteristic
347C2#1185411yescharacteristic
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
#150972
#87012
#85582
#71312
#71042