Finite Group Databasebeta

← #124659

Subgroups — #124659

Order 512.

568 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 92 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#14274C211yesmaximal, characteristic
2#8699C211yesmaximal, characteristic
3#7910C211yesmaximal, characteristic
4#7104C214yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#432—21no—
7C2×#391—21no—
8C2×#391—21no—
9C2×#382C411yescharacteristic
10C2×#376C2211yescharacteristic
11#1817—21no—
12#1445—21no—
13#1445—21no—
14#1445C412yes—
15#1445C2212yes—
16#1280—21no—
17#1280—21no—
18#1280C2212yes—
19#1280C2212yes—
20C23×Q8—21no—
21C2×C4⋊Q8—24no—
22C2×C4⋊Q8—41no—
23C2×C4⋊Q8—41no—
24C2×C4⋊Q8—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C412yes—
28C22×C42C2×C411yescharacteristic
29C22×C42D411yescharacteristic
30C22×C42D411yescharacteristic
31#432—24no—
32#432—42no—
33C2×C4⋊C8—41no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×#97—41no—
37#391—44no—
38#391—42no—
39#391—42no—
40#382—42no—
41#376—41no—
42#376—41no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49C2×#95C2×C411yescharacteristic
50C2×#95D411yescharacteristic
51C2×#95D411yescharacteristic
52C2×#95C2311yescharacteristic
53#336—41no—
54#336—41no—
55C22×Q8—24no—
56C22×Q8—42no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8—41no—
60C22×Q8—41no—
61C22×Q8—42no—
62C22×Q8—42no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C22×C8—42no—
73C22×C8—42no—
74C4⋊Q8—48no—
75C4⋊Q8—48no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4M1614yes—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—41no—
97C2×C42D812yes—
98C2×C42SD1612yes—
99C4⋊C8—44no—
100C22⋊C8—24no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—42no—
104#97—44no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—42no—
110#95—44no—
111#95—42no—
112#95—44no—
113#95—42no—
114#95—42no—
115#95—44no—
116C24C2×C22⋊C411yescharacteristic
117C2×Q8—44no—
118C2×Q8—44no—
119C2×Q8—48no—
120C2×Q8—48no—
121C2×Q8—84no—
122C2×Q8—82no—
123C2×Q8—82no—
124C2×Q8—84no—
125C2×Q8—82no—
126C2×Q8—82no—
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—21no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
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—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4#9911yescharacteristic
171C22×C4#9911yescharacteristic
172C22×C4#10012yes—
173C22×C4#10212yes—
174C22×C4#10212yes—
175C22×C4#10412yes—
176C22×C4C2×M1612yes—
177C22×C4C2×D811yescharacteristic
178C22×C4C2×SD1611yescharacteristic
179C22×C4AGL(1,ℤ/8)12yes—
180C2×C8—44no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8—82no—
186C4⋊C4—44no—
187C4⋊C4—48no—
188C4⋊C4—44no—
189C4⋊C4—84no—
190C42—24no—
191C42—24no—
192C42—22no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—42no—
197C42—42no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23#40711yescharacteristic
203C23C2×#9911yescharacteristic
204C23C2×#10011yescharacteristic
205C23C2×#10211yescharacteristic
206C23#41811yescharacteristic
207C23C2×#10411yescharacteristic
208C23#42111yescharacteristic
209Q8—88no—
210Q8—88no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#32712yes—
266C2×C4#32712yes—
267C8—84no—
268C8—162no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22#146211yescharacteristic
282C22#148611yescharacteristic
283C22#149011yescharacteristic
284C22C2×#32711yescharacteristic
285C22#151011yescharacteristic
286C22#152711yescharacteristic
287C22#154511yescharacteristic
288C22#212311yescharacteristic
289C22#212311yescharacteristic
290C22#212911yescharacteristic
291C22#213511yescharacteristic
292C4—22no—
293C4—22no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—42no—
300C4—81no—
301C4—84no—
302C4—162no—
303C2—21no—
304C2—21no—
305C2—21no—
306C2—21no—
307C2#760611yescharacteristic
308C2#1150411yescharacteristic
309C2#1151011yescharacteristic
310C2#1169411yescharacteristic
311C2#1170011yescharacteristic
312C2#1180111yescharacteristic
313C2#1180711yescharacteristic
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×#376 ◁ 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
#142742
#86992
#79102
#71042