Finite Group Databasebeta

← #135313

Subgroups — #135313

Order 512.

642 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 104 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#13926C211yesmaximal, characteristic
2#10763C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7251C214yesmaximal
5C22×C4×Q8C2211yescharacteristic
6C22×#101—21no—
7C2×#391—21no—
8C2×#384C411yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10#1817—21no—
11C2×#343C2212yes—
12#1728—21no—
13C2×#324—21no—
14C2×#324C2212yes—
15#1441C412yes—
16#1441C2212yes—
17C4⋊#95—21no—
18C4⋊#95—21no—
19C4⋊#95—21no—
20C23×Q8D411yescharacteristic
21C22×M16—22no—
22C2×C4×Q8—24no—
23C2×C4×Q8—41no—
24C2×C4×Q8—41no—
25C2×C4×Q8—42no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4D411yescharacteristic
32C22×C4⋊C4C2311yescharacteristic
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C2×#101—24no—
38C2×#101—42no—
39C2×#101—42no—
40C2×C22⋊C8—22no—
41#391—42no—
42#391—42no—
43#384—41no—
44#384—41no—
45C4×C4⋊C4—42no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50C2×#95C2×C412yes—
51#343—24no—
52#336—41no—
53#336—41no—
54#336—41no—
55#336—41no—
56#324—24no—
57#324—42no—
58C22×Q8—22no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—41no—
63C22×Q8—41no—
64C22×Q8D812yes—
65C22×Q8SD1612yes—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—24no—
76C2×M16—42no—
77C2×M16—42no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—42no—
82C22×C8—42no—
83C4×Q8—48no—
84C4×Q8—48no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—82no—
105C2×C4⋊C4#4114yes—
106C2×C42—22no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—22no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—42no—
119C2×C42—42no—
120C2×C42—42no—
121#101—48no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124#95—42no—
125#95—44no—
126#95—42no—
127#95—42no—
128#95—82no—
129#95—82no—
130C24C2×C22⋊C411yescharacteristic
131C2×Q8—24no—
132C2×Q8—24no—
133C2×Q8—24no—
134C2×Q8—24no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—44no—
140C2×Q8—44no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#9912yes—
187C22×C4#9911yescharacteristic
188C22×C4#9911yescharacteristic
189C22×C4#10212yes—
190C22×C4#10212yes—
191C22×C4#10412yes—
192C22×C4#11712yes—
193C22×C4#12312yes—
194C22×C4#12312yes—
195C22×C4C2×D811yescharacteristic
196C22×C4C2×SD1611yescharacteristic
197C22×C4#13712yes—
198M16—44no—
199M16—44no—
200M16—82no—
201M16—82no—
202C2×C8—44no—
203C2×C8—82no—
204C2×C8—82no—
205C2×C8—82no—
206C2×C8—82no—
207C2×C8—82no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—48no—
211C4⋊C4—44no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C4⋊C4—84no—
215C4⋊C4—82no—
216C42—24no—
217C42—24no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—84no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23#38111yescharacteristic
228C23C2×#9911yescharacteristic
229C23C2×#9911yescharacteristic
230C23C2×#10211yescharacteristic
231C23#41911yescharacteristic
232C23C2×#10411yescharacteristic
233C23#42111yescharacteristic
234Q8—48no—
235Q8—48no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#32714yes—
293C2×C4#35512yes—
294C2×C4#35612yes—
295C2×C4#48012yes—
296C2×C4#48312yes—
297C8—84no—
298C8—162no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22C2×#32711yescharacteristic
312C22#150811yescharacteristic
313C22#151111yescharacteristic
314C22#151911yescharacteristic
315C22#181911yescharacteristic
316C22#182311yescharacteristic
317C22#182611yescharacteristic
318C22#212211yescharacteristic
319C22#212311yescharacteristic
320C22C2×#35511yescharacteristic
321C22C2×#35611yescharacteristic
322C4—22no—
323C4—24no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—84no—
331C4—81no—
332C4—162no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2#871211yescharacteristic
338C2#1179411yescharacteristic
339C2#1179511yescharacteristic
340C2#1188211yescharacteristic
341C2#1188311yescharacteristic
342C2#1285511yescharacteristic
343C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#139262
#107632
#83032
#72512