Finite Group Databasebeta

← #115772

Subgroups — #115772

Order 512.

674 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 142 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#21955C211yesmaximal, characteristic
2C2×#1534C212yesmaximal
3#14160C211yesmaximal, characteristic
4C2×#1285C212yesmaximal
5#7912C211yesmaximal, characteristic
6C22×#125C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1940—24no—
12#1797—22no—
13#1534—24no—
14#1534—24no—
15C2×#330—24no—
16#1441—21no—
17#1441—21no—
18#1436—22no—
19C4⋊#95C412yes—
20C4⋊#95C412yes—
21#1285—24no—
22#1280—21no—
23#1280—21no—
24C2×#125—22no—
25C2×#125—22no—
26C2×#125—21no—
27C2×#125—21no—
28C2×#125—22no—
29C2×#125—22no—
30C2×#125D412yes—
31C2×#125Q812yes—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C412yes—
38C22×C42C2×C412yes—
39C22×C42D412yes—
40C22×C42C2311yescharacteristic
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432—22no—
48#432D414yes—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—24no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×#97—22no—
55C2×#97D414yes—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—21no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95D411yescharacteristic
66C2×#95D411yescharacteristic
67#336—24no—
68#330—216no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4C22⋊C412yes—
73C23×C4C22⋊C412yes—
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C22×C8—24no—
78C22×C8—42no—
79#125—24no—
80#125—24no—
81#125—24no—
82#125—24no—
83#125—42no—
84#125—42no—
85#125—42no—
86#125—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C4⋊C412yes—
109C2×C4⋊C4C4⋊C412yes—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—24no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—82no—
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42C2×D412yes—
119C2×C42C2×Q811yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4112yes—
122C2×C42#4112yes—
123C2×C42#4111yescharacteristic
124C4⋊C8—24no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—84no—
129C22⋊C8—24no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#97—24no—
134#97—24no—
135#97—24no—
136#97—42no—
137#95—22no—
138#95—22no—
139#95—44no—
140#95—44no—
141#95—41no—
142#95—41no—
143#95—82no—
144#95—82no—
145C24C2×C22⋊C411yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4#10114yes—
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4C4×D412yes—
183C22×C4C4×D412yes—
184C22×C4#12011yescharacteristic
185C22×C4#12112yes—
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12112yes—
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4#12311yescharacteristic
192C22×C4#12411yescharacteristic
193C22×C4#12711yescharacteristic
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—44no—
197C2×C8—44no—
198C2×C8—84no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—48no—
202C4⋊C4—48no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C42—44no—
212C42—82no—
213C42—84no—
214C42#12012yes—
215C42#12112yes—
216C42C22⋊Q812yes—
217C42C22⋊Q812yes—
218C42#12312yes—
219C42#12312yes—
220C42#12412yes—
221C42#12612yes—
222C23—22no—
223C23—21no—
224C23—21no—
225C23#38011yescharacteristic
226C23#38611yescharacteristic
227C23#39011yescharacteristic
228C23C2×#10112yes—
229C23#41312yes—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#38512yes—
277C2×C4#38812yes—
278C2×C4#39211yescharacteristic
279C2×C4#39311yescharacteristic
280C2×C4#39412yes—
281C2×C4#39612yes—
282C2×C4#39711yescharacteristic
283C2×C4#39911yescharacteristic
284C8—84no—
285C8—84no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22#143911yescharacteristic
295C22#180112yes—
296C22#188412yes—
297C22#190112yes—
298C22#191111yescharacteristic
299C22#191211yescharacteristic
300C22#196611yescharacteristic
301C22#196711yescharacteristic
302C4—22no—
303C4—22no—
304C4—22no—
305C4—22no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—44no—
311C4—84no—
312C4—82no—
313C4—82no—
314C4—82no—
315C2—21no—
316C2—22no—
317C2—21no—
318C2#944912yes—
319C2#945012yes—
320C2#1104511yescharacteristic
321C2#1105611yescharacteristic
322C2#1107211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219552
C2×#15342
#141602
C2×#12852
#79122