Finite Group Databasebeta

← #115663

Subgroups — #115663

Order 512.

910 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C2×#2565C211yesmaximal, characteristic
2#17449C212yesmaximal
3#17394C211yesmaximal, characteristic
4#17394C211yesmaximal, characteristic
5#7950C212yesmaximal
6C22×#125C2211yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2920C2212yes—
9#2920C2212yes—
10#2565—24no—
11#2565—24no—
12C2×#398—22no—
13C2×#391C411yescharacteristic
14C2×#391C411yescharacteristic
15C2×#387—22no—
16C2×#384C411yescharacteristic
17C2×#382C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1491—24no—
20#1487—24no—
21#1479—24no—
22#1470—24no—
23#1470—24no—
24#1463—22no—
25#1463—22no—
26#1285—24no—
27C23×Q8C2×C411yescharacteristic
28C2×#125—22no—
29C2×#125—22no—
30C2×#125—22no—
31C2×#125—22no—
32C2×#125—22no—
33C2×#125—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8D414yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D412yes—
48C22×C42—21no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52#432—22no—
53#432—22no—
54#432—22no—
55#432D414yes—
56#431—22no—
57#431—22no—
58#431—22no—
59#431D414yes—
60#423—22no—
61#423—22no—
62#407—22no—
63#407—22no—
64C4×M16—22no—
65C4×M16—22no—
66#398—216no—
67#391—22no—
68#391—22no—
69#391—22no—
70#391—22no—
71#387—216no—
72#384—22no—
73#384—22no—
74#382—22no—
75#382—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4D414yes—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82Q8⋊C8—216no—
83#324—28no—
84#324—28no—
85C22×Q8—24no—
86C22×Q8—21no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—42no—
90C22×Q8C22⋊C412yes—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—42no—
101C2×M16—42no—
102#125—42no—
103#125—42no—
104#125—42no—
105#125—42no—
106#125—42no—
107#125—42no—
108#125—42no—
109#125—42no—
110C4×Q8—24no—
111C4×Q8—24no—
112C4×Q8—24no—
113C4×Q8—24no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C22⋊C412yes—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—24no—
149C2×C42—41no—
150C2×C42—42no—
151C2×C42—41no—
152C2×C42—42no—
153C2×C42—42no—
154C2×C42C22⋊C412yes—
155C2×C42C22⋊C412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42#4112yes—
161C2×C42#4112yes—
162C4⋊C8—24no—
163C4⋊C8—24no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170#97—24no—
171#97—42no—
172C4×C8—24no—
173C4×C8—42no—
174#95—28no—
175#95—28no—
176#95—28no—
177#95—24no—
178C24C2×C22⋊C411yescharacteristic
179C2×Q8—24no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—24no—
183C2×Q8—48no—
184C2×Q8—48no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8#10114yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C4×D412yes—
220C22×C4C4×D412yes—
221C22×C4#12011yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4#12012yes—
224C22×C4#12012yes—
225C22×C4#12112yes—
226C22×C4#12112yes—
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229M16—84no—
230M16—84no—
231C2×C8—44no—
232C2×C8—44no—
233C4⋊C4—24no—
234C4⋊C4—48no—
235C4⋊C4—416no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C42—42no—
248C42—42no—
249C42—44no—
250C42—44no—
251C42—42no—
252C42—42no—
253C42#12012yes—
254C42#12112yes—
255C42#12112yes—
256C42#12112yes—
257C42#12112yes—
258C42#12312yes—
259C42#12412yes—
260C42#12712yes—
261C23—22no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23#37911yescharacteristic
266C23#38611yescharacteristic
267C23#38611yescharacteristic
268Q8—28no—
269Q8—48no—
270Q8—44no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—84no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4#38612yes—
309C2×C4#39012yes—
310C2×C4#39212yes—
311C2×C4#39212yes—
312C2×C4#39412yes—
313C2×C4#39412yes—
314C2×C4C2×#10112yes—
315C2×C4#41312yes—
316C8—84no—
317C8—84no—
318C22—24no—
319C22—24no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22#143811yescharacteristic
327C22#178411yescharacteristic
328C22#178511yescharacteristic
329C22#188412yes—
330C22#190912yes—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—44no—
337C4—44no—
338C4—82no—
339C4—82no—
340C4—82no—
341C4—82no—
342C4#190114yes—
343C2—21no—
344C2—22no—
345C2—22no—
346C2—21no—
347C2#1096811yescharacteristic
348C2#1097911yescharacteristic
349C2#1098811yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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
C2×#25652
#174492
#173942
#79502