Finite Group Databasebeta

← #192154

Subgroups — #192154

Order 512.

1182 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 258 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#34186C211yesmaximal, characteristic
2#21986C212yesmaximal, complemented
3C2×#1466C211yesmaximal, characteristic
4#14345C211yesmaximal, characteristic
5#8858C218yesmaximal, complemented
6#8018C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C2×#432C2212yes—
9#2357—24no—
10#2354—22no—
11#2354—22no—
12#2312C2214yes—
13#2309—21no—
14#2285C2212yes—
15C2×#383C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#2280—22no—
18#2274—21no—
19#1949—24no—
20#1853C2214yes—
21#1852C2214yes—
22#1847—22no—
23#1836—22no—
24#1466C418yescomplemented
25C2×#323C2214yes—
26#1445—21no—
27#1439C2214yes—
28#1438—21no—
29C22⋊#95—22no—
30#1294C2218yes—
31C23×D4D411yescharacteristic
32C24×C4D412yes—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—24no—
36C2×C4×D4C2×C414yes—
37C2×C4×D4Q814yes—
38C2×#117—22no—
39C2×#117—24no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4C2312yes—
42C22×C22⋊C4—24no—
43C22×C22⋊C4C2312yes—
44C22×C42C2311yescharacteristic
45#432—24no—
46#432C2×C418yes—
47C2×#107—24no—
48C2×#106—24no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8C2314yes—
51C2×#97—22no—
52#390—24no—
53#390—24no—
54#388—28no—
55#388—28no—
56#384D412yes—
57#384D412yes—
58#383—24no—
59#382—22no—
60#382—22no—
61#382—22no—
62#381—24no—
63#380—22no—
64#380—22no—
65#380—22no—
66#380C2314yes—
67#379—24no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4D414yes—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—24no—
79C2×#95C2312yes—
80#340—24no—
81#340—24no—
82#323C2×C4116yes—
83C25C2×D412yes—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—44no—
87C22×D4C4⋊C414yes—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—84no—
94C23×C4C4⋊C418yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2411yescharacteristic
99C22×C8—24no—
100C4×D4—28no—
101C4×D4—28no—
102C4×D4—28no—
103C4×D4—28no—
104#117—48no—
105#117—88no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—82no—
115C2×C4⋊C4C4⋊C414yes—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×D414yes—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—48no—
125C2×C22⋊C4—84no—
126C2×C22⋊C4—84no—
127C2×C22⋊C4C22×C414yes—
128C2×C22⋊C4C2×Q814yes—
129C2×C42—24no—
130C2×C42—28no—
131C2×C42C22×C412yes—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42C2×Q812yes—
135C2×C42#4114yes—
136#107—84no—
137#106—84no—
138C4⋊C8—28no—
139C22⋊C8—24no—
140C22⋊C8C22×C418yes—
141#97—28no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—41no—
148#95—42no—
149#95—41no—
150#95—42no—
151#95—42no—
152#95—84no—
153C24—22no—
154C24—24no—
155C24—22no—
156C24—24no—
157C24—42no—
158C24—44no—
159C24—42no—
160C24C2×C4⋊C414yes—
161C24C22×D411yescharacteristic
162C2×D4—28no—
163C2×D4—48no—
164C2×D4—48no—
165C2×D4—44no—
166C2×D4—44no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—28no—
171C22×C4—28no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—48no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—84no—
190C22×C4—88no—
191C22×C4—88no—
192C22×C4C2×C4⋊C412yes—
193C22×C4C2×C4⋊C414yes—
194C22×C4C2×C4⋊C412yes—
195C22×C4#12011yescharacteristic
196C22×C4#12012yes—
197C22×C4#12011yescharacteristic
198C22×C4#12312yes—
199C22×C4#12312yes—
200C22×C4C23×C411yescharacteristic
201C22×C4C22×D412yes—
202C22×C4C22×Q811yescharacteristic
203C22×C4C2×#4112yes—
204C2×C8—28no—
205C2×C8—44no—
206C2×C8—44no—
207C4⋊C4—28no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C22⋊C4—28no—
217C22⋊C4—28no—
218C22⋊C4—48no—
219C22⋊C4—84no—
220C22⋊C4—84no—
221C22⋊C4—88no—
222C22⋊C4—816no—
223C42—48no—
224C42—88no—
225C42C4×D414yes—
226C42C4×D414yes—
227C42C22⋊Q814yes—
228C42C22⋊Q814yes—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—24no—
233C23—24no—
234C23—24no—
235C23—44no—
236C23—44no—
237C23—48no—
238C23—48no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—42no—
243C23—42no—
244C23—82no—
245C23—82no—
246C23—84no—
247C23—84no—
248C23C22×C4⋊C411yescharacteristic
249C23C2×#12011yescharacteristic
250C23C2×#12311yescharacteristic
251C23#57514yes—
252D4—416no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—416no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—84no—
283C2×C4—84no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#38514yes—
289C2×C4#38512yes—
290C2×C4#38512yes—
291C2×C4C2×C4×D412yes—
292C2×C4C2×C22⋊Q812yes—
293C8—48no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—24no—
299C22—24no—
300C22—48no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—84no—
305C22—88no—
306C22—84no—
307C22—84no—
308C22C2×#38511yescharacteristic
309C22#291112yes—
310C22#300512yes—
311C22#301912yes—
312C22#302711yescharacteristic
313C22#302811yescharacteristic
314C22#309312yes—
315C4—24no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—44no—
321C4—84no—
322C4—84no—
323C4—162no—
324C4—162no—
325C2—22no—
326C2—22no—
327C2—84no—
328C2#1837012yes—
329C2#1837512yes—
330C2#2084511yescharacteristic
331C2#2085611yescharacteristic
332C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21986C2semidirect
5. #8858C2semidirect
6. #8018C2semidirect
24. #1466C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#219862
C2×#14662
#143452
#88582
#80182