Finite Group Databasebeta

← #237786

Subgroups — #237786

Order 512.

980 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 198 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#34595C211yesmaximal, characteristic
2#21841C212yesmaximal
3#14155C211yesmaximal, characteristic
4#10347C214yesmaximal
5#10343C214yesmaximal
6#8021C212yesmaximal
7#7912C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2212yes—
10#2666—24no—
11#2659C2212yes—
12#2646C2212yes—
13#2627—22no—
14#2402—22no—
15#2399—22no—
16C4×C22⋊Q8—22no—
17C2×#391—22no—
18#2283C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1993—24no—
22#1988—22no—
23#1988—22no—
24#1824C2218yes—
25#1445C2212yes—
26#1445C2212yes—
27C4⋊#95—22no—
28#1294C2218yes—
29#1280C2212yes—
30#1280C2212yes—
31C23×Q8—21no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—24no—
36C2×C4⋊Q8Q814yescomplemented
37C2×C22⋊Q8—22no—
38C2×C4×Q8—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4C2312yes—
43C22×C42—21no—
44C22×C42D412yes—
45C22×C42C2311yescharacteristic
46#432—24no—
47#432—24no—
48C2×C4⋊C8—22no—
49C2×C22⋊C8C2314yes—
50C2×#97—22no—
51#400—24no—
52#400—24no—
53#398—24no—
54#398—24no—
55#398—24no—
56#398—24no—
57#395—24no—
58#395—24no—
59#391—22no—
60#391—48no—
61#389—22no—
62#387—24no—
63#385—24no—
64#383—22no—
65#382D414yes—
66#381—22no—
67#380—22no—
68#380C2314yes—
69C4×C4⋊C4—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4D412yes—
75C4×C4⋊C4D412yes—
76C4×C22⋊C4—22no—
77#376—22no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82C2×#95C2311yescharacteristic
83#340—28no—
84#340—28no—
85C22×Q8—22no—
86C22×Q8—24no—
87C22×Q8—44no—
88C22×Q8—44no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8—42no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2411yescharacteristic
99C22×C8—24no—
100C4⋊Q8—28no—
101C4⋊Q8—28no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C22⋊Q8—44no—
105C22⋊Q8—44no—
106C4×Q8—84no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4C2×Q814yes—
128C2×C22⋊C4—24no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—44no—
134C2×C42—44no—
135C2×C42—42no—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×Q812yes—
139C2×C42#4114yes—
140C2×C42#4112yes—
141C4⋊C8—44no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144#97—44no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—22no—
154#95—24no—
155#95—24no—
156#95—48no—
157#95—48no—
158C24C22×D411yescharacteristic
159C2×Q8—44no—
160C2×Q8—416no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—88no—
166C2×Q8—82no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—82no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4#12011yescharacteristic
197C22×C4#12012yes—
198C22×C4#12011yescharacteristic
199C22×C4C22⋊Q812yes—
200C22×C4C22⋊Q812yes—
201C22×C4#12412yes—
202C22×C4#12412yes—
203C22×C4#13714yes—
204C22×C4C22×D412yes—
205C22×C4C22×Q811yescharacteristic
206C22×C4C2×#4112yes—
207C22×C4C2×#4111yescharacteristic
208C2×C8—48no—
209C2×C8—44no—
210C4⋊C4—28no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C4⋊C4—44no—
218C4⋊C4—48no—
219C4⋊C4—82no—
220C4⋊C4—84no—
221C4⋊C4—82no—
222C4⋊C4—84no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C42—44no—
226C42—44no—
227C42—44no—
228C42—82no—
229C42—84no—
230C42C22⋊Q814yes—
231C42C22⋊Q814yes—
232C42#12414yes—
233C42#12414yes—
234C23—21no—
235C23—21no—
236C23—22no—
237C23C2×#12011yescharacteristic
238C23C2×C22⋊Q811yescharacteristic
239C23C2×#12411yescharacteristic
240C23C2×#13712yes—
241C23#57512yes—
242Q8—88no—
243Q8—88no—
244Q8—88no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—48no—
263C2×C4—44no—
264C2×C4—48no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4#39314yes—
284C2×C4#39312yes—
285C2×C4#39312yes—
286C2×C4#45312yes—
287C2×C4#45412yes—
288C2×C4C2×C22⋊Q812yes—
289C2×C4C2×#12412yes—
290C8—84no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—21no—
298C22—21no—
299C22C2×#39311yescharacteristic
300C22#300712yes—
301C22C2×#45311yescharacteristic
302C22C2×#45411yescharacteristic
303C22#302812yes—
304C22#307812yes—
305C22#313911yescharacteristic
306C22#314011yescharacteristic
307C4—24no—
308C4—22no—
309C4—22no—
310C4—22no—
311C4—44no—
312C4—84no—
313C4—84no—
314C4—82no—
315C4—84no—
316C4—84no—
317C4#202714yes—
318C2—21no—
319C2—22no—
320C2—21no—
321C2#1834312yes—
322C2#1834512yes—
323C2#2345511yescharacteristic
324C2C2×#202711yescharacteristic
325C2#2347011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×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
36. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#345952
#218412
#141552
#103472
#103432
#80212
#79122