Finite Group Databasebeta

← #213615

Subgroups — #213615

Order 512.

1260 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 238 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#36625C211yesmaximal, characteristic
2C2×#1749C212yesmaximal
3#9121C218yesmaximal, complemented
4#8721C212yesmaximal, complemented
5#8338C212yesmaximal, complemented
6C2×#533C2211yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—22no—
9C22×#97C2214yes—
10#2857C412yes—
11#2857C412yes—
12#2855—22no—
13#2851C2214yes—
14#2848—22no—
15C4×C22⋊Q8—22no—
16C4×C22⋊Q8—22no—
17#2296C412yes—
18#2285C2211yescharacteristic
19#2280C412yes—
20#1749—22no—
21#1749—22no—
22#1749C2214yes—
23#1749C2214yes—
24C2×#336—22no—
25C2×#330C2218yescomplemented
26#1295C2218yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C23×C8—24no—
30#533—44no—
31C2×C4⋊Q8C2×C412yes—
32C2×#125C2312yes—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C4×Q8—22no—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C22×C42—44no—
40C22×C42C2×C412yes—
41C22×C42C2×C411yescharacteristic
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47C2×C22⋊C8—216no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×#97—24no—
51C2×#97—48no—
52C2×#97C2318yes—
53#391—22no—
54#391—22no—
55#389—22no—
56#389—22no—
57#389C2×C414yes—
58#389C2×C414yes—
59#387—24no—
60#387—24no—
61#385—24no—
62#385—24no—
63#385—24no—
64#384C2×C411yescharacteristic
65#384C2×C411yescharacteristic
66#382C2312yes—
67#381—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380C2×C414yes—
72#380C2×C414yes—
73#379—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C22⋊C4—24no—
78#336—24no—
79#336—24no—
80#330—48no—
81C25C22⋊C411yescharacteristic
82C22×Q8—22no—
83C23×C4—24no—
84C23×C4—28no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—44no—
88C23×C4—44no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C412yes—
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D412yes—
99C22×C8—216no—
100C22×C8—28no—
101C22×C8—48no—
102C22×C8—48no—
103C4⋊Q8—44no—
104#125—28no—
105C22⋊Q8—48no—
106C22⋊Q8—48no—
107C22⋊Q8—48no—
108C4×Q8—48no—
109#117—44no—
110#117—44no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4C22×C414yes—
118C2×C4⋊C4C22×C412yes—
119C2×C4⋊C4C22×C412yes—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C42—24no—
125C2×C42—28no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—42no—
129C2×C42—44no—
130C2×C42—44no—
131C2×C42—44no—
132C2×C42—44no—
133C2×C42C22⋊C411yescharacteristic
134C2×C42C22⋊C411yescharacteristic
135C2×C42C22⋊C412yes—
136C2×C42C22⋊C412yes—
137C2×C42C22×C411yescharacteristic
138C2×C42C22×C412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×D412yes—
142C2×C42C2411yescharacteristic
143C22⋊C8—216no—
144C22⋊C8—216no—
145#97—216no—
146#97—48no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152C24—24no—
153C24—24no—
154C24—41no—
155C24—41no—
156C24#10114yes—
157C24C2×C22⋊C411yescharacteristic
158C24C2×C22⋊C412yes—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—432no—
195C22×C4—416no—
196C22×C4#10418yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C411yescharacteristic
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C23×C411yescharacteristic
203C22×C4C22×D412yes—
204C2×C8—216no—
205C2×C8—432no—
206C2×C8—48no—
207C2×C8—48no—
208C4⋊C4—28no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—48no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—48no—
220C22⋊C4—48no—
221C42—24no—
222C42—22no—
223C42—24no—
224C42—24no—
225C42—24no—
226C42—41no—
227C42—41no—
228C42—42no—
229C42—42no—
230C42—44no—
231C42—44no—
232C42H(5,𝔽2)14yes—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—216no—
242C23—44no—
243C23—48no—
244C23—44no—
245C23—44no—
246C23C2×#10112yes—
247C23C2×#10114yes—
248C23C2×#10414yes—
249C23#42114yes—
250C23C22×C22⋊C411yescharacteristic
251Q8—88no—
252C2×C4—216no—
253C2×C4—28no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4#41312yes—
286C2×C4C2×#10414yes—
287C2×C4C2×#10414yes—
288C2×C4#51812yes—
289C2×C4#53414yes—
290C8—416no—
291C22—24no—
292C22—24no—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—416no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22#153118yes—
304C22#230811yescharacteristic
305C22C22×#10111yescharacteristic
306C22C2×#41311yescharacteristic
307C22C22×#10412yes—
308C22C2×#42112yes—
309C4—24no—
310C4—24no—
311C4—24no—
312C4—22no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—84no—
317C4—82no—
318C4—82no—
319C2—28no—
320C2—42no—
321C2#1763811yescharacteristic
322C2C2×#153112yes—
323C2#1767811yescharacteristic
324C2#2175311yescharacteristic
325C2#2177312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9121C2semidirect
4. #8721C2semidirect
5. #8338C2semidirect
25. C2×#330C22semidirect
26. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#17492
#91212
#87212
#83382