Finite Group Databasebeta

← #134107

Subgroups — #134107

Order 512.

1146 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2542C211yesmaximal, characteristic
2#11512C212yesmaximal, complemented
3#11505C212yesmaximal
4#8551C212yesmaximal, complemented
5C22×C4⋊Q8C2211yescharacteristic
6C22×C22⋊Q8—22no—
7C22×C4×D4C411yescharacteristic
8#2542—24no—
9#2542—24no—
10C2×#399—22no—
11C2×#390C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#384C411yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C411yescharacteristic
16#1784—22no—
17#1463—22no—
18#1463C2214yes—
19#1325—24no—
20#1318—24no—
21C23×Q8D412yes—
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×C22⋊Q8—216no—
29C2×C22⋊Q8—416no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4—42no—
33C2×C4×D4—42no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C2×#101—44no—
48#407—24no—
49#399—216no—
50#390—22no—
51#390—22no—
52#390—22no—
53#390D414yes—
54#384—22no—
55#384—22no—
56#384—22no—
57#384D414yes—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95—21no—
61#324—24no—
62#324—24no—
63#324—44no—
64C25—21no—
65C22×Q8—24no—
66C22×Q8—24no—
67C22×Q8—42no—
68C22×Q8—48no—
69C22×D4—24no—
70C22×D4—21no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—42no—
74C22×D4C22⋊C412yes—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—24no—
78C23×C4—21no—
79C23×C4—42no—
80C23×C4—48no—
81C23×C4—42no—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D412yes—
87C2×M16—44no—
88C4⋊Q8—48no—
89C4⋊Q8—48no—
90C22⋊Q8—464no—
91C4×D4—48no—
92C4×D4—48no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—48no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4C22⋊C412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—48no—
119C2×C22⋊C4—48no—
120C2×C22⋊C4—42no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42C22⋊C412yes—
130C2×C42C22⋊C412yes—
131#101—88no—
132C22⋊C8—44no—
133C22⋊C8—44no—
134#95—24no—
135C24—24no—
136C24—22no—
137C24—44no—
138C24—42no—
139C24—44no—
140C24—41no—
141C24—41no—
142C24C2×C22⋊C411yescharacteristic
143C2×Q8—48no—
144C2×Q8—416no—
145C2×Q8—48no—
146C2×Q8—84no—
147C2×Q8—88no—
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—48no—
154C2×D4—48no—
155C2×D4—42no—
156C2×D4#10114yescomplemented
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—48no—
178C22×C4—416no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—88no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4#9912yes—
192C22×C4#9912yes—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4#12014yes—
196M16—88no—
197C2×C8—84no—
198C4⋊C4—216no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—48no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—416no—
205C4⋊C4—416no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C22⋊C4—48no—
209C22⋊C4—416no—
210C22⋊C4—416no—
211C42—24no—
212C42—24no—
213C42—24no—
214C42—42no—
215C42—42no—
216C42—42no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23—24no—
222C23—44no—
223C23—44no—
224C23—44no—
225C23—416no—
226C23—44no—
227C23—42no—
228C23—42no—
229C23—82no—
230C23—88no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23#37911yescharacteristic
235C23C2×#9911yescharacteristic
236C23C2×#9911yescharacteristic
237Q8—816no—
238D4—44no—
239D4—44no—
240D4—48no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—816no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4#35314yes—
277C2×C4C2×#10112yes—
278C2×C4#41312yes—
279C8—164no—
280C22—24no—
281C22—24no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—48no—
289C22—44no—
290C22—84no—
291C22—88no—
292C22—88no—
293C22—82no—
294C22—82no—
295C22#178111yescharacteristic
296C22#178411yescharacteristic
297C22#178511yescharacteristic
298C22C2×#35312yes—
299C22#213312yes—
300C4—22no—
301C4—22no—
302C4—24no—
303C4—44no—
304C4—44no—
305C4—44no—
306C4—42no—
307C4—82no—
308C4—82no—
309C2—21no—
310C2—21no—
311C2—22no—
312C2—22no—
313C2—84no—
314C2#855011yescharacteristic
315C2#1269311yescharacteristic
316C2#1270211yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11512C2semidirect
4. #8551C2semidirect
156. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#115122
#115052
#85512