Finite Group Databasebeta

← #129334

Subgroups — #129334

Order 512.

850 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 122 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#34209C211yesmaximal, characteristic
2#11447C212yesmaximal
3#11428C212yesmaximal
4#8323C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C2×#519C411yescharacteristic
7C2×#519C2211yescharacteristic
8#2365—24no—
9#2344—24no—
10#2327—22no—
11#2310C411yescharacteristic
12#2310C2211yescharacteristic
13C4×C4⋊Q8—22no—
14#2285C2211yescharacteristic
15C4×#117C411yescharacteristic
16#1479—24no—
17#1458C2214yes—
18#1325—22no—
19#1324—22no—
20#1317—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2×C411yescharacteristic
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—21no—
27C2×C4⋊Q8—21no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8C814yescomplemented
30#519—44no—
31#519—44no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8D414yes—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117C2×C411yescharacteristic
39C2×#117C2×C411yescharacteristic
40C2×#117C2×C411yescharacteristic
41C2×#117C2311yescharacteristic
42C22×C4⋊C4—22no—
43C22×C42C2×C411yescharacteristic
44#431—24no—
45#391—22no—
46#391—22no—
47#391—22no—
48#391—22no—
49#389—28no—
50#387—28no—
51#387—28no—
52#384D412yes—
53#382D412yes—
54#381—22no—
55#381D414yes—
56#380—22no—
57C4×C4⋊C4—28no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C4×C22⋊C4—22no—
61#376—21no—
62C43—21no—
63Q8⋊C8—48no—
64#324—24no—
65#323—24no—
66C22×Q8—21no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×Q8—21no—
72C22×Q8—21no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8C22⋊C412yes—
78C22×Q8C2×C812yes—
79C22×Q8C2×C812yes—
80C22×Q8M1612yes—
81C22×Q8M1612yes—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C22⋊C412yes—
85C23×C4C22×C411yescharacteristic
86C4⋊Q8—24no—
87C4⋊Q8—24no—
88C4⋊Q8—28no—
89C4⋊Q8—28no—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4×Q8—28no—
93C4×Q8—44no—
94C4×Q8—48no—
95C4×Q8—48no—
96C4×Q8—44no—
97C4×Q8—44no—
98#117—24no—
99#117—42no—
100#117—48no—
101#117—48no—
102#117—42no—
103#117—42no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4C22⋊C412yes—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4C22⋊C412yes—
118C2×C22⋊C4C2×D412yes—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—24no—
128C2×C42C22⋊C411yescharacteristic
129C2×C42C22⋊C411yescharacteristic
130C2×C42C2×C812yes—
131C2×C42M1612yes—
132C2×C42C2×D412yes—
133C4⋊C8—44no—
134C22⋊C8—44no—
135C4×C8—44no—
136#95—24no—
137#95—28no—
138#95—24no—
139#95—28no—
140C24C2×C22⋊C411yescharacteristic
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8C22⋊C814yes—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C22⋊C814yes—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4#12014yes—
188C22×C4C22×C811yescharacteristic
189C22×C4C2×M1611yescharacteristic
190C22×C4C2×M1611yescharacteristic
191C22×C4C2×M1611yescharacteristic
192C2×C8—84no—
193C4⋊C4—216no—
194C4⋊C4—216no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—48no—
198C4⋊C4—42no—
199C4⋊C4—48no—
200C4⋊C4—48no—
201C4⋊C4—42no—
202C4⋊C4#9914yes—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C42—24no—
208C42—22no—
209C42—21no—
210C42—21no—
211C42—22no—
212C42—24no—
213C42—48no—
214C42—48no—
215C42—42no—
216C42—48no—
217C42—42no—
218C42C22⋊C812yes—
219C42C22⋊C812yes—
220C23—21no—
221C23—21no—
222C23—42no—
223C23#37911yescharacteristic
224C23C2×C22⋊C811yescharacteristic
225C23#40711yescharacteristic
226Q8—28no—
227Q8—44no—
228Q8—44no—
229Q8—44no—
230Q8—44no—
231Q8—44no—
232Q8—44no—
233Q8—44no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4C2×C22⋊C811yescharacteristic
265C2×C4C2×C22⋊C811yescharacteristic
266C2×C4#40711yescharacteristic
267C2×C4#40711yescharacteristic
268C2×C4C2×#9912yes—
269C2×C4#41312yes—
270C8—164no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22#177911yescharacteristic
279C22#178111yescharacteristic
280C22#178511yescharacteristic
281C4—24no—
282C4—21no—
283C4—21no—
284C4—42no—
285C4—44no—
286C4—44no—
287C4—44no—
288C4—44no—
289C4—82no—
290C4—82no—
291C4#145912yes—
292C4#146512yes—
293C2—21no—
294C2—21no—
295C2—41no—
296C2—41no—
297C2#830111yescharacteristic
298C2#1271712yes—
subgroupG/H
center, socleC22#1781
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
29. C2×C4⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
#342092
#114472
#114282
#83232