Finite Group Databasebeta

← #97035

Subgroups — #97035

Order 512.

832 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2640C211yesmaximal, characteristic
2#8699C212yesmaximal, complemented
3#7595C212yesmaximal, complemented
4#7592C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×#124C411yescharacteristic
7#2640—24no—
8#2640—24no—
9C2×#397—22no—
10C2×#393—22no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#376C2211yescharacteristic
16#1817C2214yes—
17#1280—22no—
18#1280—22no—
19#1272—24no—
20#1270—24no—
21C23×Q8—21no—
22C23×Q8C2×C411yescharacteristic
23C23×D4C2×C411yescharacteristic
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×#124—22no—
29C2×#124—22no—
30C2×#124—42no—
31C2×#124—42no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—22no—
38C22×C22⋊C4—22no—
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2311yescharacteristic
42C2×C22⋊C8—24no—
43#397—216no—
44#393—216no—
45#391—22no—
46#391—22no—
47#391—22no—
48#391D414yes—
49#390—22no—
50#390—22no—
51#390—22no—
52#390D414yes—
53#376—22no—
54#376—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95D412yes—
58C2×#95D412yes—
59#336—24no—
60#336—24no—
61C25—21no—
62C22×Q8—24no—
63C22×Q8—21no—
64C22×Q8—24no—
65C22×Q8—42no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—42no—
69C22×Q8—42no—
70C22×Q8—84no—
71C22×Q8C22⋊C412yes—
72C22×D4—21no—
73C22×D4—24no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—42no—
77C22×D4C22⋊C412yes—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C22⋊C412yes—
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D412yes—
86C22×C8—44no—
87C4⋊Q8—48no—
88C4⋊Q8—48no—
89#124—48no—
90#124—48no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4C22⋊C412yes—
101C2×C4⋊C4C22⋊C412yes—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—28no—
105C2×C22⋊C4—28no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—84no—
111C2×C22⋊C4—84no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42—42no—
120C2×C42C2×D412yes—
121C2×C42C2×D412yes—
122C22⋊C8—48no—
123#95—28no—
124#95—28no—
125#95—24no—
126#95—24no—
127C24—22no—
128C24—44no—
129C24—44no—
130C24—42no—
131C24—42no—
132C24—41no—
133C24—41no—
134C24C2×C22⋊C411yescharacteristic
135C2×Q8—24no—
136C2×Q8—42no—
137C2×Q8—48no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—84no—
143C2×Q8—84no—
144C2×Q8—88no—
145C2×Q8—88no—
146C2×Q8—88no—
147C2×D4—24no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—84no—
154C2×D4—84no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4#10012yes—
185C22×C4#10012yes—
186C22×C4C2×C22⋊C411yescharacteristic
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#12014yes—
189C22×C4#13114yes—
190C2×C8—84no—
191C2×C8—84no—
192C2×C8—84no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C22⋊C4—88no—
201C22⋊C4—88no—
202C42—24no—
203C42—24no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42—42no—
208C23—22no—
209C23—22no—
210C23—44no—
211C23—44no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—88no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23—82no—
220C23—88no—
221C23—82no—
222C23—82no—
223C23#37911yescharacteristic
224C23#40812yes—
225C23#41012yes—
226C23C2×#10011yescharacteristic
227C23C2×#10011yescharacteristic
228Q8—84no—
229Q8—84no—
230Q8—816no—
231D4—84no—
232D4—84no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—84no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—84no—
268C2×C4#40812yes—
269C2×C4#40812yes—
270C8—164no—
271C22—24no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—24no—
277C22—84no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—84no—
282C22—164no—
283C22—164no—
284C22#146912yes—
285C22#146912yes—
286C22#178011yescharacteristic
287C22#178211yescharacteristic
288C22#178311yescharacteristic
289C22#179512yes—
290C22#179612yes—
291C4—22no—
292C4—22no—
293C4—44no—
294C4—42no—
295C4—42no—
296C4—82no—
297C4—82no—
298C4—82no—
299C4—82no—
300C2—22no—
301C2—22no—
302C2—162no—
303C2#830511yescharacteristic
304C2#832512yes—
305C2#851912yes—
306C2#852211yescharacteristic
307C2#852411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8699C2semidirect
3. #7595C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
#86992
#75952
#75922