Finite Group Databasebeta

← #134017

Subgroups — #134017

Order 512.

1078 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1078 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×#2658C211yesmaximal, characteristic
2#12700C212yesmaximal, complemented
3#12699C212yesmaximal
4#8551C212yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C4⋊Q8C2211yescharacteristic
7C22×C4⋊Q8C2211yescharacteristic
8C22×#124C411yescharacteristic
9C22×#124C411yescharacteristic
10C22×#124C2211yescharacteristic
11#2658—24no—
12#2658—24no—
13C2×#399—24no—
14C2×#376C411yescharacteristic
15C2×#356—24no—
16C2×#355—24no—
17#1784C2214yes—
18#1463—24no—
19C23×Q8C2×C411yescharacteristic
20C23×Q8C2×C411yescharacteristic
21C23×Q8C2311yescharacteristic
22C23×D4C2×C411yescharacteristic
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—44no—
29C2×C4⋊Q8—44no—
30C2×C4⋊Q8—44no—
31C2×C4⋊Q8D414yes—
32C2×#124—22no—
33C2×#124—22no—
34C2×#124—22no—
35C2×#124—22no—
36C2×#124—22no—
37C2×#124—44no—
38C2×#124—44no—
39C2×#124—44no—
40C2×#124D414yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D414yes—
44C22×C22⋊C4—22no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4—22no—
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C2×#101—24no—
51C2×#101—24no—
52#407—24no—
53#399—232no—
54#376—22no—
55#376—22no—
56C2×#95—22no—
57#356—48no—
58#355—48no—
59#324—48no—
60C25—21no—
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—21no—
64C22×Q8—42no—
65C22×Q8—42no—
66C22×Q8—42no—
67C22×Q8—42no—
68C22×Q8—42no—
69C22×Q8—42no—
70C22×Q8—42no—
71C22×Q8—42no—
72C22×Q8—42no—
73C22×Q8C22⋊C412yes—
74C22×Q8C2×D412yes—
75C22×Q8C2×D412yes—
76C22×D4—21no—
77C22×D4—42no—
78C22×D4—42no—
79C22×D4—42no—
80C22×D4C22⋊C412yes—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C22⋊C412yes—
85C23×C4C22⋊C412yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D412yes—
88C2×M16—44no—
89C4⋊Q8—28no—
90C4⋊Q8—44no—
91C4⋊Q8—48no—
92C4⋊Q8—48no—
93C4⋊Q8—48no—
94C4⋊Q8—48no—
95C4⋊Q8—44no—
96C4⋊Q8—44no—
97#124—28no—
98#124—44no—
99#124—48no—
100#124—48no—
101#124—48no—
102#124—48no—
103#124—44no—
104#124—44no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—28no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—44no—
110C2×C4⋊C4—48no—
111C2×C4⋊C4—48no—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—28no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—48no—
116C2×C22⋊C4—48no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—44no—
119C2×C42—24no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C412yes—
128#101—48no—
129#101—48no—
130C22⋊C8—48no—
131#95—28no—
132C24—22no—
133C24—42no—
134C24—42no—
135C24—48no—
136C24—42no—
137C24C2×C22⋊C411yescharacteristic
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—88no—
157C2×Q8—88no—
158C2×Q8—88no—
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—88no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4#9914yes—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C2×C22⋊C411yescharacteristic
192C22×C4#12014yes—
193M16—84no—
194M16—84no—
195C2×C8—84no—
196C4⋊C4—416no—
197C4⋊C4—416no—
198C4⋊C4—416no—
199C22⋊C4—416no—
200C22⋊C4—416no—
201C22⋊C4—416no—
202C42—24no—
203C42—24no—
204C42—24no—
205C42—24no—
206C42—22no—
207C42—22no—
208C42#9914yes—
209C42#9914yes—
210C23—24no—
211C23—21no—
212C23—21no—
213C23—48no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—84no—
218C23—88no—
219C23—88no—
220C23—84no—
221C23—84no—
222C23#37911yescharacteristic
223C23C2×#9912yes—
224Q8—84no—
225Q8—84no—
226Q8—84no—
227Q8—84no—
228Q8—84no—
229Q8—84no—
230D4—84no—
231D4—84no—
232C2×C4—28no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—28no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—48no—
245C2×C4—48no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4C2×#9912yes—
260C2×C4C2×#9912yes—
261C8—164no—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—84no—
268C22—84no—
269C22—82no—
270C22—84no—
271C22—168no—
272C22#178111yescharacteristic
273C22#178111yescharacteristic
274C22#178111yescharacteristic
275C22#213314yes—
276C4—22no—
277C4—22no—
278C4—44no—
279C4—44no—
280C4—44no—
281C4—44no—
282C4—82no—
283C4—82no—
284C2—21no—
285C2—21no—
286C2—24no—
287C2—162no—
288C2#854611yescharacteristic
289C2#1270212yes—
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12700C2semidirect
4. #8551C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26582
#127002
#126992
#85512