Finite Group Databasebeta

← #134144

Subgroups — #134144

Order 512.

854 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2861C211yesmaximal, characteristic
2#11506C212yesmaximal, complemented
3#11505C212yesmaximal
4#8554C212yesmaximal, complemented
5C22×#125C2211yescharacteristic
6C22×#124C411yescharacteristic
7#2861—24no—
8#2861—24no—
9C2×#391C411yescharacteristic
10C2×#391C2211yescharacteristic
11C2×#390C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#388—22no—
14C2×#388—22no—
15C2×C43C411yescharacteristic
16#1821—22no—
17#1463—22no—
18#1463C2214yes—
19#1319—24no—
20#1318—24no—
21C23×Q8C2×C411yescharacteristic
22C23×D4C2×C411yescharacteristic
23C2×#125—22no—
24C2×#125—22no—
25C2×#125—42no—
26C2×#125—42no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124—42no—
30C2×#124—42no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D412yes—
33C22×C4⋊C4D412yes—
34C22×C4⋊C4C2311yescharacteristic
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—22no—
38C22×C22⋊C4—22no—
39C22×C42—21no—
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44#407—24no—
45#391—22no—
46#391—22no—
47#391—22no—
48#391D414yes—
49#390—22no—
50#390—22no—
51#390—22no—
52#390D414yes—
53#388—216no—
54#388—216no—
55C2×#95—22no—
56C2×#95—22no—
57C43—22no—
58C43—22no—
59#343—44no—
60#324—24no—
61#324—24no—
62#324—44no—
63C25—21no—
64C22×Q8—21no—
65C22×Q8—42no—
66C22×Q8—42no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8C22⋊C412yes—
70C22×D4—21no—
71C22×D4—42no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4—42no—
75C22×D4C22⋊C412yes—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22⋊C412yes—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D412yes—
84C2×M16—44no—
85#125—48no—
86#125—48no—
87#124—48no—
88#124—48no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4C2×D412yes—
102C2×C4⋊C4C2×D412yes—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—28no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—48no—
108C2×C22⋊C4—48no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—44no—
122C2×C42—44no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C22⋊C8—44no—
128C22⋊C8—44no—
129#95—28no—
130#95—28no—
131C24—22no—
132C24—44no—
133C24—44no—
134C24—42no—
135C24—42no—
136C24—41no—
137C24—41no—
138C24C2×C22⋊C411yescharacteristic
139C2×Q8—24no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—84no—
146C2×Q8—84no—
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—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4#9912yes—
186C22×C4#9912yes—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#12014yes—
190M16—88no—
191C2×C8—84no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—48no—
195C4⋊C4—48no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C22⋊C4—416no—
199C22⋊C4—416no—
200C22⋊C4—48no—
201C22⋊C4—48no—
202C42—24no—
203C42—24no—
204C42—42no—
205C42—42no—
206C42—48no—
207C42—48no—
208C42—42no—
209C42—42no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23—44no—
215C23—44no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—82no—
220C23—82no—
221C23—88no—
222C23—88no—
223C23—82no—
224C23—82no—
225C23—82no—
226C23—82no—
227C23#37911yescharacteristic
228C23C2×#9911yescharacteristic
229C23C2×#9911yescharacteristic
230Q8—84no—
231Q8—84no—
232D4—84no—
233D4—84no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—48no—
250C2×C4—48no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4#41312yes—
270C2×C4#41312yes—
271C8—164no—
272C22—24no—
273C22—24no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—84no—
281C22—84no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—164no—
286C22—164no—
287C22#178111yescharacteristic
288C22#178511yescharacteristic
289C22#178511yescharacteristic
290C22#212612yes—
291C22#213212yes—
292C4—22no—
293C4—22no—
294C4—44no—
295C4—44no—
296C4—44no—
297C4—42no—
298C4—42no—
299C4—82no—
300C4—82no—
301C2—21no—
302C2—21no—
303C2—22no—
304C2—22no—
305C2—162no—
306C2#855211yescharacteristic
307C2#1269511yescharacteristic
308C2#1270111yescharacteristic
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×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11506C2semidirect
4. #8554C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#115062
#115052
#85542