Finite Group Databasebeta

← #134145

Subgroups — #134145

Order 512.

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

normal subgroups only

This group has 846 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×#2640C211yesmaximal, characteristic
2#11506C212yesmaximal, complemented
3#11505C212yesmaximal
4#8551C212yesmaximal, complemented
5C22×C4⋊Q8C2211yescharacteristic
6C22×#124C411yescharacteristic
7#2640—24no—
8#2640—24no—
9C2×#397—22no—
10C2×#393—22no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#390C411yescharacteristic
14C2×#390C2211yescharacteristic
15C2×#376C411yescharacteristic
16#1784—22no—
17#1463—22no—
18#1463C2214yes—
19#1319—24no—
20#1318—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8D412yes—
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⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D412yes—
34C22×C4⋊C4C2311yescharacteristic
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—22no—
38C22×C22⋊C4—22no—
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C2×#101—44no—
43#407—24no—
44#397—216no—
45#393—216no—
46#391—22no—
47#391—22no—
48#391—22no—
49#391D414yes—
50#390—22no—
51#390—22no—
52#390—22no—
53#390D414yes—
54#376—22no—
55#376—22no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—22no—
60#324—24no—
61#324—24no—
62#324—44no—
63C25—21no—
64C22×Q8—24no—
65C22×Q8—21no—
66C22×Q8—24no—
67C22×Q8—42no—
68C22×Q8—42no—
69C22×Q8—42no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—84no—
73C22×Q8C22⋊C412yes—
74C22×D4—21no—
75C22×D4—42no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4—42no—
79C22×D4C22⋊C412yes—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D412yes—
88C2×M16—44no—
89C4⋊Q8—48no—
90C4⋊Q8—48no—
91#124—48no—
92#124—48no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4C2×D412yes—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—28no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—84no—
113C2×C22⋊C4—84no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—21no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42C22⋊C412yes—
123C2×C42C22⋊C412yes—
124#101—88no—
125C22⋊C8—44no—
126C22⋊C8—44no—
127#95—24no—
128#95—24no—
129#95—28no—
130#95—28no—
131C24—22no—
132C24—44no—
133C24—42no—
134C24—42no—
135C24—44no—
136C24—41no—
137C24—41no—
138C24C2×C22⋊C411yescharacteristic
139C2×Q8—24no—
140C2×Q8—48no—
141C2×Q8—48no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—84no—
148C2×Q8—84no—
149C2×Q8—84no—
150C2×Q8—88no—
151C2×Q8—88no—
152C2×D4—24no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—84no—
159C2×D4—84no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4#9912yes—
191C22×C4#9912yes—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4#12014yes—
195M16—88no—
196C2×C8—84no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—48no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C22⋊C4—48no—
203C22⋊C4—48no—
204C22⋊C4—88no—
205C22⋊C4—88no—
206C42—24no—
207C42—24no—
208C42—42no—
209C42—42no—
210C42—42no—
211C42—42no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—44no—
217C23—44no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—82no—
222C23—82no—
223C23—88no—
224C23—88no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—82no—
229C23#37911yescharacteristic
230C23C2×#9911yescharacteristic
231C23C2×#9911yescharacteristic
232Q8—84no—
233Q8—84no—
234Q8—816no—
235D4—84no—
236D4—84no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—84no—
268C2×C4—84no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4#41312yes—
273C2×C4#41312yes—
274C8—164no—
275C22—24no—
276C22—24no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—84no—
284C22—84no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22—164no—
289C22—164no—
290C22#178111yescharacteristic
291C22#178511yescharacteristic
292C22#178511yescharacteristic
293C22#212712yes—
294C22#213312yes—
295C4—22no—
296C4—22no—
297C4—44no—
298C4—42no—
299C4—42no—
300C4—82no—
301C4—82no—
302C4—82no—
303C4—82no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2—22no—
308C2—162no—
309C2#855211yescharacteristic
310C2#1269611yescharacteristic
311C2#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. #11506C2semidirect
4. #8551C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
#115062
#115052
#85512