Finite Group Databasebeta

← #116916

Subgroups — #116916

Order 512.

778 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 134 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#21788C211yesmaximal, characteristic
2#20738C212yesmaximal
3#17818C211yesmaximal, characteristic, complemented
4#7981C211yesmaximal, characteristic, complemented
5#7952C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2211yescharacteristic
9C2×#431C2211yescharacteristic
10#2285C2212yes—
11#1955C414yescomplemented
12#1954—22no—
13#1824—22no—
14#1802—24no—
15#1786—22no—
16#1779—21no—
17#1779—21no—
18#1762—24no—
19#1573—24no—
20#1568—22no—
21#1289—21no—
22#1289—21no—
23#1285—24no—
24C23×D4D411yescharacteristic
25C24×C4—21no—
26C2×C4×D4—21no—
27C2×C4×D4—22no—
28C2×C4×D4—22no—
29C2×C4×D4—21no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4C2×C412yes—
33C2×C4×D4Q812yes—
34C2×#117—22no—
35C22×C4⋊C4D411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C42C2311yescharacteristic
38#432—22no—
39#432—22no—
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#431—22no—
45#431—22no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8C2×C412yes—
48C2×C4⋊C8D414yes—
49C2×#102—28no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8D411yescharacteristic
55C2×C22⋊C8D411yescharacteristic
56C2×#97—22no—
57C2×#97C2×C412yes—
58C2×C4×C8D412yes—
59#384—22no—
60#382—22no—
61#380—22no—
62#380—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4D414yes—
65#340—28no—
66#335—24no—
67#334—24no—
68C25—21no—
69C22×D4—22no—
70C22×D4—21no—
71C22×D4—22no—
72C22×D4—44no—
73C22×D4C4⋊C412yes—
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—84no—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4#4111yescharacteristic
83C22×C8—22no—
84C22×C8—24no—
85C22×C8C22⋊C412yes—
86C22×C8C22⋊C412yes—
87C4×D4—24no—
88C4×D4—24no—
89C4×D4—24no—
90C4×D4—24no—
91C4×D4—41no—
92C4×D4—42no—
93C4×D4—41no—
94C4×D4—41no—
95C4×D4—42no—
96C4×D4—41no—
97#117—82no—
98#117—82no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4C4⋊C412yes—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C42—24no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D412yes—
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42#4112yes—
118C2×C42#4112yes—
119C2×C42#4111yescharacteristic
120C4⋊C8—24no—
121C4⋊C8—24no—
122C4⋊C8—24no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125#102—416no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—84no—
132C22⋊C8—84no—
133#97—24no—
134#97—42no—
135#97—42no—
136C4×C8—22no—
137C4×C8—22no—
138#95—24no—
139#95—42no—
140C24—22no—
141C24—44no—
142C24—42no—
143C24—42no—
144C24—41no—
145C24—41no—
146C24—82no—
147C24#12111yescharacteristic
148C2×D4—28no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—44no—
152C2×D4—42no—
153C2×D4—44no—
154C2×D4—48no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—88no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—88no—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C2×C4⋊C411yescharacteristic
177C22×C4#11711yescharacteristic
178C22×C4#12012yes—
179C22×C4#12112yes—
180C22×C4#12111yescharacteristic
181C22×C4#12112yes—
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4#12312yes—
185C22×C4#12312yes—
186C22×C4#12412yes—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—44no—
192C2×C8—48no—
193C2×C8—42no—
194C2×C8—44no—
195C4⋊C4—24no—
196C4⋊C4—48no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—84no—
200C4⋊C4—84no—
201C22⋊C4—44no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C42—84no—
207C42C4×D412yes—
208C42C4×D412yes—
209C42#12011yescharacteristic
210C42#12112yes—
211C42C22⋊Q812yes—
212C42C22⋊Q812yes—
213C42#12311yescharacteristic
214C42#12411yescharacteristic
215C42#12612yes—
216C42#12711yescharacteristic
217C23—24no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—48no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—84no—
228C23—88no—
229C23—84no—
230C23—84no—
231C23—82no—
232C23—82no—
233C23#38011yescharacteristic
234C23#39412yes—
235C23#47112yes—
236C23#47211yescharacteristic
237C23#47311yescharacteristic
238D4—28no—
239D4—44no—
240D4—44no—
241D4—44no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4—88no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4#38512yes—
269C2×C4#38611yescharacteristic
270C2×C4#38812yes—
271C2×C4#39011yescharacteristic
272C2×C4#39211yescharacteristic
273C2×C4#39311yescharacteristic
274C2×C4#39612yes—
275C2×C4#39711yescharacteristic
276C2×C4#39911yescharacteristic
277C8—42no—
278C8—42no—
279C8—82no—
280C8—84no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—21no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—44no—
289C22—44no—
290C22—44no—
291C22—82no—
292C22—82no—
293C22—88no—
294C22—88no—
295C22—82no—
296C22—82no—
297C22#143911yescharacteristic
298C22#181511yescharacteristic
299C22#181611yescharacteristic
300C22#205112yes—
301C22#205212yes—
302C4—21no—
303C4—22no—
304C4—22no—
305C4—21no—
306C4—22no—
307C4—21no—
308C4—21no—
309C4—82no—
310C4—164no—
311C4#191012yes—
312C4#201012yes—
313C2—22no—
314C2—21no—
315C2—21no—
316C2—22no—
317C2—84no—
318C2#1107311yescharacteristic
319C2#1110011yescharacteristic
320C2#1110511yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17818C2semidirect
4. #7981C2semidirect
5. #7952C2semidirect
11. #1955C4semidirect

Maximal subgroups

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

subgroupindex
#217882
#207382
#178182
#79812
#79522