Finite Group Databasebeta

← #134311

Subgroups — #134311

Order 512.

538 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 118 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#13779C211yesmaximal, characteristic
2C2×#1325C212yesmaximal
3#8560C212yesmaximal
4#7189C212yesmaximal
5C2×#384C2211yescharacteristic
6C2×#382C411yescharacteristic
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×#376C411yescharacteristic
10#1797—22no—
11C2×#324C2214yes—
12#1436—22no—
13#1436—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16#1325—24no—
17#1280—22no—
18C22×C4⋊C4—22no—
19C22×C4⋊C4C2×C411yescharacteristic
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4C2×C411yescharacteristic
22C22×C4⋊C4C2311yescharacteristic
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C22×C42C2×C411yescharacteristic
26C22×C42C2×C411yescharacteristic
27C22×C42D412yes—
28C2×C4⋊C8—42no—
29C2×C22⋊C8—24no—
30#384—22no—
31#384D412yes—
32#384Q812yes—
33#382—22no—
34#382C814yescomplemented
35C4×C4⋊C4—42no—
36C4×C4⋊C4—42no—
37#376—22no—
38#376—22no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95D412yes—
46#336—42no—
47#324—24no—
48#324—24no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4C22⋊C412yes—
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22×C411yescharacteristic
57C23×C4C2×D412yes—
58C22×C8—44no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—82no—
72C2×C4⋊C4—82no—
73C2×C4⋊C4C4⋊C412yes—
74C2×C4⋊C4C2×C812yes—
75C2×C4⋊C4C2×C812yes—
76C2×C4⋊C4M1612yes—
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C4⋊C4C2×Q811yescharacteristic
79C2×C4⋊C4#4112yes—
80C2×C42—22no—
81C2×C42—22no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—22no—
88C2×C42—21no—
89C2×C42—22no—
90C2×C42—41no—
91C2×C42—41no—
92C2×C42—42no—
93C2×C42—82no—
94C2×C42—82no—
95C2×C42C4⋊C412yes—
96C2×C42C2×C812yes—
97C2×C42M1612yes—
98C2×C42M1612yes—
99C4⋊C8—84no—
100C22⋊C8—44no—
101C22⋊C8—44no—
102#95—24no—
103#95—24no—
104#95—28no—
105#95—42no—
106#95—42no—
107#95—82no—
108#95—82no—
109C24C2×C22⋊C411yescharacteristic
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4C22⋊C814yes—
155C22×C4#9912yes—
156C22×C4#10112yes—
157C22×C4C2×C4⋊C411yescharacteristic
158C22×C4#11711yescharacteristic
159C22×C4#12112yes—
160C22×C4C22⋊Q812yes—
161C22×C4C22×C811yescharacteristic
162C22×C4C2×M1611yescharacteristic
163C22×C4C2×M1611yescharacteristic
164C22×C4C2×M1611yescharacteristic
165C2×C8—84no—
166C2×C8—84no—
167C2×C8—84no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—84no—
178C4⋊C4—84no—
179C42—24no—
180C42—24no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—42no—
185C42—44no—
186C42—42no—
187C42—82no—
188C42—84no—
189C42C4⋊C814yescomplemented
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23#38011yescharacteristic
195C23C2×C22⋊C811yescharacteristic
196C23#40711yescharacteristic
197C23C2×#9911yescharacteristic
198C23#41011yescharacteristic
199C23C2×#10111yescharacteristic
200C23#41311yescharacteristic
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4C2×C4⋊C811yescharacteristic
260C2×C4#42311yescharacteristic
261C2×C4#43111yescharacteristic
262C2×C4#43211yescharacteristic
263C8—164no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22#145911yescharacteristic
277C22#146111yescharacteristic
278C22#146311yescharacteristic
279C22#146411yescharacteristic
280C22#179711yescharacteristic
281C22#179911yescharacteristic
282C22#180111yescharacteristic
283C22#212711yescharacteristic
284C22#212811yescharacteristic
285C22#213311yescharacteristic
286C22#213511yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—44no—
292C4—44no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—82no—
298C4—82no—
299C4—82no—
300C2—21no—
301C2—21no—
302C2—21no—
303C2—21no—
304C2#856011yescharacteristic
305C2#1148011yescharacteristic
306C2#1148111yescharacteristic
307C2#1151411yescharacteristic
308C2#1151611yescharacteristic
309C2#1279911yescharacteristic
310C2#1280411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
34. #382C8semidirect
189. C42C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#137792
C2×#13252
#85602
#71892