Finite Group Databasebeta

← #128338

Subgroups — #128338

Order 512.

568 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 150 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
3C4×#324C212yesmaximal
4#7189C212yesmaximal
5C2×#384C2211yescharacteristic
6C2×#382C411yescharacteristic
7C2×C4×C4⋊C4C411yescharacteristic
8C2×C4×C4⋊C4C2211yescharacteristic
9C2×#376C411yescharacteristic
10C4×C22⋊C8—22no—
11C2×#324C2214yes—
12#1436—22no—
13#1436—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16#1325C418yescomplemented
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×C22⋊C8—24no—
29C2×C4×C8—42no—
30#384—22no—
31#384C2×C414yes—
32#382—22no—
33#382C814yescomplemented
34C4×C4⋊C4—42no—
35C4×C4⋊C4—42no—
36#376—22no—
37#376C814yescomplemented
38C2×#95—21no—
39C2×#95—22no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95D412yes—
45#336—42no—
46#324—24no—
47#324C2×C418yes—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4—21no—
52C23×C4C22⋊C412yes—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C22⋊C411yescharacteristic
55C23×C4C22×C411yescharacteristic
56C23×C4C2×D412yes—
57C22×C8—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—28no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
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—82no—
70C2×C4⋊C4—82no—
71C2×C4⋊C4C4212yes—
72C2×C4⋊C4C2×C812yes—
73C2×C4⋊C4C2×C812yes—
74C2×C4⋊C4M1612yes—
75C2×C4⋊C4C22×C412yes—
76C2×C4⋊C4#4112yes—
77C2×C42—22no—
78C2×C42—22no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42—21no—
84C2×C42—22no—
85C2×C42—41no—
86C2×C42—41no—
87C2×C42—42no—
88C2×C42—82no—
89C2×C42—82no—
90C2×C42C4212yes—
91C2×C42C2×C812yes—
92C2×C42C2×C812yes—
93C2×C42C2×C812yes—
94C2×C42C2×C812yes—
95C2×C42M1612yes—
96C22⋊C8—28no—
97C22⋊C8—44no—
98C4×C8—84no—
99#95—24no—
100#95—24no—
101#95—28no—
102#95—42no—
103#95—42no—
104#95—82no—
105#95—82no—
106C24C2×C22⋊C411yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C22⋊C812yes—
150C22×C4C22⋊C814yes—
151C22×C4C22⋊C812yes—
152C22×C4#9912yes—
153C22×C4#10112yes—
154C22×C4C2×C4211yescharacteristic
155C22×C4#11711yescharacteristic
156C22×C4C4×D414yes—
157C22×C4C22×C811yescharacteristic
158C22×C4C22×C811yescharacteristic
159C22×C4C2×M1611yescharacteristic
160C22×C4C2×M1611yescharacteristic
161C2×C8—48no—
162C2×C8—84no—
163C2×C8—84no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—88no—
173C42—24no—
174C42—24no—
175C42—22no—
176C42—22no—
177C42—22no—
178C42—42no—
179C42—44no—
180C42—42no—
181C42—82no—
182C42—84no—
183C42C4×C814yescomplemented
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23C4×C22⋊C411yescharacteristic
189C23C2×C22⋊C811yescharacteristic
190C23C2×C22⋊C811yescharacteristic
191C23C2×#9911yescharacteristic
192C23#41011yescharacteristic
193C23C2×#10111yescharacteristic
194C23#41311yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
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—22no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4#32414yes—
251C2×C4C2×C4×C811yescharacteristic
252C2×C4C4×M1611yescharacteristic
253C2×C4#43111yescharacteristic
254C2×C4#43111yescharacteristic
255C8—88no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22C2×#32411yescharacteristic
269C22#145811yescharacteristic
270C22#145911yescharacteristic
271C22#146111yescharacteristic
272C22C4×C22⋊C811yescharacteristic
273C22C4×#9911yescharacteristic
274C22C4×#10111yescharacteristic
275C22#212711yescharacteristic
276C22#212811yescharacteristic
277C22#213311yescharacteristic
278C22#213511yescharacteristic
279C4—24no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—44no—
285C4—44no—
286C4—42no—
287C4—44no—
288C4—82no—
289C4—82no—
290C4—82no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—21no—
295C2C4×#32411yescharacteristic
296C2#1144811yescharacteristic
297C2#1145011yescharacteristic
298C2#1148011yescharacteristic
299C2#1148111yescharacteristic
300C2#1262811yescharacteristic
301C2#1263111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC2×C4C2×C4×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×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
16. #1325C4semidirect
33. #382C8semidirect
37. #376C8semidirect
183. 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
C4×#3242
#71892