Finite Group Databasebeta

← #134273

Subgroups — #134273

Order 512.

590 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 130 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×#1753C211yesmaximal, characteristic
2#12783C212yesmaximal, complemented
3C2×#1314C212yesmaximal
4#7183C212yesmaximal, complemented
5C23×M16C411yescharacteristic
6C22×C22⋊C8C2211yescharacteristic
7C22×#97C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10C2×C4⋊C16—22no—
11C2×C22⋊C16C2214yes—
12#1753—22no—
13#1753—22no—
14C2×#336C411yescharacteristic
15C2×#336C2211yescharacteristic
16#1374—22no—
17#1314—24no—
18C24×C4C2×C411yescharacteristic
19C22×M16—22no—
20C22×M16—21no—
21C22×M16—21no—
22C22×M16—22no—
23C22×M16—42no—
24C22×M16—42no—
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8C2311yescharacteristic
29C22×C22⋊C4C812yes—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—24no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8D412yes—
36C2×C22⋊C8Q812yes—
37C2×#97—21no—
38C2×#97—21no—
39C2×#97—42no—
40C2×C4×C8—42no—
41C2×C4×C8D412yes—
42C2×C4×C8D412yes—
43C4×C22⋊C4—22no—
44C4×C22⋊C4—21no—
45C4×C22⋊C4—21no—
46C2×#95C812yes—
47C4⋊C16—44no—
48C22⋊C16—24no—
49C22⋊C16—24no—
50#336—22no—
51#336—22no—
52C25C2×C811yescharacteristic
53C23×C4—21no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4C4⋊C412yes—
62C23×C4C2×C812yes—
63C23×C4C2×C812yes—
64C23×C4C2×C811yescharacteristic
65C23×C4M1612yes—
66C23×C4C22×C411yescharacteristic
67C2×M16—24no—
68C2×M16—24no—
69C2×M16—48no—
70C2×M16—48no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8C4⋊C412yes—
92C22×C8M1612yes—
93C22×C8C2×D412yes—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×Q811yescharacteristic
96C22×C8#4112yes—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—44no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42C22⋊C412yes—
107C2×C42M1611yescharacteristic
108C2×C42M1611yescharacteristic
109C2×C16—44no—
110C2×C16—44no—
111C2×C16—44no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114#97—42no—
115#97—42no—
116C4×C8—24no—
117C4×C8—42no—
118#95—22no—
119#95—22no—
120C24—21no—
121C24—21no—
122C24—24no—
123C24—42no—
124C24—41no—
125C24—41no—
126C24C4⋊C812yes—
127C24C22×C811yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C22⋊C812yes—
162C22×C4C22⋊C812yes—
163C22×C4C4⋊C812yes—
164C22×C4C2×C22⋊C411yescharacteristic
165C22×C4C2×C4⋊C411yescharacteristic
166C22×C4#11711yescharacteristic
167C22×C4C2×M1611yescharacteristic
168C22×C4C2×M1611yescharacteristic
169C22×C4C2×M1611yescharacteristic
170M16—48no—
171M16—48no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—21no—
178C2×C8—22no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8#9912yes—
194C2×C8#10012yes—
195C2×C8#12112yes—
196C2×C8C22⋊Q812yes—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C42—22no—
200C42—22no—
201C42—41no—
202C42—41no—
203C42—42no—
204C42—41no—
205C42—41no—
206C16—84no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—24no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—41no—
220C23—41no—
221C23—84no—
222C23—84no—
223C23C2×C22⋊C811yescharacteristic
224C23C2×C4⋊C811yescharacteristic
225C23#43111yescharacteristic
226C23#50414yes—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—48no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—84no—
258C2×C4#34914yes—
259C2×C4#38011yescharacteristic
260C2×C4#40711yescharacteristic
261C2×C4C2×#9911yescharacteristic
262C2×C4#41011yescharacteristic
263C2×C4C2×#10011yescharacteristic
264C2×C4#41311yescharacteristic
265C2×C4#42311yescharacteristic
266C2×C4#43211yescharacteristic
267C8—22no—
268C8—22no—
269C8—22no—
270C8—22no—
271C8—42no—
272C8—42no—
273C22—21no—
274C22—21no—
275C22—24no—
276C22—24no—
277C22—21no—
278C22—21no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—84no—
283C22—84no—
284C22—82no—
285C22—82no—
286C22#145911yescharacteristic
287C22#146011yescharacteristic
288C22#179711yescharacteristic
289C22#211312yes—
290C22C2×#34912yes—
291C22#215112yes—
292C22#216412yes—
293C4—21no—
294C4—21no—
295C4—24no—
296C4—42no—
297C4—42no—
298C4—82no—
299C4#146211yescharacteristic
300C4#146411yescharacteristic
301C4#179911yescharacteristic
302C4#180011yescharacteristic
303C2—22no—
304C2—22no—
305C2—82no—
306C2#855911yescharacteristic
307C2#1146712yes—
308C2#1148912yes—
309C2#1278411yescharacteristic
310C2#1278511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C22×C8
Frattini, cosocleC23×C8C23
socleC23C2×C22⋊C8
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
4. #7183C2semidirect

Maximal subgroups

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

subgroupindex
C2×#17532
#127832
C2×#13142
#71832