Finite Group Databasebeta

← #136809

Subgroups — #136809

Order 512.

550 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 92 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#33487C211yesmaximal, characteristic, complemented
2#21442C211yesmaximal, characteristic
3#12683C212yesmaximal, complemented
4#11465C211yesmaximal, characteristic, complemented
5#7202C212yesmaximal, complemented
6C22×#131C2211yescharacteristic
7C2×C8×D4—21no—
8C2×C8×D4C2211yescharacteristic
9#2388C411yescharacteristic
10#2168—24no—
11#2163—21no—
12C2×#350—24no—
13#2112C2212yes—
14C2×C22⋊C16—21no—
15C2×C22⋊C16C2212yes—
16#1870—21no—
17#1870C411yescharacteristic
18#1851—21no—
19#1851C411yescharacteristic
20#1780C411yescharacteristic
21C8×C22⋊C4—21no—
22C8×C22⋊C4C2211yescharacteristic
23#1322—21no—
24#1314—21no—
25C22×#41C2×C411yescharacteristic
26C2×#131—22no—
27C2×#131—22no—
28C2×#131—44no—
29C22×M16—21no—
30C22×M16C2×C411yescharacteristic
31C23×C8—21no—
32C23×C8D411yescharacteristic
33C23×C8C2311yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2×C411yescharacteristic
36C2×M32D414yescomplemented
37C22×C16—22no—
38C22×C16—22no—
39C8×D4—22no—
40C8×D4—22no—
41C8×D4—44no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8D411yescharacteristic
52#395C811yescharacteristic
53#394—21no—
54#394C811yescharacteristic
55#393C811yescharacteristic
56#392—21no—
57#392C811yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C411yescharacteristic
60C4⋊C16—22no—
61#350—48no—
62C22⋊C16—22no—
63C22⋊C16—22no—
64C22⋊C16—22no—
65C22⋊C16—22no—
66C22⋊C16—22no—
67#346—22no—
68#336—21no—
69#336—21no—
70#336—21no—
71C2×#41—22no—
72C2×#41—22no—
73C2×#41—42no—
74C2×#41—42no—
75C22×Q8C2×C811yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4C2×C811yescharacteristic
79C22×D4M1611yescharacteristic
80C23×C4—21no—
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83#131—28no—
84#131—44no—
85#131—44no—
86#131—44no—
87C2×M16—22no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—24no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22⋊C411yescharacteristic
104C22×C8C2×D412yes—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8#4112yes—
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—42no—
111C4×D4—42no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4C2×C811yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4C2×C811yescharacteristic
120C2×C22⋊C4M1611yescharacteristic
121C2×C42—21no—
122C2×C42C22⋊C411yescharacteristic
123M32—44no—
124C2×C16—24no—
125C2×C16—44no—
126C2×C16—42no—
127C2×C16—82no—
128C4⋊C8—22no—
129C4⋊C8—42no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—42no—
139#95—21no—
140#95—21no—
141#95C2×C811yescharacteristic
142#95C2×C811yescharacteristic
143C24—21no—
144C24C22⋊C811yescharacteristic
145C24C2×M1611yescharacteristic
146#41—24no—
147#41—24no—
148#41—42no—
149#41—42no—
150#41—44no—
151#41—44no—
152C2×Q8—22no—
153C2×Q8—42no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—41no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4C22⋊C811yescharacteristic
181C22×C4C22⋊C811yescharacteristic
182C22×C4C22⋊C811yescharacteristic
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D412yes—
185C22×C4C22×C811yescharacteristic
186C22×C4#13112yes—
187M16—24no—
188M16—48no—
189M16—42no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—44no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—44no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—81no—
208C2×C8#12011yescharacteristic
209C2×C8#12112yes—
210C2×C8#12311yescharacteristic
211C4⋊C4—22no—
212C4⋊C4—42no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C42—21no—
221C42—21no—
222C42—41no—
223C42—41no—
224C16—44no—
225C16—82no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23#35012yescomplemented
242C23C2×C22⋊C811yescharacteristic
243C23#43512yes—
244Q8—24no—
245Q8—42no—
246D4—24no—
247D4—42no—
248D4—44no—
249D4—44no—
250D4—44no—
251D4—48no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4#38611yescharacteristic
282C2×C4#40811yescharacteristic
283C2×C4C8×D412yes—
284C8—22no—
285C8—21no—
286C8—42no—
287C8—41no—
288C8—41no—
289C8—81no—
290C8B(4,𝔽2)12yes—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22#186911yescharacteristic
311C22C2×#35011yescharacteristic
312C22#211711yescharacteristic
313C4—21no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4#188911yescharacteristic
322C4#189111yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2—42no—
327C2—82no—
328C2#896511yescharacteristic
329C2#1297811yescharacteristic
330C2#1298211yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC22×C4C22×C8
Frattini, cosocleC23×C8C23
socleC22#1869
prime radical#2388C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33487C2semidirect
3. #12683C2semidirect
4. #11465C2semidirect
5. #7202C2semidirect
36. C2×M32D4semidirect
241. C23#350semidirect

Maximal subgroups

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

subgroupindex
#334872
#214422
#126832
#114652
#72022