Finite Group Databasebeta

← #140676

Subgroups — #140676

Order 512.

454 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 84 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#21049C211yesmaximal, characteristic
2#12791C211yesmaximal, characteristic, complemented
3#12607C211yesmaximal, characteristic, complemented
4C2×#1327C212yesmaximal, complemented
5C2×#1320C212yesmaximal
6C2×#423C2211yescharacteristic
7C2×#407C411yescharacteristic
8C2×#407C2211yescharacteristic
9C2×C4×M16C2211yescharacteristic
10C2×C4×C22⋊C4C411yescharacteristic
11C2×#364—21no—
12C2×#349C2212yes—
13C2×#349C2212yes—
14C2×#347—21no—
15#1924—21no—
16#1924—21no—
17#1923—21no—
18#1923—21no—
19#1728C411yescharacteristic
20#1728C411yescharacteristic
21#1376—21no—
22#1375—21no—
23#1327—24no—
24#1320—24no—
25C24×C4C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C22×M16C2×C411yescharacteristic
29C22×M16C2311yescharacteristic
30C22×C22⋊C4—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C2×M32—22no—
34C2×M32—22no—
35#423—22no—
36#423—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39#407—22no—
40#407—22no—
41#407—42no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C4×M16—42no—
47C2×#97D411yescharacteristic
48C2×#97D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C2×#95—21no—
56#364—42no—
57#349—22no—
58#349—22no—
59#349—22no—
60#349D414yes—
61#347—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66C25M1611yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4M1611yescharacteristic
78C23×C4C22×C411yescharacteristic
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16C2×D412yes—
91C2×M16#4112yes—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—44no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111C2×C42M1611yescharacteristic
112C2×C42M1611yescharacteristic
113M32—24no—
114M32—42no—
115M32—42no—
116M32—42no—
117C2×C16—42no—
118C2×C16—42no—
119C4⋊C8—42no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127#97—22no—
128#97—41no—
129C4×C8—22no—
130C4×C8—41no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135C24—21no—
136C24—22no—
137C24—42no—
138C24—41no—
139C24—41no—
140C24—81no—
141C24C2×M1611yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C4×D412yes—
175C22×C4C2×M1611yescharacteristic
176C22×C4#13112yes—
177M16—42no—
178M16—42no—
179M16—44no—
180M16—82no—
181M16—82no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8#9912yes—
198C2×C8#12112yes—
199C2×C8#12411yescharacteristic
200C2×C8#12711yescharacteristic
201C22⋊C4—24no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—22no—
210C42—22no—
211C42—42no—
212C42—41no—
213C42—41no—
214C16—44no—
215C16—82no—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—42no—
225C23—42no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—82no—
230C23—84no—
231C23—84no—
232C23—82no—
233C23#40711yescharacteristic
234C23#43512yes—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4—84no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4#39011yescharacteristic
265C2×C4#40811yescharacteristic
266C2×C4C2×#9911yescharacteristic
267C2×C4#41011yescharacteristic
268C2×C4#41311yescharacteristic
269C2×C4#41311yescharacteristic
270C2×C4#43512yes—
271C8—22no—
272C8—21no—
273C8—21no—
274C8—22no—
275C8—41no—
276C8—81no—
277C8—81no—
278C22—21no—
279C22—21no—
280C22—22no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—81no—
289C22—81no—
290C22—84no—
291C22—84no—
292C22—81no—
293C22—81no—
294C22#146411yescharacteristic
295C22#146511yescharacteristic
296C22#195511yescharacteristic
297C4—22no—
298C4—42no—
299C4—42no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4#146811yescharacteristic
304C4#147311yescharacteristic
305C4#196411yescharacteristic
306C4#196711yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2—82no—
311C2#965211yescharacteristic
312C2#1152111yescharacteristic
313C2#1152211yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC22×M16C23
socleC22#1464
prime radicalC25M16
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12791C2semidirect
3. #12607C2semidirect
4. C2×#1327C2semidirect

Maximal subgroups

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

subgroupindex
#210492
#127912
#126072
C2×#13272
C2×#13202