Finite Group Databasebeta

← #137606

Subgroups — #137606

Order 512.

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

normal subgroups only

This group has 465 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#33324C211yesmaximal, characteristic, complemented
2#21494C211yesmaximal, characteristic
3#12786C211yesmaximal, characteristic, complemented
4#11587C212yesmaximal, complemented
5#7169C212yesmaximal, complemented
6C22×M32—21no—
7C2×#436—21no—
8C2×#436C2211yescharacteristic
9C2×#405C2211yescharacteristic
10#2392C411yescharacteristic
11#2150C2212yes—
12C2×#349—21no—
13#2108—24no—
14C4×M32—21no—
15C4×M32C2212yes—
16#1955C411yescharacteristic
17#1940C411yescharacteristic
18#1923—21no—
19#1869—21no—
20#1798C411yescharacteristic
21C8×C4⋊C4C2211yescharacteristic
22#1753—21no—
23#1377—21no—
24#1330—24no—
25C22×M16—21no—
26C22×M16D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2×C411yescharacteristic
30C2×#117C2×C411yescharacteristic
31C2×M32—22no—
32C2×M32—24no—
33C2×M32—24no—
34C2×M32—22no—
35C2×M32—22no—
36C22×C16—22no—
37#436—22no—
38#436—22no—
39#436—42no—
40#436—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48#405—22no—
49#405—22no—
50C2×#97C2×C411yescharacteristic
51C2×#97D411yescharacteristic
52C2×C4×C8—21no—
53C2×C4×C8D411yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#398C811yescharacteristic
56#397C811yescharacteristic
57#396—21no—
58#394—21no—
59#394C811yescharacteristic
60#392C811yescharacteristic
61C4×C4⋊C4C2×C411yescharacteristic
62C4×C22⋊C4—21no—
63C4⋊C16—22no—
64C4⋊C16—22no—
65#349—42no—
66#346—22no—
67#346D414yescomplemented
68C4×C16—22no—
69C4×C16—22no—
70#336—21no—
71#336—21no—
72#336—21no—
73C22×D4—21no—
74C22×D4C2×C811yescharacteristic
75C23×C4—21no—
76C23×C4C22⋊C411yescharacteristic
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—42no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—42no—
83C2×M16D812yes—
84C2×M16SD1612yes—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—42no—
98C4×D4—42no—
99#117—22no—
100#117—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C2×C811yescharacteristic
104C2×C4⋊C4C2×C811yescharacteristic
105C2×C4⋊C4M1611yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4C2×C811yescharacteristic
112C2×C22⋊C4M1611yescharacteristic
113C2×C42—21no—
114C2×C42C22⋊C411yescharacteristic
115C2×C42C22×C411yescharacteristic
116M32—28no—
117M32—24no—
118M32—24no—
119M32—42no—
120M32—42no—
121C2×C16—28no—
122C2×C16—24no—
123C2×C16—42no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129C22⋊C8—42no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132#97—22no—
133#97—22no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—41no—
137C4×C8—41no—
138C4×C8C2×D412yes—
139C4×C8#4112yes—
140#95—21no—
141#95—21no—
142#95C2×C811yescharacteristic
143#95C2×C811yescharacteristic
144C24—21no—
145C24C22⋊C811yescharacteristic
146C2×D4—22no—
147C2×D4—42no—
148C2×D4—42no—
149C2×D4—81no—
150C2×D4—81no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C22⋊C811yescharacteristic
165C22×C4C22⋊C811yescharacteristic
166C22×C4C22⋊C811yescharacteristic
167C22×C4#10212yes—
168C22×C4#10412yes—
169C22×C4C2×C22⋊C411yescharacteristic
170C22×C4C22×C811yescharacteristic
171C22×C4C2×M1611yescharacteristic
172M16—24no—
173M16—24no—
174M16—82no—
175M16—82no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—24no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8#10212yes—
192C2×C8#10412yes—
193C2×C8#12011yescharacteristic
194C2×C8#12112yes—
195C2×C8#12311yescharacteristic
196C2×C8C2×D811yescharacteristic
197C2×C8C2×SD1611yescharacteristic
198C2×C8#13512yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42C4×D412yes—
216C42#13112yes—
217C16—28no—
218C16—82no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23D4⋊C812yes—
231C23C2×C22⋊C811yescharacteristic
232D4—82no—
233D4—84no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
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—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4D4⋊C812yes—
260C2×C4#38611yescharacteristic
261C2×C4#40811yescharacteristic
262C2×C4C2×#10211yescharacteristic
263C2×C4#41611yescharacteristic
264C2×C4C2×#10411yescharacteristic
265C2×C4C2×#10411yescharacteristic
266C2×C4C8×D412yes—
267C2×C4#43512yes—
268C8—22no—
269C8—21no—
270C8—22no—
271C8—21no—
272C8—81no—
273C8—81no—
274C8#46512yes—
275C8#46612yes—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22C2×D4⋊C811yescharacteristic
291C22#147611yescharacteristic
292C22#186911yescharacteristic
293C4—22no—
294C4—22no—
295C4—22no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4#149211yescharacteristic
302C4#149411yescharacteristic
303C4#189311yescharacteristic
304C4#189611yescharacteristic
305C2—21no—
306C2—21no—
307C2—42no—
308C2—161no—
309C2#907211yescharacteristic
310C2#1298211yescharacteristic
311C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radical#392C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33324C2semidirect
3. #12786C2semidirect
4. #11587C2semidirect
5. #7169C2semidirect
67. #346D4semidirect

Maximal subgroups

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

subgroupindex
#333242
#214942
#127862
#115872
#71692