Finite Group Databasebeta

← #149915

Subgroups — #149915

Order 512.

375 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 56 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#27906C211yesmaximal, characteristic
2#11512C211yesmaximal, characteristic, complemented
3C2×#1325C211yesmaximal, characteristic, complemented
4#7371C212yesmaximal, complemented
5#7354C212yesmaximal, complemented
6C2×#481—21no—
7C2×#481—21no—
8#2856C411yescharacteristic
9C2×#384C2211yescharacteristic
10#2094—21no—
11#2094—21no—
12#1957C411yescharacteristic
13#1957C2211yescharacteristic
14#1844C411yescharacteristic
15#1844C2211yescharacteristic
16C4×C22⋊C8C411yescharacteristic
17#1463—21no—
18C2×#324—21no—
19#1325—21no—
20#1325—21no—
21#1325C2212yes—
22#1325C2212yes—
23C2×#127C2×C411yescharacteristic
24C2×#121—21no—
25C2×#121—41no—
26C22×C4⋊C4—41no—
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2×C411yescharacteristic
30#481—42no—
31#481—42no—
32#481—42no—
33#481—42no—
34C2×#106—21no—
35C2×#106—21no—
36C2×#106—21no—
37C2×#106—21no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×#102—21no—
42#407—41no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—41no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47#390—21no—
48#390—41no—
49#384—41no—
50#384C2×C411yescharacteristic
51#384C2×C411yescharacteristic
52#384D411yescharacteristic
53#384D411yescharacteristic
54#384C2311yescharacteristic
55#336—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#324—41no—
60C22×D4—21no—
61C22×D4—41no—
62C22×D4—41no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C2×M16—81no—
68C22×C8—21no—
69C22×C8—21no—
70C22×C8—81no—
71#127—22no—
72#127—41no—
73#121—41no—
74#121—41no—
75#121—41no—
76#121—41no—
77#121—84no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—84no—
91C2×C4⋊C4C22⋊C411yescharacteristic
92C2×C4⋊C4C22⋊C411yescharacteristic
93C2×C4⋊C4C2×D411yescharacteristic
94C2×C4⋊C4C2×D411yescharacteristic
95C2×C22⋊C4—42no—
96C2×C22⋊C4—41no—
97C2×C42—22no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22×C411yescharacteristic
102#106—42no—
103#106—42no—
104#106—42no—
105#106—42no—
106#102—42no—
107#102—42no—
108#102—42no—
109#102—42no—
110C22⋊C8—22no—
111C22⋊C8—22no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—81no—
115C22⋊C8—81no—
116C4×C8—22no—
117C4×C8—41no—
118C24—41no—
119C24#9911yescharacteristic
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—82no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4—84no—
130C2×D4—81no—
131C2×D4—81no—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4#9911yescharacteristic
154C22×C4C2×C22⋊C411yescharacteristic
155M16—162no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—82no—
163C2×C8—82no—
164C2×C8—162no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C22⋊C4—84no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—42no—
185C42#13112yes—
186C23—21no—
187C23—22no—
188C23—21no—
189C23—41no—
190C23—82no—
191C23—82no—
192C23—81no—
193C23—81no—
194C23—81no—
195C23#35311yescharacteristic
196C23#35311yescharacteristic
197C23C2×#9911yescharacteristic
198D4—81no—
199D4—81no—
200D4—81no—
201D4—81no—
202D4—164no—
203D4—162no—
204D4—162no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—48no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—81no—
233C2×C4—161no—
234C2×C4—161no—
235C2×C4#40811yescharacteristic
236C2×C4#41311yescharacteristic
237C8—42no—
238C8—42no—
239C8—162no—
240C8—162no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—42no—
249C22—42no—
250C22—162no—
251C22—162no—
252C22—161no—
253C22—161no—
254C22—161no—
255C22#141012yes—
256C22#146811yescharacteristic
257C22C2×#35311yescharacteristic
258C22#213311yescharacteristic
259C4—24no—
260C4—21no—
261C4—21no—
262C4—42no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—161no—
269C2—21no—
270C2—21no—
271C2—22no—
272C2—41no—
273C2—321no—
274C2#1154911yescharacteristic
275C2#1358411yescharacteristic
276C2C2×#141011yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radical#2856C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11512C2semidirect
3. C2×#1325C2semidirect
4. #7371C2semidirect
5. #7354C2semidirect

Maximal subgroups

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

subgroupindex
#279062
#115122
C2×#13252
#73712
#73542