Finite Group Databasebeta

← #150013

Subgroups — #150013

Order 512.

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

normal subgroups only

This group has 297 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#28517C211yesmaximal, characteristic
2#11530C211yesmaximal, characteristic
3#11477C211yesmaximal, characteristic
4#7372C212yesmaximal, complemented
5#7372C212yesmaximal, complemented
6C2×#479—21no—
7C2×#478—21no—
8#2859C411yescharacteristic
9C4×#124C2211yescharacteristic
10#2097—21no—
11#2092—21no—
12#1959C411yescharacteristic
13#1959C411yescharacteristic
14#1847C2211yescharacteristic
15#1847C2211yescharacteristic
16#1767C411yescharacteristic
17#1467—21no—
18#1459—21no—
19#1324C2212yes—
20#1324C2212yes—
21#1319—21no—
22#1318—21no—
23C2×#125—21no—
24C2×#124C2×C411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#479—42no—
28#479—42no—
29#478—42no—
30#478—42no—
31#432—41no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#106—21no—
35C2×#106—21no—
36C2×C4⋊C8—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×C22⋊C8—41no—
42C2×#97C2×C411yescharacteristic
43C2×#97C2×C411yescharacteristic
44#391D411yescharacteristic
45#390D411yescharacteristic
46#389—41no—
47#389—41no—
48#388—41no—
49#384C2×C411yescharacteristic
50#384C2×C411yescharacteristic
51#382—21no—
52#382C2311yescharacteristic
53C4×C22⋊C4—41no—
54C43C2×C411yescharacteristic
55#336—21no—
56#324—22no—
57#324—22no—
58#324—41no—
59#323—41no—
60C22×Q8C22⋊C411yescharacteristic
61C22×D4C22⋊C411yescharacteristic
62C23×C4—21no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—81no—
66#125—41no—
67#125—41no—
68#125—41no—
69#125—41no—
70#124—22no—
71#124—22no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×D4—41no—
75C4×D4—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22×C411yescharacteristic
94#107—42no—
95#107—42no—
96#106—42no—
97#106—42no—
98C4⋊C8—22no—
99C4⋊C8—22no—
100C4⋊C8—81no—
101#103—42no—
102#103—42no—
103#102—42no—
104#102—42no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—81no—
108#97—22no—
109#97—22no—
110#97—81no—
111#95—41no—
112#95—41no—
113#95—42no—
114C24—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×D4—21no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4#10112yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—82no—
133C22×C4#9911yescharacteristic
134C22×C4#9911yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—82no—
143C2×C8—82no—
144C2×C8—162no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C4⋊C4—81no—
154C4⋊C4—82no—
155C4⋊C4—82no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C22⋊C4—44no—
159C22⋊C4—81no—
160C22⋊C4—82no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—41no—
166C42—41no—
167C42—81no—
168C42—81no—
169C42#13112yes—
170C23—22no—
171C23—41no—
172C23—41no—
173C23—81no—
174C23—81no—
175C23C2×#9911yescharacteristic
176Q8—42no—
177Q8—81no—
178D4—42no—
179D4—81no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—161no—
204C2×C4#35311yescharacteristic
205C2×C4#35311yescharacteristic
206C2×C4#40811yescharacteristic
207C2×C4C2×#10111yescharacteristic
208C8—42no—
209C8—42no—
210C8—162no—
211C8—162no—
212C22—21no—
213C22—21no—
214C22—42no—
215C22—82no—
216C22—81no—
217C22—81no—
218C22—161no—
219C22#147211yescharacteristic
220C22C2×#35311yescharacteristic
221C22#213511yescharacteristic
222C4—21no—
223C4—21no—
224C4—21no—
225C4—21no—
226C4—42no—
227C4—41no—
228C4—81no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4—161no—
233C4—161no—
234C2—21no—
235C2—21no—
236C2—82no—
237C2#1157011yescharacteristic
238C2#1357711yescharacteristic
239C2#1357811yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7372C2semidirect
5. #7372C2semidirect

Maximal subgroups

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

subgroupindex
#285172
#115302
#114772
#73722