Finite Group Databasebeta

← #153062

Subgroups — #153062

Order 512.

312 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 68 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#29140C211yesmaximal, characteristic
2#12798C211yesmaximal, characteristic
3C4×#355C211yesmaximal, characteristic
4#7502C214yesmaximal
5#2859C411yescharacteristic, complemented
6C4×#124C2211yescharacteristic
7C2×#355C2212yes—
8C2×#354C2212yes—
9#2026—21no—
10#2025—21no—
11#1958C2212yes—
12#1835C412yescomplemented
13#1803—21no—
14#1802—21no—
15#1799—21no—
16C4×C4⋊C8C411yescharacteristic, complemented
17C4×#101—21no—
18#1394—21no—
19#1393—21no—
20C2×#125—21no—
21C2×#124C2311yescharacteristic
22C2×C4×Q8D411yescharacteristic
23C2×C4×D4D411yescharacteristic
24C2×C4⋊C8—21no—
25C2×#103—22no—
26C2×#102—22no—
27C2×#101—22no—
28C2×#99—22no—
29C4×M16—41no—
30C2×C4×C8C2×C412yes—
31#391D411yescharacteristic
32#390D411yescharacteristic
33#389—41no—
34#389—41no—
35#388—41no—
36#384—21no—
37#382C2×C412yes—
38#382C2×C411yescharacteristic
39#380—41no—
40C4×C22⋊C4—41no—
41C43C2×C411yescharacteristic
42#355—22no—
43#355Q814yes—
44#354—24no—
45#343—41no—
46#342—41no—
47#340—22no—
48#340—22no—
49#336—21no—
50C22×Q8C2×D411yescharacteristic
51C22×D4C2×D411yescharacteristic
52C23×C4—21no—
53C2×M16—42no—
54C22×C8—22no—
55#125—82no—
56#124—41no—
57#124C2×Q812yes—
58#124#4112yes—
59C4×Q8—41no—
60C4×Q8—41no—
61C4×D4—41no—
62C4×D4—41no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—81no—
69C2×C4⋊C4C22⋊C411yescharacteristic
70C2×C4⋊C4C22⋊C411yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—41no—
73C2×C22⋊C4—41no—
74C2×C22⋊C4—42no—
75C2×C42—21no—
76C2×C42—21no—
77C2×C42—21no—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22×C411yescharacteristic
81C4⋊C8—41no—
82C4⋊C8—41no—
83#103—82no—
84#102—82no—
85#101—24no—
86#101—42no—
87#99—42no—
88#99—42no—
89#97—81no—
90C4×C8—42no—
91C4×C8—81no—
92#95—41no—
93#95—41no—
94#95—42no—
95#95—81no—
96C24—21no—
97C2×Q8—21no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×Q8C22⋊Q812yes—
101C2×D4—21no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4#12411yescharacteristic
105C2×D4C4⋊Q811yescharacteristic
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4—82no—
116C22×C4#9911yescharacteristic
117C22×C4#9911yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119M16—44no—
120M16—82no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—42no—
124C2×C8—82no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—81no—
130C4⋊C4—82no—
131C4⋊C4—82no—
132C4⋊C4—84no—
133C4⋊C4—162no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—81no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C22⋊C4—82no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—41no—
144C42—42no—
145C42—41no—
146C42—41no—
147C42—81no—
148C42C4×Q812yes—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23C2×#9911yescharacteristic
156Q8—42no—
157Q8—81no—
158D4—41no—
159D4—41no—
160D4—81no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—161no—
184C2×C4—162no—
185C2×C4—162no—
186C2×C4—162no—
187C2×C4#35411yescharacteristic
188C2×C4#35411yescharacteristic
189C2×C4#39111yescharacteristic
190C2×C4#41011yescharacteristic
191C8—82no—
192C8—84no—
193C22—21no—
194C22—21no—
195C22—41no—
196C22—41no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—161no—
202C22#199611yescharacteristic
203C22C2×#35411yescharacteristic
204C22#212811yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—42no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4—162no—
215C4—322no—
216C4#141412yes—
217C2—21no—
218C2—21no—
219C2—81no—
220C2—81no—
221C2#1329911yescharacteristic
222C2C2×#141411yescharacteristic
223C2#1360411yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocleC2×#124C23
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2859C4semidirect
12. #1835C4semidirect
16. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#291402
#127982
C4×#3552
#75022