Finite Group Databasebeta

← #152233

Subgroups — #152233

Order 512.

420 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 70 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#21692C211yesmaximal, characteristic
2C2×#1410C212yesmaximal
3#12802C212yesmaximal, complemented
4#7499C212yesmaximal, complemented
5C2×#492C2211yescharacteristic
6C2×#440—22no—
7C4×C4⋊Q8C2211yescharacteristic
8C4×#127C411yescharacteristic
9C2×#356C2214yes—
10#1973—22no—
11#1957C411yescharacteristic
12#1956C2211yescharacteristic
13#1845C411yescharacteristic
14#1801—22no—
15C4×#97C411yescharacteristic
16#1410—24no—
17#1394—22no—
18C22×SD16—22no—
19C2×C4⋊Q8C2311yescharacteristic
20C2×#127C2×C411yescharacteristic
21C2×C4×Q8D412yes—
22C2×C4×D4—21no—
23C2×C4×D4—41no—
24#492—22no—
25#492D412yes—
26#492Q812yes—
27#440—48no—
28C2×#107—22no—
29#423—42no—
30C2×#103—22no—
31C2×#102—22no—
32C2×#101—24no—
33C2×#97—21no—
34C2×#97—21no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#391D412yes—
38#390—21no—
39#390—41no—
40#389—41no—
41#384C2×C411yescharacteristic
42#384C2×C411yescharacteristic
43C4×C4⋊C4—41no—
44C43C2×C411yescharacteristic
45#356—24no—
46#356—24no—
47#343—42no—
48C22×Q8C2×D412yes—
49C22×D4—21no—
50C22×D4—41no—
51C23×C4—41no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—44no—
55C2×SD16—44no—
56C2×M16—44no—
57C22×C8—21no—
58C22×C8—21no—
59C4⋊Q8—41no—
60C4⋊Q8C2×D411yescharacteristic
61C4⋊Q8C2×Q811yescharacteristic
62C4⋊Q8#4112yes—
63#127—21no—
64#127—41no—
65#127C4⋊C412yes—
66C4×Q8—44no—
67C4×D4—44no—
68C4×D4—84no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4C22⋊C412yes—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—41no—
77C2×C42—21no—
78C2×C42—22no—
79C2×C42—21no—
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22⋊C411yescharacteristic
82C2×C42C22×C411yescharacteristic
83#107—44no—
84C4⋊C8—84no—
85#103—44no—
86#102—44no—
87#101—44no—
88#101—44no—
89#97—24no—
90#97—42no—
91C4×C8—21no—
92C4×C8—41no—
93C4×C8C4⋊C412yes—
94#95—42no—
95C24—41no—
96C2×Q8—22no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8#12112yes—
100C2×Q8C22⋊Q812yes—
101C2×D4—22no—
102C2×D4—42no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—41no—
107C2×D4—82no—
108C2×D4—82no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—41no—
114C22×C4—82no—
115C22×C4—84no—
116C22×C4#9911yescharacteristic
117C22×C4#9911yescharacteristic
118C22×C4C2×C22⋊C411yescharacteristic
119SD16—48no—
120SD16—48no—
121M16—84no—
122M16—84no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—84no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C4⋊C4—82no—
137C4⋊C4—82no—
138C22⋊C4—84no—
139C22⋊C4—82no—
140C42—22no—
141C42—21no—
142C42—41no—
143C42—42no—
144C42—42no—
145C42—44no—
146C42C2×C4⋊C411yescharacteristic
147C42#11711yescharacteristic
148C23—42no—
149C23—82no—
150C23—82no—
151C23—81no—
152C23—81no—
153C23C2×#9911yescharacteristic
154Q8—42no—
155Q8—42no—
156Q8—82no—
157D4—44no—
158D4—82no—
159D4—84no—
160D4—84no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—81no—
181C2×C4—164no—
182C2×C4#35312yes—
183C2×C4#38011yescharacteristic
184C2×C4#41011yescharacteristic
185C8—24no—
186C8—42no—
187C8—164no—
188C22—21no—
189C22—21no—
190C22—84no—
191C22—82no—
192C22—162no—
193C22—161no—
194C22—161no—
195C22#179911yescharacteristic
196C22C2×#35311yescharacteristic
197C22#212611yescharacteristic
198C4—22no—
199C4—22no—
200C4—22no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—42no—
205C4—42no—
206C4—81no—
207C4—82no—
208C4—161no—
209C2—21no—
210C2—21no—
211C2—162no—
212C2#1279711yescharacteristic
213C2#1359011yescharacteristic
214C2#1359111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4⋊Q8C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12802C2semidirect
4. #7499C2semidirect

Maximal subgroups

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

subgroupindex
#216922
C2×#14102
#128022
#74992