Finite Group Databasebeta

← #150840

Subgroups — #150840

Order 512.

363 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 64 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#28440C211yesmaximal, characteristic
2#12802C211yesmaximal, characteristic
3#11478C211yesmaximal, characteristic
4#7405C214yesmaximal
5C2×#494—21no—
6C2×#462—21no—
7C2×#462—21no—
8#2868C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C2×#356—21no—
11#2016C412yes—
12#2016C2212yes—
13#1956—21no—
14#1801—21no—
15C4×C4⋊C8C411yescharacteristic
16#1461—21no—
17#1394C2212yes—
18#1325—21no—
19#1319C2212yes—
20C22×Q16—22no—
21C22×Q16—22no—
22C2×C4⋊Q8—21no—
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2×C412yes—
25C2×C4⋊Q8D411yescharacteristic
26C2×C4×Q8D411yescharacteristic
27C2×C4×Q8D411yescharacteristic
28#494—42no—
29#494—42no—
30#462—42no—
31#462—42no—
32#462—42no—
33#462—42no—
34#431—41no—
35#423—41no—
36C2×C4⋊C8—21no—
37C2×#103—22no—
38C2×#103—22no—
39C2×#101—22no—
40C2×C4×C8—21no—
41#391C2×C411yescharacteristic
42#391C2311yescharacteristic
43#389—41no—
44#384—41no—
45#384—41no—
46#384D411yescharacteristic
47C4×C4⋊C4—41no—
48C43C2×C411yescharacteristic
49#356—42no—
50#343—41no—
51#336C2×C412yes—
52#324—22no—
53#324—41no—
54C22×Q8—22no—
55C22×Q8C22⋊C411yescharacteristic
56C22×Q8C2×D411yescharacteristic
57C2×Q16—44no—
58C2×Q16—44no—
59C2×Q16—44no—
60C2×Q16—42no—
61C2×Q16—42no—
62C2×Q16—82no—
63C2×Q16—82no—
64C2×M16—42no—
65C22×C8—22no—
66C4⋊Q8—41no—
67C4⋊Q8—42no—
68C4⋊Q8—41no—
69C4⋊Q8—41no—
70C4⋊Q8—41no—
71C4⋊Q8—41no—
72C4⋊Q8—42no—
73C4⋊Q8—82no—
74C4⋊Q8D812yes—
75C4⋊Q8SD1612yes—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C22⋊C411yescharacteristic
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97C4⋊C8—22no—
98C4⋊C8—22no—
99C4⋊C8—81no—
100C4⋊C8—81no—
101C4⋊C8—81no—
102#103—44no—
103#103—44no—
104#101—42no—
105#101—42no—
106C22⋊C8—42no—
107C4×C8—42no—
108C4×C8—81no—
109#95—41no—
110#95—41no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—41no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8#10212yes—
124C2×Q8C2×D811yescharacteristic
125C2×Q8C2×SD1611yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4#9911yescharacteristic
134C22×C4#9911yescharacteristic
135C22×C4C2×C22⋊C411yescharacteristic
136Q16—88no—
137Q16—84no—
138Q16—84no—
139M16—82no—
140M16—82no—
141C2×C8—24no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—82no—
145C2×C8—82no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C4⋊C4—82no—
153C4⋊C4—84no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—84no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C42—24no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42—42no—
165C42—41no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42#10212yes—
170C23C2×#9911yescharacteristic
171Q8—42no—
172Q8—41no—
173Q8—41no—
174Q8—81no—
175Q8—81no—
176Q8—82no—
177Q8—82no—
178Q8—82no—
179Q8—82no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#35311yescharacteristic
206C2×C4#35311yescharacteristic
207C2×C4C2×#10211yescharacteristic
208C2×C4#42111yescharacteristic
209C8—44no—
210C8—162no—
211C8—162no—
212C22—21no—
213C22—21no—
214C22#151911yescharacteristic
215C22#212211yescharacteristic
216C22C2×#35311yescharacteristic
217C4—24no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—41no—
226C4—81no—
227C4—81no—
228C4—162no—
229C4#141112yes—
230C2—21no—
231C2—21no—
232C2#1187711yescharacteristic
233C2#1358011yescharacteristic
234C2C2×#141111yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC23C2×#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#284402
#128022
#114782
#74052