Finite Group Databasebeta

← #150918

Subgroups — #150918

Order 512.

312 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 62 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#28513C211yesmaximal, characteristic
2#12801C211yesmaximal, characteristic
3#11477C211yesmaximal, characteristic
4#7437C212yesmaximal, complemented
5#7436C212yesmaximal, complemented
6C2×#479—21no—
7C2×#478—21no—
8#2859C411yescharacteristic
9C4×#124C2211yescharacteristic
10C2×#355—21no—
11#2096C2211yescharacteristic
12#2095C2211yescharacteristic
13#2094C411yescharacteristic
14#2093C411yescharacteristic
15#1958—21no—
16#1846—21no—
17#1801—21no—
18C4×C4⋊C8C411yescharacteristic
19#1459—21no—
20#1394C2212yes—
21#1324—21no—
22#1319C2212yes—
23C2×#125—21no—
24C2×#124D411yescharacteristic
25C2×C4×Q8D411yescharacteristic
26C2×C4×D4D411yescharacteristic
27#479—42no—
28#479—42no—
29#478—42no—
30#478—42no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#106—21no—
34C2×#106—21no—
35#423—41no—
36C2×C4⋊C8—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×#101—22no—
42C2×C22⋊C8—41no—
43C2×C4×C8—21no—
44#391C2311yescharacteristic
45#390C2×C411yescharacteristic
46#389—41no—
47#389—41no—
48#388—41no—
49#384C2×C411yescharacteristic
50#384C2×C411yescharacteristic
51#382—21no—
52#382D411yescharacteristic
53C4×C22⋊C4—41no—
54C43C2×C411yescharacteristic
55#355—42no—
56#343—41no—
57#336C2×C411yescharacteristic
58#336C2×C411yescharacteristic
59#324—22no—
60#323—41no—
61C22×Q8C2×D411yescharacteristic
62C22×D4C22⋊C411yescharacteristic
63C23×C4—21no—
64C2×M16—42no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—81no—
68#125—41no—
69#125—41no—
70#125—41no—
71#125—41no—
72#124—41no—
73#124SD1612yes—
74#124Q1612yes—
75C4×Q8—41no—
76C4×Q8—41no—
77C4×D4—41no—
78C4×D4—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4C22⋊C411yescharacteristic
87C2×C4⋊C4C2×D411yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42C22⋊C411yescharacteristic
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97#107—42no—
98#107—42no—
99#106—42no—
100#106—42no—
101C4⋊C8—22no—
102C4⋊C8—22no—
103C4⋊C8—81no—
104C4⋊C8—81no—
105#103—42no—
106#103—42no—
107#102—42no—
108#102—42no—
109#101—42no—
110#101—42no—
111C22⋊C8—42no—
112C22⋊C8—81no—
113C4×C8—42no—
114#95—41no—
115#95—41no—
116#95—42no—
117C24—21no—
118C2×Q8—21no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8C2×SD1611yescharacteristic
122C2×Q8C2×Q1611yescharacteristic
123C2×D4—21no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4#10312yes—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—82no—
137C22×C4#9911yescharacteristic
138C22×C4#9911yescharacteristic
139C22×C4C2×C22⋊C411yescharacteristic
140M16—82no—
141M16—82no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—162no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—81no—
158C4⋊C4—82no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C4⋊C4—82no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C22⋊C4—81no—
167C22⋊C4—82no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—41no—
172C42—42no—
173C42—41no—
174C42—41no—
175C42—81no—
176C42#10312yes—
177C23—22no—
178C23—41no—
179C23—41no—
180C23—81no—
181C23—81no—
182C23C2×#9911yescharacteristic
183Q8—41no—
184Q8—41no—
185Q8—81no—
186D4—42no—
187D4—81no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—82no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—161no—
213C2×C4#35311yescharacteristic
214C2×C4#35311yescharacteristic
215C2×C4C2×#10311yescharacteristic
216C2×C4#42111yescharacteristic
217C8—42no—
218C8—42no—
219C8—162no—
220C8—162no—
221C22—21no—
222C22—21no—
223C22—42no—
224C22—82no—
225C22—81no—
226C22—81no—
227C22—161no—
228C22#152011yescharacteristic
229C22#212311yescharacteristic
230C22C2×#35311yescharacteristic
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—81no—
241C4—81no—
242C4—161no—
243C4—161no—
244C2—21no—
245C2—21no—
246C2—82no—
247C2#1190111yescharacteristic
248C2#1357811yescharacteristic
249C2#1358911yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#391C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ #391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7437C2semidirect
5. #7436C2semidirect

Maximal subgroups

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

subgroupindex
#285132
#128012
#114772
#74372
#74362