Finite Group Databasebeta

← #152946

Subgroups — #152946

Order 512.

338 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 66 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#28720C211yesmaximal, characteristic
2C2×#1414C214yesmaximal
3#12798C211yesmaximal, characteristic
4C4×#355C211yesmaximal, characteristic
5C2×#488C2212yes—
6#2859C411yescharacteristic, complemented
7C4×#124C2211yescharacteristic
8C2×#355C2212yes—
9C2×#354C2212yes—
10#2072—21no—
11#2071—21no—
12#1991C412yescomplemented
13#1805—21no—
14#1804—21no—
15#1799—21no—
16C4×C4⋊C8C411yescharacteristic, complemented
17C4×#101—21no—
18#1414—28no—
19#1394—21no—
20#1393—21no—
21C2×#125C2×C412yes—
22C2×#124C2311yescharacteristic
23C2×C4×Q8D411yescharacteristic
24C2×C4×D4D411yescharacteristic
25#488—24no—
26#488—24no—
27C2×C4⋊C8—21no—
28C2×#103—22no—
29C2×#102—22no—
30C2×#101—22no—
31C2×#99—22no—
32C4×M16—41no—
33C2×C4×C8C2×C412yes—
34#391D411yescharacteristic
35#390D411yescharacteristic
36#389—41no—
37#389—41no—
38#388—41no—
39#384—21no—
40#382—21no—
41#382C2×C411yescharacteristic
42#380—41no—
43C4×C22⋊C4—41no—
44C43C2×C411yescharacteristic
45#355—22no—
46#355D414yes—
47#354—24no—
48#343—41no—
49#342—41no—
50#340—22no—
51#340—22no—
52#336—21no—
53C22×Q8C2×D411yescharacteristic
54C22×D4C2×D411yescharacteristic
55C23×C4—21no—
56C2×M16—42no—
57C22×C8—22no—
58#125—24no—
59#125—42no—
60#124—41no—
61#124C2×D412yes—
62#124#4112yes—
63C4×Q8—41no—
64C4×Q8—41no—
65C4×D4—41no—
66C4×D4—41no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—81no—
73C2×C4⋊C4C22⋊C411yescharacteristic
74C2×C4⋊C4C22⋊C411yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—42no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42C22⋊C411yescharacteristic
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22×C411yescharacteristic
85C4⋊C8—41no—
86C4⋊C8—41no—
87#103—44no—
88#102—42no—
89#102—42no—
90#101—24no—
91#101—42no—
92#99—42no—
93#99—42no—
94#97—81no—
95C4×C8—24no—
96C4×C8—81no—
97#95—41no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—81no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8#12112yes—
107C2×D4—21no—
108C2×D4—41no—
109C2×D4—41no—
110C2×D4#12411yescharacteristic
111C2×D4#12711yescharacteristic
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—82no—
121C22×C4—82no—
122C22×C4#9911yescharacteristic
123C22×C4#9911yescharacteristic
124C22×C4C2×C22⋊C411yescharacteristic
125M16—44no—
126M16—82no—
127C2×C8—42no—
128C2×C8—42no—
129C2×C8—42no—
130C2×C8—82no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—81no—
136C4⋊C4—84no—
137C4⋊C4—82no—
138C4⋊C4—82no—
139C4⋊C4—162no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—81no—
143C22⋊C4—82no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—41no—
150C42—42no—
151C42—41no—
152C42—41no—
153C42—81no—
154C42C4×D412yes—
155C23—21no—
156C23—21no—
157C23—41no—
158C23—41no—
159C23—81no—
160C23—81no—
161C23C2×#9911yescharacteristic
162Q8—42no—
163Q8—81no—
164D4—41no—
165D4—41no—
166D4—81no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—161no—
190C2×C4—162no—
191C2×C4—162no—
192C2×C4—162no—
193C2×C4#35411yescharacteristic
194C2×C4#35411yescharacteristic
195C2×C4#39011yescharacteristic
196C2×C4#41011yescharacteristic
197C8—82no—
198C8—84no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—161no—
208C22#196411yescharacteristic
209C22C2×#35411yescharacteristic
210C22#212911yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—41no—
218C4—42no—
219C4—81no—
220C4—81no—
221C4—162no—
222C4—322no—
223C2—21no—
224C2—21no—
225C2—81no—
226C2—81no—
227C2#1322411yescharacteristic
228C2#1360211yescharacteristic
229C2#1360311yescharacteristic
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
6. #2859C4semidirect
12. #1991C4semidirect
16. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#287202
C2×#14142
#127982
C4×#3552