Finite Group Databasebeta

← #116566

Subgroups — #116566

Order 512.

586 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 170 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#21301C211yesmaximal, characteristic
2C2×#1838C211yesmaximal, characteristic
3#17708C212yesmaximal
4#8079C212yesmaximal
5C2×#1275C211yesmaximal, characteristic
6C2×#533C2211yescharacteristic
7C2×#431C2212yes—
8C2×#423C2212yes—
9C2×#423C2211yescharacteristic
10C22×#97C2211yescharacteristic
11#1985—22no—
12#1985C414yes—
13#1838—24no—
14C2×#341—22no—
15#1729—22no—
16#1550—24no—
17#1549—22no—
18#1548—22no—
19#1542—24no—
20#1301—24no—
21#1275—22no—
22C22×M16—21no—
23C22×M16—22no—
24C23×C8D412yes—
25#533—22no—
26#533—22no—
27#533—22no—
28#533—22no—
29C2×C4⋊Q8D411yescharacteristic
30C2×C4⋊Q8D411yescharacteristic
31C2×#125—21no—
32C2×C22⋊Q8—21no—
33C2×C22⋊Q8Q812yes—
34C2×C4×Q8—21no—
35C2×C4×Q8C2×C412yes—
36C2×#117—21no—
37C22×C42C2311yescharacteristic
38#431—22no—
39#431—22no—
40#431—22no—
41#431D414yes—
42#423—22no—
43#423—22no—
44#423—22no—
45#423—22no—
46#423—22no—
47#423D414yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C412yes—
53C2×C22⋊C8—22no—
54C2×#97—22no—
55C2×#97—22no—
56C2×#97C2×C412yes—
57C2×#97D412yes—
58C2×C4×C8—22no—
59#341—28no—
60#336—28no—
61#333—28no—
62#332—28no—
63C22×Q8—21no—
64C22×Q8C4⋊C412yes—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×Q811yescharacteristic
67C23×C4#4111yescharacteristic
68C2×M16—24no—
69C2×M16—42no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16—44no—
73C2×M16—44no—
74C2×M16—44no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—24no—
78C22×C8—24no—
79C22×C8—22no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8C22⋊C414yes—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87#125—42no—
88#125—42no—
89C22⋊Q8—24no—
90C22⋊Q8—24no—
91C22⋊Q8—42no—
92C22⋊Q8—42no—
93C4×Q8—24no—
94C4×Q8—24no—
95C4×Q8—42no—
96C4×Q8—42no—
97#117—41no—
98#117—41no—
99#117—41no—
100#117—41no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C4⋊C412yes—
107C2×C22⋊C4—22no—
108C2×C42—22no—
109C2×C42C22×C411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42#4112yes—
115C2×C42#4112yes—
116C2×C42#4111yescharacteristic
117C4⋊C8—24no—
118C4⋊C8—24no—
119C4⋊C8—24no—
120C4⋊C8—24no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C22⋊C8—24no—
126C22⋊C8—42no—
127#97—24no—
128#97—24no—
129C4×C8—24no—
130C4×C8—42no—
131C24C22⋊Q811yescharacteristic
132C2×Q8—24no—
133C2×Q8—42no—
134C2×Q8—82no—
135C2×Q8—82no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—81no—
139C22×C4—81no—
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C2×C4⋊C411yescharacteristic
142C22×C4#11711yescharacteristic
143C22×C4#12011yescharacteristic
144C22×C4#12111yescharacteristic
145C22×C4#12111yescharacteristic
146C22×C4C22⋊Q812yes—
147C22×C4C22⋊Q812yes—
148C22×C4C22⋊Q811yescharacteristic
149C22×C4#12312yes—
150C22×C4#12311yescharacteristic
151C22×C4#12412yes—
152C22×C4#12411yescharacteristic
153C22×C4#12711yescharacteristic
154C22×C4#13514yes—
155C22×C4#13714yes—
156M16—48no—
157M16—84no—
158M16—82no—
159M16—82no—
160C2×C8—24no—
161C2×C8—24no—
162C2×C8—24no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—84no—
170C2×C8—82no—
171C2×C8—82no—
172C4⋊C4—24no—
173C4⋊C4—44no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C42—42no—
183C42—42no—
184C42C4×D412yes—
185C42C4×D412yes—
186C42#12012yes—
187C42#12112yes—
188C42#12112yes—
189C42#12112yes—
190C42#12312yes—
191C42#12612yes—
192C23—41no—
193C23—41no—
194C23#38011yescharacteristic
195C23#39311yescharacteristic
196C23#39911yescharacteristic
197C23#47914yes—
198Q8—84no—
199Q8—84no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4#38512yes—
213C2×C4#38611yescharacteristic
214C2×C4#38812yes—
215C2×C4#39011yescharacteristic
216C2×C4#39211yescharacteristic
217C2×C4#39412yes—
218C2×C4#39612yes—
219C2×C4#39711yescharacteristic
220C2×C4#41612yes—
221C2×C4#41912yes—
222C2×C4#45214yes—
223C2×C4#46414yes—
224C2×C4#48514yes—
225C2×C4#48812yes—
226C2×C4#49112yes—
227C2×C4#49512yes—
228C2×C4#49712yes—
229C8—44no—
230C8—84no—
231C8—82no—
232C8—84no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22#143911yescharacteristic
241C22#180512yes—
242C22#202612yes—
243C22#209712yes—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—82no—
249C4—82no—
250C4#187312yes—
251C4#195811yescharacteristic
252C4#196011yescharacteristic
253C4#201512yes—
254C4#207412yes—
255C2—22no—
256C2—22no—
257C2—22no—
258C2#1105211yescharacteristic
259C2#1110311yescharacteristic
260C2#1111011yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#213012
C2×#18382
#177082
#80792
C2×#12752