Finite Group Databasebeta

← #102350

Subgroups — #102350

Order 512.

438 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 98 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#33209C211yesmaximal, characteristic
2#10822C211yesmaximal, characteristic
3#8145C211yesmaximal, characteristic
4#7688C212yesmaximal
5#7687C212yesmaximal
6C2×#500C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2857C2212yes—
9#1990C412yes—
10#1942—21no—
11#1942—21no—
12#1845C411yescharacteristic
13#1808—21no—
14#1808—21no—
15C4×C22⋊C8C2211yescharacteristic
16C4×C22⋊C8C2211yescharacteristic
17C2×#341—21no—
18C2×#341—21no—
19#1728C2212yes—
20C2×#322—21no—
21#1273—21no—
22#1272—24no—
23#1271—24no—
24C22×M16—22no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×C22⋊Q8—22no—
29C22×C42C2311yescharacteristic
30#500—42no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×C22⋊C8—21no—
36C2×C22⋊C8—21no—
37C2×C4×C8—21no—
38C2×C4×C8—21no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—22no—
45#389—22no—
46#389—22no—
47#384C2×C412yes—
48#380—22no—
49#380—22no—
50#341—24no—
51#341—42no—
52#340—22no—
53#340—22no—
54#336—22no—
55#336—22no—
56#336D412yes—
57#336D412yes—
58#322—42no—
59C22×Q8—22no—
60C23×C4C2×D411yescharacteristic
61C23×C4#4111yescharacteristic
62C23×C4#4111yescharacteristic
63C2×M16—24no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C4⋊Q8—42no—
83C4⋊Q8—42no—
84C22⋊Q8—84no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C42—41no—
93C2×C42—41no—
94C2×C42C22×C411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×D411yescharacteristic
98#107—42no—
99#107—42no—
100#107—42no—
101#107—42no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—41no—
107C4×C8—41no—
108C4×C8—41no—
109C4×C8—41no—
110C4×C8—41no—
111C4×C8—41no—
112#95—42no—
113#95—42no—
114C24#12411yescharacteristic
115C2×Q8—44no—
116C2×Q8—82no—
117C2×Q8—82no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4C2×C22⋊C411yescharacteristic
131C22×C4C4×D411yescharacteristic
132C22×C4C4×D411yescharacteristic
133C22×C4#12111yescharacteristic
134C22×C4#12111yescharacteristic
135C22×C4#12711yescharacteristic
136C22×C4#13712yes—
137C22×C4#13712yes—
138M16—44no—
139M16—44no—
140M16—82no—
141M16—82no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—24no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—42no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8#9912yes—
164C2×C8#10412yes—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—84no—
173C22⋊C4—84no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—41no—
179C42—41no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23#39011yescharacteristic
187C23#49112yes—
188Q8—164no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4C2×#9911yescharacteristic
212C2×C4#41011yescharacteristic
213C2×C4#41911yescharacteristic
214C2×C4#41911yescharacteristic
215C2×C4C2×#10411yescharacteristic
216C2×C4#42111yescharacteristic
217C2×C4#44312yes—
218C2×C4#44412yes—
219C2×C4#47212yes—
220C2×C4#47312yes—
221C2×C4#49712yes—
222C8—22no—
223C8—22no—
224C8—44no—
225C8—41no—
226C8—41no—
227C8—82no—
228C8—84no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22#196011yescharacteristic
237C22#196411yescharacteristic
238C22#197411yescharacteristic
239C22#197511yescharacteristic
240C22#197811yescharacteristic
241C22#197911yescharacteristic
242C22#198011yescharacteristic
243C4—22no—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—22no—
248C4—162no—
249C4—162no—
250C4#150611yescharacteristic
251C4#150811yescharacteristic
252C4#151111yescharacteristic
253C4#151511yescharacteristic
254C2—42no—
255C2#968311yescharacteristic
256C2#972411yescharacteristic
257C2#972711yescharacteristic
258C2#977811yescharacteristic
259C2#981411yescharacteristic
260C2#981811yescharacteristic
261C2#985911yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#332092
#108222
#81452
#76882
#76872