Finite Group Databasebeta

← #98959

Subgroups — #98959

Order 512.

414 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C8×C22⋊Q8C211yesmaximal, characteristic
2#10822C211yesmaximal, characteristic
3#8596C211yesmaximal, characteristic
4#7617C212yesmaximal
5#7045C212yesmaximal
6C2×C8×Q8C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8C4×C22⋊Q8C2211yescharacteristic
9#1984C411yescharacteristic
10#1939C411yescharacteristic
11#1916—21no—
12#1850—21no—
13#1797C2212yes—
14#1797C2211yescharacteristic
15C8×C4⋊C4—21no—
16C8×C4⋊C4C411yescharacteristic
17C8×C22⋊C4—21no—
18C2×#341—21no—
19C2×#341—21no—
20#1728C2212yes—
21C2×#335—21no—
22#1277—21no—
23#1272—22no—
24#1271—22no—
25C22×M16—22no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C22⋊Q8Q811yescharacteristic
29C2×C4×Q8C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C8×Q8—42no—
32C2×C4⋊C8—21no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C22⋊C8—21no—
38C2×C22⋊C8—22no—
39C2×C4×C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—42no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391D411yescharacteristic
47#389D411yescharacteristic
48#387—21no—
49#385—21no—
50#380Q811yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2×C411yescharacteristic
53C4×C22⋊C4—21no—
54#341—24no—
55#341—42no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—22no—
62#336—22no—
63#336—22no—
64#335—42no—
65C22×Q8C4⋊C411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×Q811yescharacteristic
68C23×C4#4111yescharacteristic
69C2×M16—24no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—42no—
84C22×C8—42no—
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C4×Q8—41no—
92C4×Q8—41no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C4⋊C411yescharacteristic
97C2×C4⋊C4C4⋊C411yescharacteristic
98C2×C4⋊C4C4⋊C411yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—41no—
104C2×C42—41no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42#4111yescharacteristic
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—42no—
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115C22⋊C8—42no—
116C22⋊C8—82no—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8—42no—
122C4×C8—42no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128C24C22⋊Q811yescharacteristic
129C2×Q8—21no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8#10812yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×C4⋊C411yescharacteristic
151C22×C4#11711yescharacteristic
152C22×C4#12111yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4C22⋊Q811yescharacteristic
155C22×C4#13512yes—
156C22×C4#13712yes—
157M16—44no—
158M16—44no—
159M16—82no—
160M16—82no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—44no—
172C2×C8—44no—
173C2×C8—44no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—82no—
185C2×C8#9912yes—
186C2×C8#10412yes—
187C4⋊C4—24no—
188C4⋊C4—21no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4#10812yes—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—41no—
206C42—41no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23#38011yescharacteristic
214C23#47912yes—
215Q8—42no—
216Q8—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4C2×#9911yescharacteristic
250C2×C4#41011yescharacteristic
251C2×C4#41611yescharacteristic
252C2×C4#41911yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C2×C4#42111yescharacteristic
255C2×C4C2×#10811yescharacteristic
256C2×C4C2×#10811yescharacteristic
257C2×C4#46412yes—
258C2×C4#47211yescharacteristic
259C2×C4#47311yescharacteristic
260C8—22no—
261C8—22no—
262C8—41no—
263C8—41no—
264C8—84no—
265C8—81no—
266C8—81no—
267C8—162no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22#179911yescharacteristic
276C22#180511yescharacteristic
277C22#180611yescharacteristic
278C22#181011yescharacteristic
279C22#181311yescharacteristic
280C22#181411yescharacteristic
281C22#181611yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—42no—
291C4—81no—
292C4—81no—
293C4#150011yescharacteristic
294C4#150811yescharacteristic
295C4#151511yescharacteristic
296C4#151611yescharacteristic
297C2—42no—
298C2#857411yescharacteristic
299C2#861211yescharacteristic
300C2#862811yescharacteristic
301C2#866111yescharacteristic
302C2#867111yescharacteristic
303C2#868911yescharacteristic
304C2#869011yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
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
C8×C22⋊Q82
#108222
#85962
#76172
#70452