Finite Group Databasebeta

← #303606

Subgroups — #303606

Order 512.

884 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 182 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
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#33069C211yesmaximal, characteristic
3#30220C211yesmaximal, characteristic
4#29150C211yesmaximal, characteristic
5#10414C214yesmaximal
6#9702C211yesmaximal, characteristic
7#9641C211yesmaximal, characteristic
8#8451C214yesmaximal
9#8175C211yesmaximal, characteristic
10C2×Q82C2212yes—
11#2868—21no—
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2567—22no—
15#2567C2212yes—
16#2551—22no—
17#2551—24no—
18Q8×C4⋊C4—22no—
19C4×C4⋊Q8—21no—
20C4×C4⋊Q8C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#2043—24no—
23#2026—22no—
24#1960—21no—
25#1956C2214yes—
26#1947C2212yes—
27#1921—22no—
28#1863—22no—
29#1836—21no—
30#1831C2211yescharacteristic
31#1803C2212yes—
32#1803C2214yes—
33#1767C2211yescharacteristic
34#1744C2211yescharacteristic
35C2×#333C2214yes—
36C2×Q8⋊C8C2212yes—
37C2×Q8⋊C8C2214yes—
38#1279C2212yes—
39Q82—28no—
40Q82—44no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8D411yescharacteristic
46C2×C4⋊Q8D414yes—
47C2×C4⋊Q8C2312yes—
48C2×#125—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8D412yes—
52C2×C4×Q8C2312yes—
53C2×C4×Q8C2311yescharacteristic
54C2×#107—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#103—22no—
58C2×#103—24no—
59C2×#103—28no—
60C2×#97—21no—
61C2×C4×C8C2312yes—
62#395—28no—
63#395—24no—
64#391—22no—
65#391D412yes—
66#389—24no—
67#389—28no—
68#389—24no—
69#389—22no—
70#389—22no—
71#389—22no—
72#384—24no—
73#384—22no—
74#384—21no—
75#384—21no—
76#384—22no—
77#384D411yescharacteristic
78#384C2312yes—
79#382—21no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4C2311yescharacteristic
86C43—21no—
87C43C2311yescharacteristic
88#340—24no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—21no—
94#333—44no—
95Q8⋊C8—24no—
96Q8⋊C8—44no—
97C22×Q8—24no—
98C22×Q8C2×D412yes—
99C22×Q8C2×D411yescharacteristic
100C22×Q8C2×D414yes—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—24no—
106C4⋊Q8—24no—
107C4⋊Q8—48no—
108C4⋊Q8—416no—
109C4⋊Q8—44no—
110C4⋊Q8—44no—
111C4⋊Q8—88no—
112C4⋊Q8—84no—
113C4⋊Q8Q1614yes—
114#125—82no—
115C4×Q8—24no—
116C4×Q8—24no—
117C4×Q8—22no—
118C4×Q8—24no—
119C4×Q8—24no—
120C4×Q8—48no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8#4114yes—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C4⋊C4C2×D412yes—
143C2×C42—22no—
144C2×C42—21no—
145C2×C42—24no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42C2×D412yes—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D412yes—
152C2×C42C2×D411yescharacteristic
153C2×C42C2411yescharacteristic
154#107—44no—
155C4⋊C8—22no—
156C4⋊C8—24no—
157C4⋊C8—22no—
158#103—44no—
159#103—84no—
160#103—88no—
161#97—42no—
162C4×C8—22no—
163C4×C8—22no—
164#95—28no—
165#95—22no—
166#95—24no—
167#95—21no—
168#95—42no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—24no—
173C2×Q8—28no—
174C2×Q8—24no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—88no—
178C2×Q8#12412yes—
179C2×Q8C2×Q1614yes—
180C2×Q8#13714yes—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4#12012yes—
189C22×C4#12012yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C2×C8—44no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C4⋊C4—28no—
202C4⋊C4—21no—
203C4⋊C4—24no—
204C4⋊C4—22no—
205C4⋊C4—28no—
206C4⋊C4—24no—
207C4⋊C4—48no—
208C4⋊C4—48no—
209C4⋊C4—48no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C4⋊C4—88no—
222C4⋊C4#12412yes—
223C4⋊C4#13714yes—
224C42—28no—
225C42—24no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—42no—
230C42—48no—
231C42—44no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42C2×Q1612yes—
236C42#13712yes—
237C42C2×#4112yes—
238C42H(5,𝔽2)12yes—
239C23C2×#12011yescharacteristic
240Q8—22no—
241Q8—24no—
242Q8—28no—
243Q8—44no—
244Q8—88no—
245Q8—168no—
246Q8#49114yes—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—48no—
266C2×C4—44no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#45114yes—
271C2×C4C2×#12411yescharacteristic
272C2×C4#53511yescharacteristic
273C2×C4#54612yes—
274C2×C4#54612yes—
275C2×C4C22×Q1611yescharacteristic
276C2×C4C2×#13711yescharacteristic
277C2×C4C2×#13712yes—
278C2×C4C2×#13712yes—
279C2×C4#57712yes—
280C8—82no—
281C8—82no—
282C8—82no—
283C22#243511yescharacteristic
284C22C2×#45111yescharacteristic
285C22#300412yes—
286C22#301011yescharacteristic
287C22#301912yes—
288C4—24no—
289C4—22no—
290C4—24no—
291C4—42no—
292C4—44no—
293C4—42no—
294C4—44no—
295C4—84no—
296C4—82no—
297C4—162no—
298C4C2×#49111yescharacteristic
299C4#313911yescharacteristic
300C4#321812yes—
301C4#329912yes—
302C4#332912yes—
303C2#1876912yes—
304C2#1881612yes—
305C2#2586111yescharacteristic
306C2#2587811yescharacteristic
307C2#2588811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#330692
#302202
#291502
#104142
#97022
#96412
#84512
#81752