Finite Group Databasebeta

← #325128

Subgroups — #325128

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29208C211yesmaximal, characteristic
3#28687C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10658C214yesmaximal
6#9898C211yesmaximal, characteristic
7#9896C211yesmaximal, characteristic
8#9613C214yesmaximal
9C4×#334C211yesmaximal, characteristic
10C2×#498—21no—
11C2×C8×Q8—21no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2084—22no—
16#2057—22no—
17#1990—21no—
18#1990—21no—
19#1990—21no—
20#1990—21no—
21#1990—21no—
22#1990—21no—
23#1984—21no—
24#1948C2214yes—
25#1947C2214yes—
26#1939—21no—
27#1939C2212yes—
28#1845—21no—
29#1842C2211yescharacteristic
30#1834C2212yes—
31#1834C2212yes—
32C8×C4⋊C4—21no—
33C8×C4⋊C4C2212yes—
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36C42×C8C2211yescharacteristic
37C2×#334C2212yes—
38#1296C2214yes—
39#1296C2214yes—
40#1277C2212yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4×Q8—21no—
49#498—44no—
50C8×Q8—44no—
51C2×#107—24no—
52C2×#106—24no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4×C8—22no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#389—21no—
70#389D412yes—
71#384—21no—
72#384—21no—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384D411yescharacteristic
78#384C2312yes—
79#384C2312yes—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2312yes—
82C43C2311yescharacteristic
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—24no—
88#340—24no—
89#336—22no—
90#336—22no—
91#336—21no—
92#336—21no—
93#334Q818yescomplemented
94C22×Q8—21no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C4⋊Q8—22no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—82no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C4⋊Q8SD1614yes—
115C4×Q8—42no—
116C4×Q8—42no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4C2×D412yes—
134C2×C4⋊C4C2×D412yes—
135C2×C42—22no—
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42C2411yescharacteristic
143#107—84no—
144#106—48no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—24no—
148C4⋊C8—44no—
149C4⋊C8—42no—
150C4⋊C8—42no—
151C4⋊C8C2×Q814yes—
152C4⋊C8C2×Q814yes—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—44no—
159C4×C8—42no—
160C4×C8—42no—
161C4×C8C2×Q814yes—
162#95—22no—
163#95—22no—
164C2×Q8—24no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—82no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×Q8—82no—
172C2×Q8—82no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4#12111yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4C22×D411yescharacteristic
183C22×C4C2×#4111yescharacteristic
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8C4⋊Q812yes—
204C2×C8C4⋊Q812yes—
205C4⋊C4—22no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4C2×SD1614yes—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—42no—
231C42—42no—
232C42C2×SD1612yes—
233C42#13512yes—
234C42#13512yes—
235C42#13512yes—
236C42AGL(1,ℤ/8)12yes—
237C42#13712yes—
238C42C22×Q812yes—
239C422−1+412yes—
240C23C2×#12111yescharacteristic
241Q8—48no—
242Q8—164no—
243Q8—164no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4#46312yes—
263C2×C4#46312yes—
264C2×C4#46412yes—
265C2×C4#46412yes—
266C2×C4#46512yes—
267C2×C4#46512yes—
268C2×C4#47212yes—
269C2×C4#47312yes—
270C2×C4C2×C4⋊Q811yescharacteristic
271C2×C4#53611yescharacteristic
272C2×C4D4×Q812yes—
273C2×C4#55712yes—
274C2×C4C22×SD1611yescharacteristic
275C2×C4C2×#13511yescharacteristic
276C2×C4C2×#13511yescharacteristic
277C2×C4C2×#13511yescharacteristic
278C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
279C2×C4C2×#13711yescharacteristic
280C2×C4#57712yes—
281C8—22no—
282C8—22no—
283C8—44no—
284C8—44no—
285C8—41no—
286C8—41no—
287C8—82no—
288C22#254611yescharacteristic
289C22C2×#46311yescharacteristic
290C22C2×#46411yescharacteristic
291C22C2×#46511yescharacteristic
292C22#305711yescharacteristic
293C22C2×#47211yescharacteristic
294C22C2×#47311yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—44no—
299C4—82no—
300C4—162no—
301C4—162no—
302C4#167814yes—
303C4#167914yes—
304C4#316011yescharacteristic
305C4#316511yescharacteristic
306C4#322811yescharacteristic
307C4#323111yescharacteristic
308C4Q8×SD1612yes—
309C4#338112yes—
310C2C2×#167911yescharacteristic
311C2C2×#167811yescharacteristic
312C2#1919611yescharacteristic
313C2#1921311yescharacteristic
314C2#2867711yescharacteristic
315C2#2868811yescharacteristic
316C2#2869911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
93. #334Q8semidirect

Maximal subgroups

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

subgroupindex
#332892
#292082
#286872
C8×C4⋊Q82
#106582
#98982
#98962
#96132
C4×#3342