Finite Group Databasebeta

← #325169

Subgroups — #325169

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30304C211yesmaximal, characteristic
2#29212C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10656C212yesmaximal
6#10650C212yesmaximal
7#9903C211yesmaximal, characteristic
8#9897C211yesmaximal, characteristic
9#9616C212yesmaximal
10#9614C212yesmaximal
11#8278C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2083—22no—
16#2059—22no—
17#2057—22no—
18#2057—22no—
19#1993—21no—
20#1990—21no—
21#1984—22no—
22#1948C2212yes—
23#1947C2212yes—
24#1947C2212yes—
25#1947C2212yes—
26#1939C2212yes—
27#1916—22no—
28#1847—21no—
29#1845—21no—
30#1845C2211yescharacteristic
31#1845C2211yescharacteristic
32#1834C2211yescharacteristic
33#1834C2211yescharacteristic
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8D411yescharacteristic
49C2×#125—21no—
50C2×C4×Q8—21no—
51C2×#107—22no—
52C2×#107—22no—
53C2×#107—22no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×#97—21no—
62C2×C4×C8—21no—
63C2×C4×C8—22no—
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#391—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389D412yes—
73#384—21no—
74#384—21no—
75#384—21no—
76#384D411yescharacteristic
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#384C2311yescharacteristic
80#384C2311yescharacteristic
81#382—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4C2312yes—
84C43C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—21no—
94#336—21no—
95#336—21no—
96#336—21no—
97#336D412yes—
98#336D412yes—
99#335Q818yescomplemented
100C22×Q8—21no—
101C22×Q8—41no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—22no—
108C4⋊Q8—82no—
109C4⋊Q8—82no—
110C4⋊Q8—82no—
111C4⋊Q8Q1614yes—
112#125—82no—
113C4×Q8—82no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D412yes—
137C2×C42—22no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42C2411yescharacteristic
145#107—82no—
146#107—82no—
147#107—82no—
148#107—82no—
149#107—82no—
150#106—82no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—24no—
154C4⋊C8—82no—
155C4⋊C8C2×Q814yes—
156C4⋊C8C2×Q814yes—
157#97—42no—
158C4×C8—22no—
159C4×C8—42no—
160C4×C8—82no—
161C4×C8C2×Q814yes—
162#95—22no—
163#95—22no—
164#95—41no—
165#95—41no—
166C2×Q8—24no—
167C2×Q8—81no—
168C2×Q8—81no—
169C2×Q8—82no—
170C2×Q8—82no—
171C2×Q8—82no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4C2×#4111yescharacteristic
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—24no—
189C2×C8—24no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8C4⋊Q814yes—
200C4⋊C4—22no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4C2×Q1614yes—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—41no—
229C42—41no—
230C42—42no—
231C42—42no—
232C42C2×Q1612yes—
233C42#13512yes—
234C42C22×Q812yes—
235C422−1+412yes—
236C23C2×#12111yescharacteristic
237Q8—84no—
238Q8—164no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4#46712yes—
270C2×C4#46712yes—
271C2×C4C2×C4⋊Q811yescharacteristic
272C2×C4#53611yescharacteristic
273C2×C4D4×Q812yes—
274C2×C4#55712yes—
275C2×C4C22×Q1611yescharacteristic
276C2×C4C2×#13511yescharacteristic
277C2×C4#57612yes—
278C2×C4#57612yes—
279C2×C4#57812yes—
280C8—24no—
281C8—44no—
282C8—44no—
283C8—42no—
284C8—82no—
285C22#254611yescharacteristic
286C22#304111yescharacteristic
287C22#304411yescharacteristic
288C22C2×#46711yescharacteristic
289C22#305811yescharacteristic
290C22#306811yescharacteristic
291C22#306911yescharacteristic
292C4—22no—
293C4—22no—
294C4—22no—
295C4—41no—
296C4—41no—
297C4—44no—
298C4—82no—
299C4—161no—
300C4—161no—
301C4—161no—
302C4—161no—
303C4#316412yes—
304C4#324312yes—
305C4#338012yes—
306C4Q8×Q1612yes—
307C2#1938711yescharacteristic
308C2#1940511yescharacteristic
309C2#1950611yescharacteristic
310C2#1953011yescharacteristic
311C2#2867611yescharacteristic
312C2#2869711yescharacteristic
313C2#2870311yescharacteristic
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
99. #335Q8semidirect

Maximal subgroups

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

subgroupindex
#303042
#292122
#292082
#223072
#106562
#106502
#99032
#98972
#96162
#96142
#82782