Finite Group Databasebeta

← #325166

Subgroups — #325166

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30304C211yesmaximal, characteristic
2#28693C211yesmaximal, characteristic
3#28689C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10655C212yesmaximal
6#10649C212yesmaximal
7#9905C211yesmaximal, characteristic
8#9899C211yesmaximal, characteristic
9#9616C212yesmaximal
10#9614C212yesmaximal
11#8278C211yesmaximal, characteristic
12C2×#501—21no—
13C2×#500—21no—
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2085—22no—
18#2083—22no—
19#2083—22no—
20#2057—22no—
21#1992—21no—
22#1990—21no—
23#1984—22no—
24#1948C2212yes—
25#1947C2212yes—
26#1947C2212yes—
27#1947C2212yes—
28#1939C2212yes—
29#1916—22no—
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—
51#501—44no—
52#500—44no—
53C2×#107—22no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#97—21no—
64C2×C4×C8—21no—
65C2×C4×C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389D412yes—
75#384—21no—
76#384—21no—
77#384—21no—
78#384D411yescharacteristic
79#384C2311yescharacteristic
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#382—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2312yes—
86C43C2311yescharacteristic
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#335Q818yescomplemented
102C22×Q8—21no—
103C22×Q8—41no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C4⋊Q8—22no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C4⋊Q8D814yes—
115#125—42no—
116#125—42no—
117C4×Q8—82no—
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—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D412yes—
141C2×C42—22no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42C2411yescharacteristic
149#107—44no—
150#107—44no—
151#107—44no—
152#107—82no—
153#107—82no—
154#106—44no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—24no—
158C4⋊C8—82no—
159C4⋊C8C2×Q814yes—
160C4⋊C8C2×Q814yes—
161#97—42no—
162C4×C8—22no—
163C4×C8—42no—
164C4×C8—82no—
165C4×C8C2×Q814yes—
166#95—22no—
167#95—22no—
168#95—41no—
169#95—41no—
170C2×Q8—24no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—82no—
174C2×Q8—82no—
175C2×Q8—82no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4C22×D411yescharacteristic
188C22×C4C2×#4111yescharacteristic
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8C4⋊Q814yes—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4C2×D814yes—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—41no—
236C42—41no—
237C42—42no—
238C42—42no—
239C42C2×D812yes—
240C42#13512yes—
241C42C22×Q812yes—
242C422−1+412yes—
243C23C2×#12111yescharacteristic
244Q8—84no—
245Q8—164no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4#46612yes—
277C2×C4#46612yes—
278C2×C4C2×C4⋊Q811yescharacteristic
279C2×C4#53611yescharacteristic
280C2×C4D4×Q812yes—
281C2×C4#55712yes—
282C2×C4C22×D811yescharacteristic
283C2×C4C2×#13511yescharacteristic
284C2×C4#57612yes—
285C2×C4#57812yes—
286C2×C4#57812yes—
287C8—24no—
288C8—44no—
289C8—44no—
290C8—42no—
291C8—82no—
292C22#254611yescharacteristic
293C22#304111yescharacteristic
294C22#304411yescharacteristic
295C22C2×#46611yescharacteristic
296C22#305911yescharacteristic
297C22#306811yescharacteristic
298C22#306911yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—44no—
305C4—82no—
306C4—161no—
307C4—161no—
308C4—161no—
309C4—161no—
310C4#316412yes—
311C4#324312yes—
312C4Q8×D812yes—
313C4#338312yes—
314C2#1938611yescharacteristic
315C2#1940411yescharacteristic
316C2#1950711yescharacteristic
317C2#1953111yescharacteristic
318C2#2867511yescharacteristic
319C2#2869711yescharacteristic
320C2#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
101. #335Q8semidirect

Maximal subgroups

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

subgroupindex
#303042
#286932
#286892
#223072
#106552
#106492
#99052
#98992
#96162
#96142
#82782