Finite Group Databasebeta

← #324829

Subgroups — #324829

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30205C211yesmaximal, characteristic
2#29224C211yesmaximal, characteristic
3#29206C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10601C212yesmaximal
6#10600C212yesmaximal
7#9903C211yesmaximal, characteristic
8#9693C211yesmaximal, characteristic
9#9567C212yesmaximal
10#9562C212yesmaximal
11#8278C211yesmaximal, characteristic
12#2868C2211yescharacteristic
13#2862C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2101—22no—
16#2072—22no—
17#2058—22no—
18#2024—22no—
19#1993—21no—
20#1990—21no—
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1988C2212yes—
24#1987C2212yes—
25#1984—22no—
26#1956C2211yescharacteristic
27#1956C2211yescharacteristic
28#1948C2212yes—
29#1948C2212yes—
30#1939C2212yes—
31#1916—22no—
32#1844—21no—
33#1836—21no—
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8D411yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4×Q8—21no—
52C2×#106—22no—
53C2×#106—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—22no—
59C2×#103—22no—
60C2×#97—21no—
61C2×C4×C8—21no—
62C2×C4×C8—22no—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—21no—
66#391—21no—
67#391—21no—
68#391—21no—
69#391—21no—
70#389—21no—
71#389D412yes—
72#384—21no—
73#384—21no—
74#384—21no—
75#384D411yescharacteristic
76#384C2311yescharacteristic
77#384C2311yescharacteristic
78#382—21no—
79#382—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4C2312yes—
82C43C2311yescharacteristic
83#340—22no—
84#340—22no—
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#335—24no—
100#333—24no—
101#332—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—41no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—82no—
117C4⋊Q8—82no—
118C4⋊Q8—82no—
119C4⋊Q8Q1614yes—
120C4×Q8—82no—
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—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C42—22no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42C2411yescharacteristic
148#106—82no—
149#106—82no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—82no—
156C4⋊C8#4114yes—
157#103—82no—
158#103—82no—
159#97—42no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—42no—
163C4×C8—82no—
164#95—22no—
165#95—22no—
166#95—41no—
167#95—41no—
168#95—41no—
169#95—41no—
170C2×Q8—24no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—81no—
177C2×Q8—81no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8—81no—
182C2×Q8—81no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C2×#4111yescharacteristic
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—24no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C4⋊C4—22no—
212C4⋊C4—24no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C4⋊C4—82no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4C2×Q1614yes—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C42—42no—
239C42—42no—
240C42C2×Q1612yes—
241C42#13512yes—
242C42C2×#4112yes—
243C422−1+412yes—
244C23C2×#12111yescharacteristic
245Q8—84no—
246Q8—82no—
247Q8—82no—
248Q8—162no—
249Q8—164no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
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—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4#46712yes—
281C2×C4#46712yes—
282C2×C4#53211yescharacteristic
283C2×C4#53611yescharacteristic
284C2×C4#54812yes—
285C2×C4#55612yes—
286C2×C4C22×Q1611yescharacteristic
287C2×C4C2×#13511yescharacteristic
288C2×C4#57712yes—
289C2×C4#57712yes—
290C2×C4#57712yes—
291C8—44no—
292C8—42no—
293C8—42no—
294C8—82no—
295C8—82no—
296C22#254511yescharacteristic
297C22#304211yescharacteristic
298C22#304311yescharacteristic
299C22C2×#46711yescharacteristic
300C22#305711yescharacteristic
301C22#306811yescharacteristic
302C22#306911yescharacteristic
303C4—22no—
304C4—22no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—44no—
309C4—82no—
310C4—161no—
311C4—161no—
312C4—161no—
313C4—161no—
314C4#317312yes—
315C4#324412yes—
316C4#336012yes—
317C4#336312yes—
318C2#1939311yescharacteristic
319C2#1939911yescharacteristic
320C2#1951211yescharacteristic
321C2#1952011yescharacteristic
322C2#2863811yescharacteristic
323C2#2866211yescharacteristic
324C2#2867211yescharacteristic
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

Maximal subgroups

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

subgroupindex
#302052
#292242
#292062
#223072
#106012
#106002
#99032
#96932
#95672
#95622
#82782