Finite Group Databasebeta

← #315294

Subgroups — #315294

Order 512.

505 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28830C211yesmaximal, characteristic
2#28682C211yesmaximal, characteristic
3#27738C212yesmaximal
4#10414C212yesmaximal
5#10352C212yesmaximal
6#9836C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9576C212yesmaximal
9#8651C211yesmaximal, characteristic
10#8286C211yesmaximal, characteristic
11#8211C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#446C2211yescharacteristic
14#2571C2212yes—
15#2567C2212yes—
16#2099—22no—
17#2086—22no—
18#2085—22no—
19#2074—22no—
20#1990C2212yes—
21#1987C2212yes—
22#1987C2212yes—
23#1956C2212yes—
24#1947C2212yes—
25#1944—21no—
26#1944—21no—
27#1941—21no—
28#1935—22no—
29#1846—21no—
30#1844C2211yescharacteristic
31#1836—22no—
32#1803C2212yes—
33C4×#106C2211yescharacteristic
34#1770C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#333C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×#322C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8C2311yescharacteristic
47#500—22no—
48#500—22no—
49#446—22no—
50#446—22no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—22no—
54C2×#106—21no—
55C2×#106—21no—
56C2×#106D411yescharacteristic
57C2×#106D411yescharacteristic
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—22no—
63C2×#103—42no—
64C2×#103D412yes—
65C2×#97—22no—
66C2×#97—21no—
67C2×C4×C8—21no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#400—42no—
72#395—42no—
73#391—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#384—22no—
79#384C2312yes—
80#382—22no—
81#382—22no—
82C4×C4⋊C4C2312yes—
83C4×C4⋊C4C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340D412yes—
93#340D412yes—
94#336—22no—
95#336—21no—
96#335—22no—
97#333—24no—
98Q8⋊C8—22no—
99#322—22no—
100C22×Q8—22no—
101C22×Q8C2×D411yescharacteristic
102C22×C8—21no—
103C22×C8—22no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8C2×D412yes—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8—21no—
116C4×Q8—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4C2×D411yescharacteristic
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—22no—
133C2×C42C2411yescharacteristic
134#107—42no—
135#107—42no—
136#107—82no—
137#106—22no—
138#106—22no—
139#106—42no—
140#106—81no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—42no—
147C4⋊C8#4112yes—
148#103—42no—
149#103—44no—
150#103—84no—
151#97—44no—
152#97—42no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—42no—
158C4×C8#4112yes—
159C4×C8#4112yes—
160#95—22no—
161#95—21no—
162#95—42no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—42no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—82no—
174C2×Q8—82no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4C22×D411yescharacteristic
183C22×C4C22×D411yescharacteristic
184C22×C4H(5,𝔽2)11yescharacteristic
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8#12412yes—
202C2×C8#12412yes—
203C2×C8#13712yes—
204C2×C8#13712yes—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—162no—
225C4⋊C4—162no—
226C4⋊C4—162no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—42no—
231C42—42no—
232C42—41no—
233C42C2×#4111yescharacteristic
234C42C2×#4111yescharacteristic
235C42C2×#4111yescharacteristic
236C422−1+411yescharacteristic
237C23D4211yescharacteristic
238Q8—42no—
239Q8—41no—
240Q8—41no—
241Q8—84no—
242Q8—162no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4C2×#12411yescharacteristic
264C2×C4#53211yescharacteristic
265C2×C4#54211yescharacteristic
266C2×C4#54711yescharacteristic
267C2×C4#54811yescharacteristic
268C2×C4#55511yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4C2×#13711yescharacteristic
271C2×C4#57512yes—
272C2×C4#57711yescharacteristic
273C2×C4#57812yes—
274C2×C4#57811yescharacteristic
275C8—22no—
276C8—42no—
277C8—42no—
278C8—81no—
279C8—81no—
280C8—82no—
281C8#49114yes—
282C22#249111yescharacteristic
283C22#326912yes—
284C22#327611yescharacteristic
285C22#327811yescharacteristic
286C22#328812yes—
287C4—21no—
288C4—21no—
289C4—82no—
290C4—82no—
291C4—82no—
292C4—81no—
293C4—81no—
294C4—162no—
295C4—162no—
296C4C2×#49111yescharacteristic
297C4#313711yescharacteristic
298C4#317711yescharacteristic
299C4#318311yescharacteristic
300C4#335212yes—
301C4#337211yescharacteristic
302C4#337411yescharacteristic
303C2#2022912yes—
304C2#2023612yes—
305C2#2777711yescharacteristic
306C2#2780611yescharacteristic
307C2#2781111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#288302
#286822
#277382
#104142
#103522
#98362
#97302
#95762
#86512
#82862
#82112