Finite Group Databasebeta

← #307360

Subgroups — #307360

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28531C211yesmaximal, characteristic
2#26279C212yesmaximal
3#22808C211yesmaximal, characteristic
4#10482C212yesmaximal
5#10414C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9656C211yesmaximal, characteristic
9#9614C211yesmaximal, characteristic
10#8974C212yesmaximal
11C8×#106C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C2×C4×Q16C2211yescharacteristic
15#2567C2212yes—
16#2355C2212yes—
17#2101—22no—
18#2095—22no—
19#2085—22no—
20#1990C2212yes—
21#1987C2212yes—
22#1984C2212yes—
23#1969—21no—
24#1969—21no—
25#1956C2212yes—
26#1947C2212yes—
27#1928—22no—
28#1919—22no—
29#1918—22no—
30#1845—21no—
31#1845C2211yescharacteristic
32#1803C2212yes—
33C4×#107—21no—
34C4×#106C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#334C2211yescharacteristic
39C2×#333C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×C82C2211yescharacteristic
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1279C2211yescharacteristic
45C22×Q16D411yescharacteristic
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8C2312yes—
48#500—22no—
49#500—22no—
50#477—24no—
51#477—42no—
52C4×Q16—22no—
53C4×Q16—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—21no—
60C2×#107D411yescharacteristic
61C2×#106D411yescharacteristic
62C2×#106D411yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—42no—
69C2×#103D412yes—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#395—42no—
76#391—22no—
77#391—22no—
78#389—22no—
79#389—22no—
80#387—42no—
81#384—22no—
82#384C2312yes—
83#382—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4C2311yescharacteristic
87C4×C4⋊C4C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340D411yescharacteristic
95#340D411yescharacteristic
96#336—22no—
97#336—21no—
98#336—22no—
99#336—21no—
100#335—22no—
101#334—22no—
102#333—42no—
103Q8⋊C8—42no—
104C82—22no—
105C22×Q8—22no—
106C22×Q8C2×D412yes—
107C2×Q16—22no—
108C2×Q16—22no—
109C2×Q16—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—42no—
143C2×C42C2411yescharacteristic
144#107—22no—
145#107—44no—
146#107—42no—
147#107—42no—
148#107—42no—
149#107—44no—
150#107—81no—
151#106—22no—
152#106Q1614yescomplemented
153C4⋊C8—21no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8—22no—
157C4⋊C8—21no—
158C4⋊C8#4112yes—
159#103—24no—
160#103—44no—
161#103—82no—
162#103—84no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—42no—
168C4×C8—42no—
169C4×C8#4112yes—
170C4×C8#4112yes—
171#95—21no—
172#95—21no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—82no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4H(5,𝔽2)11yescharacteristic
199Q16—44no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—21no—
206C2×C8—22no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8#12312yes—
220C2×C8#12312yes—
221C2×C8C2×Q1612yes—
222C2×C8#13512yes—
223C2×C8#13712yes—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—162no—
243C4⋊C4—162no—
244C4⋊C4C2×Q1612yes—
245C4⋊C4C2×Q1612yes—
246C4⋊C4#13512yes—
247C42—42no—
248C42—42no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—162no—
254C42C2×#4111yescharacteristic
255C42C2×#4111yescharacteristic
256C42C2×#4111yescharacteristic
257C42H(5,𝔽2)11yescharacteristic
258C23D4211yescharacteristic
259Q8—42no—
260Q8—42no—
261Q8—42no—
262Q8—42no—
263Q8—84no—
264Q8—162no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4C2×#12311yescharacteristic
292C2×C4#53211yescharacteristic
293C2×C4#54011yescharacteristic
294C2×C4#54611yescharacteristic
295C2×C4#55011yescharacteristic
296C2×C4#55911yescharacteristic
297C2×C4C22×Q1611yescharacteristic
298C2×C4C2×#13511yescharacteristic
299C2×C4C2×#13511yescharacteristic
300C2×C4C2×#13711yescharacteristic
301C2×C4#57711yescharacteristic
302C2×C4#57811yescharacteristic
303C2×C4#57811yescharacteristic
304C2×C4#57811yescharacteristic
305C8—22no—
306C8—22no—
307C8—42no—
308C8—41no—
309C8—42no—
310C8—81no—
311C8—82no—
312C8#48514yes—
313C22#244511yescharacteristic
314C22#327611yescharacteristic
315C22#328411yescharacteristic
316C22#328511yescharacteristic
317C22D4×Q1611yescharacteristic
318C22#329011yescharacteristic
319C22#329211yescharacteristic
320C4—21no—
321C4—21no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—82no—
326C4—82no—
327C4—81no—
328C4—162no—
329C4—162no—
330C4C2×#48511yescharacteristic
331C4#309711yescharacteristic
332C4#317011yescharacteristic
333C4#317711yescharacteristic
334C4#330211yescharacteristic
335C4#332911yescharacteristic
336C4#339311yescharacteristic
337C4#339511yescharacteristic
338C2#1993011yescharacteristic
339C2#1994011yescharacteristic
340C2#2000711yescharacteristic
341C2#2001211yescharacteristic
342C2#2623311yescharacteristic
343C2#2625511yescharacteristic
344C2#2627211yescharacteristic
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

Complements

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

NH ≅ G/Ntype
152. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#285312
#262792
#228082
#104822
#104142
#98232
#97302
#96562
#96142
#89742
C8×#1062