Finite Group Databasebeta

← #325384

Subgroups — #325384

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33291C211yesmaximal, characteristic, complemented
2#29131C212yesmaximal
3C8×#124C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10625C212yesmaximal
6#9907C212yesmaximal, complemented
7#8649C211yesmaximal, characteristic, complemented
8#8649C211yesmaximal, characteristic, complemented
9#8648C211yesmaximal, characteristic
10#8648C211yesmaximal, characteristic
11#8274C211yesmaximal, characteristic, complemented
12C2×C8×Q8C2211yescharacteristic
13C2×C8×D4C2211yescharacteristic
14#2859C2212yes—
15C4×#124C2211yescharacteristic
16#2101—22no—
17#2042—22no—
18#1991—22no—
19#1991—22no—
20#1984C2211yescharacteristic
21#1954C2211yescharacteristic
22#1936—22no—
23#1935—22no—
24#1923—22no—
25#1834C2212yes—
26#1834C2211yescharacteristic
27#1834C2211yescharacteristic
28#1831C2211yescharacteristic
29#1803C2212yes—
30#1803C2212yes—
31#1802C2212yes—
32#1802C2212yes—
33#1767C2212yes—
34C8×C22⋊C4—22no—
35C42×C8C2211yescharacteristic
36C2×#335C2211yescharacteristic
37C2×#335C2211yescharacteristic
38#1296C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1277C2212yes—
43C23×C8—21no—
44C2×#125—22no—
45C2×#124D411yescharacteristic
46C2×C4×Q8C2311yescharacteristic
47C2×C4×D4C2311yescharacteristic
48C8×Q8—22no—
49C8×Q8—22no—
50C8×D4—22no—
51C8×D4—22no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2311yescharacteristic
56C2×C4⋊C8C2311yescharacteristic
57C2×#103—22no—
58C2×#103—22no—
59C2×#102—22no—
60C2×#102—22no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64C2×C4×C8—21no—
65C2×C4×C8D412yes—
66C2×C4×C8D412yes—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391D411yescharacteristic
70#390D411yescharacteristic
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#388—22no—
76#384C2312yes—
77#384C2312yes—
78#382—22no—
79#382D411yescharacteristic
80C4×C22⋊C4—22no—
81C43C2311yescharacteristic
82#340—24no—
83#340—21no—
84#340—21no—
85#340—21no—
86#340—21no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#335—22no—
95#335—22no—
96C22×Q8C2×D411yescharacteristic
97C22×D4C2×D411yescharacteristic
98C23×C4—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—44no—
104C22×C8—42no—
105C22×C8C2×D412yes—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108#125—44no—
109#125—44no—
110#124—22no—
111#124—22no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8—21no—
115C4×Q8—21no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—21no—
134C2×C42—21no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42#4112yes—
139C2×C42C2411yescharacteristic
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—42no—
149#103—44no—
150#103—82no—
151#102—44no—
152#102—82no—
153C22⋊C8—44no—
154C22⋊C8—42no—
155#97—82no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—42no—
159C4×C8—42no—
160C4×C8—42no—
161C4×C8#4112yes—
162C4×C8#4112yes—
163#95—22no—
164#95—22no—
165#95—42no—
166#95—42no—
167C24—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8#13512yes—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4#13512yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4#12112yes—
190C22×C4#12112yes—
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C2×#4111yescharacteristic
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—84no—
210C2×C8#12412yes—
211C2×C8#12412yes—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—28no—
231C22⋊C4—44no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C42—21no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42—41no—
242C42—41no—
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C422−1+411yescharacteristic
246C422−1+411yescharacteristic
247C23—22no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23C2×#12111yescharacteristic
253Q8—22no—
254Q8—22no—
255Q8—41no—
256Q8—41no—
257D4—22no—
258D4—22no—
259D4—41no—
260D4—41no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C2×#12411yescharacteristic
285C2×C4#53611yescharacteristic
286C2×C4#54711yescharacteristic
287C2×C4#54711yescharacteristic
288C2×C4#55611yescharacteristic
289C2×C4#55611yescharacteristic
290C2×C4C2×#13511yescharacteristic
291C2×C4C2×#13511yescharacteristic
292C2×C4#57512yes—
293C2×C4#57512yes—
294C2×C4#57611yescharacteristic
295C2×C4#57811yescharacteristic
296C8—22no—
297C8—22no—
298C8—42no—
299C8—82no—
300C8—82no—
301C8—81no—
302C8—81no—
303C22—42no—
304C22—44no—
305C22—81no—
306C22—81no—
307C22—82no—
308C22#254711yescharacteristic
309C22#304012yes—
310C22#305011yescharacteristic
311C22#306111yescharacteristic
312C22#306712yes—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—162no—
321C4—162no—
322C4#313511yescharacteristic
323C4#314011yescharacteristic
324C4#324012yes—
325C4#333611yescharacteristic
326C4#333611yescharacteristic
327C4#333911yescharacteristic
328C4#333911yescharacteristic
329C2—82no—
330C2#1927812yes—
331C2#1936712yes—
332C2#2870811yescharacteristic
333C2#2872711yescharacteristic
334C2#2873311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
1. #33291C2semidirect
4. #10626C2semidirect
6. #9907C2semidirect
7. #8649C2semidirect
8. #8649C2semidirect
11. #8274C2semidirect

Maximal subgroups

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

subgroupindex
#332912
#291312
C8×#1242
#106262
#106252
#99072
#86492
#86482
#82742