Finite Group Databasebeta

← #323354

Subgroups — #323354

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33310C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29218C211yesmaximal, characteristic
4#29208C211yesmaximal, characteristic
5#29141C211yesmaximal, characteristic
6C4×#487C211yesmaximal, characteristic
7#10681C212yesmaximal
8#10675C212yesmaximal
9#9826C212yesmaximal
10#8821C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#498—22no—
13C2×#487C2211yescharacteristic
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18#2100—22no—
19#2100—22no—
20#2057—22no—
21#2024—22no—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2211yescharacteristic
26#1987—22no—
27#1987—22no—
28#1956C2212yes—
29#1956C2211yescharacteristic
30#1947—22no—
31#1933—22no—
32#1845C2212yes—
33#1844—22no—
34#1844C2212yes—
35#1844C2212yes—
36#1834C2212yes—
37#1834C2212yes—
38#1834C2211yescharacteristic
39#1834C2212yes—
40#1803—22no—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×#103—22no—
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57#498—48no—
58#487—22no—
59#487Q814yescomplemented
60C2×#107—24no—
61C2×#106—24no—
62C2×#106—24no—
63C2×C4⋊C8D412yes—
64C2×C4⋊C8D412yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—24no—
68C2×C4×C8D412yes—
69C2×C4×C8D412yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—21no—
75#391—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#384C2312yes—
89#384C2312yes—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C43C2311yescharacteristic
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—24no—
101#340—24no—
102#340—22no—
103#336D412yes—
104#336D412yes—
105#322—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×C8C2×D414yes—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113C22×C8C2×D412yes—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—44no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124C4⋊Q8—82no—
125C4⋊Q8—82no—
126C4⋊Q8C2×Q812yes—
127C4⋊Q8C2×Q812yes—
128C4×Q8—44no—
129C4×Q8—44no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2411yescharacteristic
163#107—84no—
164#106—48no—
165#106—48no—
166C4⋊C8—24no—
167C4⋊C8—24no—
168#103—28no—
169#103—44no—
170#103—84no—
171C4×C8—24no—
172C4×C8—24no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8C2×Q812yes—
180C4×C8#4112yes—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4#12711yescharacteristic
206C22×C4#12711yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4#12711yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8C22⋊Q814yes—
223C2×C8AGL(1,ℤ/8)14yes—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—82no—
256C42—82no—
257C42#13514yes—
258C42#13714yes—
259C42#13714yes—
260C42C22×Q811yescharacteristic
261C42C2×#4111yescharacteristic
262C422−1+411yescharacteristic
263C422−1+411yescharacteristic
264C23C2×#12711yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—82no—
268Q8—82no—
269Q8—164no—
270Q8—164no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4#49512yes—
296C2×C4#49512yes—
297C2×C4#49712yes—
298C2×C4#49712yes—
299C2×C4#49712yes—
300C2×C4#49712yes—
301C2×C4C2×C22⋊Q811yescharacteristic
302C2×C4#53611yescharacteristic
303C2×C4#54711yescharacteristic
304C2×C4#54711yescharacteristic
305C2×C4D4×Q811yescharacteristic
306C2×C4D4×Q811yescharacteristic
307C2×C4C2×#13512yes—
308C2×C4C2×AGL(1,ℤ/8)12yes—
309C2×C4C2×#13712yes—
310C2×C4C2×#13712yes—
311C2×C4#57712yes—
312C2×C4#57712yes—
313C8—24no—
314C8—24no—
315C8—44no—
316C8—44no—
317C22#254211yescharacteristic
318C22C2×#49511yescharacteristic
319C22C2×#49711yescharacteristic
320C22C2×#49711yescharacteristic
321C22#315111yescharacteristic
322C22#315411yescharacteristic
323C22#315411yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—82no—
328C4—82no—
329C4—161no—
330C4—162no—
331C4—162no—
332C4—161no—
333C4#171814yes—
334C4#307912yes—
335C4#323812yes—
336C4#334012yes—
337C4#335612yes—
338C4#338112yes—
339C4#338712yes—
340C2C2×#171811yescharacteristic
341C2#1962511yescharacteristic
342C2#1967911yescharacteristic
343C2#1970311yescharacteristic
344C2#2851911yescharacteristic
345C2#2852911yescharacteristic
346C2#2854211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
59. #487Q8semidirect

Maximal subgroups

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

subgroupindex
#333102
#292182
#292082
#291412
C4×#4872
#106812
#106752
#98262
#88212
#82612