Finite Group Databasebeta

← #352741

Subgroups — #352741

Order 512.

682 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33294C211yesmaximal, characteristic
2#33293C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic
4#33225C211yesmaximal, characteristic
5#30305C211yesmaximal, characteristic
6#30279C211yesmaximal, characteristic
7#9840C212yesmaximal
8#9836C212yesmaximal
9#9831C212yesmaximal
10#9826C212yesmaximal
11#7842C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#487—22no—
14C2×#487C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18#2859C2211yescharacteristic
19#2095—22no—
20#2095—22no—
21#2084—22no—
22#2058—22no—
23#1993—22no—
24#1992—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1956—22no—
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2212yes—
36#1933—22no—
37#1933—22no—
38#1844C2212yes—
39#1836—22no—
40#1836—22no—
41#1834C2212yes—
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×#322C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2312yes—
55C2×#125—21no—
56#500—24no—
57#500—24no—
58#487—24no—
59#487—24no—
60#487—48no—
61C2×#107—24no—
62C2×#106—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#97D412yes—
69C2×#97D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8D412yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384C2311yescharacteristic
92#384C2312yes—
93#384C2311yescharacteristic
94#382—21no—
95#382—21no—
96#382—21no—
97C43C2311yescharacteristic
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—22no—
104#340—22no—
105#336D412yes—
106#336D412yes—
107#322—24no—
108#322—24no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—41no—
113C22×C8C2×D414yes—
114C22×C8C2×D412yes—
115C22×C8C2×D414yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—82no—
130#125—44no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
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×C4⋊C4—41no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2411yescharacteristic
162#107—48no—
163#106—84no—
164#103—44no—
165#103—48no—
166#103—48no—
167#103—44no—
168#103—48no—
169#97—24no—
170#97—24no—
171C4×C8—24no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—42no—
177C4×C8#4114yes—
178C4×C8#4114yes—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—44no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—42no—
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—24no—
215C2×C8—24no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—22no—
222C2×C8#12418yes—
223C2×C8AGL(1,ℤ/8)14yes—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42#13714yes—
256C42#13714yes—
257C42C2×#4112yes—
258C42C2×#4112yes—
259C23C2×#12711yescharacteristic
260Q8—84no—
261Q8—84no—
262Q8—82no—
263Q8—84no—
264Q8—82no—
265Q8—164no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4#49712yes—
293C2×C4#49712yes—
294C2×C4#49712yes—
295C2×C4#49712yes—
296C2×C4C2×#12412yes—
297C2×C4#53211yescharacteristic
298C2×C4#53211yescharacteristic
299C2×C4#53212yes—
300C2×C4C2×AGL(1,ℤ/8)12yes—
301C2×C4C2×#13712yes—
302C2×C4C2×#13712yes—
303C2×C4#57512yes—
304C2×C4#57712yes—
305C2×C4#57812yes—
306C8—28no—
307C8—24no—
308C8—42no—
309C8—44no—
310C22#285811yescharacteristic
311C22C2×#49711yescharacteristic
312C22C2×#49711yescharacteristic
313C22#315111yescharacteristic
314C22#315311yescharacteristic
315C22#315411yescharacteristic
316C22#315611yescharacteristic
317C4—22no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—161no—
322C4—162no—
323C4—162no—
324C4—161no—
325C4—161no—
326C4—161no—
327C4#313714yes—
328C4#317112yes—
329C4#318012yes—
330C4#318514yes—
331C2#1967911yescharacteristic
332C2#1968111yescharacteristic
333C2#1968411yescharacteristic
334C2#1968511yescharacteristic
335C2#3322911yescharacteristic
336C2#3325111yescharacteristic
337C2#3325511yescharacteristic
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

Maximal subgroups

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

subgroupindex
#332942
#332932
#332782
#332252
#303052
#302792
#98402
#98362
#98312
#98262
#78422