Finite Group Databasebeta

← #352852

Subgroups — #352852

Order 512.

666 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#33225C211yesmaximal, characteristic
3#30304C211yesmaximal, characteristic
4#30280C211yesmaximal, characteristic
5#30228C211yesmaximal, characteristic
6#30205C211yesmaximal, characteristic
7#9841C212yesmaximal
8#9840C212yesmaximal
9#9833C212yesmaximal
10#9828C212yesmaximal
11#7843C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#487—22no—
14C2×#487C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18#2859C2211yescharacteristic
19#2099—22no—
20#2099—22no—
21#2096—22no—
22#2096—22no—
23#2083—22no—
24#2072—22no—
25#2057—22no—
26#2024—22no—
27#1991—22no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2212yes—
33#1990C2212yes—
34#1990C2212yes—
35#1956—22no—
36#1956C2212yes—
37#1956C2212yes—
38#1956C2212yes—
39#1845C2212yes—
40#1835—22no—
41#1834C2212yes—
42#1744C2211yescharacteristic
43#1744C2211yescharacteristic
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#498—24no—
57#498—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—24no—
67C2×C4×C8D412yes—
68C2×C4×C8D412yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2312yes—
72C2×C4×C8C2312yes—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384C2311yescharacteristic
89#384C2312yes—
90#384C2311yescharacteristic
91#382—21no—
92#382—21no—
93#382—21no—
94C43C2311yescharacteristic
95#340—24no—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#336D412yes—
104#336D412yes—
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×D412yes—
116C22×C8C2×D414yes—
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—82no—
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—84no—
163#106—48no—
164#103—48no—
165#103—48no—
166#103—48no—
167#103—84no—
168C4×C8—24no—
169C4×C8—24no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8#4114yes—
175C4×C8#4114yes—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—82no—
193C2×Q8—82no—
194C2×Q8—82no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4#12711yescharacteristic
202C22×C4#12711yescharacteristic
203C22×C4#12711yescharacteristic
204C22×C4#12711yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C22×D411yescharacteristic
208C2×C8—24no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8#12418yes—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—44no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42AGL(1,ℤ/8)14yes—
253C42#13714yes—
254C42C2×#4112yes—
255C42C2×#4112yes—
256C23C2×#12711yescharacteristic
257Q8—84no—
258Q8—84no—
259Q8—84no—
260Q8—84no—
261Q8—164no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4#49612yes—
289C2×C4#49612yes—
290C2×C4#49712yes—
291C2×C4#49712yes—
292C2×C4C2×#12412yes—
293C2×C4#53212yes—
294C2×C4#53211yescharacteristic
295C2×C4#53211yescharacteristic
296C2×C4C2×AGL(1,ℤ/8)12yes—
297C2×C4C2×#13712yes—
298C2×C4#57712yes—
299C2×C4#57712yes—
300C2×C4#57812yes—
301C2×C4#57812yes—
302C8—28no—
303C8—44no—
304C8—42no—
305C8—42no—
306C22#285811yescharacteristic
307C22C2×#49611yescharacteristic
308C22C2×#49711yescharacteristic
309C22#315411yescharacteristic
310C22#315411yescharacteristic
311C22#315611yescharacteristic
312C22#315611yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—161no—
318C4—161no—
319C4—162no—
320C4—161no—
321C4—162no—
322C4—161no—
323C4#313714yes—
324C4#317814yes—
325C4#318212yes—
326C4#318212yes—
327C2#1969711yescharacteristic
328C2#1969911yescharacteristic
329C2#1970611yescharacteristic
330C2#1970611yescharacteristic
331C2#3322911yescharacteristic
332C2#3324411yescharacteristic
333C2#3325211yescharacteristic
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
#332902
#332252
#303042
#302802
#302282
#302052
#98412
#98402
#98332
#98282
#78432