Finite Group Databasebeta

← #352755

Subgroups — #352755

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33314C211yesmaximal, characteristic
2#33297C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#33278C211yesmaximal, characteristic
5#30220C211yesmaximal, characteristic
6#30205C211yesmaximal, characteristic
7#9834C212yesmaximal
8#9831C212yesmaximal
9#9828C212yesmaximal
10#9828C212yesmaximal
11#7842C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#498C2212yes—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17#2859C2211yescharacteristic
18#2072—22no—
19#2043—22no—
20#2043—22no—
21#2024—22no—
22#1993—22no—
23#1992—22no—
24#1990—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1956C2212yes—
31#1956C2212yes—
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2212yes—
35#1933—22no—
36#1933—22no—
37#1847—22no—
38#1847—22no—
39#1834—22no—
40#1834C2212yes—
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#498—24no—
59#498—24no—
60C2×#107—24no—
61C2×#106—24no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#97D412yes—
68C2×#97D412yes—
69C2×C4×C8D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8C2312yes—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2312yes—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2311yescharacteristic
91#384C2311yescharacteristic
92#384C2312yes—
93#382—21no—
94#382—21no—
95#382—21no—
96C43C2311yescharacteristic
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#336D412yes—
105#336D412yes—
106#322—24no—
107#322—24no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—41no—
112C22×C8C2×D414yes—
113C22×C8C2×D412yes—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—82no—
127C4⋊Q8—82no—
128C4⋊Q8—82no—
129#125—44no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
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—41no—
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×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2411yescharacteristic
161#107—48no—
162#106—48no—
163#103—44no—
164#103—44no—
165#103—84no—
166#103—84no—
167#103—84no—
168#97—24no—
169#97—24no—
170C4×C8—24no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—42no—
176C4×C8#4114yes—
177C4×C8#4114yes—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183#95—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#12711yescharacteristic
205C22×C4#12711yescharacteristic
206C22×C4#12711yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C2×C8—22no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8#12418yes—
222C2×C8#13714yes—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42#13714yes—
255C42#13714yes—
256C42C2×#4112yes—
257C42C2×#4112yes—
258C23C2×#12711yescharacteristic
259Q8—82no—
260Q8—84no—
261Q8—84no—
262Q8—84no—
263Q8—82no—
264Q8—164no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4#49712yes—
292C2×C4#49712yes—
293C2×C4#49712yes—
294C2×C4#49712yes—
295C2×C4C2×#12412yes—
296C2×C4#53211yescharacteristic
297C2×C4#53212yes—
298C2×C4#53211yescharacteristic
299C2×C4C2×#13712yes—
300C2×C4C2×#13712yes—
301C2×C4C2×#13712yes—
302C2×C4#57512yes—
303C2×C4#57612yes—
304C2×C4#57712yes—
305C8—28no—
306C8—24no—
307C8—42no—
308C8—44no—
309C22#285811yescharacteristic
310C22C2×#49711yescharacteristic
311C22C2×#49711yescharacteristic
312C22#315211yescharacteristic
313C22#315311yescharacteristic
314C22#315411yescharacteristic
315C22#315511yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—162no—
321C4—161no—
322C4—162no—
323C4—161no—
324C4—161no—
325C4—161no—
326C4#313914yes—
327C4#317212yes—
328C4#317914yes—
329C4#318112yes—
330C2#1968011yescharacteristic
331C2#1968211yescharacteristic
332C2#1968311yescharacteristic
333C2#1968511yescharacteristic
334C2#3323611yescharacteristic
335C2#3324311yescharacteristic
336C2#3325011yescharacteristic
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
#333142
#332972
#332892
#332782
#302202
#302052
#98342
#98312
#98282
#78422