Finite Group Databasebeta

← #325721

Subgroups — #325721

Order 512.

536 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#30279C211yesmaximal, characteristic
2#29224C211yesmaximal, characteristic
3#29218C211yesmaximal, characteristic
4#22311C211yesmaximal, characteristic
5#10543C212yesmaximal
6#10541C212yesmaximal
7#9745C212yesmaximal
8#9739C212yesmaximal
9#9702C211yesmaximal, characteristic
10#9693C211yesmaximal, characteristic
11#8277C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#487C2211yescharacteristic
15C2×#487C2211yescharacteristic
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2101—22no—
20#2100—22no—
21#2095—22no—
22#2095—22no—
23#1993—21no—
24#1990—21no—
25#1988C2212yes—
26#1987C2212yes—
27#1987C2212yes—
28#1987C2212yes—
29#1985—22no—
30#1939C2212yes—
31#1917—22no—
32#1836—21no—
33#1834—21no—
34#1831C2211yescharacteristic
35#1770C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53#500—24no—
54#500—42no—
55#487—24no—
56#487—42no—
57C2×#107—22no—
58C2×#107—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×#97—21no—
66C2×#97—22no—
67C2×C4×C8—21no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#389—21no—
76#389D412yes—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384D411yescharacteristic
82#382—21no—
83#382—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2312yes—
86C43C2311yescharacteristic
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336D412yes—
102#336D412yes—
103#334—24no—
104#333—42no—
105#333—42no—
106#333—42no—
107#332—42no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—82no—
129C4⋊Q8—82no—
130C4×Q8—82no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C42—22no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—44no—
158#107—44no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—82no—
166#103—44no—
167#103—44no—
168#97—42no—
169#97—82no—
170C4×C8—22no—
171C4×C8—42no—
172C4×C8#4114yes—
173#95—22no—
174#95—22no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—81no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8—81no—
194C2×Q8—81no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8#12414yes—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—41no—
249C42—41no—
250C42—42no—
251C42—42no—
252C42#13712yes—
253C42#13712yes—
254C42C2×#4112yes—
255C42H(5,𝔽2)12yes—
256C23C2×#12111yescharacteristic
257Q8—84no—
258Q8—82no—
259Q8—82no—
260Q8—162no—
261Q8—162no—
262Q8—162no—
263C2×C4—21no—
264C2×C4—21no—
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—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4#47012yes—
294C2×C4#47012yes—
295C2×C4C2×#12411yescharacteristic
296C2×C4#53511yescharacteristic
297C2×C4#55012yes—
298C2×C4#55912yes—
299C2×C4C2×#13711yescharacteristic
300C2×C4C2×#13711yescharacteristic
301C2×C4#57712yes—
302C2×C4#57812yes—
303C2×C4#57812yes—
304C8—24no—
305C8—42no—
306C8—82no—
307C8—82no—
308C8—82no—
309C22#254811yescharacteristic
310C22#304211yescharacteristic
311C22#304411yescharacteristic
312C22C2×#47011yescharacteristic
313C22#305911yescharacteristic
314C22#306811yescharacteristic
315C22#306811yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—161no—
324C4—161no—
325C4—161no—
326C4—161no—
327C4#313712yes—
328C4#322512yes—
329C4#339812yes—
330C4#339812yes—
331C2#1942911yescharacteristic
332C2#1944111yescharacteristic
333C2#1951511yescharacteristic
334C2#1953111yescharacteristic
335C2#2874411yescharacteristic
336C2#2875811yescharacteristic
337C2#2876811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
#302792
#292242
#292182
#223112
#105432
#105412
#97452
#97392
#97022
#96932
#82772