Finite Group Databasebeta

← #320141

Subgroups — #320141

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33225C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#29141C211yesmaximal, characteristic
4#28717C211yesmaximal, characteristic
5#28687C211yesmaximal, characteristic
6C4×#500C211yesmaximal, characteristic
7#10684C212yesmaximal
8#10314C212yesmaximal
9#9821C212yesmaximal
10#9819C212yesmaximal
11C8×C4⋊C8C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#498C2212yes—
14C2×#487—22no—
15C2×#487C2212yes—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2084—22no—
21#2072—22no—
22#2058—22no—
23#2024—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1956—22no—
30#1956C2212yes—
31#1956C2212yes—
32#1956C2212yes—
33#1948—22no—
34#1948—22no—
35#1942—22no—
36#1845C2211yescharacteristic
37#1844C2212yes—
38#1834C2212yes—
39#1803—22no—
40#1803—22no—
41C4×#107—22no—
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×C82C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57#500—22no—
58#500—22no—
59#498—24no—
60#498—42no—
61#487—24no—
62#487—42no—
63#487—48no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#106—24no—
67C2×#106—24no—
68C2×C4⋊C8D412yes—
69C2×C4⋊C8D412yes—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×C4×C8D412yes—
75C2×C4×C8D412yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391—21no—
81#391—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384C2311yescharacteristic
94#384C2311yescharacteristic
95#384C2312yes—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C43C2311yescharacteristic
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—22no—
106#340—22no—
107#336D412yes—
108#336D412yes—
109C82—42no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—41no—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C22×C8C2×D414yes—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—22no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—82no—
132C4⋊Q8—82no—
133C4⋊Q8#4112yes—
134C4⋊Q8#4112yes—
135C4×Q8—44no—
136C4×Q8—44no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2411yescharacteristic
170#107—28no—
171#107—44no—
172#106—48no—
173#106—84no—
174C4⋊C8—24no—
175C4⋊C8Q1618yes—
176#103—48no—
177#103—48no—
178#103—48no—
179#103—84no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—21no—
188C4×C8—42no—
189C4×C8SD1618yes—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—44no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—44no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4#12711yescharacteristic
216C22×C4#12711yescharacteristic
217C22×C4#12711yescharacteristic
218C22×C4#12711yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8C2×SD1614yes—
232C2×C8C2×SD1614yes—
233C2×C8C2×Q1618yes—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—82no—
268C42—82no—
269C42C2×SD1614yes—
270C42C2×Q1614yes—
271C42#13514yes—
272C42C2×#4111yescharacteristic
273C42C2×#4111yescharacteristic
274C42H(5,𝔽2)11yescharacteristic
275C422−1+411yescharacteristic
276C23C2×#12711yescharacteristic
277Q8—44no—
278Q8—44no—
279Q8—82no—
280Q8—82no—
281Q8—84no—
282Q8—84no—
283Q8—164no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4#49212yes—
309C2×C4#49212yes—
310C2×C4#49412yes—
311C2×C4#49412yes—
312C2×C4#49512yes—
313C2×C4#49512yes—
314C2×C4#52511yescharacteristic
315C2×C4#53711yescharacteristic
316C2×C4#54611yescharacteristic
317C2×C4#54611yescharacteristic
318C2×C4#54811yescharacteristic
319C2×C4#54811yescharacteristic
320C2×C4C22×SD1612yes—
321C2×C4C22×Q1612yes—
322C2×C4C2×#13512yes—
323C2×C4#57512yes—
324C2×C4#57712yes—
325C2×C4#57712yes—
326C8—24no—
327C8—24no—
328C8—28no—
329C8—44no—
330C22#254011yescharacteristic
331C22C2×#49211yescharacteristic
332C22C2×#49411yescharacteristic
333C22C2×#49511yescharacteristic
334C22#315311yescharacteristic
335C22#315411yescharacteristic
336C22#315411yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—82no—
341C4—82no—
342C4—161no—
343C4—161no—
344C4—162no—
345C4—162no—
346C4#170714yes—
347C4#310312yes—
348C4#324612yes—
349C4#329912yes—
350C4#330312yes—
351C4#336012yes—
352C4#336312yes—
353C2C2×#170711yescharacteristic
354C2#1963411yescharacteristic
355C2#1964011yescharacteristic
356C2#1965111yescharacteristic
357C2#2837411yescharacteristic
358C2#2839811yescharacteristic
359C2#2840111yescharacteristic
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
#332252
#292062
#291412
#287172
#286872
C4×#5002
#106842
#103142
#98212
#98192
C8×C4⋊C82