Finite Group Databasebeta

← #315187

Subgroups — #315187

Order 512.

516 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#28828C211yesmaximal, characteristic
2#28676C211yesmaximal, characteristic
3#27738C212yesmaximal
4#10415C212yesmaximal
5#10352C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9575C212yesmaximal
9#8645C211yesmaximal, characteristic
10#8282C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2571C2212yes—
15#2567C2212yes—
16#2099—22no—
17#2087—22no—
18#2085—22no—
19#2074—22no—
20#1990C2212yes—
21#1987C2212yes—
22#1987C2212yes—
23#1956C2212yes—
24#1948C2212yes—
25#1942—21no—
26#1942—21no—
27#1942—21no—
28#1935—22no—
29#1846—21no—
30#1845C2211yescharacteristic
31#1836—22no—
32#1803C2212yes—
33C4×#107C2211yescharacteristic
34#1770C2212yes—
35C8×C4⋊C4C2211yescharacteristic
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#333C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×C4×Q8C2311yescharacteristic
47#500—22no—
48#500—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107D411yescharacteristic
56C2×#107D411yescharacteristic
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—22no—
63C2×#103—42no—
64C2×#103D411yescharacteristic
65C2×#103D411yescharacteristic
66C2×#97—22no—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#400—42no—
73#395—42no—
74#391—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#384—22no—
80#384C2312yes—
81#382—22no—
82#382—22no—
83C4×C4⋊C4C2312yes—
84C4×C4⋊C4C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340D412yes—
94#340D411yescharacteristic
95#340D411yescharacteristic
96#336—22no—
97#336—21no—
98#335—22no—
99#333—24no—
100Q8⋊C8—22no—
101C82—22no—
102C22×Q8—22no—
103C22×Q8C2×D411yescharacteristic
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—21no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—22no—
137C2×C42C2411yescharacteristic
138#107—22no—
139#107—22no—
140#107—42no—
141#107—42no—
142#107—42no—
143#107—81no—
144#106—82no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—42no—
151C4⋊C8#4112yes—
152#103—22no—
153#103—22no—
154#103—44no—
155#103—84no—
156#97—44no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—42no—
162C4×C8—42no—
163C4×C8#4112yes—
164C4×C8#4112yes—
165#95—22no—
166#95—21no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—42no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8#13512yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4H(5,𝔽2)11yescharacteristic
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8#12412yes—
213C2×C8#12412yes—
214C2×C8#13512yes—
215C2×C8#13712yes—
216C2×C8#13712yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—82no—
238C4⋊C4—162no—
239C4⋊C4—162no—
240C4⋊C4—162no—
241C42—41no—
242C42—41no—
243C42—42no—
244C42—41no—
245C42—41no—
246C42—42no—
247C42C2×#4111yescharacteristic
248C42C2×#4111yescharacteristic
249C42C2×#4111yescharacteristic
250C422−1+411yescharacteristic
251C23D4211yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—41no—
255Q8—41no—
256Q8—84no—
257Q8—162no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4C2×#12411yescharacteristic
285C2×C4#53211yescharacteristic
286C2×C4#54211yescharacteristic
287C2×C4#54711yescharacteristic
288C2×C4#54811yescharacteristic
289C2×C4#55511yescharacteristic
290C2×C4C2×#13511yescharacteristic
291C2×C4C2×#13511yescharacteristic
292C2×C4C2×#13711yescharacteristic
293C2×C4C2×#13711yescharacteristic
294C2×C4#57711yescharacteristic
295C2×C4#57811yescharacteristic
296C2×C4#57811yescharacteristic
297C2×C4#57811yescharacteristic
298C8—22no—
299C8—22no—
300C8—42no—
301C8—41no—
302C8—42no—
303C8—81no—
304C8—82no—
305C8#49114yes—
306C22#249111yescharacteristic
307C22#327611yescharacteristic
308C22#327811yescharacteristic
309C22#328311yescharacteristic
310C22#328511yescharacteristic
311C22#329011yescharacteristic
312C22#329211yescharacteristic
313C4—21no—
314C4—21no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4—162no—
323C4C2×#49111yescharacteristic
324C4#313711yescharacteristic
325C4#316811yescharacteristic
326C4#317711yescharacteristic
327C4#333611yescharacteristic
328C4#333611yescharacteristic
329C4#337211yescharacteristic
330C4#337411yescharacteristic
331C2#1994011yescharacteristic
332C2#1994211yescharacteristic
333C2#1995711yescharacteristic
334C2#1995911yescharacteristic
335C2#2777711yescharacteristic
336C2#2778911yescharacteristic
337C2#2780911yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ 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
#288282
#286762
#277382
#104152
#103522
#98232
#97302
#95752
#86452
#82822
C8×#1032