Finite Group Databasebeta

← #315190

Subgroups — #315190

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28827C211yesmaximal, characteristic
2#28676C211yesmaximal, characteristic
3#27824C212yesmaximal
4#10414C212yesmaximal
5#10355C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9576C212yesmaximal
9#8645C211yesmaximal, characteristic
10#8282C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2567C2212yes—
15#2552C2212yes—
16#2099—22no—
17#2087—22no—
18#2083—22no—
19#2026—22no—
20#1992—22no—
21#1990C2212yes—
22#1987C2212yes—
23#1987C2212yes—
24#1956C2212yes—
25#1947C2212yes—
26#1942—21no—
27#1942—21no—
28#1942—21no—
29#1935—22no—
30#1845—21no—
31#1845C2211yescharacteristic
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×#125—22no—
47C2×C4×Q8C2311yescharacteristic
48#500—22no—
49#500—22no—
50C8×Q8—22no—
51C8×Q8—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#107D411yescharacteristic
58C2×#107D411yescharacteristic
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—22no—
64C2×#103—42no—
65C2×#103D411yescharacteristic
66C2×#103D411yescharacteristic
67C2×#97—22no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#398—42no—
74#395—42no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#384—22no—
81#384—22no—
82#384C2312yes—
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—21no—
97#336—22no—
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—
116#125—44no—
117#125—82no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C42—21no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42C2411yescharacteristic
142#107—22no—
143#107—22no—
144#107—42no—
145#107—42no—
146#107—42no—
147#107—82no—
148#107—81no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—21no—
152C4⋊C8—21no—
153C4⋊C8—21no—
154C4⋊C8—42no—
155C4⋊C8#4112yes—
156#103—22no—
157#103—82no—
158#103—84no—
159#103Q1614yes—
160#97—82no—
161C4×C8—21no—
162C4×C8—21no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—42no—
166C4×C8—42no—
167C4×C8#4112yes—
168C4×C8#4112yes—
169#95—22no—
170#95—21no—
171#95—42no—
172#95—42no—
173#95—42no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×Q8C2×Q1612yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4H(5,𝔽2)11yescharacteristic
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—21no—
201C2×C8—22no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—42no—
214C2×C8#12412yes—
215C2×C8#12412yes—
216C2×C8C2×Q1612yes—
217C2×C8#13712yes—
218C2×C8#13712yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
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—84no—
238C4⋊C4—84no—
239C4⋊C4—82no—
240C4⋊C4—162no—
241C4⋊C4—162no—
242C4⋊C4—162no—
243C4⋊C4C2×Q1612yes—
244C4⋊C4#13512yes—
245C42—41no—
246C42—41no—
247C42—42no—
248C42—41no—
249C42—42no—
250C42—41no—
251C42C2×#4111yescharacteristic
252C42C2×#4111yescharacteristic
253C42C2×#4111yescharacteristic
254C422−1+411yescharacteristic
255C23D4211yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—41no—
259Q8—41no—
260Q8—84no—
261Q8—162no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4C2×#12411yescharacteristic
289C2×C4#53211yescharacteristic
290C2×C4#54211yescharacteristic
291C2×C4#54711yescharacteristic
292C2×C4#54811yescharacteristic
293C2×C4#55511yescharacteristic
294C2×C4C22×Q1611yescharacteristic
295C2×C4C2×#13511yescharacteristic
296C2×C4C2×#13711yescharacteristic
297C2×C4C2×#13711yescharacteristic
298C2×C4#57611yescharacteristic
299C2×C4#57711yescharacteristic
300C2×C4#57811yescharacteristic
301C2×C4#57811yescharacteristic
302C8—22no—
303C8—22no—
304C8—42no—
305C8—41no—
306C8—42no—
307C8—81no—
308C8—82no—
309C8#49114yes—
310C22#249111yescharacteristic
311C22#327611yescharacteristic
312C22#327811yescharacteristic
313C22#328011yescharacteristic
314C22D4×Q1611yescharacteristic
315C22#329011yescharacteristic
316C22#329211yescharacteristic
317C4—21no—
318C4—21no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—81no—
323C4—82no—
324C4—82no—
325C4—162no—
326C4—162no—
327C4C2×#49111yescharacteristic
328C4#313711yescharacteristic
329C4#317011yescharacteristic
330C4#317511yescharacteristic
331C4#333811yescharacteristic
332C4#334211yescharacteristic
333C4#337211yescharacteristic
334C4#337411yescharacteristic
335C2#1990311yescharacteristic
336C2#1990911yescharacteristic
337C2#1993011yescharacteristic
338C2#1993211yescharacteristic
339C2#2777711yescharacteristic
340C2#2779211yescharacteristic
341C2#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
#288272
#286762
#278242
#104142
#103552
#98232
#97302
#95762
#86452
#82822
C8×#1032