Finite Group Databasebeta

← #315030

Subgroups — #315030

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28813C211yesmaximal, characteristic
2#28676C211yesmaximal, characteristic
3#27815C212yesmaximal
4#10414C212yesmaximal
5#10349C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9568C212yesmaximal
9#8645C211yesmaximal, characteristic
10C8×#107C211yesmaximal, characteristic
11C8×#103C211yesmaximal, characteristic
12C2×#500—21no—
13C2×#500C2211yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15#2567C2212yes—
16#2551C2212yes—
17#2100—22no—
18#2087—22no—
19#2057—22no—
20#2024—22no—
21#1990—22no—
22#1990C2212yes—
23#1987C2212yes—
24#1987C2212yes—
25#1956C2212yes—
26#1947C2212yes—
27#1942—21no—
28#1942—21no—
29#1942—21no—
30#1933—22no—
31#1845C2211yescharacteristic
32#1803C2212yes—
33C4×#107C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36C4×#103C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#333C2212yes—
39C2×Q8⋊C8C2211yescharacteristic
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2212yes—
44#1277C2211yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8C2312yes—
48C2×C4×Q8C2311yescharacteristic
49#500—22no—
50#500—22no—
51#500—44no—
52C8×Q8—22no—
53C8×Q8—22no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—22no—
58C2×#107—21no—
59C2×#107D411yescharacteristic
60C2×#107D411yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×#103—22no—
66C2×#103—42no—
67C2×#103D411yescharacteristic
68C2×#103D411yescharacteristic
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#395—42no—
76#395—42no—
77#391—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384—22no—
83#384C2312yes—
84C4×C4⋊C4C2312yes—
85C4×C4⋊C4C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340D412yes—
95#340D411yescharacteristic
96#340D411yescharacteristic
97#336—21no—
98#336—22no—
99#335—22no—
100#333—24no—
101Q8⋊C8—22no—
102C82—22no—
103C22×Q8—22no—
104C22×Q8—42no—
105C22×Q8C2×D411yescharacteristic
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—21no—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—44no—
119C4⋊Q8—44no—
120C4⋊Q8—82no—
121C4⋊Q8—82no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×D412yes—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—22no—
144C2×C42C2411yescharacteristic
145#107—42no—
146#107—42no—
147#107—42no—
148#107—44no—
149#107—81no—
150#107SD1614yes—
151#107Q1614yes—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—42no—
158C4⋊C8#4112yes—
159#103—22no—
160#103—82no—
161#103—84no—
162#103Q1614yes—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—21no—
167C4×C8—42no—
168C4×C8—42no—
169C4×C8—82no—
170C4×C8#4112yes—
171C4×C8#4112yes—
172#95—22no—
173#95—21no—
174#95—42no—
175#95—42no—
176#95—42no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—84no—
185C2×Q8—84no—
186C2×Q8—82no—
187C2×Q8—82no—
188C2×Q8—162no—
189C2×Q8C2×Q1612yes—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4H(5,𝔽2)11yescharacteristic
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—22no—
206C2×C8—21no—
207C2×C8—22no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—42no—
220C2×C8#12412yes—
221C2×C8#12412yes—
222C2×C8C2×SD1612yes—
223C2×C8C2×Q1612yes—
224C2×C8C2×Q1612yes—
225C4⋊C4—21no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—82no—
244C4⋊C4—162no—
245C4⋊C4—162no—
246C4⋊C4C2×SD1614yes—
247C4⋊C4C2×Q1612yes—
248C4⋊C4C2×Q1614yes—
249C4⋊C4#13512yes—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—42no—
254C42—42no—
255C42—41no—
256C42C2×#4111yescharacteristic
257C42C2×#4111yescharacteristic
258C42C2×#4111yescharacteristic
259C422−1+411yescharacteristic
260C23D4211yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—84no—
266Q8—162no—
267Q8—168no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4C2×#12411yescharacteristic
295C2×C4#53211yescharacteristic
296C2×C4#54211yescharacteristic
297C2×C4#54711yescharacteristic
298C2×C4#54811yescharacteristic
299C2×C4#55511yescharacteristic
300C2×C4C22×SD1611yescharacteristic
301C2×C4C22×Q1611yescharacteristic
302C2×C4C22×Q1611yescharacteristic
303C2×C4C2×#13511yescharacteristic
304C2×C4#57611yescharacteristic
305C2×C4#57711yescharacteristic
306C2×C4#57811yescharacteristic
307C2×C4#57811yescharacteristic
308C8—22no—
309C8—22no—
310C8—42no—
311C8—41no—
312C8—42no—
313C8—81no—
314C8—82no—
315C8#48714yes—
316C22#249111yescharacteristic
317C22#328011yescharacteristic
318C22D4×SD1611yescharacteristic
319C22D4×Q1611yescharacteristic
320C22D4×Q1611yescharacteristic
321C22#329011yescharacteristic
322C22#329211yescharacteristic
323C4—21no—
324C4—21no—
325C4—44no—
326C4—42no—
327C4—42no—
328C4—44no—
329C4—42no—
330C4—81no—
331C4—162no—
332C4—162no—
333C4C2×#48711yescharacteristic
334C4#313711yescharacteristic
335C4#317011yescharacteristic
336C4#317511yescharacteristic
337C4#333811yescharacteristic
338C4#334211yescharacteristic
339C4#335911yescharacteristic
340C4#336311yescharacteristic
341C2#1997611yescharacteristic
342C2SD16×Q1611yescharacteristic
343C2#1999311yescharacteristic
344C2Q16211yescharacteristic
345C2#2776911yescharacteristic
346C2#2779211yescharacteristic
347C2#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
#288132
#286762
#278152
#104142
#103492
#98232
#97302
#95682
#86452
C8×#1072
C8×#1032