Finite Group Databasebeta

← #324801

Subgroups — #324801

Order 512.

542 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#33278C211yesmaximal, characteristic
2#29224C211yesmaximal, characteristic
3#29212C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10601C212yesmaximal
6#10592C212yesmaximal
7#9902C211yesmaximal, characteristic
8#9701C211yesmaximal, characteristic
9#9568C212yesmaximal
10#9562C212yesmaximal
11#8274C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2862C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2101—22no—
17#2059—22no—
18#1993—21no—
19#1993—21no—
20#1990C2211yescharacteristic
21#1990C2211yescharacteristic
22#1988C2212yes—
23#1987C2212yes—
24#1984—21no—
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1948C2212yes—
28#1947C2212yes—
29#1939—21no—
30#1939C2212yes—
31#1933—21no—
32#1933—21no—
33#1933—21no—
34#1933—21no—
35#1847—21no—
36#1842C2211yescharacteristic
37#1836—21no—
38C8×C4⋊C4—21no—
39C8×C4⋊C4C2212yes—
40#1767C2211yescharacteristic
41#1767C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×#335C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×#125—21no—
53C2×C4×Q8—21no—
54C8×Q8—44no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—21no—
66C2×#103—21no—
67C2×#97—21no—
68C2×#97—21no—
69C2×C4×C8—22no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389D412yes—
87#384D411yescharacteristic
88#384C2311yescharacteristic
89#384C2311yescharacteristic
90#382—21no—
91#382—21no—
92#382—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2312yes—
95C43C2311yescharacteristic
96#340—21no—
97#340—21no—
98#340—21no—
99#340—21no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—22no—
107#336—22no—
108#336—21no—
109#336—21no—
110#335—24no—
111#333—24no—
112#332—24no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8Q1614yes—
129#125—82no—
130C4×Q8—42no—
131C4×Q8—42no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
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—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D412yes—
150C2×C42—22no—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42C2411yescharacteristic
158#107—82no—
159#106—82no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—44no—
168C4⋊C8#4114yes—
169#103—42no—
170#103—42no—
171#103—42no—
172#103—42no—
173#97—42no—
174#97—42no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—41no—
180C4×C8—41no—
181C4×C8—44no—
182#95—22no—
183#95—22no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190C2×Q8—24no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—81no—
201C2×Q8—81no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C2×#4111yescharacteristic
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—24no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8AGL(1,ℤ/8)12yes—
235C2×C8#13712yes—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4C2×Q1614yes—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42—41no—
265C42—41no—
266C42—42no—
267C42—42no—
268C42C2×Q1612yes—
269C42#13512yes—
270C42C2×#4112yes—
271C422−1+412yes—
272C23C2×#12111yescharacteristic
273Q8—48no—
274Q8—81no—
275Q8—81no—
276Q8—81no—
277Q8—81no—
278Q8—162no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4#46712yes—
310C2×C4#46712yes—
311C2×C4#53211yescharacteristic
312C2×C4#53611yescharacteristic
313C2×C4#54812yes—
314C2×C4#55612yes—
315C2×C4C22×Q1611yescharacteristic
316C2×C4C2×#13511yescharacteristic
317C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
318C2×C4C2×#13711yescharacteristic
319C2×C4#57511yescharacteristic
320C2×C4#57511yescharacteristic
321C2×C4#57712yes—
322C8—22no—
323C8—22no—
324C8—44no—
325C8—41no—
326C8—41no—
327C8—82no—
328C8—82no—
329C22#254511yescharacteristic
330C22#303611yescharacteristic
331C22#303711yescharacteristic
332C22C2×#46711yescharacteristic
333C22#305711yescharacteristic
334C22#306711yescharacteristic
335C22#306711yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—41no—
341C4—41no—
342C4—82no—
343C4—161no—
344C4—161no—
345C4—161no—
346C4—161no—
347C4#317211yescharacteristic
348C4#318011yescharacteristic
349C4#323911yescharacteristic
350C4#324011yescharacteristic
351C4#336012yes—
352C4#336312yes—
353C2#1925611yescharacteristic
354C2#1926211yescharacteristic
355C2#1933411yescharacteristic
356C2#1934011yescharacteristic
357C2#2863811yescharacteristic
358C2#2865811yescharacteristic
359C2#2867111yescharacteristic
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
#332782
#292242
#292122
C8×C4⋊Q82
#106012
#105922
#99022
#97012
#95682
#95622
#82742