Finite Group Databasebeta

← #324794

Subgroups — #324794

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#29347C211yesmaximal, characteristic
3#29312C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10590C212yesmaximal
6#10589C212yesmaximal
7#9900C211yesmaximal, characteristic
8#9697C211yesmaximal, characteristic
9#9568C212yesmaximal
10#9566C212yesmaximal
11C4×#334C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2862C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2083—21no—
17#2072—21no—
18#2057—21no—
19#2024—21no—
20#1991—21no—
21#1991—21no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2212yes—
27#1984—21no—
28#1981C2212yes—
29#1947C2212yes—
30#1947C2212yes—
31#1942—22no—
32#1939—21no—
33#1939C2212yes—
34#1933—22no—
35#1842C2211yescharacteristic
36#1835—21no—
37#1835—21no—
38C8×C4⋊C4—21no—
39C8×C4⋊C4C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×#334C2212yes—
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×#107—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#103—22no—
64C2×#103—22no—
65C2×C4×C8—22no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389D412yes—
85#384D411yescharacteristic
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#382—21no—
89#382—21no—
90#382—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2312yes—
93C43C2311yescharacteristic
94#340—21no—
95#340—21no—
96#340—21no—
97#340—21no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
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#334—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—24no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128#125—42no—
129#125—42no—
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—44no—
159#107—82no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—44no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168C4⋊C8#4114yes—
169#103—44no—
170#103—82no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—44no—
177C4×C8—81no—
178C4×C8—81no—
179C4×C8Q1614yes—
180#95—22no—
181#95—22no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C2×Q8—24no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—81no—
197C2×Q8—81no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4111yescharacteristic
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—24no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8C2×Q1612yes—
231C2×C8C2×Q1612yes—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4#13514yes—
257C42—22no—
258C42—21no—
259C42—21no—
260C42—41no—
261C42—41no—
262C42—42no—
263C42—42no—
264C42C2×Q1612yes—
265C42#13512yes—
266C42AGL(1,ℤ/8)12yes—
267C42#13712yes—
268C42C2×#4112yes—
269C422−1+412yes—
270C23C2×#12111yescharacteristic
271Q8—48no—
272Q8—82no—
273Q8—82no—
274Q8—162no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4#46212yes—
304C2×C4#46212yes—
305C2×C4#46312yes—
306C2×C4#46312yes—
307C2×C4#53211yescharacteristic
308C2×C4#53611yescharacteristic
309C2×C4#54812yes—
310C2×C4#55612yes—
311C2×C4C22×Q1611yescharacteristic
312C2×C4C2×#13512yes—
313C2×C4C2×#13511yescharacteristic
314C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
315C2×C4C2×#13711yescharacteristic
316C2×C4#57711yescharacteristic
317C2×C4#57711yescharacteristic
318C8—22no—
319C8—22no—
320C8—44no—
321C8—41no—
322C8—41no—
323C8—82no—
324C8—82no—
325C22#254511yescharacteristic
326C22C2×#46311yescharacteristic
327C22C2×#46211yescharacteristic
328C22#305011yescharacteristic
329C22#306011yescharacteristic
330C22#306711yescharacteristic
331C22#306711yescharacteristic
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—44no—
338C4—82no—
339C4—161no—
340C4—161no—
341C4—161no—
342C4—161no—
343C4#317011yescharacteristic
344C4#318211yescharacteristic
345C4#323011yescharacteristic
346C4#323111yescharacteristic
347C4#336012yes—
348C4#336512yes—
349C2#1911011yescharacteristic
350C2#1911811yescharacteristic
351C2#1920011yescharacteristic
352C2#1921011yescharacteristic
353C2#2864211yescharacteristic
354C2#2865611yescharacteristic
355C2#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
#332902
#293472
#293122
C8×C4⋊Q82
#105902
#105892
#99002
#96972
#95682
#95662
C4×#3342