Finite Group Databasebeta

← #351027

Subgroups — #351027

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33232C211yesmaximal, characteristic
2#29996C211yesmaximal, characteristic
3#29866C211yesmaximal, characteristic
4#20521C211yesmaximal, characteristic
5#9899C211yesmaximal, characteristic
6#9694C211yesmaximal, characteristic
7#9570C212yesmaximal
8#9562C212yesmaximal
9#9228C212yesmaximal
10#9220C212yesmaximal
11#8786C211yesmaximal, characteristic
12C2×#491—21no—
13C2×#491—21no—
14C2×#475C2212yes—
15C2×#446C2211yescharacteristic
16C2×C8×Q8C2211yescharacteristic
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2295C2211yescharacteristic
20#2088—22no—
21#2084—22no—
22#2072—22no—
23#2066—22no—
24#1988C2212yes—
25#1960—21no—
26#1960—21no—
27#1956C2212yes—
28#1956C2212yes—
29#1948C2212yes—
30#1842C2211yescharacteristic
31#1831C2211yescharacteristic
32#1803C2212yes—
33#1771—21no—
34#1770—21no—
35#1770C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37#1768—21no—
38C4×C4⋊C8C2211yescharacteristic
39#1726—21no—
40C4×#97C2211yescharacteristic
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2212yes—
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2312yes—
50C2×#125—21no—
51C2×C4×Q8C2311yescharacteristic
52#491—44no—
53#491—44no—
54#475—24no—
55#475—24no—
56#446—22no—
57#446—22no—
58C8×Q8—22no—
59C8×Q8—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97D411yescharacteristic
78C2×#97D411yescharacteristic
79C2×#97D411yescharacteristic
80C2×#97D411yescharacteristic
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86#391D412yes—
87#389—22no—
88#389—22no—
89#389D412yes—
90#387—22no—
91#384—22no—
92#384—22no—
93#384C2312yes—
94#382—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4C2311yescharacteristic
97#376—21no—
98C43C2311yescharacteristic
99#340—22no—
100#340—22no—
101#340—24no—
102#336—21no—
103#336—21no—
104#336—22no—
105#336—21no—
106#332—24no—
107#332—24no—
108C22×Q8—22no—
109C22×Q8C2×D411yescharacteristic
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—42no—
126#125—42no—
127#125—42no—
128C4×Q8—21no—
129C4×Q8—21no—
130C4×Q8#4112yes—
131C4×Q8#4112yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C42—21no—
145C2×C42—21no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42#4112yes—
150C2×C42C2411yescharacteristic
151#106—44no—
152#106—82no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—82no—
162C4⋊C8#4112yes—
163C4⋊C8#4112yes—
164#103—44no—
165#103—44no—
166#103—44no—
167#103—44no—
168#103—44no—
169#103—82no—
170#97—24no—
171#97—22no—
172#97—41no—
173#97—41no—
174#97—42no—
175#97SD1614yes—
176C4×C8—24no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—42no—
182C4×C8—42no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—21no—
187#95—42no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8#12412yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4#12112yes—
204C22×C4#12112yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8C2×SD1612yes—
231C2×C8C2×SD1612yes—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—84no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C4⋊C4#12412yes—
251C42—22no—
252C42—22no—
253C42—21no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42—42no—
258C42—41no—
259C42—41no—
260C42C2×SD1612yes—
261C42AGL(1,ℤ/8)12yes—
262C42C2×#4111yescharacteristic
263C42C2×#4111yescharacteristic
264C42C2×#4111yescharacteristic
265C42C2×#4111yescharacteristic
266C23C2×#12111yescharacteristic
267Q8—24no—
268Q8—41no—
269Q8—41no—
270Q8—82no—
271Q8—82no—
272Q8—82no—
273Q8—82no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#46014yes—
294C2×C4#52511yescharacteristic
295C2×C4#52511yescharacteristic
296C2×C4C2×#12411yescharacteristic
297C2×C4#52911yescharacteristic
298C2×C4#52911yescharacteristic
299C2×C4#53211yescharacteristic
300C2×C4C22×SD1611yescharacteristic
301C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
302C2×C4#57511yescharacteristic
303C2×C4#57511yescharacteristic
304C2×C4#57712yes—
305C2×C4#57812yes—
306C8—22no—
307C8—22no—
308C8—41no—
309C8—41no—
310C8—42no—
311C8—42no—
312C8—82no—
313C8—82no—
314C22#285011yescharacteristic
315C22C2×#46011yescharacteristic
316C22#304011yescharacteristic
317C22#306312yes—
318C22#306912yes—
319C4—24no—
320C4—21no—
321C4—21no—
322C4—42no—
323C4—42no—
324C4—42no—
325C4—82no—
326C4—162no—
327C4—162no—
328C4#310311yescharacteristic
329C4#310311yescharacteristic
330C4#311511yescharacteristic
331C4#311511yescharacteristic
332C4#313712yes—
333C4#317411yescharacteristic
334C4#317711yescharacteristic
335C2#1923912yes—
336C2#1938112yes—
337C2#3302511yescharacteristic
338C2#3305611yescharacteristic
339C2#3306511yescharacteristic
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
#332322
#299962
#298662
#205212
#98992
#96942
#95702
#95622
#92282
#92202
#87862