Finite Group Databasebeta

← #351033

Subgroups — #351033

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33294C211yesmaximal, characteristic
2#29999C211yesmaximal, characteristic
3#29869C211yesmaximal, characteristic
4#20521C211yesmaximal, characteristic
5#9897C211yesmaximal, characteristic
6#9691C211yesmaximal, characteristic
7#9576C212yesmaximal
8#9568C212yesmaximal
9#9224C212yesmaximal
10#9216C212yesmaximal
11#8786C211yesmaximal, characteristic
12C2×#477C2212yes—
13C2×#446C2211yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2295C2211yescharacteristic
18#2087—22no—
19#2083—22no—
20#2072—22no—
21#2068—22no—
22#1992—21no—
23#1992—21no—
24#1990C2212yes—
25#1990C2212yes—
26#1987C2212yes—
27#1947C2212yes—
28#1842C2211yescharacteristic
29#1836—21no—
30#1836—21no—
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×#333C2212yes—
42C2×#333C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2212yes—
46#1277C2212yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2312yes—
49C2×#125—21no—
50C2×C4×Q8C2311yescharacteristic
51#477—24no—
52#477—24no—
53#446—22no—
54#446—22no—
55C8×Q8—22no—
56C8×Q8—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97D411yescharacteristic
71C2×#97D411yescharacteristic
72C2×#97D411yescharacteristic
73C2×#97D411yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391D412yes—
80#389—22no—
81#389—22no—
82#389D412yes—
83#387—22no—
84#384—22no—
85#384—21no—
86#384—21no—
87#384C2312yes—
88#382—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2311yescharacteristic
91#376—21no—
92C43C2311yescharacteristic
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—24no—
100#336—21no—
101#336—21no—
102#336—22no—
103#336—21no—
104#333—24no—
105#333—24no—
106C22×Q8—22no—
107C22×Q8C2×D411yescharacteristic
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—84no—
121#125—42no—
122#125—42no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8#4112yes—
126C4×Q8#4112yes—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4C2×D412yes—
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C42—21no—
140C2×C42—21no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2×D411yescharacteristic
144C2×C42#4112yes—
145C2×C42C2411yescharacteristic
146#107—44no—
147#107—82no—
148C4⋊C8—24no—
149C4⋊C8—24no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—82no—
157C4⋊C8#4112yes—
158C4⋊C8#4112yes—
159#103—44no—
160#103—82no—
161#97—24no—
162#97—22no—
163#97—41no—
164#97—41no—
165#97—42no—
166#97Q1614yes—
167C4×C8—24no—
168C4×C8—21no—
169C4×C8—21no—
170C4×C8—21no—
171C4×C8—21no—
172C4×C8—42no—
173C4×C8—42no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—21no—
178#95—42no—
179C2×Q8—21no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—84no—
185C2×Q8#12412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4#12112yes—
192C22×C4#12112yes—
193C22×C4C22×D411yescharacteristic
194C22×C4C22×D411yescharacteristic
195C22×C4C2×#4111yescharacteristic
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8C2×Q1612yes—
219C2×C8C2×Q1612yes—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C4⋊C4#12412yes—
239C42—22no—
240C42—22no—
241C42—21no—
242C42—21no—
243C42—22no—
244C42—22no—
245C42—42no—
246C42—41no—
247C42—41no—
248C42C2×Q1612yes—
249C42#13712yes—
250C42C2×#4111yescharacteristic
251C42C2×#4111yescharacteristic
252C42C2×#4111yescharacteristic
253C42C2×#4111yescharacteristic
254C23C2×#12111yescharacteristic
255Q8—24no—
256Q8—41no—
257Q8—41no—
258Q8—162no—
259Q8—162no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—84no—
277C2×C4—82no—
278C2×C4#46214yes—
279C2×C4#52511yescharacteristic
280C2×C4#52511yescharacteristic
281C2×C4C2×#12411yescharacteristic
282C2×C4#52911yescharacteristic
283C2×C4#52911yescharacteristic
284C2×C4#53211yescharacteristic
285C2×C4C22×Q1611yescharacteristic
286C2×C4C2×#13711yescharacteristic
287C2×C4#57511yescharacteristic
288C2×C4#57511yescharacteristic
289C2×C4#57712yes—
290C2×C4#57812yes—
291C8—22no—
292C8—22no—
293C8—41no—
294C8—41no—
295C8—42no—
296C8—42no—
297C8—82no—
298C8—82no—
299C22#285011yescharacteristic
300C22C2×#46211yescharacteristic
301C22#304011yescharacteristic
302C22#306312yes—
303C22#306812yes—
304C4—21no—
305C4—21no—
306C4—24no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—82no—
311C4—162no—
312C4—162no—
313C4#310411yescharacteristic
314C4#310411yescharacteristic
315C4#311611yescharacteristic
316C4#311611yescharacteristic
317C4#313712yes—
318C4#317411yescharacteristic
319C4#317711yescharacteristic
320C2#1924112yes—
321C2#1938212yes—
322C2#3302711yescharacteristic
323C2#3305611yescharacteristic
324C2#3306411yescharacteristic
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
#332942
#299992
#298692
#205212
#98972
#96912
#95762
#95682
#92242
#92162
#87862