Finite Group Databasebeta

← #215721

Subgroups — #215721

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21924C212yesmaximal
3#9907C212yesmaximal, complemented
4#9904C212yesmaximal, complemented
5#9900C212yesmaximal, complemented
6#9897C212yesmaximal, complemented
7#8728C212yesmaximal
8#8725C211yesmaximal, characteristic
9#8723C211yesmaximal, characteristic
10#2859C2212yes—
11#2859C2212yes—
12#2647C414yes—
13#2571—22no—
14#2552—22no—
15#2442—22no—
16#2442—22no—
17#2317C412yes—
18#2316C412yes—
19#2285C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#1924—22no—
23#1923—22no—
24#1917—22no—
25#1916—22no—
26#1842C2212yes—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32#1296C2212yes—
33#1296C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2211yescharacteristic
38#1296C2212yes—
39#1296C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42C2×#125—22no—
43C2×#125—22no—
44C2×#117C2×C412yes—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2×C412yes—
49C22×C42C2311yescharacteristic
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×#97—22no—
59C2×C4×C8—22no—
60#400—24no—
61#400—24no—
62#398—24no—
63#398—24no—
64#389—24no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389C2×C414yes—
70#389C2×C414yes—
71#389C2×C414yes—
72#389C2×C414yes—
73#385—24no—
74#384—22no—
75#384—22no—
76#384C2×C412yes—
77#384C2311yescharacteristic
78#384C2311yescharacteristic
79#384C2312yes—
80#382—22no—
81#382—22no—
82#382—22no—
83#382C2×C412yes—
84#380—22no—
85#380—22no—
86#380—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4D412yes—
91C4×C22⋊C4D412yes—
92#376C2×C412yes—
93C43C2312yes—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99#336—22no—
100C23×C4—22no—
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D412yes—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107#125—82no—
108#125—82no—
109#125—82no—
110#125—82no—
111#117—42no—
112#117—42no—
113#117—44no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—82no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4C22⋊C412yes—
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22×C412yes—
131C2×C4⋊C4C22×C412yes—
132C2×C4⋊C4C22×C414yes—
133C2×C22⋊C4—22no—
134C2×C22⋊C4C22⋊C412yes—
135C2×C22⋊C4C22⋊C411yescharacteristic
136C2×C22⋊C4C22⋊C411yescharacteristic
137C2×C42—21no—
138C2×C42—22no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42C22×C411yescharacteristic
142C2×C42C22×C412yes—
143C2×C42C22×C412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2411yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—82no—
159C4⋊C8—82no—
160C22⋊C8—82no—
161C22⋊C8—82no—
162#97—82no—
163C4×C8—82no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95C22⋊C412yes—
170#95C22⋊C412yes—
171#95C22⋊C412yes—
172#95C22⋊C412yes—
173C24C2×C22⋊C411yescharacteristic
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—84no—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C2×C22⋊C412yes—
193C22×C4C2×C22⋊C412yes—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C23×C411yescharacteristic
196C22×C4C22×D412yes—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C4⋊C4—24no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C22⋊C4—24no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C42—21no—
234C42—21no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—42no—
246C42—42no—
247C42—44no—
248C422−1+411yescharacteristic
249C422−1+412yes—
250C422−1+411yescharacteristic
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23#41312yes—
256C23C22×C22⋊C411yescharacteristic
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—164no—
290C2×C4#41312yes—
291C2×C4#51911yescharacteristic
292C2×C4#51911yescharacteristic
293C2×C4#53612yes—
294C2×C4#53612yes—
295C2×C4#57612yes—
296C2×C4#57712yes—
297C2×C4#57712yes—
298C2×C4#57812yes—
299C8—82no—
300C8—82no—
301C8—82no—
302C8—82no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22#231011yescharacteristic
309C22C2×#41311yescharacteristic
310C22C2×#41311yescharacteristic
311C22#289712yes—
312C22#289812yes—
313C4—21no—
314C4—21no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—82no—
322C4—82no—
323C4—84no—
324C4#324112yes—
325C4#324212yes—
326C4#324312yes—
327C4#324412yes—
328C2—42no—
329C2#1777111yescharacteristic
330C2#1778011yescharacteristic
331C2#1778112yes—
332C2#2183011yescharacteristic
333C2#2184312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #9907C2semidirect
4. #9904C2semidirect
5. #9900C2semidirect
6. #9897C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#352862
#219242
#99072
#99042
#99002
#98972
#87282
#87252
#87232