Finite Group Databasebeta

← #215677

Subgroups — #215677

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic
2#21435C212yesmaximal
3#9893C212yesmaximal, complemented
4#9893C212yesmaximal, complemented
5#9892C212yesmaximal, complemented
6#9892C212yesmaximal, complemented
7#8722C212yesmaximal
8#8719C211yesmaximal, characteristic
9#8719C211yesmaximal, characteristic
10#2864C2212yes—
11#2864C2212yes—
12#2647C412yes—
13#2647C412yes—
14#2570—22no—
15#2570—22no—
16#2569—22no—
17#2569—22no—
18#2317C412yes—
19#2316C412yes—
20#2285C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1924—22no—
24#1923—22no—
25#1917—22no—
26#1916—22no—
27#1767C2212yes—
28#1767C2212yes—
29#1749C2212yes—
30#1749C2212yes—
31#1745C2212yes—
32C4×#97C2212yes—
33#1295C2211yescharacteristic
34#1295C2211yescharacteristic
35#1295C2212yes—
36#1295C2212yes—
37#1295C2211yescharacteristic
38#1295C2211yescharacteristic
39#1295C2212yes—
40#1295C2212yes—
41#1295C2212yes—
42#1295C2212yes—
43C2×#117C2×C412yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C42C2311yescharacteristic
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×#97—22no—
53C2×#97C2312yes—
54C2×#97C2312yes—
55C2×#97C2312yes—
56C2×#97C2312yes—
57C2×C4×C8—22no—
58#400—22no—
59#400—22no—
60#400—22no—
61#400—22no—
62#398—22no—
63#398—22no—
64#398—22no—
65#398—22no—
66#389—22no—
67#389—22no—
68#389C2×C412yes—
69#389C2×C412yes—
70#389C2×C412yes—
71#389C2×C412yes—
72#389C2×C412yes—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#385—22no—
85#385—22no—
86#384—22no—
87#384C2×C411yescharacteristic
88#384C2×C411yescharacteristic
89#382—22no—
90#382—22no—
91#382C2×C412yes—
92#382C2311yescharacteristic
93#382C2311yescharacteristic
94#382C2312yes—
95#380—22no—
96#380—22no—
97#380—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4D412yes—
101C4×C22⋊C4D412yes—
102C4×C22⋊C4D412yes—
103#376C2×C412yes—
104C2×#95—22no—
105C43C2312yes—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112C23×C4—22no—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119#117—42no—
120#117—42no—
121#117—44no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C22×C412yes—
138C2×C4⋊C4C22×C412yes—
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C22⋊C411yescharacteristic
144C2×C22⋊C4C22⋊C411yescharacteristic
145C2×C42—21no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42C22×C411yescharacteristic
150C2×C42C22×C412yes—
151C2×C42C22×C412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2411yescharacteristic
158C4⋊C8—44no—
159C4⋊C8—44no—
160C22⋊C8—44no—
161C22⋊C8—44no—
162#97—22no—
163#97—22no—
164#97—22no—
165#97—22no—
166#97—22no—
167#97—22no—
168#97—22no—
169#97—22no—
170#97—44no—
171C4×C8—44no—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95C22⋊C412yes—
187#95C22⋊C412yes—
188#95C22⋊C412yes—
189#95C22⋊C412yes—
190C24C2×C22⋊C411yescharacteristic
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C23×C411yescharacteristic
213C22×C4C22×D412yes—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C4⋊C4—24no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C22⋊C4—24no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—44no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42—44no—
259C422−1+411yescharacteristic
260C422−1+412yes—
261C422−1+411yescharacteristic
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23#41312yes—
267C23C22×C22⋊C411yescharacteristic
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
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—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—164no—
306C2×C4#41312yes—
307C2×C4#51911yescharacteristic
308C2×C4#51911yescharacteristic
309C2×C4#53612yes—
310C2×C4#53612yes—
311C8—44no—
312C8—44no—
313C8—82no—
314C8—82no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22#231011yescharacteristic
321C22C2×#41311yescharacteristic
322C22C2×#41311yescharacteristic
323C22#290112yes—
324C22#290112yes—
325C4—21no—
326C4—21no—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—82no—
336C4—82no—
337C4—82no—
338C4—82no—
339C2—42no—
340C2#1769011yescharacteristic
341C2#1769011yescharacteristic
342C2#1769112yes—
343C2#2183011yescharacteristic
344C2#2184612yes—
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. #9893C2semidirect
4. #9893C2semidirect
5. #9892C2semidirect
6. #9892C2semidirect

Maximal subgroups

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

subgroupindex
#353392
#214352
#98932
#98922
#87222
#87192