Finite Group Databasebeta

← #217091

Subgroups — #217091

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9902C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9691C211yesmaximal, characteristic, complemented
8#9136C212yesmaximal
9#9135C212yesmaximal
10#8594C212yesmaximal, complemented
11#8590C212yesmaximal, complemented
12C2×#432C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16#2857C2212yes—
17#2647—21no—
18#2629—21no—
19#2507—22no—
20#2497—22no—
21#2446C414yes—
22#2429C414yes—
23#2330—21no—
24#2314—21no—
25C4×C4⋊Q8C2212yes—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1851—22no—
29#1842C2211yescharacteristic
30#1842C2211yescharacteristic
31#1840—21no—
32#1831C2212yes—
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1818—21no—
36#1797C2212yes—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1750—21no—
40#1746—21no—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—21no—
50C2×C4⋊Q8C2×C414yes—
51C2×C4⋊Q8C2312yes—
52C2×#125—21no—
53C2×C22⋊Q8C2×C414yes—
54C2×C4×Q8C2×C414yes—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C42C2311yescharacteristic
58#432—42no—
59#423—42no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C22⋊C8—22no—
70C2×C22⋊C8—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74#400—24no—
75#398—24no—
76#398—24no—
77#396—24no—
78#391C2×C414yes—
79#391C2×C414yes—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—24no—
86#389C2×C414yes—
87#388—24no—
88#387—24no—
89#387—24no—
90#384—21no—
91#384—21no—
92#384—22no—
93#384C2312yes—
94#382—21no—
95#380—22no—
96#380—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4D412yes—
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4D412yes—
103#376D411yescharacteristic
104#376D411yescharacteristic
105C43C2311yescharacteristic
106C43C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—21no—
110#336—21no—
111#336—22no—
112#336—22no—
113#336—22no—
114#333—42no—
115#333—42no—
116#333—42no—
117#333—42no—
118C22×Q8C22×C414yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—42no—
123C2×M16—82no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—84no—
135#125—44no—
136C22⋊Q8—84no—
137C4×Q8—84no—
138#117—22no—
139#117—22no—
140#117—42no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C22⋊C4—24no—
154C2×C22⋊C4C22⋊C412yes—
155C2×C22⋊C4C22⋊C412yes—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C412yes—
161C2×C42C22×C412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2411yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—42no—
180C4⋊C8—82no—
181C4⋊C8—82no—
182C22⋊C8—44no—
183C22⋊C8—42no—
184#97—42no—
185#97—42no—
186C4×C8—42no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95C22⋊C412yes—
193#95C22⋊C412yes—
194#95C22⋊C412yes—
195#95C22⋊C412yes—
196C24C2×C22⋊C411yescharacteristic
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—44no—
200C2×Q8—44no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C23×C411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223M16—84no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—48no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—84no—
256C42—22no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—21no—
261C42—21no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—42no—
269C42—42no—
270C42—84no—
271C42#13712yes—
272C42#13712yes—
273C42H(5,𝔽2)12yes—
274C422−1+412yes—
275C23—21no—
276C23—22no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23C22×C22⋊C411yescharacteristic
281Q8—84no—
282Q8—84no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4#41914yes—
315C2×C4#52012yes—
316C2×C4#53511yescharacteristic
317C2×C4#53611yescharacteristic
318C2×C4#53712yes—
319C2×C4C2×#13711yescharacteristic
320C2×C4C2×#13711yescharacteristic
321C2×C4#57512yes—
322C2×C4#57712yes—
323C2×C4#57812yes—
324C8—81no—
325C8—81no—
326C8—81no—
327C8—81no—
328C8—82no—
329C8—82no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22#231111yescharacteristic
336C22#288812yes—
337C22C2×#41911yescharacteristic
338C22#289611yescharacteristic
339C22#289812yes—
340C4—24no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—84no—
349C4—84no—
350C4#322211yescharacteristic
351C4#322311yescharacteristic
352C4#324212yes—
353C4#325212yes—
354C4#325212yes—
355C2—81no—
356C2#1776012yes—
357C2#1776812yes—
358C2#2187311yescharacteristic
359C2#2188211yescharacteristic
360C2#2189211yescharacteristic
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
4. #9902C2semidirect
5. #9897C2semidirect
6. #9702C2semidirect
7. #9691C2semidirect
10. #8594C2semidirect
11. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#219782
#207862
#99022
#98972
#97022
#96912
#91362
#91352
#85942
#85902