Finite Group Databasebeta

← #217082

Subgroups — #217082

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9902C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9705C211yesmaximal, characteristic, complemented
7#9694C211yesmaximal, characteristic, complemented
8#9130C212yesmaximal
9#9129C212yesmaximal
10#8593C212yesmaximal, complemented
11#8591C212yesmaximal, complemented
12C2×#541—21no—
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2857C2212yes—
18#2642—21no—
19#2621—21no—
20#2506—22no—
21#2506C414yes—
22#2496—22no—
23#2496C414yes—
24#2327—21no—
25C4×C4⋊Q8C2212yes—
26C4×#117C2211yescharacteristic
27#1939—22no—
28#1850—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×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C23×C8—21no—
50#541—48no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2312yes—
54C2×#126—24no—
55C2×#125—24no—
56C2×#125—21no—
57C2×C22⋊Q8—22no—
58C2×C4×Q8—22no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C42C2311yescharacteristic
62#431—42no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—42no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C22⋊C8—22no—
74C2×C22⋊C8—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#400—24no—
79#400—24no—
80#398—24no—
81#397—24no—
82#391—22no—
83#391—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389C2×C414yes—
88#389C2×C414yes—
89#389C2×C414yes—
90#387—24no—
91#387—24no—
92#384—22no—
93#384C2×C414yes—
94#384C2312yes—
95#382—21no—
96#380—22no—
97#380C2×C414yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C414yes—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4D412yes—
104#376D411yescharacteristic
105#376D411yescharacteristic
106C43C2311yescharacteristic
107C43C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336—22no—
113#336—22no—
114#336—22no—
115#332—42no—
116#332—42no—
117#332—42no—
118#332—42no—
119C22×Q8—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—42no—
130C22×C8—82no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—44no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138#126—416no—
139#125—44no—
140#125—416no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143C4×Q8—44no—
144C4×Q8—44no—
145#117—24no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C22×C414yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—24no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2411yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—42no—
187C4⋊C8—82no—
188C4⋊C8—82no—
189C22⋊C8—42no—
190C22⋊C8—44no—
191#97—42no—
192C4×C8—42no—
193C4×C8—42no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95C22⋊C412yes—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203C24C2×C22⋊C411yescharacteristic
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4AGL(1,ℤ/8)14yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—84no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—48no—
266C42—22no—
267C42—21no—
268C42—21no—
269C42—22no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—42no—
279C42—42no—
280C42—48no—
281C42#13512yes—
282C42#13512yes—
283C42H(5,𝔽2)12yes—
284C422−1+412yes—
285C23—21no—
286C23—22no—
287C23—21no—
288C23—22no—
289C23#41812yes—
290C23C22×C22⋊C411yescharacteristic
291Q8—82no—
292Q8—82no—
293Q8—82no—
294Q8—82no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—82no—
325C2×C4#41614yes—
326C2×C4#41812yes—
327C2×C4#52012yes—
328C2×C4#53511yescharacteristic
329C2×C4#53611yescharacteristic
330C2×C4#53712yes—
331C2×C4C2×#13511yescharacteristic
332C2×C4C2×#13511yescharacteristic
333C2×C4C2×AGL(1,ℤ/8)12yes—
334C2×C4#57712yes—
335C2×C4#57812yes—
336C8—81no—
337C8—81no—
338C8—81no—
339C8—81no—
340C8—82no—
341C8—82no—
342C22—24no—
343C22—22no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22#231111yescharacteristic
348C22#288912yes—
349C22C2×#41611yescharacteristic
350C22C2×#41811yescharacteristic
351C22#289812yes—
352C4—24no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—84no—
361C4—84no—
362C4#322211yescharacteristic
363C4#322311yescharacteristic
364C4#324212yes—
365C4#324712yes—
366C4#324712yes—
367C2—42no—
368C2#1772512yes—
369C2#1775912yes—
370C2#2187311yescharacteristic
371C2#2188311yescharacteristic
372C2#2188811yescharacteristic
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. #9705C2semidirect
7. #9694C2semidirect
10. #8593C2semidirect
11. #8591C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#219722
#207862
#99022
#98972
#97052
#96942
#91302
#91292
#85932
#85912