Finite Group Databasebeta

← #217103

Subgroups — #217103

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9902C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9705C211yesmaximal, characteristic, complemented
7#9694C211yesmaximal, characteristic, complemented
8#9138C212yesmaximal
9#9137C212yesmaximal
10#8589C212yesmaximal, complemented
11#8586C212yesmaximal, complemented
12C2×#541—21no—
13C2×#433—21no—
14C2×#432C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2868C2211yescharacteristic
17#2859C2211yescharacteristic
18#2857C2212yes—
19#2642—21no—
20#2621—21no—
21#2506—22no—
22#2506C414yes—
23#2496—22no—
24#2496C414yes—
25#2327—21no—
26C4×C4⋊Q8C2212yes—
27C4×#117C2211yescharacteristic
28#1940—22no—
29#1851—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1832—21no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1798—21no—
37#1797C2212yes—
38#1768—21no—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—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#433—44no—
63#432—42no—
64#423—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C22⋊C8—22no—
77C2×C22⋊C8—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81#400—24no—
82#400—24no—
83#398—24no—
84#397—24no—
85#391—22no—
86#391—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389C2×C414yes—
91#389C2×C414yes—
92#389C2×C414yes—
93#387—24no—
94#387—24no—
95#384—22no—
96#384C2×C414yes—
97#384C2312yes—
98#382—21no—
99#380—22no—
100#380C2×C414yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C414yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4D412yes—
107#376D411yescharacteristic
108#376D411yescharacteristic
109C43C2311yescharacteristic
110C43C2311yescharacteristic
111#336—22no—
112#336—21no—
113#336—21no—
114#336—22no—
115#336—22no—
116#332—42no—
117#332—42no—
118#332—42no—
119#332—42no—
120C22×Q8—22no—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—42no—
125C2×M16—82no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—44no—
137C4⋊Q8—42no—
138C4⋊Q8—42no—
139#126—416no—
140#125—44no—
141#125—416no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C4×Q8—44no—
145C4×Q8—44no—
146#117—24no—
147#117—22no—
148#117—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C414yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2411yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—42no—
188C4⋊C8—44no—
189C4⋊C8—44no—
190C4⋊C8—82no—
191C4⋊C8—82no—
192C22⋊C8—42no—
193C22⋊C8—44no—
194#97—42no—
195#97—42no—
196C4×C8—42no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95C22⋊C412yes—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206C24C2×C22⋊C411yescharacteristic
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235M16—84no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—48no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—48no—
269C42—22no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—21no—
274C42—22no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—42no—
281C42—42no—
282C42—48no—
283C42AGL(1,ℤ/8)12yes—
284C42AGL(1,ℤ/8)14yes—
285C42#13712yes—
286C42H(5,𝔽2)12yes—
287C422−1+412yes—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23C22×C22⋊C411yescharacteristic
294Q8—82no—
295Q8—82no—
296Q8—82no—
297Q8—82no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—82no—
328C2×C4#41714yes—
329C2×C4#41814yes—
330C2×C4#52012yes—
331C2×C4#53511yescharacteristic
332C2×C4#53611yescharacteristic
333C2×C4#53712yes—
334C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
335C2×C4C2×AGL(1,ℤ/8)12yes—
336C2×C4C2×#13711yescharacteristic
337C2×C4#57712yes—
338C2×C4#57812yes—
339C8—81no—
340C8—81no—
341C8—81no—
342C8—81no—
343C8—82no—
344C8—82no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22#231111yescharacteristic
351C22#288912yes—
352C22C2×#41811yescharacteristic
353C22C2×#41711yescharacteristic
354C22#289812yes—
355C4—24no—
356C4—24no—
357C4—22no—
358C4—42no—
359C4—42no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—84no—
364C4—84no—
365C4#322211yescharacteristic
366C4#322311yescharacteristic
367C4#324212yes—
368C4#325312yes—
369C4#325412yes—
370C2—81no—
371C2#1775912yes—
372C2#1776312yes—
373C2#2187311yescharacteristic
374C2#2188311yescharacteristic
375C2#2189311yescharacteristic
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. #8589C2semidirect
11. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#219782
#208122
#99022
#98972
#97052
#96942
#91382
#91372
#85892
#85862