Finite Group Databasebeta

← #97504

Subgroups — #97504

Order 512.

864 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59822C211yesmaximal, characteristic
2#9638C212yesmaximal, complemented
3#9637C211yesmaximal, characteristic, complemented
4#8587C211yesmaximal, characteristic, complemented
5#8585C212yesmaximal, complemented
6C2×#555—21no—
7#2733C414yes—
8#2548—22no—
9#2541—22no—
10#2537—21no—
11#2414—24no—
12#2354—22no—
13C4×C4⋊Q8C2211yescharacteristic
14C4×#127C2212yes—
15#1842C2211yescharacteristic
16#1842C2212yes—
17C42×C8C2211yescharacteristic
18C2×#333—21no—
19C2×#331—22no—
20C2×Q8⋊C8—22no—
21C2×D4⋊C8—24no—
22#1296—22no—
23#1296—21no—
24#555—44no—
25#555—42no—
26#555—42no—
27C2×C4⋊Q8D411yescharacteristic
28C2×#127—22no—
29C2×#127D412yes—
30C2×#124—24no—
31C2×#121—24no—
32C2×#121—24no—
33C2×C4×Q8D412yes—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4D414yes—
37C2×#117—24no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—21no—
43C2×C4×C8—22no—
44C2×C4×C8—21no—
45#394—24no—
46#394—24no—
47#392—28no—
48#391C2×C412yes—
49#390—24no—
50#390—24no—
51#390—22no—
52#390—22no—
53#390C2×C414yes—
54#389—22no—
55#388—28no—
56#384D412yes—
57#384D411yescharacteristic
58#380—24no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—22no—
61C43C2311yescharacteristic
62#333—42no—
63#331—44no—
64Q8⋊C8—44no—
65D4⋊C8—48no—
66C22×Q8C22⋊C412yes—
67C22×D4—24no—
68C22×D4—24no—
69C22×D4—22no—
70C22×D4—22no—
71C22×D4C22⋊C414yes—
72C23×C4—24no—
73C22×C8—41no—
74C22×C8—42no—
75C22×C8—41no—
76C4⋊Q8—22no—
77C4⋊Q8D812yes—
78C4⋊Q8SD1612yes—
79#127—22no—
80#127—22no—
81#127—22no—
82#127—22no—
83#127—82no—
84#127—82no—
85#124—48no—
86#124—48no—
87#121—48no—
88#121—48no—
89#121—84no—
90#121—84no—
91C4×Q8—42no—
92C4×Q8—42no—
93C4×D4—44no—
94C4×D4—44no—
95C4×D4—42no—
96C4×D4—42no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102C4×D4—42no—
103#117—48no—
104#117—48no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C22⋊C414yes—
111C2×C4⋊C4C22⋊C412yes—
112C2×C22⋊C4—28no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4—24no—
117C2×C42—24no—
118C2×C42—22no—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C4⋊C8—44no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128C4⋊C8—42no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4×C8—24no—
133C4×C8—22no—
134C4×C8—42no—
135C4×C8—41no—
136#95—24no—
137#95—24no—
138C24—24no—
139C2×Q8—22no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8#10214yes—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—44no—
146C2×D4—44no—
147C2×D4—48no—
148C2×D4—42no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—88no—
161C2×D4—84no—
162C2×D4—84no—
163C2×D4—84no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—44no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C22⋊C412yes—
175C22×C4#12011yescharacteristic
176C22×C4#12012yes—
177C22×C4#12011yescharacteristic
178C2×C8—44no—
179C2×C8—42no—
180C2×C8—82no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—81no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C22⋊C4—416no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—84no—
202C22⋊C4—88no—
203C42—22no—
204C42—21no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—48no—
209C42—42no—
210C42—42no—
211C42C2×D811yescharacteristic
212C42C2×SD1611yescharacteristic
213C42#13512yes—
214C42#13512yes—
215C42AGL(1,ℤ/8)12yes—
216C42AGL(1,ℤ/8)12yes—
217C42#13712yes—
218C23—44no—
219C23—44no—
220C23—44no—
221C23—44no—
222C23—44no—
223C23—44no—
224C23—44no—
225C23#37911yescharacteristic
226Q8—44no—
227Q8—82no—
228D4—48no—
229D4—84no—
230D4—82no—
231D4—82no—
232D4—82no—
233D4—82no—
234D4—82no—
235D4—82no—
236D4—82no—
237D4—82no—
238D4—168no—
239D4—168no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—24no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4—84no—
258C2×C4C2×#10211yescharacteristic
259C2×C4#41612yes—
260C2×C4#41712yes—
261C2×C4#41811yescharacteristic
262C2×C4#44712yes—
263C2×C4#44912yes—
264C2×C4#44912yes—
265C2×C4#45012yes—
266C2×C4#45212yes—
267C2×C4#45212yes—
268C2×C4#45312yes—
269C2×C4#45311yescharacteristic
270C2×C4#45512yes—
271C2×C4#45511yescharacteristic
272C8—44no—
273C8—162no—
274C8—161no—
275C22—84no—
276C22—84no—
277C22—84no—
278C22—84no—
279C22—84no—
280C22—84no—
281C22—84no—
282C22#178611yescharacteristic
283C22#178812yes—
284C22#179111yescharacteristic
285C22#179211yescharacteristic
286C22#179212yes—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—41no—
292C4—44no—
293C4—82no—
294C4—84no—
295C4#162112yes—
296C4#162411yescharacteristic
297C4#163112yes—
298C4#163411yescharacteristic
299C4#164112yes—
300C4#164212yes—
301C4#164612yes—
302C2—164no—
303C2#835311yescharacteristic
304C2#835812yes—
305C2#844412yes—
306C2#845211yescharacteristic
307C2#853011yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radical#59822C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9638C2semidirect
3. #9637C2semidirect
4. #8587C2semidirect
5. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#598222
#96382
#96372
#85872
#85852