Finite Group Databasebeta

← #99942

Subgroups — #99942

Order 512.

397 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 112 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#22307C211yesmaximal, characteristic
2#22246C211yesmaximal, characteristic
3#9641C211yesmaximal, characteristic
4#7591C211yesmaximal, characteristic
5#7588C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic
7#7014C211yesmaximal, characteristic
8C2×#498C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×C4⋊Q8C2211yescharacteristic
11#1990C411yescharacteristic
12#1990C411yescharacteristic
13#1984—22no—
14#1939—21no—
15#1920—22no—
16#1919—22no—
17#1916—22no—
18#1834C411yescharacteristic
19#1831C2211yescharacteristic
20#1831C2211yescharacteristic
21C8×C4⋊C4—21no—
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25C2×Q8⋊C8—21no—
26C2×Q8⋊C8—21no—
27#1271—22no—
28#1271—22no—
29C2×C4⋊Q8C2×C411yescharacteristic
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C4×Q8—21no—
32C2×C4×Q8—21no—
33#498—22no—
34#498C814yescomplemented
35C2×#106—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×C4×C8—21no—
40C2×C4×C8—22no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#391—21no—
46#391—21no—
47#389—21no—
48#389—21no—
49#389—21no—
50#389D412yes—
51#384C2×C411yescharacteristic
52#384C2×C411yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—21no—
56C4×C4⋊C4D412yes—
57C43C2311yescharacteristic
58#340—22no—
59#340—22no—
60#340—22no—
61#336—22no—
62#336—22no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336D412yes—
73Q8⋊C8—42no—
74Q8⋊C8—42no—
75C22×Q8—21no—
76C22×Q8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8C2×C812yes—
90C4⋊Q8C2×C812yes—
91C4⋊Q8M1612yes—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×Q8—41no—
96C4×Q8—82no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C22⋊C412yes—
109C2×C42—22no—
110C2×C42—41no—
111C2×C42C22×C411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42#4111yescharacteristic
117C2×C42#4111yescharacteristic
118#106—28no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—82no—
124C4×C8—22no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8—41no—
132C4×C8—82no—
133C4×C8C2×C812yes—
134C4×C8SD1612yes—
135#95—22no—
136#95—22no—
137#95—41no—
138#95—41no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—81no—
148C2×Q8—81no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C4×D411yescharacteristic
155C22×C4C4×D411yescharacteristic
156C22×C4#12011yescharacteristic
157C22×C4#12111yescharacteristic
158C22×C4#12111yescharacteristic
159C22×C4#12311yescharacteristic
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—81no—
195C4⋊C4—82no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4C22⋊C814yes—
200C42—21no—
201C42—21no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—41no—
206C42—42no—
207C42—42no—
208C42—81no—
209C42—81no—
210C42C22×C811yescharacteristic
211C42C2×M1611yescharacteristic
212C42#13111yescharacteristic
213C42#13111yescharacteristic
214C42C2×SD1611yescharacteristic
215C42#13511yescharacteristic
216C42AGL(1,ℤ/8)11yescharacteristic
217C42#13711yescharacteristic
218C23#38611yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—82no—
222Q8—82no—
223Q8—81no—
224Q8—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4C2×C22⋊C811yescharacteristic
252C2×C4#40811yescharacteristic
253C2×C4C8×D411yescharacteristic
254C2×C4C8×D411yescharacteristic
255C2×C4#43511yescharacteristic
256C2×C4#43511yescharacteristic
257C2×C4C4×SD1611yescharacteristic
258C2×C4#44011yescharacteristic
259C2×C4#44311yescharacteristic
260C2×C4#44411yescharacteristic
261C2×C4#44811yescharacteristic
262C2×C4#44911yescharacteristic
263C2×C4#46011yescharacteristic
264C2×C4#46411yescharacteristic
265C8—42no—
266C8—42no—
267C8—82no—
268C8—81no—
269C8—81no—
270C8—82no—
271C8—161no—
272C22#186911yescharacteristic
273C22#187611yescharacteristic
274C22#188011yescharacteristic
275C22#190211yescharacteristic
276C22#190411yescharacteristic
277C22#190911yescharacteristic
278C22#191211yescharacteristic
279C4—21no—
280C4—21no—
281C4—21no—
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—41no—
288C4—41no—
289C4—44no—
290C4—82no—
291C4—82no—
292C4—161no—
293C4C8×SD1611yescharacteristic
294C4SD16⋊C811yescharacteristic
295C4#158311yescharacteristic
296C4#158811yescharacteristic
297C2#894811yescharacteristic
298C2#908711yescharacteristic
299C2#910011yescharacteristic
300C2#927911yescharacteristic
301C2#928611yescharacteristic
302C2#929211yescharacteristic
303C2#930211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC42C22×C8
Frattini, cosocleC43C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
34. #498C8semidirect

Maximal subgroups

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

subgroupindex
#223072
#222462
#96412
#75912
#75882
#70182
#70142