Finite Group Databasebeta

← #99539

Subgroups — #99539

Order 512.

480 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 104 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#23346C211yesmaximal, characteristic
2C4×#497C211yesmaximal, characteristic
3#9873C212yesmaximal, complemented
4#7567C211yesmaximal, characteristic
5#7030C212yesmaximal, complemented
6C2×#497C411yescharacteristic
7C2×C4×Q16—21no—
8C2×C4×SD16—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#1984—21no—
12#1973—21no—
13#1971—21no—
14#1960C411yescharacteristic
15#1959C411yescharacteristic
16#1923—22no—
17#1869—21no—
18#1850—22no—
19#1847C411yescharacteristic
20#1831C2212yes—
21#1744C2212yes—
22C4×#97C2211yescharacteristic
23#1279—22no—
24#1279—22no—
25#1272—21no—
26#1270—21no—
27C22×Q16—21no—
28C22×SD16—21no—
29C23×C8—21no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#124C2×C411yescharacteristic
32C2×C4×Q8—21no—
33C2×C4×Q8D411yescharacteristic
34C2×C4×D4D411yescharacteristic
35#497—22no—
36#497C814yescomplemented
37C4×Q16—42no—
38C4×Q16—42no—
39C4×SD16—42no—
40C4×SD16—42no—
41C2×#107—21no—
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D412yes—
45C2×C4⋊C8D412yes—
46C2×#103—21no—
47C2×#103—21no—
48C2×#103—21no—
49C2×#102—21no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—21no—
52C2×#97—21no—
53C2×#97C2×C411yescharacteristic
54C2×#97C2×C411yescharacteristic
55C2×C4×C8—21no—
56C2×C4×C8—22no—
57C2×C4×C8—21no—
58#391—21no—
59#391D411yescharacteristic
60#390D411yescharacteristic
61#389—22no—
62#388—22no—
63#384C2×C411yescharacteristic
64#382C2×C411yescharacteristic
65C4×C4⋊C4—22no—
66C4×C22⋊C4—22no—
67C43C2311yescharacteristic
68#336—22no—
69#336—22no—
70#336—22no—
71#336—22no—
72#336—22no—
73C22×Q8—21no—
74C22×Q8C22⋊C411yescharacteristic
75C22×D4C22⋊C411yescharacteristic
76C23×C4—21no—
77C2×Q16—24no—
78C2×Q16—42no—
79C2×Q16—42no—
80C2×SD16—24no—
81C2×SD16—42no—
82C2×SD16—42no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—42no—
88C22×C8—81no—
89C22×C8—82no—
90C4⋊Q8—21no—
91C4⋊Q8—22no—
92C4⋊Q8C2×C812yescomplemented
93#124—22no—
94#124C2×C812yes—
95#124M1612yes—
96C4×Q8—41no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100C4×Q8—41no—
101C4×Q8—41no—
102C4×D4—41no—
103C4×D4—41no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C4⋊C4C22⋊C411yescharacteristic
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—21no—
114C2×C42—21no—
115C2×C42—41no—
116C2×C42C22×C411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C2×C42C2×D411yescharacteristic
120C2×C42#4112yes—
121#107—42no—
122#106—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C4⋊C8—81no—
126#103—42no—
127#103—42no—
128#103—42no—
129#102—42no—
130C22⋊C8—81no—
131C22⋊C8—82no—
132#97—21no—
133#97—22no—
134#97—42no—
135#97C2×C812yescomplemented
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—24no—
139C4×C8—42no—
140C4×C8—81no—
141#95—22no—
142#95—22no—
143#95—42no—
144C24—21no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8C22⋊C812yes—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C22⋊C812yes—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4C2×C22⋊C411yescharacteristic
169C22×C4C4×D412yes—
170C22×C4#12011yescharacteristic
171C22×C4#12112yes—
172C22×C4#12311yescharacteristic
173Q16—48no—
174SD16—48no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—44no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8—84no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—21no—
206C42—21no—
207C42—22no—
208C42—22no—
209C42—41no—
210C42—42no—
211C42—41no—
212C42—41no—
213C42—81no—
214C42—81no—
215C42C22×C811yescharacteristic
216C42C2×M1611yescharacteristic
217C42#13112yes—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23#38611yescharacteristic
224Q8—42no—
225Q8—42no—
226Q8—42no—
227Q8—81no—
228Q8—81no—
229Q8—81no—
230D4—42no—
231D4—81no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4C2×C22⋊C811yescharacteristic
257C2×C4#40811yescharacteristic
258C2×C4C8×D412yes—
259C2×C4#43512yes—
260C2×C4#45311yescharacteristic
261C2×C4#45311yescharacteristic
262C2×C4#45512yes—
263C2×C4#45612yes—
264C8—24no—
265C8—42no—
266C8—84no—
267C8—82no—
268C8—82no—
269C22—44no—
270C22—42no—
271C22—82no—
272C22—81no—
273C22—81no—
274C22#186911yescharacteristic
275C22#188112yes—
276C22#188212yes—
277C22#188411yescharacteristic
278C22#188711yescharacteristic
279C4—22no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—42no—
284C4—41no—
285C4—42no—
286C4—81no—
287C4—81no—
288C4—82no—
289C4—161no—
290C4#165911yescharacteristic
291C4#166011yescharacteristic
292C4#166112yes—
293C2—82no—
294C2#895312yes—
295C2#895911yescharacteristic
296C2#899812yes—
297C2#900311yescharacteristic
298C2#900711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC23#386
prime radicalC22×D4C22⋊C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9873C2semidirect
5. #7030C2semidirect
36. #497C8semidirect
92. C4⋊Q8C2×C8semidirect
135. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#233462
C4×#4972
#98732
#75672
#70302