Finite Group Databasebeta

← #168752

Subgroups — #168752

Order 512.

920 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C2×#2811C211yesmaximal, characteristic
2#29953C212yesmaximal
3#28304C211yesmaximal, characteristic
4#7837C214yesmaximal, complemented
5#7759C214yesmaximal, complemented
6#7643C211yesmaximal, characteristic
7#7575C212yesmaximal
8C2×#484—21no—
9C2×#451—21no—
10#2857C2211yescharacteristic
11#2811—24no—
12#2811—22no—
13#2811C2214yes—
14C2×#395—21no—
15C2×#395—22no—
16C2×#391—21no—
17C2×#391—22no—
18C2×#391C2212yes—
19C2×#391C2211yescharacteristic
20#2095C2214yes—
21#2095C2212yes—
22#2068—24no—
23#2024C2214yes—
24#2016C2212yes—
25#1969—22no—
26#1852—22no—
27#1797C2212yes—
28#1787—22no—
29C4×C22⋊C8C2211yescharacteristic
30#1272C2218yes—
31#1272C2214yes—
32C23×Q8—21no—
33C23×Q8D411yescharacteristic
34C23×Q8D412yes—
35C22×Q16—22no—
36C2×C4⋊Q8C2311yescharacteristic
37C2×C22⋊Q8—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44#484—44no—
45#451—44no—
46C2×#107—22no—
47C2×#107—24no—
48C2×C4⋊C8—22no—
49C2×#103—22no—
50C2×#103—22no—
51C2×#103—24no—
52C2×#103—24no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—22no—
55C2×C4×C8—21no—
56#395—24no—
57#395—28no—
58#395—28no—
59#395—24no—
60#391—24no—
61#391—28no—
62#391—28no—
63#391—24no—
64#391—22no—
65#391—22no—
66#391—22no—
67#391—21no—
68#391D414yes—
69#391D412yes—
70#391C2312yes—
71#391C2314yes—
72#389—22no—
73#384C2311yescharacteristic
74#380—22no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80#340—24no—
81#336C2314yes—
82#336C2312yes—
83C22×Q8—24no—
84C22×Q8—24no—
85C22×Q8—21no—
86C22×Q8—22no—
87C22×Q8—24no—
88C22×Q8—21no—
89C22×Q8—42no—
90C22×Q8—82no—
91C22×Q8—81no—
92C22×Q8—82no—
93C22×Q8—84no—
94C22×Q8—82no—
95C22×Q8Q1618yescomplemented
96C22×Q8C2×D414yes—
97C22×Q8C2×D412yes—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C2×Q16—44no—
105C2×Q16—44no—
106C2×Q16—82no—
107C22×C8—22no—
108C22×C8—24no—
109C4⋊Q8—81no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4—84no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D412yes—
126C2×C22⋊C4—41no—
127C2×C42—24no—
128C2×C42—81no—
129C2×C42—82no—
130C2×C42C2×D412yes—
131C2×C42C2×D411yescharacteristic
132C2×C42C2411yescharacteristic
133#107—44no—
134#107—84no—
135C4⋊C8—82no—
136#103—44no—
137#103—44no—
138#103—84no—
139#103—84no—
140C22⋊C8—24no—
141C22⋊C8—44no—
142C4×C8—81no—
143#95—24no—
144#95—28no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—42no—
152C24#12011yescharacteristic
153C2×Q8—24no—
154C2×Q8—24no—
155C2×Q8—22no—
156C2×Q8—28no—
157C2×Q8—42no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—81no—
166C2×Q8—82no—
167C2×Q8—82no—
168C2×Q8—81no—
169C2×Q8—88no—
170C2×Q8—88no—
171C2×Q8—88no—
172C2×Q8—84no—
173C2×Q8—88no—
174C2×Q8—816no—
175C2×Q8C2×Q1618yes—
176C2×Q8#13714yes—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—28no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4#12011yescharacteristic
200C22×C4#12012yes—
201C22×C4C2×Q1614yes—
202C22×C4#13714yes—
203C22×C4C22×D412yes—
204C22×C4C22×D411yescharacteristic
205Q16—88no—
206C2×C8—24no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—41no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—41no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—88no—
223C22⋊C4—82no—
224C42—41no—
225C42—42no—
226C42—41no—
227C42—84no—
228C23—21no—
229C23—22no—
230C23—21no—
231C23#45114yes—
232C23#45312yes—
233C23C2×#12011yescharacteristic
234Q8—44no—
235Q8—48no—
236Q8—84no—
237Q8—82no—
238Q8—88no—
239Q8—82no—
240Q8—82no—
241Q8—84no—
242Q8—816no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—24no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—416no—
258C2×C4—48no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—84no—
269C2×C4—88no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4#45114yes—
276C2×C4C22×Q1612yes—
277C2×C4C2×#13712yes—
278C2×C4C2×#13712yes—
279C8—44no—
280C8—84no—
281C22—22no—
282C22—24no—
283C22—24no—
284C22—22no—
285C22#163314yes—
286C22C2×#45112yes—
287C22#300411yescharacteristic
288C22C2×#45311yescharacteristic
289C22#301912yes—
290C22#302611yescharacteristic
291C4—22no—
292C4—22no—
293C4—24no—
294C4—22no—
295C4—48no—
296C4—41no—
297C4—44no—
298C4—82no—
299C4—84no—
300C4—161no—
301C4—161no—
302C2—24no—
303C2#1828712yes—
304C2#1831811yescharacteristic
305C2C2×#163311yescharacteristic
306C2#1876912yes—
307C2#1899211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7837C2semidirect
5. #7759C2semidirect
95. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
#299532
#283042
#78372
#77592
#76432
#75752