Finite Group Databasebeta

← #133207

Subgroups — #133207

Order 512.

900 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2623C211yesmaximal, characteristic
2#11809C214yesmaximal, complemented
3#8553C211yesmaximal, characteristic
4#8543C211yesmaximal, characteristic
5C22×#123C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic, complemented
7#2623—28no—
8C2×#400C412yescomplemented
9C2×#399C2212yes—
10C2×#398—21no—
11C2×#397—21no—
12C2×#386—21no—
13C2×#385—21no—
14C2×#381C411yescharacteristic, complemented
15#1821—21no—
16#1819—21no—
17#1819C2212yes—
18#1781—21no—
19#1462—21no—
20#1462C2212yes—
21#1352—28no—
22C23×D4D411yescharacteristic
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×#123—24no—
26C2×#123—42no—
27C2×#123—42no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4—24no—
31C22×C4⋊C4—42no—
32C22×C4⋊C4—42no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4D411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—22no—
38C22×C22⋊C4D411yescharacteristic
39C22×C22⋊C4D411yescharacteristic
40C22×C22⋊C4C2311yescharacteristic
41C2×#99—42no—
42#407—22no—
43#400—24no—
44#400—24no—
45#399—24no—
46#399D418yes—
47#398—28no—
48#397—28no—
49#386—28no—
50#385—28no—
51#381—24no—
52#379—22no—
53C2×#95—21no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95C2×C412yes—
57#343—42no—
58#342—24no—
59#342—42no—
60#323—24no—
61#323—42no—
62C25C22⋊C411yescharacteristic
63C25C2×D411yescharacteristic
64C22×D4—22no—
65C22×D4—22no—
66C22×D4—41no—
67C22×D4—44no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—24no—
74C23×C4—44no—
75C23×C4—42no—
76C23×C4—42no—
77C23×C4—84no—
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—42no—
84#123—416no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—416no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—84no—
100C2×C4⋊C4C22⋊C414yes—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—28no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—44no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—84no—
116C2×C22⋊C4C2×D414yes—
117#99—84no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120#95—24no—
121#95—28no—
122#95—28no—
123#95—22no—
124#95C22⋊C414yes—
125C24—24no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—42no—
131C24—42no—
132C24—42no—
133C24—42no—
134C24—42no—
135C24#9912yes—
136C24C2×C22⋊C411yescharacteristic
137C24#12012yes—
138C2×D4—44no—
139C2×D4—48no—
140C2×D4—44no—
141C2×D4—82no—
142C2×D4—84no—
143C22×C4—28no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—48no—
159C22×C4—48no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—48no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—88no—
176C22×C4—88no—
177C22×C4—88no—
178C22×C4—82no—
179C22×C4#9912yes—
180C22×C4C2×C22⋊C412yes—
181C22×C4#12012yes—
182M16—84no—
183C2×C8—82no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—88no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—48no—
194C22⋊C4—88no—
195C22⋊C4—84no—
196C22⋊C4—84no—
197C22⋊C4—84no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—416no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23—88no—
223C23#35214yes—
224C23#37911yescharacteristic
225C23C2×#9911yescharacteristic
226C23C2×#9911yescharacteristic
227D4—88no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—416no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—44no—
251C2×C4—81no—
252C2×C4—88no—
253C2×C4—84no—
254C2×C4—816no—
255C2×C4—84no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—162no—
260C2×C4—162no—
261C2×C4—162no—
262C8—162no—
263C22—28no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—24no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—48no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—84no—
281C22—162no—
282C22—162no—
283C22#178111yescharacteristic
284C22#179612yes—
285C22C2×#35212yes—
286C22#212812yes—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—84no—
291C4—81no—
292C4—81no—
293C4—84no—
294C4—322no—
295C2—22no—
296C2—22no—
297C2—22no—
298C2—44no—
299C2—82no—
300C2#852011yescharacteristic
301C2#1268811yescharacteristic
302C2#1269711yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11809C2semidirect
6. C23×C4⋊C4C4semidirect
8. C2×#400C4semidirect
14. C2×#381C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26232
#118092
#85532
#85432