Finite Group Databasebeta

← #11794301

Subgroups — #11794301

Order 1152.

471 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 100 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#10589585C211yesmaximal, characteristic
2C9×#385C211yesmaximal, characteristic
3#10588890C212yesmaximal
4#10588204C212yesmaximal
5#10588203C211yesmaximal, characteristic
6#69594—31nomaximal
7C23×C4×C9C2211yescharacteristic
8C2×C9×C4⋊C4—21no—
9C2×C9×C22⋊C4C2211yescharacteristic
10C2×C9×C22⋊C4C2211yescharacteristic
11C2×#3676C2212yes—
12C2×#3676C2212yes—
13#63486—21no—
14#63486—21no—
15C2×Dic9⋊C4—22no—
16#63440—22no—
17#63440—22no—
18C9×#95—21no—
19C9⋊#95—22no—
20#5483—31no—
21C3×#385D311yescharacteristic
22#5210—32no—
23#4527—32no—
24#4526—31no—
25C24×C9C2311yescharacteristic
26C22×C4×C9—21no—
27C22×C4×C9—21no—
28C22×C4×C9—22no—
29C22×C4×C9—42no—
30C22×C4×C9D411yescharacteristic
31C22×C4×C9D411yescharacteristic
32C22×Dic9—22no—
33C22×Dic9D412yes—
34C22×Dic9D412yes—
35C2×Dic18—21no—
36C2×Dic18—21no—
37C9×C4⋊C4—42no—
38C9×C22⋊C4—22no—
39C9×C22⋊C4—21no—
40C9×C22⋊C4Q812yescomplemented
41#3676—22no—
42#3676—22no—
43#3676—22no—
44#3676Q814yescomplemented
45C4⋊Dic9—21no—
46C4⋊Dic9—21no—
47Dic9⋊C4—22no—
48Dic9⋊C4—22no—
49Dic9⋊C4—44no—
50#2062—91noSylow
51C23×C3×C4C2×D311yescharacteristic
52C2×C3×C4⋊C4—21no—
53C2×C3×C22⋊C4C2×D311yescharacteristic
54C2×C3×C22⋊C4C2×D311yescharacteristic
55C2×#217—32no—
56C2×#217—32no—
57#924—61no—
58#924—61no—
59C2×Dic3⋊C4—62no—
60#878—62no—
61#878—62no—
62C3×#95—21no—
63C3⋊#95—62no—
64C23×C9—41no—
65C23×C9—41no—
66C23×C9C2×D412yes—
67C23×C9C2×D411yescharacteristic
68C23×C9C2×Q811yescharacteristic
69C23×C9C2×Q812yes—
70C23×C9#4111yescharacteristic
71C2×C4×C9—24no—
72C2×C4×C9—22no—
73C2×C4×C9—21no—
74C2×C4×C9—21no—
75C2×C4×C9—22no—
76C2×C4×C9—41no—
77C2×C4×C9—41no—
78C2×C4×C9—42no—
79C2×C4×C9—41no—
80C2×C4×C9—41no—
81C2×C4×C9—41no—
82C2×C4×C9—41no—
83C2×Dic9—22no—
84C2×Dic9—22no—
85C2×Dic9—24no—
86C2×Dic9—42no—
87C2×Dic9—42no—
88C2×Dic9—42no—
89C2×Dic9—42no—
90Dic18—81no—
91Dic18—81no—
92#543—91no—
93#385—92no—
94#385D911yescharacteristic
95#342—91no—
96#342—92no—
97C24×C3C22×D311yescharacteristic
98C22×C3×C4—22no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—42no—
102C22×C3×C4#6711yescharacteristic
103C22×C3×C4#6711yescharacteristic
104C22×Dic3—32no—
105C22×Dic3—32no—
106C22×Dic3—62no—
107C2×Dic6—61no—
108C2×Dic6—61no—
109C3×C4⋊C4—42no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4Dic612yes—
113#217—34no—
114#217—62no—
115#217—62no—
116#217—62no—
117C4⋊Dic3—61no—
118C4⋊Dic3—61no—
119Dic3⋊C4—62no—
120Dic3⋊C4—62no—
121Dic3⋊C4—124no—
122C22×C9—22no—
123C22×C9—21no—
124C22×C9—22no—
125C22×C9—21no—
126C22×C9—41no—
127C22×C9—42no—
128C22×C9—41no—
129C22×C9#12011yescharacteristic
130C22×C9C22⋊Q812yes—
131C22×C9C22⋊Q811yescharacteristic
132C22×C9C4⋊Q812yes—
133C22×C9C4⋊Q811yescharacteristic
134C4×C9—42no—
135C4×C9—41no—
136C4×C9—42no—
137C4×C9—41no—
138Dic9—44no—
139Dic9—42no—
140Dic9—42no—
141C23×C4C2×D911yescharacteristic
142C22⋊Q8—182no—
143C22⋊Q8—182no—
144C22⋊Q8—181no—
145C22⋊Q8—181no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—182no—
148C2×C22⋊C4—92no—
149C2×C22⋊C4—92no—
150C2×C22⋊C4C2×D911yescharacteristic
151C2×C22⋊C4C2×D911yescharacteristic
152#95—21no—
153#95—182no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3C2×Dic611yescharacteristic
157C23×C3#23511yescharacteristic
158C23×C3D3×D412yes—
159C23×C3D3×Q812yes—
160C23×C3C2×#6711yescharacteristic
161C2×C3×C4—24no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×Dic3—64no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—122no—
177C2×Dic3—122no—
178C2×Dic3—122no—
179C2×Dic3—122no—
180Dic6—241no—
181Dic6—241no—
182C2×C9—22no—
183C2×C9—21no—
184C2×C9—22no—
185C2×C9—21no—
186C2×C9#39511yescharacteristic
187C2×C9B(4,𝔽2)11yescharacteristic
188C2×C9#45811yescharacteristic
189C24C22×D911yescharacteristic
190C2×Q8—181no—
191C2×Q8—181no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—92no—
197C22×C4—92no—
198C22×C4—182no—
199C22×C4#61111yescharacteristic
200C22×C4#61111yescharacteristic
201C4⋊C4—42no—
202C4⋊C4—182no—
203C4⋊C4—182no—
204C4⋊C4—181no—
205C4⋊C4—181no—
206C4⋊C4—364no—
207C22⋊C4—21no—
208C22⋊C4—22no—
209C22⋊C4—94no—
210C22⋊C4—182no—
211C22⋊C4—182no—
212C22⋊C4—182no—
213C22⋊C4Dic1812yescomplemented
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—22no—
217C22×C3—21no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—42no—
221C22×C3#88812yes—
222C22×C3#89612yes—
223C22×C3#92411yescharacteristic
224C22×C3#93711yescharacteristic
225C22×C3#94411yescharacteristic
226C3×C4—42no—
227C3×C4—41no—
228C3×C4—42no—
229C3×C4—41no—
230Dic3—124no—
231Dic3—122no—
232Dic3—122no—
233C9#206211yescharacteristic, complemented, Sylow
234C23—41no—
235C23—41no—
236C23C2×Dic1811yescharacteristic
237C23#369711yescharacteristic
238C23D4×D912yes—
239C23Q8×D912yes—
240C23C2×#61111yescharacteristic
241Q8—721no—
242Q8—721no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—182no—
256C2×C4—182no—
257C2×C4—184no—
258C2×C4—362no—
259C2×C4—362no—
260C2×C4—362no—
261C2×C4—362no—
262C2×C3—22no—
263C2×C3—21no—
264C2×C3—22no—
265C2×C3—21no—
266C2×C3#497911yescharacteristic
267C2×C3#503211yescharacteristic
268C2×C3#505311yescharacteristic
269C22—21no—
270C22—22no—
271C22—21no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22#6345012yes—
277C22#6345812yes—
278C22#6348611yescharacteristic
279C22#6349911yescharacteristic
280C22#6350611yescharacteristic
281C4—42no—
282C4—42no—
283C4—41no—
284C4—41no—
285C4—364no—
286C4—362no—
287C4—362no—
288C3#6959411yescharacteristic
289C2—21no—
290C2—22no—
291C2—21no—
292C2—22no—
293C2#1058865911yescharacteristic
294C2#1058871211yescharacteristic
295C2#1058873311yescharacteristic
subgroupG/H
centerC22#63499
commutator, cosocleC24×C9C23
Frattini, prime radicalC24×C3C22×D3
FittingC9×#385C2
socleC22×C3#937
hypercenter#385D9
hypocenterC9#2062
LCSG ▷ C24×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C24 ◁ #385

Complements

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

NH ≅ G/Ntype
40. C9×C22⋊C4Q8semidirect
44. #3676Q8semidirect
213. C22⋊C4Dic18semidirect
233. C9#2062semidirect

Maximal subgroups

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

subgroupindex
#105895852
C9×#3852
#105888902
#105882042
#105882032
#695943