Finite Group Databasebeta

← #11794269

Subgroups — #11794269

Order 1152.

690 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 181 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#10595990C211yesmaximal, characteristic
2C32×#428C211yesmaximal, characteristic
3#10592676C211yesmaximal, characteristic
4#10592663C211yesmaximal, characteristic
5#10589421C211yesmaximal, characteristic
6#10589420C211yesmaximal, characteristic
7#10589398C211yesmaximal, characteristic
8#70667—34nomaximal
9C2×C32×M16C2211yescharacteristic
10C32×#117C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12#64135C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C4×C3⋊Dic3C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16#64099—21no—
17#64097—21no—
18#64087—22no—
19#64079—21no—
20#64078—21no—
21C32×#107—21no—
22C32×#106—21no—
23#63658—22no—
24#63635—21no—
25#63634—21no—
26#5519—34no—
27C3×#428D314yes—
28#4977—34no—
29#4964—34no—
30#4555—34no—
31#4554—34no—
32#4532—34no—
33C22×C32×C4—21no—
34C22×C32×C4C2311yescharacteristic
35C22×C3⋊Dic3—21no—
36C22×C3⋊Dic3D412yes—
37C2×#634—22no—
38C32×M16—21no—
39C32×M16Q812yescomplemented
40C2×C32×C8—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4D411yescharacteristic
44C32×C4⋊C4D411yescharacteristic
45C32×C22⋊C4—21no—
46C32×C42—21no—
47#3757—21no—
48C4⋊(C3⋊Dic3)—22no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)D411yescharacteristic
52C4⋊(C3⋊Dic3)D411yescharacteristic
53#3750—22no—
54#3750—22no—
55C4×C3⋊Dic3—21no—
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3Q812yescomplemented
58#3748—21no—
59#3748Q812yescomplemented
60C2×C3⋊(C3⋊C8)—21no—
61#2061—91noSylow
62C2×C3×M16C2×D314yes—
63C3×#117C2×D314yes—
64C2×C3×C4⋊C4C2×D314yes—
65#926—34no—
66C2×C4⋊Dic3—34no—
67C2×C4×Dic3—34no—
68C2×#208—34no—
69#890—64no—
70#888—64no—
71#878—68no—
72#870—64no—
73#869—64no—
74C3×#107—24no—
75C3×#106—24no—
76C3⋊#95—68no—
77#808—64no—
78#807—64no—
79C23×C32C2×D411yescharacteristic
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4—42no—
83C2×C32×C4—41no—
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×Q811yescharacteristic
87C2×C32×C4C2×Q811yescharacteristic
88C2×C32×C4C2×Q811yescharacteristic
89C2×C32×C4#4111yescharacteristic
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—41no—
95C2×C3⋊Dic3—41no—
96C2×C3⋊Dic3—42no—
97#634—82no—
98C32×C8—22no—
99C3⋊(C3⋊C8)—22no—
100#544—91no—
101#428—91no—
102#428C3⋊D311yescharacteristic
103#382—91no—
104#343—91no—
105#339—91no—
106#339—91no—
107C22×C3×C4—24no—
108C22×C3×C4C22×D314yes—
109C22×Dic3—38no—
110C22×Dic3—64no—
111C2×Dic6—68no—
112C3×M16—24no—
113C3×M16Dic618yescomplemented
114C2×C3×C8—24no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4#6714yes—
118C3×C4⋊C4#6714yes—
119C3×C22⋊C4—24no—
120C3×C42—24no—
121#217—64no—
122C4⋊Dic3—34no—
123C4⋊Dic3—34no—
124C4⋊Dic3—68no—
125C4⋊Dic3—64no—
126C4⋊Dic3—64no—
127Dic3⋊C4—68no—
128Dic3⋊C4—68no—
129C4×Dic3—38no—
130C4×Dic3—64no—
131C4×Dic3—64no—
132#208—38no—
133#208—64no—
134C2×C3⋊C8—64no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32#12011yescharacteristic
138C22×C32C22⋊Q811yescharacteristic
139C22×C32C4⋊Q811yescharacteristic
140C32×C4—41no—
141C32×C4—41no—
142C32×C4—81no—
143C32×C4C22⋊Q811yescharacteristic
144C32×C4C22⋊Q811yescharacteristic
145C32×C4C4⋊Q811yescharacteristic
146C32×C4C4⋊Q811yescharacteristic
147C3⋊Dic3—42no—
148C3⋊Dic3—41no—
149C3⋊Dic3—41no—
150C3⋊Dic3—42no—
151C2×M16—91no—
152C2×M16C2×C3⋊D311yescharacteristic
153C4⋊Q8—181no—
154C4⋊Q8—181no—
155#125—181no—
156#125—181no—
157C22⋊Q8—182no—
158#117—91no—
159#117C2×C3⋊D311yescharacteristic
160C2×C4⋊C4—91no—
161C2×C4⋊C4C2×C3⋊D311yescharacteristic
162C2×C42—91no—
163#107—21no—
164#107—181no—
165#106—21no—
166#106—181no—
167#95—182no—
168C23×C3D3×D414yes—
169C2×C3×C4—24no—
170C2×C3×C4—24no—
171C2×C3×C4—48no—
172C2×C3×C4—44no—
173C2×C3×C4C2×Dic614yes—
174C2×C3×C4#23514yes—
175C2×C3×C4D3×D414yes—
176C2×C3×C4D3×Q814yes—
177C2×C3×C4D3×Q814yes—
178C2×C3×C4C2×#6714yes—
179C2×Dic3—68no—
180C2×Dic3—64no—
181C2×Dic3—64no—
182C2×Dic3—68no—
183C2×Dic3—124no—
184C2×Dic3—124no—
185C2×Dic3—128no—
186Dic6—248no—
187C3×C8—28no—
188C3⋊C8—68no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32#39511yescharacteristic
192C2×C32#45411yescharacteristic
193C2×C32#45511yescharacteristic
194C2×Q8—182no—
195C22×C4—21no—
196C22×C4—92no—
197C22×C4—181no—
198C22×C4C22×C3⋊D311yescharacteristic
199M16—21no—
200M16—92no—
201M16—181no—
202M16#63412yescomplemented
203C2×C8—21no—
204C2×C8—181no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C4⋊C4—182no—
210C4⋊C4—182no—
211C4⋊C4—182no—
212C4⋊C4—181no—
213C4⋊C4—181no—
214C4⋊C4#63811yescharacteristic
215C4⋊C4#63811yescharacteristic
216C22⋊C4—21no—
217C22⋊C4—181no—
218C42—21no—
219C42—92no—
220C42—181no—
221C42—181no—
222C22×C3—24no—
223C22×C3—24no—
224C22×C3#88814yes—
225C22×C3#89614yes—
226C22×C3#93714yes—
227C3×C4—44no—
228C3×C4—44no—
229C3×C4—84no—
230C3×C4#88814yes—
231C3×C4#89614yes—
232C3×C4#92414yes—
233C3×C4#94414yes—
234Dic3—128no—
235Dic3—124no—
236Dic3—124no—
237Dic3—128no—
238C32#206111yescharacteristic, complemented, Sylow
239C23D4×C3⋊D311yescharacteristic
240Q8—722no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—182no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4—182no—
249C2×C4—361no—
250C2×C4—361no—
251C2×C4—362no—
252C2×C4C2×#63411yescharacteristic
253C2×C4#382811yescharacteristic
254C2×C4D4×C3⋊D311yescharacteristic
255C2×C4Q8×C3⋊D311yescharacteristic
256C2×C4Q8×C3⋊D311yescharacteristic
257C2×C4C2×#63811yescharacteristic
258C8—22no—
259C8—182no—
260C2×C3—24no—
261C2×C3—24no—
262C2×C3#497914yes—
263C2×C3#520114yes—
264C2×C3#520214yes—
265C22—21no—
266C22—21no—
267C22#6409711yescharacteristic
268C22#6410511yescharacteristic
269C22#6414611yescharacteristic
270C4—41no—
271C4—41no—
272C4—81no—
273C4—362no—
274C4—361no—
275C4—362no—
276C4—361no—
277C4#6409711yescharacteristic
278C4#6410511yescharacteristic
279C4#6413311yescharacteristic
280C4#6415311yescharacteristic
281C3#7066714yescomplemented
282C2—21no—
283C2—21no—
284C2#1059267811yescharacteristic
285C2#1059290011yescharacteristic
286C2#1059290111yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#428C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenter#428C3⋊D3
hypocenterC32#2061
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
39. C32×M16Q8semidirect
57. C4×C3⋊Dic3Q8semidirect
59. #3748Q8semidirect
113. C3×M16Dic6semidirect
202. M16#634semidirect
238. C32#2061semidirect
281. C3#70667semidirect

Maximal subgroups

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

subgroupindex
#105959902
C32×#4282
#105926762
#105926632
#105894212
#105894202
#105893982
#706673