Finite Group Databasebeta

← #18010581

Subgroups — #18010581

Order 4736.

404 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#3866C211yesmaximal, characteristic
2C2×#11924599C212yesmaximal, complemented
3D37⋊(C22⋊C8)C211yesmaximal, characteristic
4#14558406C212yesmaximal, complemented
5C22⋊Q8×D37C211yesmaximal, characteristic
6C23×#3866—21no—
7C2×D37⋊C8C2212yes—
8C2×C4⋊#3866—21no—
9C2×C4⋊#3866—24no—
10C2×C4⋊#3866C2212yes—
11#11924599—24no—
12#11924593—21no—
13C22⋊#64438—21no—
14C2×Q8×D37C2211yescharacteristic
15C22×C4×D37C2211yescharacteristic
16#11924516C411yescharacteristic, complemented
17#11924515C411yescharacteristic, complemented
18#11924514—21no—
19#11924513—21no—
20#11924512C411yescharacteristic, complemented
21C4⋊C4×D37—21no—
22C4⋊C4×D37C2211yescharacteristic
23C22⋊C4×D37—21no—
24C22⋊Q8×C37C411yescharacteristic, complemented
25C23×D37D411yescharacteristic
26C22×#3866—24no—
27C22×#3866—22no—
28C22×#3866—42no—
29C22×#3866—42no—
30C2×Q8×C37C2×C411yescharacteristic, complemented
31C22×C4×C37C2×C411yescharacteristic
32C22×Dic37D411yescharacteristic
33Q8×D37—22no—
34Q8×D37—22no—
35C2×C4×D37—22no—
36C2×C4×D37—22no—
37C2×C4×D37—21no—
38C2×C4×D37—21no—
39C2×C4×D37C2×C411yescharacteristic
40C2×C4×D37D411yescharacteristic
41C2×C4×D37D411yescharacteristic
42C2×C4×D37C2311yescharacteristic
43C2×Dic74—21no—
44C2×Dic74C2×C411yescharacteristic, complemented
45C2×#64438—22no—
46C4⋊#3866—24no—
47C4⋊#3866—22no—
48C4⋊#3866D414yes—
49D37⋊C8—22no—
50D37⋊C8—22no—
51C4⋊C4×C37—21no—
52C4⋊C4×C37C2×C411yescharacteristic, complemented
53C22⋊C4×C37—21no—
54C22⋊Dic37—21no—
55#10596938—21no—
56#10596938—21no—
57C4⋊Dic37—21no—
58C4⋊Dic37C2×C411yescharacteristic, complemented
59#10596936—21no—
60#10596936—21no—
61#10596936—21no—
62C23×C37C22⋊C411yescharacteristic
63C22×D37—21no—
64C22×D37—21no—
65C22×D37—21no—
66C22×D37—21no—
67C22×D37C22⋊C411yescharacteristic
68C22×D37SD1612yes—
69C22×D37Q1612yes—
70C22×D37C2×D411yescharacteristic
71C2×#3866—24no—
72C2×#3866—48no—
73C2×#3866—42no—
74C2×#3866—42no—
75Q8×C37—22no—
76C2×C4×C37—22no—
77C2×C4×C37—21no—
78C2×C4×C37—21no—
79C2×C4×C37C22⋊C411yescharacteristic
80C2×C4×C37C22×C411yescharacteristic
81C2×Dic37—22no—
82C2×Dic37—21no—
83C2×Dic37—21no—
84C2×Dic37C22⋊C411yescharacteristic
85C2×Dic37C2×D411yescharacteristic
86C4×D37—21no—
87C4×D37—21no—
88C4×D37—22no—
89C4×D37—22no—
90C4×D37—41no—
91C4×D37—41no—
92C4×D37—41no—
93C4×D37—41no—
94C4×D37C2×D412yes—
95C4×D37#4112yes—
96Dic74—22no—
97Dic74—42no—
98#64438—42no—
99C22×C37—21no—
100C22×C37—21no—
101C22×C37#10312yescomplemented
102C22×C37C2×C22⋊C411yescharacteristic
103C2×D37—21no—
104C2×D37—21no—
105C2×D37—22no—
106C2×D37—22no—
107C2×D37—22no—
108C2×D37—22no—
109C2×D37#10312yescomplemented
110C2×D37#12011yescharacteristic
111C2×D37#12311yescharacteristic
112C2×D37C2×SD1611yescharacteristic
113C2×D37C2×Q1611yescharacteristic
114C2×D37AGL(1,ℤ/8)11yescharacteristic
115C2×D37#13711yescharacteristic
116#3866—44no—
117C4×C37—21no—
118C4×C37—41no—
119C4×C37—41no—
120C4×C37C4×D412yes—
121Dic37—21no—
122Dic37—41no—
123Dic37—41no—
124Dic37#12112yes—
125#1872—371nomaximal, Sylow
126C2×C37—22no—
127C2×C37#38611yescharacteristic
128C2×C37C2×#10311yescharacteristic
129C2×C37#41711yescharacteristic
130D37—22no—
131D37#45011yescharacteristic
132D37#45111yescharacteristic
133D37#48311yescharacteristic
134D37#48411yescharacteristic
135C2×C22⋊Q8—371no—
136C22×C4⋊C4—371no—
137C2×#103—372no—
138C2×C22⋊C8—371no—
139#340—372no—
140C37#187211yescharacteristic, complemented, Sylow
141C22×Q8—371no—
142C23×C4—371no—
143C23×C4—741no—
144C22×C8—372no—
145C22⋊Q8—371no—
146C22⋊Q8—371no—
147C22⋊Q8—371no—
148C22⋊Q8—741no—
149C22⋊Q8—741no—
150C22⋊Q8#386611yescharacteristic, complemented
151C2×C4⋊C4—372no—
152C2×C4⋊C4—371no—
153C2×C4⋊C4—741no—
154C2×C4⋊C4—744no—
155C2×C4⋊C4—741no—
156C2×C22⋊C4—741no—
157#103—744no—
158C22⋊C8—741no—
159C22⋊C8—741no—
160C24—371no—
161C2×Q8—371no—
162C2×Q8—741no—
163C2×Q8—742no—
164C2×Q8—742no—
165C2×Q8C2×#386611yescharacteristic, complemented
166C22×C4—371no—
167C22×C4—371no—
168C22×C4—371no—
169C22×C4—371no—
170C22×C4—371no—
171C22×C4—742no—
172C22×C4—742no—
173C22×C4—741no—
174C22×C4—741no—
175C22×C4—742no—
176C22×C4—744no—
177C22×C4—1482no—
178C22×C4—1482no—
179C22×C4C2×#386611yescharacteristic
180C2×C8—742no—
181C2×C8—742no—
182C2×C8—742no—
183C4⋊C4—21no—
184C4⋊C4—374no—
185C4⋊C4—371no—
186C4⋊C4—741no—
187C4⋊C4—741no—
188C4⋊C4—741no—
189C4⋊C4—742no—
190C4⋊C4—744no—
191C4⋊C4—741no—
192C4⋊C4C2×#386611yescharacteristic, complemented
193C22⋊C4—21no—
194C22⋊C4—741no—
195C22⋊C4—741no—
196C22⋊C4—741no—
197C23—371no—
198C23—372no—
199C23—372no—
200C23—371no—
201C23—741no—
202C23—741no—
203C23—741no—
204C23—741no—
205C23C22⋊#386611yescharacteristic
206Q8—22no—
207Q8—742no—
208Q8—1482no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—371no—
213C2×C4—372no—
214C2×C4—372no—
215C2×C4—371no—
216C2×C4—742no—
217C2×C4—741no—
218C2×C4—741no—
219C2×C4—742no—
220C2×C4—742no—
221C2×C4—744no—
222C2×C4—741no—
223C2×C4—741no—
224C2×C4—1481no—
225C2×C4—1481no—
226C2×C4—1481no—
227C2×C4—1481no—
228C2×C4—1482no—
229C2×C4—1482no—
230C2×C4—1488no—
231C2×C4C22⋊#386611yescharacteristic
232C2×C4C22×#386611yescharacteristic
233C8—1482no—
234C22—21no—
235C22—21no—
236C22—371no—
237C22—371no—
238C22—371no—
239C22—371no—
240C22—371no—
241C22—371no—
242C22—372no—
243C22—742no—
244C22—742no—
245C22—742no—
246C22—742no—
247C22—741no—
248C22—741no—
249C22#1192459912yescomplemented
250C22C2×C22⋊#386611yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—372no—
255C4—741no—
256C4—1484no—
257C4—1481no—
258C4—1481no—
259C4D4×#386612yes—
260C2—22no—
261C2—371no—
262C2—371no—
263C2—371no—
264C2—371no—
265C2—742no—
266C2#1455848111yescharacteristic
267C2C2×#1192459911yescharacteristic
268C2#1455852411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#3866
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
FittingC22⋊Q8×C37C4
socleC22×C37C2×C22⋊C4
prime radicalC23×D37D4
cosocleC2×C4×D37C23
hypercenterC22⋊Q8#3866
hypocenterC37#1872
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11924599C2semidirect
4. #14558406C2semidirect
16. #11924516C4semidirect
17. #11924515C4semidirect
20. #11924512C4semidirect
24. C22⋊Q8×C37C4semidirect
30. C2×Q8×C37C2×C4semidirect
44. C2×Dic74C2×C4semidirect
52. C4⋊C4×C37C2×C4semidirect
58. C4⋊Dic37C2×C4semidirect
101. C22×C37#103semidirect
109. C2×D37#103semidirect
140. C37#1872semidirect
150. C22⋊Q8#3866semidirect
165. C2×Q8C2×#3866semidirect
192. C4⋊C4C2×#3866semidirect
249. C22#11924599semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#38662
C2×#119245992
D37⋊(C22⋊C8)2
#145584062
C22⋊Q8×D372
#187237