Finite Group Databasebeta

← #37571444

Subgroups — #37571444

Order 24448.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#27133288C211yesmaximal, characteristic, complemented
2#27132816C211yesmaximal, characteristic, complemented
3#27132813C211yesmaximal, characteristic, complemented
4#27132573C211yesmaximal, characteristic
5#27132399C211yesmaximal, characteristic, complemented
6#27132386C211yesmaximal, characteristic, complemented
7#328×C191C211yesmaximal, characteristic, complemented
8C22×D764—21no—
9C22×D764C2211yescharacteristic
10C2×#15690276—21no—
11C2×#15690274—21no—
12#20259302—21no—
13#20259301C2211yescharacteristic
14#20259300—21no—
15#20259300—21no—
16#20259299—21no—
17#20259298—21no—
18C2×(C2×D191)⋊C4—21no—
19C2×(C2×D191)⋊C4C2211yescharacteristic
20C22⋊C4×D191—21no—
21#20259244—21no—
22#20259242—21no—
23C22⋊(C191⋊C8)C2211yescharacteristic
24C22⋊Q8×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic
26#95×C191C2211yescharacteristic
27C23×D191—21no—
28C23×D191D411yescharacteristic
29C2×Dic382—21no—
30C2×Dic382D411yescharacteristic, complemented
31C2×D764—21no—
32C2×D764—21no—
33C2×D764—42no—
34C2×D764—42no—
35C2×D764—42no—
36C2×D764—42no—
37C22×Dic191D411yescharacteristic
38C2×C4×D191—21no—
39C2×C4×D191—21no—
40#15690276—42no—
41#15690274—42no—
42C2×C191⋊C8—21no—
43C4⋊Dic191—21no—
44C4⋊Dic191D411yescharacteristic, complemented
45Dic191⋊C4—21no—
46Dic191⋊C4—21no—
47Dic191⋊C4—21no—
48C22⋊Dic191—21no—
49(C2×D191)⋊C4—21no—
50(C2×D191)⋊C4—21no—
51(C2×D191)⋊C4—21no—
52(C2×D191)⋊C4—21no—
53(C2×D191)⋊C4—42no—
54C2×Q8×C191D411yescharacteristic, complemented
55C22×C4×C191—21no—
56C22×C4×C191C2311yescharacteristic
57C2×C8×C191—21no—
58C4⋊C4×C191—21no—
59C4⋊C4×C191D411yescharacteristic, complemented
60C22⋊C4×C191—21no—
61C22×D191—21no—
62C22×D191—21no—
63C22×D191—22no—
64C22×D191—41no—
65C22×D191—41no—
66C22×D191—41no—
67C22×D191—41no—
68C22×D191—41no—
69C22×D191—41no—
70C22×D191—41no—
71C22×D191—41no—
72C22×D191SD1612yescomplemented
73Dic382—41no—
74Dic382—81no—
75D764—22no—
76D764—22no—
77D764—41no—
78D764—41no—
79D764—48no—
80C2×Dic191—21no—
81C2×Dic191—21no—
82C2×Dic191—21no—
83C2×Dic191—41no—
84C4×D191—81no—
85C4×D191—81no—
86C191⋊C8—41no—
87C23×C191C2×D411yescharacteristic
88Q8×C191—41no—
89C2×C4×C191—22no—
90C2×C4×C191—21no—
91C2×C4×C191—21no—
92C2×C4×C191—41no—
93C2×C4×C191—41no—
94C2×C4×C191—41no—
95C2×C4×C191C2×D411yescharacteristic
96C2×C4×C191C2×D411yescharacteristic
97C8×C191—41no—
98C22×C191—21no—
99C22×C191—21no—
100C22×C191#12011yescharacteristic
101C22×C191C2×SD1611yescharacteristic
102C22×C191#13711yescharacteristic
103C2×D191—22no—
104C2×D191—41no—
105C2×D191—41no—
106C2×D191—42no—
107C2×D191—42no—
108C2×D191—42no—
109C2×D191—42no—
110C2×D191—81no—
111C2×D191—81no—
112C2×D191—81no—
113C2×D191—81no—
114C2×D191—81no—
115C4×C191—21no—
116C4×C191—21no—
117C4×C191—42no—
118C4×C191—41no—
119C4×C191—41no—
120Dic191—41no—
121Dic191—41no—
122Dic191—41no—
123C2×C191—21no—
124C2×C191—21no—
125C2×C191#44811yescharacteristic
126C2×C191#45311yescharacteristic
127C2×C191B(4,𝔽2)11yescharacteristic
128D191—42no—
129D191—82no—
130C191#159611yescharacteristic, complemented, Sylow
131#1596—1911nomaximal, Sylow
132#543—1911no—
133#450—1911no—
134#450—1911no—
135#392—1911no—
136#328—1911no—
137#328D19111yescharacteristic, complemented
138#323—1911no—
139C22×D4—3821no—
140C22×D4—1911no—
141C2×SD16—3821no—
142C2×SD16—3821no—
143C22⋊Q8—3821no—
144C22⋊Q8—3821no—
145C22⋊Q8—3821no—
146C22⋊Q8—3821no—
147C22⋊Q8—3821no—
148C22⋊Q8—1911no—
149C22⋊Q8C2×D19111yescharacteristic
150C2×C22⋊C4—3821no—
151C2×C22⋊C4—3821no—
152C2×C22⋊C4—1911no—
153#102—3821no—
154#102—3821no—
155C22⋊C8—1911no—
156C22⋊C8C2×D19111yescharacteristic
157#95C2×D19111yescharacteristic
158C24—3821no—
159C24—1911no—
160C2×Q8—3821no—
161C2×Q8—1911no—
162C2×Q8D76411yescharacteristic, complemented
163C2×D4—3821no—
164C2×D4—3821no—
165C2×D4—7642no—
166C2×D4—7642no—
167C2×D4—7642no—
168C2×D4—7642no—
169C22×C4—21no—
170C22×C4—3821no—
171C22×C4—3821no—
172C22×C4—1911no—
173C22×C4C22×D19111yescharacteristic
174SD16—7642no—
175SD16—7642no—
176C2×C8—21no—
177C2×C8—3821no—
178C4⋊C4—21no—
179C4⋊C4—3821no—
180C4⋊C4—3821no—
181C4⋊C4—3821no—
182C4⋊C4—3821no—
183C4⋊C4—1911no—
184C4⋊C4D76411yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—3821no—
187C22⋊C4—3821no—
188C22⋊C4—3821no—
189C22⋊C4—3821no—
190C22⋊C4—3821no—
191C22⋊C4—7642no—
192C23—3822no—
193C23—3821no—
194C23—3821no—
195C23—1912no—
196C23—7641no—
197C23—7641no—
198C23—7641no—
199C23—7641no—
200C23—7641no—
201C23—7641no—
202C23—7641no—
203C23—7641no—
204C23D4×D19111yescharacteristic
205Q8—41no—
206Q8—15281no—
207Q8—7641no—
208D4—3822no—
209D4—3822no—
210D4—7641no—
211D4—7641no—
212D4—7648no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3821no—
220C2×C4—3821no—
221C2×C4—3821no—
222C2×C4—15281no—
223C2×C4—15281no—
224C2×C4—7641no—
225C2×C4D4×D19111yescharacteristic
226C2×C4C2×D76411yescharacteristic
227C8—41no—
228C8—7641no—
229C22—21no—
230C22—21no—
231C22—3822no—
232C22—15281no—
233C22—15281no—
234C22—15281no—
235C22—15281no—
236C22—15281no—
237C22—7642no—
238C22—7642no—
239C22—7642no—
240C22—7642no—
241C22—7641no—
242C22—7641no—
243C22#2025929211yescharacteristic
244C22SD16×D19111yescharacteristic
245C22#2025935411yescharacteristic
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—7641no—
252C4—7641no—
253C4—7641no—
254C2—21no—
255C2—21no—
256C2—15282no—
257C2—7642no—
258C2#2713280311yescharacteristic
259C2#2713283211yescharacteristic
260C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#328×C191C2
socleC22×C191#120
prime radical#27132573C2
hypercenter#328D191
hypocenterC191#1596
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27133288C2semidirect
2. #27132816C2semidirect
3. #27132813C2semidirect
5. #27132399C2semidirect
6. #27132386C2semidirect
7. #328×C191C2semidirect
30. C2×Dic382D4semidirect
44. C4⋊Dic191D4semidirect
54. C2×Q8×C191D4semidirect
59. C4⋊C4×C191D4semidirect
72. C22×D191SD16semidirect
130. C191#1596semidirect
137. #328D191semidirect
162. C2×Q8D764semidirect
184. C4⋊C4D764semidirect

Maximal subgroups

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

subgroupindex
#271332882
#271328162
#271328132
#271325732
#271323992
#271323862
#328×C1912
#1596191