Finite Group Databasebeta

← #30666244

Subgroups — #30666244

Order 16768.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#22553787C211yesmaximal, characteristic
2#22553192C211yesmaximal, characteristic
3#22553187C212yesmaximal, complemented
4#22553078C212yesmaximal, complemented
5C2×#99×C131C211yesmaximal, characteristic
6C23×Dic131C2211yescharacteristic
7D4×Dic131—22no—
8D4×Dic131—22no—
9#16862205—21no—
10C2×C4⋊Dic131—22no—
11C2×C22⋊Dic131—21no—
12C2×C22⋊Dic131—21no—
13C2×C22⋊Dic131C2211yescharacteristic
14C2×C22⋊Dic131C2212yes—
15C131⋊#95—22no—
16C131⋊#95—21no—
17C22×D4×C131C2211yescharacteristic
18C2×C22⋊C4×C131C2212yes—
19#99×C131C414yescomplemented
20C22×Dic131—21no—
21C22×Dic131—21no—
22C22×Dic131—21no—
23C22×Dic131—22no—
24C22×Dic131—21no—
25C22×Dic131—42no—
26C22×Dic131D412yes—
27C22×Dic131D412yes—
28C4⋊Dic131—21no—
29C4⋊Dic131—21no—
30C4⋊Dic131—44no—
31C22⋊Dic131—22no—
32C22⋊Dic131—22no—
33C22⋊Dic131—22no—
34C22⋊Dic131—22no—
35C22⋊Dic131—42no—
36C4×Dic131—22no—
37C24×C131D411yescharacteristic
38C24×C131C2311yescharacteristic
39C2×D4×C131—22no—
40C2×D4×C131—22no—
41C2×D4×C131C2×C412yes—
42C2×D4×C131Q812yes—
43C22×C4×C131—22no—
44C22×C4×C131D411yescharacteristic
45C22⋊C4×C131—22no—
46C22⋊C4×C131C2×C414yescomplemented
47C2×Dic131—22no—
48C2×Dic131—24no—
49C2×Dic131—22no—
50C2×Dic131—22no—
51C2×Dic131—42no—
52C2×Dic131—41no—
53C2×Dic131—41no—
54C2×Dic131—42no—
55C2×Dic131—42no—
56C2×Dic131—41no—
57C2×Dic131—41no—
58C2×Dic131—41no—
59C2×Dic131—81no—
60C2×Dic131—81no—
61C23×C131—21no—
62C23×C131—21no—
63C23×C131—41no—
64C23×C131—41no—
65C23×C131—41no—
66C23×C131—41no—
67C23×C131C4⋊C412yescomplemented
68C23×C131C22×C411yescharacteristic
69C23×C131C2×D412yes—
70C23×C131C2×D411yescharacteristic
71C23×C131C2×Q811yescharacteristic
72C23×C131#4112yes—
73D4×C131—24no—
74D4×C131—24no—
75C2×C4×C131—21no—
76C2×C4×C131—21no—
77C2×C4×C131—24no—
78C2×C4×C131—42no—
79C2×C4×C131—42no—
80C2×C4×C131C4⋊C412yescomplemented
81C22×C131—22no—
82C22×C131—21no—
83C22×C131—22no—
84C22×C131—21no—
85C22×C131—22no—
86C22×C131—41no—
87C22×C131—41no—
88C22×C131—42no—
89C22×C131—42no—
90C22×C131—41no—
91C22×C131—41no—
92C22×C131—81no—
93C22×C131C2×C4⋊C411yescharacteristic
94C22×C131C4×D412yes—
95C22×C131#12011yescharacteristic
96C22×C131C22⋊Q812yes—
97C22×C131#12311yescharacteristic
98C4×C131—22no—
99C4×C131—44no—
100Dic131—42no—
101Dic131—42no—
102Dic131—42no—
103Dic131—81no—
104C2×C131—21no—
105C2×C131—22no—
106C2×C131—21no—
107C2×C131—22no—
108C2×C131—42no—
109C2×C131#38511yescharacteristic
110C2×C131B(4,𝔽2)11yescharacteristic
111C2×C131#45811yescharacteristic
112C131#186711yescharacteristic, complemented, Sylow
113#1867—1311nomaximal, Sylow
114#518—1311no—
115C2×#99D13111yescharacteristic
116#386—1311no—
117#385—1312no—
118#342—1312no—
119C22×D4C2×D13111yescharacteristic
120C23×C4—1311no—
121C4×D4—2622no—
122C4×D4—2622no—
123#117—2621no—
124C2×C4⋊C4—2622no—
125C2×C22⋊C4—2621no—
126C2×C22⋊C4—2621no—
127C2×C22⋊C4—1311no—
128C2×C22⋊C4—1312no—
129C2×C22⋊C4C2×D13112yes—
130#99Dic13114yescomplemented
131#95—2622no—
132#95—2621no—
133C24D52411yescharacteristic
134C24C22×D13111yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4C2×Dic13112yes—
138C2×D4Dic26212yes—
139C22×C4—22no—
140C22×C4—2622no—
141C22×C4—2621no—
142C22×C4—2621no—
143C22×C4—2621no—
144C22×C4—2621no—
145C22×C4—5242no—
146C22×C4—1312no—
147C22×C4—1312no—
148C22×C4D52411yescharacteristic
149C4⋊C4—2621no—
150C4⋊C4—2621no—
151C4⋊C4—5244no—
152C22⋊C4—22no—
153C22⋊C4—2622no—
154C22⋊C4—2622no—
155C22⋊C4—2622no—
156C22⋊C4—2622no—
157C22⋊C4—5242no—
158C22⋊C4C2×Dic13114yescomplemented
159C42—2622no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23C4⋊Dic13112yescomplemented
167C23C22×Dic13111yescharacteristic
168C23D4×D13112yes—
169C23C2×D52411yescharacteristic
170C23C2×Dic26211yescharacteristic
171C23#1452351212yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—2622no—
180C2×C4—2624no—
181C2×C4—2622no—
182C2×C4—2622no—
183C2×C4—5242no—
184C2×C4—5241no—
185C2×C4—5241no—
186C2×C4—5242no—
187C2×C4—5242no—
188C2×C4—5241no—
189C2×C4—5241no—
190C2×C4—5241no—
191C2×C4—10481no—
192C2×C4—10481no—
193C2×C4C4⋊Dic13112yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×C4⋊Dic13111yescharacteristic
207C22D4×Dic13112yes—
208C22#1686221611yescharacteristic
209C22#1686222312yes—
210C22#1686223011yescharacteristic
211C4—22no—
212C4—44no—
213C4—5242no—
214C4—5242no—
215C4—5242no—
216C4—10481no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2255318711yescharacteristic
223C2#2255350011yescharacteristic
224C2#2255350311yescharacteristic
subgroupG/H
centerC22#16862216
commutatorC23×C131C22×C4
FrattiniC24C22×D131
FittingC2×#99×C131C2
socleC22×C131#120
prime radicalC22×D4×C131C22
cosocleC24×C131C23
hypercenterC2×#99D131
hypocenterC131#1867
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #22553187C2semidirect
4. #22553078C2semidirect
19. #99×C131C4semidirect
46. C22⋊C4×C131C2×C4semidirect
67. C23×C131C4⋊C4semidirect
80. C2×C4×C131C4⋊C4semidirect
112. C131#1867semidirect
130. #99Dic131semidirect
158. C22⋊C4C2×Dic131semidirect
166. C23C4⋊Dic131semidirect
193. C2×C4C4⋊Dic131semidirect

Maximal subgroups

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

subgroupindex
#225537872
#225531922
#225531872
#225530782
C2×#99×C1312
#1867131