Finite Group Databasebeta

← #30664483

Subgroups — #30664483

Order 16768.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#22553843C211yesmaximal, characteristic
2#22553366C211yesmaximal, characteristic, complemented
3#22553163C211yesmaximal, characteristic, complemented
4#22553050C211yesmaximal, characteristic, complemented
5#22553048C211yesmaximal, characteristic, complemented
6#22553046C211yesmaximal, characteristic
7#327×C131C211yesmaximal, characteristic
8#16862239—21no—
9#16862237C411yescharacteristic, complemented
10#16862233C411yescharacteristic, complemented
11#16862227—21no—
12#16862225C2211yescharacteristic
13#16862211C411yescharacteristic, complemented
14D4×Dic131—21no—
15D4×Dic131C411yescharacteristic, complemented
16#16862202—21no—
17C2×Dic131⋊C4—21no—
18C2×C4×Dic131—21no—
19C2×C4×Dic131C2211yescharacteristic
20Dic131⋊C8—21no—
21C22⋊(C131⋊C8)C2211yescharacteristic
22#16862156—21no—
23C131⋊#95C2211yescharacteristic
24#121×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C2×Dic262C2×C411yescharacteristic
28C22×Dic131—21no—
29C22×Dic131—21no—
30C22×Dic131D411yescharacteristic
31C22×Dic131D411yescharacteristic
32C2×C131⋊C8—21no—
33C4⋊Dic131—21no—
34C4⋊Dic131C2×C411yescharacteristic
35Dic131⋊C4—24no—
36Dic131⋊C4—21no—
37Dic131⋊C4—21no—
38Dic131⋊C4—21no—
39C22⋊Dic131—21no—
40C22⋊Dic131—21no—
41C22⋊Dic131—21no—
42C22⋊Dic131—21no—
43C4×Dic131—21no—
44C4×Dic131—42no—
45C4×Dic131C2×C411yescharacteristic
46C4×Dic131C2×C411yescharacteristic
47C4×Dic131D411yescharacteristic
48C4×Dic131D411yescharacteristic
49C2×D4×C131—21no—
50C2×D4×C131C2×C411yescharacteristic
51C22×C4×C131—21no—
52C22×C4×C131C2311yescharacteristic
53C2×C8×C131—21no—
54C4⋊C4×C131C2×C411yescharacteristic
55C22⋊C4×C131—21no—
56Dic262—41no—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—22no—
60C2×Dic131—22no—
61C2×Dic131—21no—
62C2×Dic131—21no—
63C2×Dic131—41no—
64C2×Dic131—41no—
65C2×Dic131—42no—
66C2×Dic131C22⋊C411yescharacteristic
67C2×Dic131C22⋊C411yescharacteristic
68C2×Dic131C22⋊C411yescharacteristic
69C2×Dic131C22⋊C411yescharacteristic
70C131⋊C8—41no—
71C23×C131—21no—
72C23×C131C2×D411yescharacteristic
73D4×C131—41no—
74D4×C131—41no—
75D4×C131—41no—
76C2×C4×C131—22no—
77C2×C4×C131—21no—
78C2×C4×C131—41no—
79C2×C4×C131—41no—
80C2×C4×C131—41no—
81C2×C4×C131C22×C411yescharacteristic
82C2×C4×C131C2×D411yescharacteristic
83C8×C131—41no—
84C22×C131—21no—
85C22×C131—21no—
86C22×C131—41no—
87C22×C131—41no—
88C22×C131—41no—
89C22×C131C2×C22⋊C411yescharacteristic
90C22×C131AGL(1,ℤ/8)11yescharacteristic
91C22×C131#13711yescharacteristic
92C4×C131—21no—
93C4×C131—21no—
94C4×C131—42no—
95C4×C131—41no—
96Dic131—21no—
97Dic131—21no—
98Dic131—21no—
99Dic131—42no—
100Dic131—41no—
101Dic131—41no—
102Dic131#10412yes—
103C2×C131—21no—
104C2×C131—21no—
105C2×C131—81no—
106C2×C131#41011yescharacteristic
107C2×C131#41711yescharacteristic
108C2×C131C2×#10411yescharacteristic
109C131#150911yescharacteristic, complemented, Sylow
110#1509—1311nomaximal, Sylow
111#529—1311no—
112#432—1311no—
113C4×C4⋊C4—1311no—
114#328—1311no—
115#328—1311no—
116#327—1311no—
117#327D13111yescharacteristic
118#126—2621no—
119#125—1311no—
120#124—1311no—
121#123—2621no—
122C22⋊Q8—1311no—
123#121C2×D13111yescharacteristic
124C4×Q8—1311no—
125C4×D4—2621no—
126C4×D4—1311no—
127#117—2621no—
128C2×C4⋊C4—2621no—
129C2×C42—2621no—
130C2×C42—1311no—
131C4⋊C8—2621no—
132C22⋊C8—1311no—
133C22⋊C8C2×D13111yescharacteristic
134#97—2621no—
135#95—1311no—
136#95C2×D13111yescharacteristic
137C2×Q8—1311no—
138C2×D4—21no—
139C2×D4C4×D13111yescharacteristic
140C22×C4—21no—
141C22×C4—2621no—
142C22×C4—2621no—
143C22×C4—1311no—
144C22×C4—1311no—
145C22×C4C22×D13111yescharacteristic
146C2×C8—21no—
147C2×C8—2621no—
148C4⋊C4—2624no—
149C4⋊C4—2621no—
150C4⋊C4—2621no—
151C4⋊C4—2621no—
152C4⋊C4—2621no—
153C4⋊C4—1311no—
154C4⋊C4C4×D13111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2621no—
157C22⋊C4—2621no—
158C22⋊C4—2621no—
159C22⋊C4—2621no—
160C42—2621no—
161C42—5242no—
162C42—1311no—
163C42—1311no—
164C42—1311no—
165C42—1311no—
166C23—21no—
167C23D4×D13111yescharacteristic
168Q8—5241no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—2621no—
178C2×C4—2621no—
179C2×C4—2622no—
180C2×C4—2622no—
181C2×C4—2621no—
182C2×C4—2621no—
183C2×C4—5241no—
184C2×C4—5241no—
185C2×C4—5242no—
186C2×C4—1311no—
187C2×C4—1311no—
188C2×C4—1311no—
189C2×C4—1311no—
190C2×C4C2×C4×D13111yescharacteristic
191C2×C4D4×D13111yescharacteristic
192C8—41no—
193C8—5241no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D13111yescharacteristic
200C22#1686227011yescharacteristic
201C22#1686227511yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—2621no—
207C4—2621no—
208C4—2621no—
209C4—5242no—
210C4—5241no—
211C4—5241no—
212C4—1312no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2255328411yescharacteristic
217C2#2255331011yescharacteristic
218C2#104×D13111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#327×C131C2
socleC22×C131C2×C22⋊C4
prime radical#121×C131C22
cosocleC22×C4×C131C23
hypercenter#327D131
hypocenterC131#1509
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #22553366C2semidirect
3. #22553163C2semidirect
4. #22553050C2semidirect
5. #22553048C2semidirect
9. #16862237C4semidirect
10. #16862233C4semidirect
13. #16862211C4semidirect
15. D4×Dic131C4semidirect
109. C131#1509semidirect

Maximal subgroups

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

subgroupindex
#225538432
#225533662
#225531632
#225530502
#225530482
#225530462
#327×C1312
#1509131