Finite Group Databasebeta

← #30664950

Subgroups — #30664950

Order 16768.

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

normal subgroups only

This group has 228 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#22553937C211yesmaximal, characteristic
2#22553642C211yesmaximal, characteristic, complemented
3#22553238C211yesmaximal, characteristic
4#22553050C211yesmaximal, characteristic, complemented
5#22553049C211yesmaximal, characteristic, complemented
6#22553035C211yesmaximal, characteristic
7#483×C131C211yesmaximal, characteristic, complemented
8#16862226—21no—
9#16862225C2211yescharacteristic
10#16862224—21no—
11#16862224—21no—
12#16862223—21no—
13#16862222—21no—
14C2×(C2×D131)⋊C4—21no—
15C2×(C2×D131)⋊C4C2211yescharacteristic
16C22⋊C4×D131—21no—
17#16862177—21no—
18#16862169—21no—
19C22⋊(C131⋊C8)C2211yescharacteristic
20C131⋊#95—21no—
21C131⋊#95C2211yescharacteristic
22C22⋊Q8×C131C2211yescharacteristic
23C2×C4⋊C4×C131C2211yescharacteristic
24#106×C131—21no—
25#103×C131—21no—
26C22⋊C8×C131C2211yescharacteristic
27C23×D131D411yescharacteristic
28C2×Dic262—21no—
29C2×Dic262D411yescharacteristic
30C22×Dic131—21no—
31C22×Dic131D411yescharacteristic
32C2×C4×D131—21no—
33C2×C4×D131—21no—
34C2×C131⋊C8—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131D411yescharacteristic
37Dic131⋊C4—21no—
38Dic131⋊C4—21no—
39Dic131⋊C4—21no—
40C22⋊Dic131—21no—
41(C2×D131)⋊C4—21no—
42(C2×D131)⋊C4—21no—
43(C2×D131)⋊C4—21no—
44(C2×D131)⋊C4—21no—
45(C2×D131)⋊C4—42no—
46C2×Q8×C131D411yescharacteristic
47C22×C4×C131—21no—
48C22×C4×C131C2311yescharacteristic
49C2×C8×C131—21no—
50C4⋊C4×C131—21no—
51C4⋊C4×C131—21no—
52C4⋊C4×C131—21no—
53C4⋊C4×C131D411yescharacteristic
54C22⋊C4×C131—21no—
55C22×D131—21no—
56C22×D131—21no—
57C22×D131—41no—
58C22×D131—41no—
59C22×D131SD1612yes—
60Dic262—41no—
61Dic262—81no—
62C2×Dic131—21no—
63C2×Dic131—21no—
64C2×Dic131—21no—
65C2×Dic131—41no—
66C2×Dic131—41no—
67C2×Dic131—41no—
68C2×Dic131—41no—
69C4×D131—81no—
70C4×D131—81no—
71C131⋊C8—41no—
72C23×C131C2×D411yescharacteristic
73Q8×C131—41no—
74C2×C4×C131—21no—
75C2×C4×C131—21no—
76C2×C4×C131—41no—
77C2×C4×C131—41no—
78C2×C4×C131—41no—
79C2×C4×C131—41no—
80C2×C4×C131C2×D411yescharacteristic
81C2×C4×C131C2×D411yescharacteristic
82C8×C131—41no—
83C22×C131—21no—
84C22×C131—21no—
85C22×C131#12011yescharacteristic
86C22×C131C2×SD1611yescharacteristic
87C22×C131#13711yescharacteristic
88C2×D131—22no—
89C2×D131—41no—
90C2×D131—41no—
91C2×D131—42no—
92C2×D131—81no—
93C4×C131—21no—
94C4×C131—21no—
95C4×C131—41no—
96C4×C131—41no—
97C4×C131—81no—
98Dic131—41no—
99Dic131—41no—
100Dic131—41no—
101Dic131—81no—
102C2×C131—21no—
103C2×C131—21no—
104C2×C131#44811yescharacteristic
105C2×C131#45511yescharacteristic
106C2×C131#45811yescharacteristic
107D131—42no—
108C131#160811yescharacteristic, complemented, Sylow
109#1608—1311nomaximal, Sylow
110#543—1311no—
111#483—1311no—
112#483D13111yescharacteristic, complemented
113#397—1311no—
114#328—1311no—
115#328—1311no—
116#323—1311no—
117C22⋊Q8—2621no—
118C22⋊Q8—2621no—
119C22⋊Q8—2621no—
120C22⋊Q8—2621no—
121C22⋊Q8—2621no—
122C22⋊Q8—1311no—
123C22⋊Q8C2×D13111yescharacteristic
124C2×C4⋊C4C2×D13111yescharacteristic
125C2×C22⋊C4—2621no—
126C2×C22⋊C4—2621no—
127C2×C22⋊C4—1311no—
128#106—21no—
129#106—2621no—
130#103—21no—
131#103—2621no—
132C22⋊C8—1311no—
133C22⋊C8C2×D13111yescharacteristic
134#95—2621no—
135#95—1311no—
136C24—1311no—
137C2×Q8—2621no—
138C2×Q8—1311no—
139C2×Q8C22⋊D13111yescharacteristic
140C22×C4—21no—
141C22×C4—2621no—
142C22×C4—2621no—
143C22×C4—2621no—
144C22×C4—1311no—
145C22×C4C22×D13111yescharacteristic
146C2×C8—21no—
147C2×C8—2621no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—2621no—
152C4⋊C4—2621no—
153C4⋊C4—2621no—
154C4⋊C4—2621no—
155C4⋊C4—1311no—
156C4⋊C4C22⋊D13111yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—2621no—
159C22⋊C4—2621no—
160C22⋊C4—2621no—
161C22⋊C4—2621no—
162C22⋊C4—2621no—
163C22⋊C4—5242no—
164C23—2621no—
165C23—2621no—
166C23—5241no—
167C23—5241no—
168C23—1312no—
169C23D4×D13111yescharacteristic
170Q8—41no—
171Q8—5241no—
172Q8—10481no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—2621no—
180C2×C4—2621no—
181C2×C4—2621no—
182C2×C4—5241no—
183C2×C4—5241no—
184C2×C4—5241no—
185C2×C4—5241no—
186C2×C4—10481no—
187C2×C4—10481no—
188C2×C4D4×D13111yescharacteristic
189C2×C4C2×C22⋊D13111yescharacteristic
190C8—41no—
191C8—5241no—
192C22—21no—
193C22—21no—
194C22—2622no—
195C22—5242no—
196C22—5241no—
197C22—5241no—
198C22—10481no—
199C22#1686221411yescharacteristic
200C22SD16×D13111yescharacteristic
201C22#1686227511yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—5241no—
208C4—5241no—
209C4—5241no—
210C4—10481no—
211C2—21no—
212C2—21no—
213C2—5242no—
214C2#2255344911yescharacteristic
215C2#2255348811yescharacteristic
216C2#2255350111yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC22×C4×C131C23
FrattiniC22×C4C22×D131
Fitting#483×C131C2
socleC22×C131#120
prime radicalC23×D131D4
hypercenter#483D131
hypocenterC131#1608
LCSG ▷ C22×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
2. #22553642C2semidirect
4. #22553050C2semidirect
5. #22553049C2semidirect
7. #483×C131C2semidirect
108. C131#1608semidirect
112. #483D131semidirect

Maximal subgroups

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

subgroupindex
#225539372
#225536422
#225532382
#225530502
#225530492
#225530352
#483×C1312
#1608131