Finite Group Databasebeta

← #30669296

Subgroups — #30669296

Order 16768.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14523513C211yesmaximal, characteristic
2C2×#16862229C212yesmaximal, complemented
3#22553232C212yesmaximal, complemented
4#22553229C212yesmaximal, complemented
5#22553191C212yesmaximal, complemented
6#22553183C214yesmaximal, complemented
7#22553175C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C131C211yesmaximal, characteristic, complemented
9C2×#14523513—28no—
10C2×#14523513—24no—
11C22×Dic262C2211yescharacteristic
12C22×D524C2211yescharacteristic
13C22×C22⋊D131C2212yes—
14C22×C4×D131C2212yes—
15#16862229—28no—
16C2×Dic131⋊C4C2214yes—
17C2×C22⋊Dic131C2212yes—
18C2×(C2×D131)⋊C4C2214yes—
19C2×(C2×D131)⋊C4C2214yes—
20C131⋊#95C2212yes—
21C131⋊#95C2212yes—
22C131⋊#95C2214yes—
23C23×C4×C131C2211yescharacteristic
24C23×C4×C131C2211yescharacteristic
25C23×C4×C131C2211yescharacteristic
26C2×C4⋊C4×C131—24no—
27C2×C4⋊C4×C131C2212yes—
28C2×C4⋊C4×C131C2212yes—
29C23×D131C2312yes—
30#14523513—232no—
31C2×Dic262—24no—
32C2×Dic262—22no—
33C2×D524—24no—
34C2×D524—22no—
35C2×C22⋊D131—24no—
36C2×C22⋊D131—24no—
37C2×C22⋊D131D418yes—
38C22×Dic131C2314yes—
39C22×Dic131C2312yes—
40C2×C4×D131—24no—
41C2×C4×D131—24no—
42C2×C4×D131—24no—
43Dic131⋊C4—28no—
44C22⋊Dic131—24no—
45(C2×D131)⋊C4—28no—
46(C2×D131)⋊C4—28no—
47C24×C131C2311yescharacteristic
48C22×C4×C131—24no—
49C22×C4×C131—22no—
50C22×C4×C131—24no—
51C22×C4×C131—24no—
52C22×C4×C131—22no—
53C22×C4×C131D414yes—
54C22×C4×C131C2312yes—
55C22×C4×C131C2311yescharacteristic
56C22×C4×C131C2311yescharacteristic
57C22×C4×C131C2312yes—
58C4⋊C4×C131—24no—
59C4⋊C4×C131—24no—
60C22×D131—22no—
61C22×D131—22no—
62C22×D131—24no—
63C22×D131—24no—
64C22×D131C2×D414yes—
65Dic262—28no—
66D524—28no—
67C22⋊D131—216no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—24no—
71C2×Dic131—24no—
72C2×Dic131—24no—
73C2×Dic131C2×D414yes—
74C4×D131—216no—
75C23×C131—22no—
76C23×C131—22no—
77C23×C131C2×D414yes—
78C23×C131C2×D412yes—
79C23×C131C2411yescharacteristic
80C2×C4×C131—28no—
81C2×C4×C131—28no—
82C2×C4×C131—22no—
83C2×C4×C131—22no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—22no—
87C2×C4×C131—24no—
88C2×C4×C131—44no—
89C2×C4×C131C2×D412yes—
90C2×C4×C131C2×D412yes—
91C2×C4×C131#4114yescomplemented
92C22×C131—22no—
93C22×C131—22no—
94C22×C131—28no—
95C22×C131#12014yes—
96C22×C131C22×D412yes—
97C22×C131C22×D411yescharacteristic
98C22×C131C2×#4112yes—
99C22×C1312−1+412yes—
100C2×D131—24no—
101C2×D131—28no—
102C2×D131—42no—
103C2×D131—42no—
104C2×D131—44no—
105C4×C131—22no—
106C4×C131—22no—
107C4×C131—24no—
108C4×C131—42no—
109Dic131—24no—
110Dic131—44no—
111C2×C131—24no—
112C2×C131C2×#12011yescharacteristic
113C2×C131#52511yescharacteristic
114C2×C131#53611yescharacteristic
115C2×C131#54714yes—
116D131—44no—
117C131#240611yescharacteristic, complemented, Sylow
118#2406—1311nomaximal, Sylow
119C22×#41—1311no—
120C2×#123—1312no—
121C22×C4⋊C4D13111yescharacteristic, complemented
122#395—1312no—
123#394—1312no—
124#386—1312no—
125#385—1314no—
126#381—1311no—
127C2×#41—2628no—
128C2×#41—2624no—
129C22×Q8—1311no—
130C22×D4—1312no—
131C22×D4—1311no—
132C23×C4—1312no—
133C23×C4C2×D13111yescharacteristic
134C23×C4C2×D13111yescharacteristic
135C23×C4C2×D13111yescharacteristic
136#123—2628no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—1314no—
139C2×C4⋊C4C2×D13112yes—
140C2×C4⋊C4C2×D13112yes—
141C2×C22⋊C4—1312no—
142C2×C22⋊C4—1314no—
143C2×C22⋊C4—1314no—
144#95—1312no—
145#95—1312no—
146#95—1314no—
147C24—1312no—
148C24C22×D13111yescharacteristic
149#41—26232no—
150C2×Q8—2624no—
151C2×Q8—2622no—
152C2×D4—2624no—
153C2×D4—2624no—
154C2×D4—2624no—
155C2×D4—2622no—
156C2×D4—1318no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—2624no—
163C22×C4—2624no—
164C22×C4—2624no—
165C22×C4—1314no—
166C22×C4—1312no—
167C22×C4C22⋊D13114yes—
168C22×C4C22×D13112yes—
169C22×C4C22×D13111yescharacteristic
170C22×C4C22×D13111yescharacteristic
171C22×C4C22×D13112yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—2628no—
175C22⋊C4—2628no—
176C22⋊C4—2624no—
177C22⋊C4—2628no—
178C23—22no—
179C23—22no—
180C23—2622no—
181C23—2622no—
182C23—2624no—
183C23—2624no—
184C23—1314no—
185C23D4×D13114yes—
186C23C2×C22⋊D13112yes—
187C23C23×D13111yescharacteristic
188Q8—2628no—
189D4—2628no—
190D4—26216no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—26216no—
201C2×C4—2622no—
202C2×C4—2622no—
203C2×C4—2624no—
204C2×C4—2624no—
205C2×C4—2624no—
206C2×C4—1314no—
207C2×C4C2×C22⋊D13112yes—
208C2×C4C2×C22⋊D13112yes—
209C2×C4#1452351314yescomplemented
210C22—22no—
211C22—28no—
212C22—22no—
213C22—2628no—
214C22—2624no—
215C22—5242no—
216C22—5244no—
217C22—5242no—
218C22#1686221414yes—
219C22C2×D4×D13112yes—
220C22C22×C22⋊D13111yescharacteristic
221C22C2×#1452351312yes—
222C22#1686229412yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—2624no—
228C4—5244no—
229C2—24no—
230C2—5244no—
231C2C2×#1686221411yescharacteristic
232C2#2255382311yescharacteristic
233C2#2255388611yescharacteristic
234C2#2255396314yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C4⋊C4×C131C2
prime radicalC22×#14523513C2
hypercenterC22×C4⋊C4D131
hypocenterC131#2406
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16862229C2semidirect
3. #22553232C2semidirect
4. #22553229C2semidirect
5. #22553191C2semidirect
6. #22553183C2semidirect
7. #22553175C2semidirect
8. C22×C4⋊C4×C131C2semidirect
91. C2×C4×C131#41semidirect
117. C131#2406semidirect
121. C22×C4⋊C4D131semidirect
209. C2×C4#14523513semidirect

Maximal subgroups

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

subgroupindex
C22×#145235132
C2×#168622292
#225532322
#225532292
#225531912
#225531832
#225531752
C22×C4⋊C4×C1312
#2406131