Finite Group Databasebeta

← #30670249

Subgroups — #30670249

Order 16768.

512 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×#16862204C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic
3#22553250C211yesmaximal, characteristic, complemented
4#22553244C211yesmaximal, characteristic, complemented
5#22553242C211yesmaximal, characteristic, complemented
6#22553199C212yesmaximal, complemented
7#22553195C212yesmaximal, complemented
8#22553186C212yesmaximal
9#22553174C212yesmaximal
10#22553174C211yesmaximal, characteristic
11#400×C131C211yesmaximal, characteristic, complemented
12C22×C4×D131C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15#16862204—24no—
16C2×C4⋊Dic131C2212yes—
17C2×C4⋊Dic131C2211yescharacteristic
18C2×C4⋊Dic131C2211yescharacteristic
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C4⋊C4×D131—24no—
23C2×(C2×D131)⋊C4C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×C4×Dic131C2212yes—
26C131⋊#95C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2211yescharacteristic
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2211yescharacteristic
34#95×C131C2212yes—
35#95×C131C2211yescharacteristic
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C22×Dic131C2311yescharacteristic
39C22×Dic131C2312yes—
40C22×Dic131C2312yes—
41C22×Dic131C2311yescharacteristic
42C22×Dic131C2311yescharacteristic
43C2×C4×D131—22no—
44C2×C4×D131—24no—
45C2×C4×D131—22no—
46C2×C4×D131—22no—
47C2×C4×D131—24no—
48C2×C4×D131—22no—
49C2×C4×D131D414yes—
50C4⋊Dic131—22no—
51C4⋊Dic131—24no—
52C4⋊Dic131—22no—
53Dic131⋊C4—22no—
54Dic131⋊C4—22no—
55Dic131⋊C4—24no—
56(C2×D131)⋊C4—24no—
57(C2×D131)⋊C4—24no—
58C4×Dic131—24no—
59C22×C4×C131C2311yescharacteristic
60C22×C4×C131C2312yes—
61C22×C4×C131C2312yes—
62C22×C4×C131C2311yescharacteristic
63C22×C4×C131C2311yescharacteristic
64C4⋊C4×C131—22no—
65C4⋊C4×C131—22no—
66C22×D131—21no—
67C22×D131—21no—
68C22×D131—22no—
69C22×D131C2×D412yes—
70C22×D131#4114yes—
71C2×Dic131—21no—
72C2×Dic131—22no—
73C2×Dic131—21no—
74C2×Dic131—22no—
75C2×Dic131—21no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131C2×D412yes—
83C2×Dic131#4112yes—
84C4×D131—28no—
85C4×D131—28no—
86C4×D131—44no—
87C23×C131C2411yescharacteristic
88C2×C4×C131—21no—
89C2×C4×C131—22no—
90C2×C4×C131—21no—
91C2×C4×C131—22no—
92C2×C4×C131—21no—
93C2×C4×C131—22no—
94C2×C4×C131—22no—
95C2×C4×C131—22no—
96C2×C4×C131—24no—
97C2×C4×C131—22no—
98C2×C4×C131C2×D412yes—
99C2×C4×C131#4112yes—
100C22×C131C22×D411yescharacteristic
101C22×C131C2×#4112yes—
102C22×C131C2×#4111yescharacteristic
103C22×C131C2×#4111yescharacteristic
104C22×C1312−1+412yes—
105C2×D131—22no—
106C2×D131—22no—
107C2×D131—24no—
108C2×D131—24no—
109C2×D131#12314yes—
110C4×C131—22no—
111C4×C131—22no—
112C4×C131—41no—
113C4×C131—42no—
114C4×C131—42no—
115Dic131—22no—
116Dic131—22no—
117Dic131—41no—
118Dic131—42no—
119Dic131—42no—
120C2×C131C2×#12311yescharacteristic
121C2×C131#52511yescharacteristic
122C2×C131#53611yescharacteristic
123C2×C131#55212yes—
124C2×C131#55212yes—
125D131—24no—
126C131#248411yescharacteristic, complemented, Sylow
127#2484—1311nomaximal, Sylow
128C2×#117—1311no—
129C22×C4⋊C4—1311no—
130#400—1311no—
131#400D13111yescharacteristic, complemented
132#398—1311no—
133#398—1311no—
134#387—1312no—
135#387—1312no—
136#385—1312no—
137#381—1311no—
138#381—1312no—
139C23×C4—1311no—
140C23×C4—1311no—
141C23×C4—1311no—
142#117—2624no—
143C2×C4⋊C4—2624no—
144C2×C4⋊C4—1311no—
145C2×C4⋊C4—1311no—
146C2×C4⋊C4—1311no—
147C2×C4⋊C4—1311no—
148C2×C4⋊C4—1312no—
149C2×C4⋊C4—1312no—
150C2×C4⋊C4C2×D13111yescharacteristic
151C2×C4⋊C4C2×D13111yescharacteristic
152C2×C22⋊C4—1312no—
153C2×C22⋊C4—1312no—
154C2×C42—1312no—
155#95—1312no—
156#95—1312no—
157#95—1312no—
158#95—1312no—
159#95—1311no—
160#95—1312no—
161#95C2×D13112yes—
162#95C2×D13111yescharacteristic
163#95C2×D13112yes—
164C24—1311no—
165C22×C4—2622no—
166C22×C4—2624no—
167C22×C4—2622no—
168C22×C4—2622no—
169C22×C4—2624no—
170C22×C4—2622no—
171C22×C4—1312no—
172C22×C4—1311no—
173C22×C4—1312no—
174C22×C4—1311no—
175C22×C4—1311no—
176C22×C4—1314no—
177C22×C4C22×D13112yes—
178C22×C4C22×D13112yes—
179C22×C4C22×D13111yescharacteristic
180C22×C4C22×D13111yescharacteristic
181C22×C4C22×D13111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—2622no—
185C4⋊C4—2622no—
186C4⋊C4—2624no—
187C4⋊C4—2622no—
188C4⋊C4—2622no—
189C4⋊C4—2624no—
190C22⋊C4—2624no—
191C22⋊C4—2624no—
192C42—2624no—
193C23—2621no—
194C23—2621no—
195C23—2622no—
196C23—1312no—
197C23—1314no—
198C23C23×D13111yescharacteristic
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—2628no—
210C2×C4—2628no—
211C2×C4—2621no—
212C2×C4—2622no—
213C2×C4—2621no—
214C2×C4—2622no—
215C2×C4—2621no—
216C2×C4—2622no—
217C2×C4—2622no—
218C2×C4—2622no—
219C2×C4—2622no—
220C2×C4—2622no—
221C2×C4—2622no—
222C2×C4—5244no—
223C2×C4—1312no—
224C2×C4—1312no—
225C2×C4D4×D13112yes—
226C2×C4#1452351412yes—
227C22—2622no—
228C22—2624no—
229C22—2624no—
230C22—2622no—
231C22—1314no—
232C22C2×D4×D13111yescharacteristic
233C22#41×D13111yescharacteristic
234C22#41×D13112yes—
235C22C2×#1452351411yescharacteristic
236C22#1686229312yes—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—42no—
241C4—41no—
242C4—2622no—
243C4—2622no—
244C4—5242no—
245C4—5241no—
246C4—5242no—
247C2—2624no—
248C2#123×D13111yescharacteristic
249C2#2255382511yescharacteristic
250C2#2255388211yescharacteristic
251C2#2255399812yes—
252C2#2255400012yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#400×C131C2
prime radicalC23×D131C23
hypercenter#400D131
hypocenterC131#2484
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #22553250C2semidirect
4. #22553244C2semidirect
5. #22553242C2semidirect
6. #22553199C2semidirect
7. #22553195C2semidirect
11. #400×C131C2semidirect
126. C131#2484semidirect
131. #400D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622042
C2×C4⋊C4×D1312
#225532502
#225532442
#225532422
#225531992
#225531952
#225531862
#225531742
#400×C1312
#2484131