Finite Group Databasebeta

← #30664503

Subgroups — #30664503

Order 16768.

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

normal subgroups only

This group has 286 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#22553825C211yesmaximal, characteristic
2#22553274C211yesmaximal, characteristic, complemented
3#22553171C211yesmaximal, characteristic, complemented
4#22553049C211yesmaximal, characteristic, complemented
5#22553047C211yesmaximal, characteristic, complemented
6#22553045C211yesmaximal, characteristic, complemented
7#328×C131C211yesmaximal, characteristic, complemented
8C2×#14523514C411yescharacteristic, complemented
9C22×C4×D131C2211yescharacteristic
10C2×#14523496—21no—
11#16862231—21no—
12#16862222C411yescharacteristic, complemented
13#16862221C411yescharacteristic, complemented
14#16862220C2211yescharacteristic
15#16862216—21no—
16#16862208—21no—
17(C22⋊D131)⋊C4—21no—
18#16862204C411yescharacteristic, complemented
19C2×Dic131⋊C4—21no—
20C2×(C2×D131)⋊C4—21no—
21C22⋊(C131⋊C8)C2211yescharacteristic
22(C2×D131)⋊C8—21no—
23C131⋊#95C2211yescharacteristic
24C22⋊Q8×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C23×D131D411yescharacteristic
28#14523514—42no—
29C2×D524—21no—
30C2×D524—21no—
31C2×D524C2×C411yescharacteristic
32C2×C22⋊D131—21no—
33C22×Dic131—21no—
34C22×Dic131D411yescharacteristic
35C2×C4×D131—21no—
36C2×C4×D131—42no—
37C2×C4×D131C2×C411yescharacteristic
38C2×C4×D131C2×C411yescharacteristic
39C2×C4×D131D411yescharacteristic
40C2×C4×D131D411yescharacteristic
41#14523496—42no—
42C2×C131⋊C8—21no—
43C4⋊Dic131C2×C411yescharacteristic
44Dic131⋊C4—21no—
45Dic131⋊C4—42no—
46(C2×D131)⋊C4—21no—
47(C2×D131)⋊C4—21no—
48(C2×D131)⋊C4—21no—
49(C2×D131)⋊C4—21no—
50(C2×D131)⋊C4—42no—
51C4×Dic131—21no—
52C2×Q8×C131C2×C411yescharacteristic
53C22×C4×C131—21no—
54C22×C4×C131C2311yescharacteristic
55C2×C8×C131—21no—
56C4⋊C4×C131—21no—
57C4⋊C4×C131C2×C411yescharacteristic
58C22⋊C4×C131—21no—
59C22×D131—21no—
60C22×D131—21no—
61C22×D131—21no—
62C22×D131—41no—
63C22×D131—41no—
64C22×D131C22⋊C411yescharacteristic
65C22×D131C22⋊C411yescharacteristic
66D524—41no—
67D524—42no—
68D524—42no—
69C22⋊D131—41no—
70C22⋊D131—41no—
71C2×Dic131—22no—
72C2×Dic131—21no—
73C2×Dic131—41no—
74C2×Dic131—41no—
75C2×Dic131—41no—
76C2×Dic131C22⋊C411yescharacteristic
77C2×Dic131C22⋊C411yescharacteristic
78C4×D131—22no—
79C4×D131—22no—
80C4×D131—41no—
81C4×D131—41no—
82C4×D131—42no—
83C131⋊C8—41no—
84C23×C131C2×D411yescharacteristic
85Q8×C131—41no—
86C2×C4×C131—21no—
87C2×C4×C131—21no—
88C2×C4×C131—41no—
89C2×C4×C131—41no—
90C2×C4×C131—41no—
91C2×C4×C131—41no—
92C2×C4×C131C22×C411yescharacteristic
93C2×C4×C131C2×D411yescharacteristic
94C8×C131—41no—
95C22×C131—21no—
96C22×C131—21no—
97C22×C131C2×C22⋊C411yescharacteristic
98C22×C131AGL(1,ℤ/8)11yescharacteristic
99C22×C131#13711yescharacteristic
100C2×D131—21no—
101C2×D131—21no—
102C2×D131—21no—
103C2×D131—21no—
104C2×D131—21no—
105C2×D131—21no—
106C2×D131—42no—
107C2×D131—42no—
108C2×D131—41no—
109C2×D131—41no—
110C2×D131—41no—
111C4×C131—21no—
112C4×C131—21no—
113C4×C131—41no—
114C4×C131—41no—
115C4×C131—81no—
116Dic131—21no—
117Dic131—41no—
118Dic131—81no—
119Dic131#9912yes—
120C2×C131—21no—
121C2×C131—21no—
122C2×C131C2×#9911yescharacteristic
123C2×C131#41711yescharacteristic
124C2×C131#42111yescharacteristic
125D131—41no—
126D131—41no—
127D131—81no—
128C131#151311yescharacteristic, complemented, Sylow
129#1513—1311nomaximal, Sylow
130#525—1311no—
131#407—1311no—
132#380—1311no—
133#328—1311no—
134#328D13111yescharacteristic, complemented
135#327—1311no—
136#327—1311no—
137C2×#41—1311no—
138C23×C4—1311no—
139C2×M16—2621no—
140#123—2621no—
141C22⋊Q8—1311no—
142C22⋊Q8C2×D13111yescharacteristic
143#121—1311no—
144#121—1311no—
145#120—2621no—
146C4×D4—2621no—
147C4×D4—2621no—
148#117—1311no—
149C2×C4⋊C4—2621no—
150C2×C22⋊C4—2621no—
151C22⋊C8—2621no—
152C22⋊C8—1311no—
153C22⋊C8C2×D13111yescharacteristic
154#95—1311no—
155#95C2×D13111yescharacteristic
156C24—1311no—
157#41—5242no—
158C2×Q8C4×D13111yescharacteristic
159C2×D4—2621no—
160C2×D4—2621no—
161C2×D4—2621no—
162C2×D4—1311no—
163C22×C4—21no—
164C22×C4—2621no—
165C22×C4—2621no—
166C22×C4—5242no—
167C22×C4—1311no—
168C22×C4—1311no—
169C22×C4—1311no—
170C22×C4—1311no—
171C22×C4—1311no—
172C22×C4C22×D13111yescharacteristic
173M16—5242no—
174C2×C8—21no—
175C2×C8—2621no—
176C4⋊C4—21no—
177C4⋊C4—2621no—
178C4⋊C4—5242no—
179C4⋊C4—1311no—
180C4⋊C4C4×D13111yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—2621no—
183C22⋊C4—2621no—
184C22⋊C4—2621no—
185C22⋊C4—2621no—
186C22⋊C4—5242no—
187C42—2621no—
188C23—2621no—
189C23—2621no—
190C23—2621no—
191C23—5241no—
192C23—5241no—
193C23—1311no—
194C23—1311no—
195C23D4×D13111yescharacteristic
196Q8—41no—
197D4—5241no—
198D4—5242no—
199D4—5242no—
200D4—5241no—
201D4—5241no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—2622no—
209C2×C4—2622no—
210C2×C4—2622no—
211C2×C4—2621no—
212C2×C4—5241no—
213C2×C4—5241no—
214C2×C4—5242no—
215C2×C4—5241no—
216C2×C4—5241no—
217C2×C4—5241no—
218C2×C4—1311no—
219C2×C4—1311no—
220C2×C4C2×C4×D13111yescharacteristic
221C2×C4D4×D13111yescharacteristic
222C8—41no—
223C8—5241no—
224C22—21no—
225C22—21no—
226C22—2621no—
227C22—2621no—
228C22—2621no—
229C22—2621no—
230C22—2621no—
231C22—2621no—
232C22—5242no—
233C22—5242no—
234C22—5241no—
235C22—5241no—
236C22—5241no—
237C22C22⋊C4×D13111yescharacteristic
238C22#1686227311yescharacteristic
239C22#1686227811yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—2621no—
245C4—81no—
246C4—5241no—
247C4—10481no—
248C4—1312no—
249C2—21no—
250C2—21no—
251C2—5241no—
252C2—5241no—
253C2—10481no—
254C2#99×D13111yescharacteristic
255C2#2255331211yescharacteristic
256C2#2255332711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#328×C131C2
socleC22×C131C2×C22⋊C4
prime radical#22553825C2
cosocleC22×C4×C131C23
hypercenter#328D131
hypocenterC131#1513
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #22553274C2semidirect
3. #22553171C2semidirect
4. #22553049C2semidirect
5. #22553047C2semidirect
6. #22553045C2semidirect
7. #328×C131C2semidirect
8. C2×#14523514C4semidirect
12. #16862222C4semidirect
13. #16862221C4semidirect
18. #16862204C4semidirect
128. C131#1513semidirect
134. #328D131semidirect

Maximal subgroups

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

subgroupindex
#225538252
#225532742
#225531712
#225530492
#225530472
#225530452
#328×C1312
#1513131