Finite Group Databasebeta

← #30670434

Subgroups — #30670434

Order 16768.

544 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 161 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×#16862224C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic
3#22553244C211yesmaximal, characteristic, complemented
4#22553243C211yesmaximal, characteristic, complemented
5#22553242C211yesmaximal, characteristic, complemented
6#22553216C212yesmaximal, complemented
7#22553186C212yesmaximal
8#22553183C212yesmaximal
9#22553181C212yesmaximal, complemented
10#22553159C211yesmaximal, characteristic
11#398×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14C22×C4×D131C2212yes—
15#16862224—24no—
16C2×C4⋊Dic131C2211yescharacteristic
17C2×C4⋊Dic131C2212yes—
18C2×Dic131⋊C4C2211yescharacteristic
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2212yes—
22C2×Dic131⋊C4C2212yes—
23C4⋊C4×D131—24no—
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2211yescharacteristic
26C2×(C2×D131)⋊C4C2212yes—
27C2×C4×Dic131C2212yes—
28C131⋊#95C2211yescharacteristic
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2211yescharacteristic
34C2×C4⋊C4×C131C2212yes—
35#95×C131C2211yescharacteristic
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C2×Dic262—22no—
39C2×Dic262—24no—
40C22×Dic131C2312yes—
41C22×Dic131C2311yescharacteristic
42C22×Dic131C2311yescharacteristic
43C22×Dic131C2312yes—
44C22×Dic131C2311yescharacteristic
45C2×C4×D131—24no—
46C2×C4×D131—24no—
47C2×C4×D131—42no—
48C2×C4×D131—42no—
49C2×C4×D131D414yes—
50C4⋊Dic131—22no—
51C4⋊Dic131—42no—
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54Dic131⋊C4—22no—
55Dic131⋊C4—42no—
56Dic131⋊C4—42no—
57(C2×D131)⋊C4—22no—
58(C2×D131)⋊C4—22no—
59(C2×D131)⋊C4Q818yescomplemented
60C4×Dic131—24no—
61C22×C4×C131C2312yes—
62C22×C4×C131C2311yescharacteristic
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2312yes—
66C4⋊C4×C131—22no—
67C4⋊C4×C131—22no—
68C4⋊C4×C131—42no—
69C22×D131—22no—
70C22×D131—21no—
71C22×D131—21no—
72C22×D131C2×D412yes—
73C22×D131C2×Q814yes—
74Dic262—44no—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—21no—
82C2×Dic131—22no—
83C2×Dic131—21no—
84C2×Dic131—22no—
85C2×Dic131C2×D412yes—
86C2×Dic131C2×Q814yes—
87C2×Dic131#4114yes—
88C4×D131—28no—
89C4×D131—48no—
90C23×C131C2411yescharacteristic
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—22no—
94C2×C4×C131—21no—
95C2×C4×C131—21no—
96C2×C4×C131—22no—
97C2×C4×C131—22no—
98C2×C4×C131—21no—
99C2×C4×C131—22no—
100C2×C4×C131—21no—
101C2×C4×C131—22no—
102C2×C4×C131C2×D412yes—
103C2×C4×C131C2×Q814yes—
104C22×C131C22×D411yescharacteristic
105C22×C131C22×Q812yes—
106C22×C131C2×#4112yes—
107C22×C131H(5,𝔽2)11yescharacteristic
108C22×C131H(5,𝔽2)11yescharacteristic
109C2×D131—22no—
110C2×D131—22no—
111C2×D131—24no—
112C2×D131—24no—
113C2×D131C4⋊Q814yes—
114C4×C131—22no—
115C4×C131—24no—
116C4×C131—41no—
117C4×C131—41no—
118C4×C131—42no—
119Dic131—24no—
120Dic131—41no—
121Dic131—42no—
122Dic131—41no—
123Dic131#12314yes—
124C2×C131C2×#12311yescharacteristic
125C2×C131C2×C4⋊Q811yescharacteristic
126C2×C131#53511yescharacteristic
127C2×C131#55412yes—
128C2×C131#55412yes—
129D131—24no—
130C131#250211yescharacteristic, complemented, Sylow
131#2502—1311nomaximal, Sylow
132C2×C22⋊Q8—1311no—
133C22×C4⋊C4—1311no—
134#398—1311no—
135#398—1311no—
136#398—1311no—
137#398D13111yescharacteristic, complemented
138#391—1312no—
139#385—1312no—
140#385—1312no—
141#384—1312no—
142C4×C22⋊C4—1311no—
143C22×Q8—1311no—
144C23×C4—1311no—
145C23×C4—1312no—
146C22⋊Q8—2624no—
147C2×C4⋊C4—2624no—
148C2×C4⋊C4—1312no—
149C2×C4⋊C4—1311no—
150C2×C4⋊C4—1311no—
151C2×C4⋊C4—1311no—
152C2×C4⋊C4—1312no—
153C2×C4⋊C4—1312no—
154C2×C4⋊C4—1312no—
155C2×C4⋊C4C2×D13111yescharacteristic
156C2×C4⋊C4C2×D13111yescharacteristic
157C2×C4⋊C4C2×D13112yes—
158C2×C22⋊C4—1311no—
159C2×C22⋊C4—1311no—
160C2×C22⋊C4—1312no—
161C2×C42—1312no—
162#95—1311no—
163#95—1312no—
164#95—1312no—
165#95—1312no—
166#95C2×D13111yescharacteristic
167#95C2×D13112yes—
168C24—1311no—
169C2×Q8—2622no—
170C2×Q8—2624no—
171C22×C4—2624no—
172C22×C4—2624no—
173C22×C4—5242no—
174C22×C4—5242no—
175C22×C4—1311no—
176C22×C4—1312no—
177C22×C4—1311no—
178C22×C4—1312no—
179C22×C4—1314no—
180C22×C4—1311no—
181C22×C4C22×D13111yescharacteristic
182C22×C4C22×D13112yes—
183C22×C4C22×D13111yescharacteristic
184C22×C4C22×D13111yescharacteristic
185C22×C4C22×D13112yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—2624no—
190C4⋊C4—2622no—
191C4⋊C4—2622no—
192C4⋊C4—2622no—
193C4⋊C4—5242no—
194C4⋊C4—5242no—
195C4⋊C4—5242no—
196C22⋊C4—2622no—
197C22⋊C4—2622no—
198C22⋊C4—1318no—
199C42—2624no—
200C23—2622no—
201C23—2621no—
202C23—2621no—
203C23—1312no—
204C23—1314no—
205C23C23×D13111yescharacteristic
206Q8—5244no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—2628no—
219C2×C4—2622no—
220C2×C4—2622no—
221C2×C4—2621no—
222C2×C4—2621no—
223C2×C4—2622no—
224C2×C4—2622no—
225C2×C4—2621no—
226C2×C4—2622no—
227C2×C4—2621no—
228C2×C4—2622no—
229C2×C4—5248no—
230C2×C4—1314no—
231C2×C4—1312no—
232C2×C4—1314no—
233C2×C4D4×D13112yes—
234C2×C4Q8×D13114yes—
235C22—2622no—
236C22—2624no—
237C22—2624no—
238C22—2622no—
239C22—1314no—
240C22C2×D4×D13111yescharacteristic
241C22C2×Q8×D13112yes—
242C22#41×D13112yes—
243C22#1686229111yescharacteristic
244C22#1686229211yescharacteristic
245C4—24no—
246C4—22no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—2624no—
251C4—5242no—
252C4—5241no—
253C4—5241no—
254C4—1314no—
255C2—2624no—
256C2#123×D13111yescharacteristic
257C2C4⋊Q8×D13111yescharacteristic
258C2#2255387811yescharacteristic
259C2#2255401612yes—
260C2#2255401812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#398×C131C2
prime radicalC23×D131C23
hypercenter#398D131
hypocenterC131#2502
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #22553244C2semidirect
4. #22553243C2semidirect
5. #22553242C2semidirect
6. #22553216C2semidirect
9. #22553181C2semidirect
11. #398×C131C2semidirect
59. (C2×D131)⋊C4Q8semidirect
130. C131#2502semidirect
137. #398D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622242
C2×C4⋊C4×D1312
#225532442
#225532432
#225532422
#225532162
#225531862
#225531832
#225531812
#225531592
#398×C1312
#2502131