Finite Group Databasebeta

← #30670451

Subgroups — #30670451

Order 16768.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862218C211yesmaximal, characteristic
2C22×Dic131⋊C4C211yesmaximal, characteristic, complemented
3#22553241C212yesmaximal, complemented
4#22553240C212yesmaximal, complemented
5#22553212C212yesmaximal, complemented
6#22553192C211yesmaximal, characteristic, complemented
7#22553191C211yesmaximal, characteristic, complemented
8#22553189C211yesmaximal, characteristic, complemented
9#22553176C212yesmaximal, complemented
10C22⋊C4×Dic131C211yesmaximal, characteristic, complemented
11#386×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C22×C22⋊D131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15C23×Dic131C2211yescharacteristic
16#16862218—24no—
17C2×Dic131⋊C4—24no—
18C2×Dic131⋊C4C2211yescharacteristic
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2212yes—
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2212yes—
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×C4×Dic131C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C22×D4×C131C2211yescharacteristic
33C23×C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C2×C22⋊D131—22no—
39C2×C22⋊D131—24no—
40C2×C22⋊D131—22no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131—22no—
43C22×Dic131—24no—
44C22×Dic131—22no—
45C22×Dic131—42no—
46C22×Dic131D414yes—
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2312yes—
50C22×Dic131C2311yescharacteristic
51C22×Dic131C2311yescharacteristic
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54Dic131⋊C4—24no—
55C22⋊Dic131—22no—
56C22⋊Dic131—22no—
57C22⋊Dic131—42no—
58(C2×D131)⋊C4—22no—
59(C2×D131)⋊C4—42no—
60(C2×D131)⋊C4—42no—
61C4×Dic131—24no—
62C24×C131C2311yescharacteristic
63C24×C131C2311yescharacteristic
64C2×D4×C131—22no—
65C2×D4×C131—22no—
66C2×D4×C131—22no—
67C22×C4×C131—22no—
68C22×C4×C131—42no—
69C22×C4×C131C2311yescharacteristic
70C22×C4×C131C2312yes—
71C22×C4×C131C2312yes—
72C22⋊C4×C131—22no—
73C22⋊C4×C131—42no—
74C22×D131—21no—
75C22×D131—22no—
76C22×D131—21no—
77C22×D131—21no—
78C22×D131—22no—
79C22⋊D131—44no—
80C22⋊D131—44no—
81C2×Dic131—22no—
82C2×Dic131—28no—
83C2×Dic131—21no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—21no—
88C2×Dic131—21no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—44no—
93C2×Dic131C2×D412yes—
94C2×Dic131C2×D412yes—
95C2×Dic131#4114yes—
96C23×C131—22no—
97C23×C131—22no—
98C23×C131—21no—
99C23×C131—21no—
100C23×C131—21no—
101C23×C131—22no—
102C23×C131—22no—
103C23×C131C2×D412yes—
104C23×C131#4112yes—
105C23×C131#4112yes—
106C23×C131C2411yescharacteristic
107D4×C131—44no—
108C2×C4×C131—22no—
109C2×C4×C131—21no—
110C2×C4×C131—22no—
111C2×C4×C131—21no—
112C2×C4×C131—21no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131—22no—
117C2×C4×C131—44no—
118C22×C131—22no—
119C22×C131—22no—
120C22×C131—24no—
121C22×C131—24no—
122C22×C131—41no—
123C22×C131—41no—
124C22×C131—41no—
125C22×C131—42no—
126C22×C131—42no—
127C22×C131#12414yes—
128C22×C131C22×D411yescharacteristic
129C22×C131C2×#4112yes—
130C22×C131C2×#4111yescharacteristic
131C22×C131C2×#4111yescharacteristic
132C22×C131H(5,𝔽2)12yes—
133C2×D131—41no—
134C2×D131—41no—
135C2×D131—41no—
136C2×D131—42no—
137C2×D131—42no—
138C4×C131—41no—
139C4×C131—42no—
140C4×C131—42no—
141Dic131—22no—
142Dic131—41no—
143Dic131—42no—
144Dic131—42no—
145Dic131#12314yes—
146C2×C131—24no—
147C2×C131—81no—
148C2×C131C2×#12311yescharacteristic
149C2×C131C2×#12411yescharacteristic
150C2×C131#53511yescharacteristic
151C2×C131#55312yes—
152C2×C131#55312yes—
153D131—81no—
154C131#250411yescharacteristic, complemented, Sylow
155#2504—1311nomaximal, Sylow
156C2×#121—1311no—
157C22×C4⋊C4—1311no—
158#397—1312no—
159#397—1312no—
160#390—1312no—
161#386—1311no—
162#386—1311no—
163#386—1311no—
164#386D13111yescharacteristic, complemented
165#382—1312no—
166C4×C22⋊C4—1311no—
167C22×D4—1311no—
168C22×D4—1311no—
169C22×D4C2×D13111yescharacteristic
170C23×C4—1311no—
171C23×C4—1311no—
172C23×C4C2×D13111yescharacteristic
173#121—2624no—
174C2×C4⋊C4—2624no—
175C2×C4⋊C4—1312no—
176C2×C4⋊C4—1312no—
177C2×C4⋊C4—1311no—
178C2×C22⋊C4—1311no—
179C2×C22⋊C4—1311no—
180C2×C22⋊C4—1311no—
181C2×C22⋊C4—1312no—
182C2×C22⋊C4—1312no—
183C2×C22⋊C4—1312no—
184C2×C22⋊C4C2×D13111yescharacteristic
185C2×C22⋊C4C2×D13112yes—
186C2×C42—1312no—
187#95—1312no—
188#95—1312no—
189#95—1312no—
190#95—1312no—
191#95C2×D13112yes—
192C24—1311no—
193C24C22×D13111yescharacteristic
194C24C22×D13111yescharacteristic
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—2622no—
199C2×D4—2622no—
200C2×D4—2622no—
201C2×D4—2622no—
202C2×D4—2624no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—2624no—
206C22×C4—2622no—
207C22×C4—5242no—
208C22×C4—1311no—
209C22×C4—1312no—
210C22×C4—1312no—
211C22×C4—1314no—
212C22×C4—1311no—
213C22×C4—1311no—
214C22×C4C22×D13111yescharacteristic
215C22×C4C22×D13112yes—
216C22×C4C22×D13112yes—
217C4⋊C4—2624no—
218C4⋊C4—2624no—
219C4⋊C4—2622no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—2622no—
223C22⋊C4—2622no—
224C22⋊C4—2622no—
225C22⋊C4—5242no—
226C22⋊C4—5242no—
227C22⋊C4—5242no—
228C42—2624no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—2621no—
237C23—2622no—
238C23—2621no—
239C23—2621no—
240C23—2622no—
241C23D4×D13112yes—
242C23#1452351212yes—
243C23#1452351312yes—
244C23C23×D13111yescharacteristic
245D4—44no—
246D4—5244no—
247D4—5244no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—2622no—
259C2×C4—2628no—
260C2×C4—2621no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—2621no—
264C2×C4—2622no—
265C2×C4—2621no—
266C2×C4—2622no—
267C2×C4—2622no—
268C2×C4—2622no—
269C2×C4—5244no—
270C2×C4—1312no—
271C2×C4—1312no—
272C2×C4—1314no—
273C22—22no—
274C22—24no—
275C22—24no—
276C22—22no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—5241no—
283C22—5242no—
284C22—5242no—
285C22—5241no—
286C22—5241no—
287C22#1686223214yes—
288C22C2×D4×D13111yescharacteristic
289C22#41×D13112yes—
290C22C2×#1452351211yescharacteristic
291C22C2×#1452351311yescharacteristic
292C22#1686229112yes—
293C4—42no—
294C4—42no—
295C4—41no—
296C4—2622no—
297C4—5242no—
298C4—5242no—
299C4—5241no—
300C4—1314no—
301C2—24no—
302C2—81no—
303C2—10481no—
304C2#123×D13111yescharacteristic
305C2C2×#1686223211yescharacteristic
306C2#2255387711yescharacteristic
307C2#2255400512yes—
308C2#2255401112yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#386×C131C2
prime radicalC2×#16862218C2
hypercenter#386D131
hypocenterC131#2504
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic131⋊C4C2semidirect
3. #22553241C2semidirect
4. #22553240C2semidirect
5. #22553212C2semidirect
6. #22553192C2semidirect
7. #22553191C2semidirect
8. #22553189C2semidirect
9. #22553176C2semidirect
10. C22⋊C4×Dic131C2semidirect
11. #386×C131C2semidirect
154. C131#2504semidirect
164. #386D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622182
C22×Dic131⋊C42
#225532412
#225532402
#225532122
#225531922
#225531912
#225531892
#225531762
C22⋊C4×Dic1312
#386×C1312
#2504131