Finite Group Databasebeta

← #30670857

Subgroups — #30670857

Order 16768.

732 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16862218C212yesmaximal, complemented
2C2×D4×Dic131C211yesmaximal, characteristic, complemented
3C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic, complemented
4#22553239C211yesmaximal, characteristic, complemented
5#22553237C211yesmaximal, characteristic, complemented
6#22553220C211yesmaximal, characteristic, complemented
7#22553192C212yesmaximal, complemented
8#22553188C212yesmaximal, complemented
9#22553176C211yesmaximal, characteristic, complemented
10#22553170C212yesmaximal, complemented
11#392×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2212yes—
13C22×C22⋊D131C2211yescharacteristic
14C23×Dic131C2212yes—
15C23×Dic131C2211yescharacteristic
16C22×C4×D131C2211yescharacteristic
17#16862218—28no—
18D4×Dic131—24no—
19(C22⋊D131)⋊C4—24no—
20C2×C4⋊Dic131C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×Dic131⋊C4C2212yes—
23C2×C22⋊Dic131C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×C22⋊Dic131C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×C4×Dic131C2211yescharacteristic
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C131⋊#95C2211yescharacteristic
33C22×D4×C131C2211yescharacteristic
34C22×D4×C131C2212yes—
35C2×C22⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2212yes—
37#95×C131C2211yescharacteristic
38C23×D131C2311yescharacteristic
39C2×C22⋊D131—24no—
40C2×C22⋊D131—22no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131—24no—
43C2×C22⋊D131—42no—
44C2×C22⋊D131D414yescomplemented
45C22×Dic131—24no—
46C22×Dic131—24no—
47C22×Dic131—22no—
48C22×Dic131—24no—
49C22×Dic131—42no—
50C22×Dic131C2311yescharacteristic
51C22×Dic131C2312yes—
52C22×Dic131C2312yes—
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2311yescharacteristic
55C2×C4×D131—22no—
56C2×C4×D131—42no—
57C4⋊Dic131—22no—
58Dic131⋊C4—24no—
59Dic131⋊C4—22no—
60C22⋊Dic131—24no—
61C22⋊Dic131—24no—
62C22⋊Dic131—42no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—22no—
65C4×Dic131D414yescomplemented
66C24×C131C2311yescharacteristic
67C24×C131C2312yes—
68C2×D4×C131—24no—
69C2×D4×C131—24no—
70C2×D4×C131—24no—
71C2×D4×C131—22no—
72C2×D4×C131—42no—
73C22×C4×C131C2311yescharacteristic
74C22×C4×C131C2312yes—
75C22×C4×C131C2311yescharacteristic
76C22⋊C4×C131—22no—
77C22⋊C4×C131—42no—
78C22×D131—21no—
79C22×D131—21no—
80C22×D131—22no—
81C22×D131—22no—
82C22×D131C2×D412yes—
83C22⋊D131—28no—
84C22⋊D131—48no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—21no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—44no—
95C2×Dic131—48no—
96C2×Dic131C2×D412yes—
97C2×Dic131C2×D412yes—
98C2×Dic131C2×D412yes—
99C2×Dic131#4112yes—
100C4×D131—44no—
101C23×C131—21no—
102C23×C131—22no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—21no—
108C23×C131—22no—
109C23×C131—22no—
110C23×C131—22no—
111C23×C131C2×D412yes—
112C23×C131#4114yes—
113C23×C131C2411yescharacteristic
114D4×C131—28no—
115D4×C131—48no—
116C2×C4×C131—22no—
117C2×C4×C131—21no—
118C2×C4×C131—22no—
119C2×C4×C131—22no—
120C2×C4×C131—22no—
121C2×C4×C131—22no—
122C2×C4×C131C2×D412yes—
123C22×C131—22no—
124C22×C131—24no—
125C22×C131—24no—
126C22×C131—24no—
127C22×C131—24no—
128C22×C131—42no—
129C22×C131—41no—
130C22×C131—42no—
131C22×C131—41no—
132C22×C131—42no—
133C22×C131—42no—
134C22×C131—42no—
135C22×C131C22×D411yescharacteristic
136C22×C131C22×D411yescharacteristic
137C22×C131C2×#4111yescharacteristic
138C22×C131C2×#4112yes—
139C22×C131H(5,𝔽2)12yes—
140C2×D131—22no—
141C2×D131—24no—
142C2×D131—41no—
143C2×D131—41no—
144C2×D131—42no—
145C4×C131—22no—
146C4×C131—41no—
147C4×C131—42no—
148Dic131—22no—
149Dic131—42no—
150Dic131—41no—
151Dic131—42no—
152Dic131#12114yes—
153C2×C131—42no—
154C2×C131—44no—
155C2×C131C2×#12111yescharacteristic
156C2×C131#52511yescharacteristic
157C2×C131#53511yescharacteristic
158C2×C131#54612yes—
159C2×C131#55312yes—
160D131—42no—
161C131#253511yescharacteristic, complemented, Sylow
162#2535—1311nomaximal, Sylow
163C2×#121—1312no—
164C2×C4×D4—1311no—
165C2×C4×D4—1311no—
166#397—1311no—
167#397—1311no—
168#392—1311no—
169#392D13111yescharacteristic, complemented
170#386—1312no—
171#386—1312no—
172#382—1311no—
173#380—1312no—
174C22×D4—1312no—
175C22×D4—1311no—
176C22×D4C2×D13111yescharacteristic
177C22×D4C2×D13112yes—
178C23×C4—1312no—
179C23×C4—1311no—
180C23×C4—1311no—
181#121—2628no—
182C4×D4—2624no—
183C4×D4—2624no—
184C2×C4⋊C4—1311no—
185C2×C4⋊C4—1311no—
186C2×C4⋊C4—1312no—
187C2×C22⋊C4—1312no—
188C2×C22⋊C4—1312no—
189C2×C22⋊C4—1311no—
190C2×C22⋊C4—1312no—
191C2×C22⋊C4—1312no—
192C2×C22⋊C4C2×D13111yescharacteristic
193C2×C22⋊C4C2×D13112yes—
194C2×C42—1311no—
195#95—1312no—
196#95—1312no—
197#95—1312no—
198#95—1311no—
199#95C2×D13111yescharacteristic
200C24—1311no—
201C24C22×D13111yescharacteristic
202C24C22×D13112yes—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—2624no—
209C2×D4—2624no—
210C2×D4—2622no—
211C2×D4—2622no—
212C2×D4—5242no—
213C2×D4—1314no—
214C22×C4—2622no—
215C22×C4—2622no—
216C22×C4—2624no—
217C22×C4—2624no—
218C22×C4—2624no—
219C22×C4—5242no—
220C22×C4—5242no—
221C22×C4—1312no—
222C22×C4—1312no—
223C22×C4—1311no—
224C22×C4—1311no—
225C22×C4—1311no—
226C22×C4C22×D13112yes—
227C22×C4C22×D13111yescharacteristic
228C22×C4C22×D13111yescharacteristic
229C4⋊C4—2622no—
230C4⋊C4—2624no—
231C4⋊C4—2622no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—2624no—
235C22⋊C4—2624no—
236C22⋊C4—2624no—
237C22⋊C4—2622no—
238C22⋊C4—5242no—
239C42—1314no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—2621no—
251C23—2621no—
252C23—2622no—
253C23—2622no—
254C23—1312no—
255C23D4×D13112yes—
256C23#1452351214yes—
257C23C23×D13111yescharacteristic
258D4—28no—
259D4—48no—
260D4—2628no—
261D4—5248no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—2622no—
269C2×C4—2622no—
270C2×C4—2621no—
271C2×C4—2622no—
272C2×C4—2622no—
273C2×C4—2622no—
274C2×C4—2622no—
275C2×C4—2622no—
276C2×C4—2622no—
277C2×C4—5244no—
278C2×C4—5244no—
279C2×C4—5248no—
280C2×C4—1312no—
281C2×C4—1312no—
282C2×C4—1312no—
283C2×C4—1312no—
284C2×C4D4×D13112yes—
285C22—24no—
286C22—24no—
287C22—24no—
288C22—22no—
289C22—24no—
290C22—41no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—2624no—
298C22—2622no—
299C22—5241no—
300C22—5242no—
301C22—5241no—
302C22C2×D4×D13111yescharacteristic
303C22C2×D4×D13111yescharacteristic
304C22#41×D13111yescharacteristic
305C22C2×#1452351212yes—
306C22#1686229112yes—
307C4—22no—
308C4—41no—
309C4—42no—
310C4—2622no—
311C4—5241no—
312C4—5242no—
313C4—5242no—
314C4—1314no—
315C2—44no—
316C2—42no—
317C2—5242no—
318C2#121×D13111yescharacteristic
319C2#2255381811yescharacteristic
320C2#2255387511yescharacteristic
321C2#2255394912yes—
322C2#2255400512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#392×C131C2
hypercenter#392D131
hypocenterC131#2535
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16862218C2semidirect
2. C2×D4×Dic131C2semidirect
3. C2×(C22⋊D131)⋊C4C2semidirect
4. #22553239C2semidirect
5. #22553237C2semidirect
6. #22553220C2semidirect
7. #22553192C2semidirect
8. #22553188C2semidirect
9. #22553176C2semidirect
10. #22553170C2semidirect
11. #392×C131C2semidirect
44. C2×C22⋊D131D4semidirect
65. C4×Dic131D4semidirect
161. C131#2535semidirect
169. #392D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622182
C2×D4×Dic1312
C2×(C22⋊D131)⋊C42
#225532392
#225532372
#225532202
#225531922
#225531882
#225531762
#225531702
#392×C1312
#2535131