Finite Group Databasebeta

← #30670894

Subgroups — #30670894

Order 16768.

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

normal subgroups only

This group has 760 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×#16862232C211yesmaximal, characteristic, complemented
2C2×#16862223C212yesmaximal, complemented
3C2×#16862218C212yesmaximal, complemented
4C2×D4×Dic131C211yesmaximal, characteristic, complemented
5#22553249C211yesmaximal, characteristic, complemented
6#22553248C211yesmaximal, characteristic, complemented
7#22553220C211yesmaximal, characteristic, complemented
8#22553215C212yesmaximal, complemented
9#22553212C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#392×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×C22⋊D131C2211yescharacteristic
14C22×C22⋊D131C2212yes—
15C23×Dic131C2212yes—
16#16862232—24no—
17#16862223—28no—
18#16862218—28no—
19D4×Dic131—24no—
20C2×C4⋊Dic131C2212yes—
21C2×C4⋊Dic131C2211yescharacteristic
22C2×Dic131⋊C4C2212yes—
23C2×Dic131⋊C4C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×C22⋊Dic131C2211yescharacteristic
26C2×C22⋊Dic131C2212yes—
27C2×C22⋊Dic131C2212yes—
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2212yes—
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×C4×Dic131C2212yes—
32C2×C4×Dic131C2211yescharacteristic
33C131⋊#95C2211yescharacteristic
34C22×D4×C131C2211yescharacteristic
35C22×D4×C131C2212yes—
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2211yescharacteristic
38#95×C131C2211yescharacteristic
39C23×D131C2311yescharacteristic
40C2×Dic262—24no—
41C2×Dic262—22no—
42C2×C22⋊D131—24no—
43C2×C22⋊D131—24no—
44C2×C22⋊D131—22no—
45C2×C22⋊D131—24no—
46C2×C22⋊D131—42no—
47C22×Dic131—24no—
48C22×Dic131—24no—
49C22×Dic131—24no—
50C22×Dic131C2312yes—
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2312yes—
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2311yescharacteristic
55C4⋊Dic131—22no—
56C4⋊Dic131—24no—
57Dic131⋊C4—24no—
58Dic131⋊C4D418yescomplemented
59C22⋊Dic131—24no—
60C22⋊Dic131—24no—
61C22⋊Dic131—24no—
62C22⋊Dic131—22no—
63(C2×D131)⋊C4—22no—
64(C2×D131)⋊C4—24no—
65(C2×D131)⋊C4—22no—
66C4×Dic131—42no—
67C4×Dic131D414yescomplemented
68C24×C131C2311yescharacteristic
69C24×C131C2312yes—
70C2×D4×C131—24no—
71C2×D4×C131—24no—
72C2×D4×C131—22no—
73C2×D4×C131—24no—
74C2×D4×C131—42no—
75C22×C4×C131C2311yescharacteristic
76C22×C4×C131C2311yescharacteristic
77C22×C4×C131C2312yes—
78C22⋊C4×C131—24no—
79C22⋊C4×C131—22no—
80C22×D131—21no—
81C22×D131—22no—
82C22×D131—22no—
83C22×D131—21no—
84C22×D131—21no—
85Dic262—44no—
86C22⋊D131—48no—
87C22⋊D131—44no—
88C2×Dic131—22no—
89C2×Dic131—21no—
90C2×Dic131—22no—
91C2×Dic131—21no—
92C2×Dic131—22no—
93C2×Dic131—21no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—22no—
97C2×Dic131—48no—
98C2×Dic131C2×D412yes—
99C2×Dic131C2×D414yes—
100C2×Dic131C2×D412yes—
101C2×Dic131C2×D414yes—
102C23×C131—22no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—22no—
108C23×C131—21no—
109C23×C131—22no—
110C23×C131—22no—
111C23×C131—21no—
112C23×C131—21no—
113C23×C131#4114yes—
114C23×C131C2411yescharacteristic
115D4×C131—28no—
116D4×C131—48no—
117C2×C4×C131—22no—
118C2×C4×C131—22no—
119C2×C4×C131—22no—
120C2×C4×C131—21no—
121C2×C4×C131—22no—
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2×D414yes—
124C22×C131—24no—
125C22×C131—24no—
126C22×C131—24no—
127C22×C131—42no—
128C22×C131—42no—
129C22×C131—42no—
130C22×C131—42no—
131C22×C131—41no—
132C22×C131—41no—
133C22×C131—41no—
134C22×C131—42no—
135C22×C131—42no—
136C22×C131C22×D411yescharacteristic
137C22×C131C22×D412yes—
138C22×C131C2×#4112yes—
139C22×C131H(5,𝔽2)11yescharacteristic
140C22×C1312−1+411yescharacteristic
141C2×D131—41no—
142C2×D131—41no—
143C2×D131—41no—
144C2×D131—42no—
145C2×D131—42no—
146C4×C131—22no—
147C4×C131—24no—
148C4×C131—41no—
149Dic131—22no—
150Dic131—24no—
151Dic131—42no—
152Dic131—41no—
153Dic131#12714yes—
154C2×C131—44no—
155C2×C131—81no—
156C2×C131#52511yescharacteristic
157C2×C131C2×#12711yescharacteristic
158C2×C131#53711yescharacteristic
159C2×C131#54612yes—
160C2×C131#54812yes—
161D131—81no—
162C131#254011yescharacteristic, complemented, Sylow
163#2540—1311nomaximal, Sylow
164C2×#124—1311no—
165C2×C22⋊Q8—1312no—
166C2×#121—1312no—
167C2×C4×D4—1311no—
168#399—1311no—
169#399—1311no—
170#392—1311no—
171#392D13111yescharacteristic, complemented
172#390—1312no—
173#390—1312no—
174C4×C4⋊C4—1311no—
175C22×Q8—1311no—
176C22×D4—1311no—
177C22×D4—1312no—
178C22×D4C2×D13111yescharacteristic
179C22×D4C2×D13112yes—
180C23×C4—1312no—
181#124—2624no—
182C22⋊Q8—2628no—
183#121—2628no—
184C4×D4—2624no—
185C2×C4⋊C4—1311no—
186C2×C4⋊C4—1312no—
187C2×C4⋊C4—1312no—
188C2×C4⋊C4—1312no—
189C2×C22⋊C4—1312no—
190C2×C22⋊C4—1312no—
191C2×C22⋊C4—1312no—
192C2×C22⋊C4—1311no—
193C2×C22⋊C4—1312no—
194C2×C22⋊C4—1311no—
195C2×C22⋊C4—1311no—
196C2×C22⋊C4C2×D13112yes—
197C2×C22⋊C4C2×D13111yescharacteristic
198C2×C42—1312no—
199C2×C42—1311no—
200#95—1311no—
201#95C2×D13111yescharacteristic
202C24—1311no—
203C24C22×D13111yescharacteristic
204C24C22×D13112yes—
205C2×Q8—2624no—
206C2×Q8—2622no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—42no—
212C2×D4—2624no—
213C2×D4—2624no—
214C2×D4—2624no—
215C2×D4—2622no—
216C2×D4—5242no—
217C22×C4—2624no—
218C22×C4—2624no—
219C22×C4—2624no—
220C22×C4—1311no—
221C22×C4—1312no—
222C22×C4—1312no—
223C22×C4—1311no—
224C22×C4—1311no—
225C22×C4C22×D13111yescharacteristic
226C22×C4C22×D13111yescharacteristic
227C22×C4C22×D13112yes—
228C4⋊C4—2624no—
229C4⋊C4—2622no—
230C4⋊C4—2624no—
231C4⋊C4—1318no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—2622no—
235C22⋊C4—2624no—
236C22⋊C4—2624no—
237C22⋊C4—2624no—
238C22⋊C4—2622no—
239C22⋊C4—2624no—
240C22⋊C4—2622no—
241C42—5242no—
242C42—1314no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—21no—
254C23—2621no—
255C23—2622no—
256C23—2622no—
257C23—2621no—
258C23—2621no—
259C23#1452351214yes—
260C23C23×D13111yescharacteristic
261Q8—5244no—
262D4—28no—
263D4—48no—
264D4—5248no—
265D4—5244no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—2622no—
272C2×C4—2621no—
273C2×C4—2622no—
274C2×C4—2621no—
275C2×C4—2622no—
276C2×C4—2621no—
277C2×C4—2622no—
278C2×C4—2622no—
279C2×C4—2622no—
280C2×C4—5248no—
281C2×C4—1312no—
282C2×C4—1314no—
283C2×C4—1312no—
284C2×C4—1314no—
285C2×C4D4×D13112yes—
286C2×C4D4×D13114yes—
287C22—24no—
288C22—24no—
289C22—24no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—5241no—
300C22—5242no—
301C22—5242no—
302C22—5241no—
303C22—5241no—
304C22C2×D4×D13111yescharacteristic
305C22C2×D4×D13112yes—
306C22C2×#1452351212yes—
307C22#1686229111yescharacteristic
308C22#1686229311yescharacteristic
309C4—24no—
310C4—22no—
311C4—41no—
312C4—2622no—
313C4—2624no—
314C4—5242no—
315C4—5241no—
316C4—1314no—
317C2—44no—
318C2—81no—
319C2—10481no—
320C2#2255381811yescharacteristic
321C2#127×D13111yescharacteristic
322C2#2255389111yescharacteristic
323C2#2255394912yes—
324C2#2255396812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#392×C131C2
hypercenter#392D131
hypocenterC131#2540
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16862232C2semidirect
2. C2×#16862223C2semidirect
3. C2×#16862218C2semidirect
4. C2×D4×Dic131C2semidirect
5. #22553249C2semidirect
6. #22553248C2semidirect
7. #22553220C2semidirect
8. #22553215C2semidirect
9. #22553212C2semidirect
10. #22553163C2semidirect
11. #392×C131C2semidirect
58. Dic131⋊C4D4semidirect
67. C4×Dic131D4semidirect
162. C131#2540semidirect
171. #392D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622322
C2×#168622232
C2×#168622182
C2×D4×Dic1312
#225532492
#225532482
#225532202
#225532152
#225532122
#225531632
#392×C1312
#2540131