Finite Group Databasebeta

← #30670896

Subgroups — #30670896

Order 16768.

760 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16862224C212yesmaximal, complemented
3C2×#16862219C212yesmaximal, complemented
4C2×#16862208C211yesmaximal, characteristic, complemented
5#22553247C211yesmaximal, characteristic, complemented
6#22553222C211yesmaximal, characteristic, complemented
7#22553221C211yesmaximal, characteristic, complemented
8#22553213C212yesmaximal, complemented
9#22553212C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×D524C2211yescharacteristic
14C22×D524C2212yes—
15C22×C22⋊D131C2211yescharacteristic
16C22×C22⋊D131C2212yes—
17C22×C4×D131C2212yes—
18#16862232—24no—
19#16862224—28no—
20#16862219—28no—
21#16862208—24no—
22C2×Dic131⋊C4C2212yes—
23C2×Dic131⋊C4C2212yes—
24C2×C22⋊Dic131C2211yescharacteristic
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2212yes—
29C2×(C2×D131)⋊C4C2212yes—
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×C4×Dic131C2212yes—
32C2×C4×Dic131C2211yescharacteristic
33C131⋊#95C2211yescharacteristic
34C2×C4⋊C4×C131C2212yes—
35C2×C4⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2211yescharacteristic
38#95×C131C2211yescharacteristic
39C23×D131C2311yescharacteristic
40C23×D131C2312yes—
41C2×Dic262—24no—
42C2×Dic262—22no—
43C2×D524—24no—
44C2×D524—24no—
45C2×D524—22no—
46C2×D524—24no—
47C2×D524—42no—
48C2×C22⋊D131—24no—
49C2×C22⋊D131—24no—
50C2×C22⋊D131—24no—
51C2×C22⋊D131—22no—
52C2×C22⋊D131—42no—
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2312yes—
55C22×Dic131C2311yescharacteristic
56C22×Dic131C2311yescharacteristic
57C2×C4×D131—24no—
58C2×C4×D131—24no—
59C2×C4×D131—24no—
60Dic131⋊C4—24no—
61Dic131⋊C4D418yescomplemented
62C22⋊Dic131—22no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—24no—
65(C2×D131)⋊C4—24no—
66(C2×D131)⋊C4—22no—
67(C2×D131)⋊C4—24no—
68(C2×D131)⋊C4—22no—
69C4×Dic131—42no—
70C4×Dic131D414yescomplemented
71C24×C131C2311yescharacteristic
72C22×C4×C131C2312yes—
73C22×C4×C131C2311yescharacteristic
74C22×C4×C131C2311yescharacteristic
75C22×C4×C131C2312yes—
76C4⋊C4×C131—22no—
77C4⋊C4×C131—24no—
78C22⋊C4×C131—24no—
79C22⋊C4×C131—22no—
80C22×D131—22no—
81C22×D131—22no—
82C22×D131—22no—
83C22×D131—22no—
84C22×D131—22no—
85C22×D131—22no—
86C22×D131—21no—
87C22×D131—22no—
88C22×D131—22no—
89C22×D131—21no—
90C22×D131—21no—
91C22×D131#4114yes—
92Dic262—44no—
93D524—28no—
94D524—48no—
95C22⋊D131—48no—
96C22⋊D131—44no—
97C2×Dic131—22no—
98C2×Dic131—21no—
99C2×Dic131—22no—
100C2×Dic131—21no—
101C2×Dic131—21no—
102C2×Dic131—22no—
103C2×Dic131C2×D412yes—
104C2×Dic131C2×D414yes—
105C2×Dic131C2×D412yes—
106C2×Dic131C2×D414yes—
107C4×D131—48no—
108C23×C131—21no—
109C23×C131—22no—
110C23×C131—22no—
111C23×C131—21no—
112C23×C131—21no—
113C23×C131C2411yescharacteristic
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131—22no—
117C2×C4×C131—22no—
118C2×C4×C131—21no—
119C2×C4×C131—22no—
120C2×C4×C131—22no—
121C2×C4×C131—22no—
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2×D414yes—
124C22×C131—41no—
125C22×C131—41no—
126C22×C131—41no—
127C22×C131—42no—
128C22×C131—42no—
129C22×C131C22×D411yescharacteristic
130C22×C131C22×D412yes—
131C22×C131C2×#4112yes—
132C22×C131H(5,𝔽2)11yescharacteristic
133C22×C1312−1+411yescharacteristic
134C2×D131—24no—
135C2×D131—24no—
136C2×D131—24no—
137C2×D131—42no—
138C2×D131—42no—
139C2×D131—42no—
140C2×D131—42no—
141C2×D131—41no—
142C2×D131—41no—
143C2×D131—41no—
144C2×D131—42no—
145C2×D131—42no—
146C4×C131—22no—
147C4×C131—24no—
148C4×C131—42no—
149C4×C131—41no—
150Dic131—22no—
151Dic131—24no—
152Dic131—41no—
153Dic131#12714yes—
154C2×C131—81no—
155C2×C131#52511yescharacteristic
156C2×C131C2×#12711yescharacteristic
157C2×C131#53711yescharacteristic
158C2×C131#54612yes—
159C2×C131#54812yes—
160D131—44no—
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#399D13111yescharacteristic, complemented
170#392—1311no—
171#392—1311no—
172#390—1312no—
173#390—1312no—
174C4×C4⋊C4—1311no—
175C22×Q8—1311no—
176C22×D4—1311no—
177C22×D4—1312no—
178C22×D4—1311no—
179C22×D4—1312no—
180C23×C4—1312no—
181#124—2624no—
182C22⋊Q8—2628no—
183#121—2628no—
184C4×D4—2624no—
185C2×C4⋊C4—1312no—
186C2×C4⋊C4—1312no—
187C2×C4⋊C4C2×D13111yescharacteristic
188C2×C4⋊C4C2×D13112yes—
189C2×C22⋊C4—1311no—
190C2×C22⋊C4—1312no—
191C2×C22⋊C4—1312no—
192C2×C22⋊C4—1312no—
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—
203C24—1312no—
204C24C22×D13111yescharacteristic
205C2×Q8—2624no—
206C2×Q8—2622no—
207C2×D4—2624no—
208C2×D4—2624no—
209C2×D4—2624no—
210C2×D4—2624no—
211C2×D4—2624no—
212C2×D4—2622no—
213C2×D4—2622no—
214C2×D4—2624no—
215C2×D4—5242no—
216C2×D4—5242no—
217C22×C4—2624no—
218C22×C4—2624no—
219C22×C4—2624no—
220C22×C4—1311no—
221C22×C4—1312no—
222C22×C4—1311no—
223C22×C4—1311no—
224C22×C4C22×D13111yescharacteristic
225C22×C4C22×D13112yes—
226C22×C4C22×D13111yescharacteristic
227C22×C4C22×D13112yes—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—2624no—
231C4⋊C4—1318no—
232C22⋊C4—24no—
233C22⋊C4—22no—
234C22⋊C4—2624no—
235C22⋊C4—2624no—
236C22⋊C4—2624no—
237C22⋊C4—2622no—
238C22⋊C4—2622no—
239C22⋊C4—2624no—
240C22⋊C4—2622no—
241C42—5242no—
242C42—1314no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—2622no—
249C23—2622no—
250C23—2622no—
251C23—2622no—
252C23—2622no—
253C23—2622no—
254C23—2621no—
255C23—2622no—
256C23—2622no—
257C23—2621no—
258C23—2621no—
259C23—1314no—
260C23C23×D13111yescharacteristic
261Q8—5244no—
262D4—2628no—
263D4—5248no—
264D4—5248no—
265D4—5244no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—2622no—
275C2×C4—2621no—
276C2×C4—2622no—
277C2×C4—2621no—
278C2×C4—2621no—
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—41no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—2624no—
293C22—2624no—
294C22—2624no—
295C22—5242no—
296C22—5242no—
297C22—5242no—
298C22—5242no—
299C22—5241no—
300C22—5242no—
301C22—5242no—
302C22—5241no—
303C22—5241no—
304C22C2×D4×D13111yescharacteristic
305C22C2×D4×D13112yes—
306C22#41×D13112yes—
307C22#1686229211yescharacteristic
308C22#1686229411yescharacteristic
309C4—24no—
310C4—22no—
311C4—42no—
312C4—41no—
313C4—2622no—
314C4—2624no—
315C4—5241no—
316C4—1314no—
317C2—81no—
318C2—5244no—
319C2—10481no—
320C2#2255382011yescharacteristic
321C2#127×D13111yescharacteristic
322C2#2255389211yescharacteristic
323C2#2255395412yes—
324C2#2255397212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
hypercenter#399D131
hypocenterC131#2540
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16862232C2semidirect
2. C2×#16862224C2semidirect
3. C2×#16862219C2semidirect
4. C2×#16862208C2semidirect
5. #22553247C2semidirect
6. #22553222C2semidirect
7. #22553221C2semidirect
8. #22553213C2semidirect
9. #22553212C2semidirect
10. #22553163C2semidirect
11. #399×C131C2semidirect
61. Dic131⋊C4D4semidirect
70. C4×Dic131D4semidirect
162. C131#2540semidirect
169. #399D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622322
C2×#168622242
C2×#168622192
C2×#168622082
#225532472
#225532222
#225532212
#225532132
#225532122
#225531632
#399×C1312
#2540131