Finite Group Databasebeta

← #30670427

Subgroups — #30670427

Order 16768.

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

normal subgroups only

This group has 708 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×#16862231C212yesmaximal, complemented
2C2×#16862228C212yesmaximal, complemented
3C2×#16862219C211yesmaximal, characteristic
4C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
5#22553249C211yesmaximal, characteristic, complemented
6#22553248C211yesmaximal, characteristic, complemented
7#22553247C211yesmaximal, characteristic, complemented
8#22553212C212yesmaximal, complemented
9#22553181C212yesmaximal, complemented
10#22553159C211yesmaximal, characteristic, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C22×D524C2211yescharacteristic
13C22×C22⋊D131C2211yescharacteristic
14C22×C22⋊D131C2211yescharacteristic
15C22×C4×D131C2212yes—
16C22×C4×D131C2211yescharacteristic
17#16862231—28no—
18#16862228—28no—
19#16862219—24no—
20C2×C4⋊Dic131C2212yes—
21C2×C4⋊Dic131C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2212yes—
24C2×Dic131⋊C4C2212yes—
25C2×Dic131⋊C4C2212yes—
26C4⋊C4×D131—24no—
27C2×C22⋊Dic131C2212yes—
28C2×(C2×D131)⋊C4C2212yes—
29C2×(C2×D131)⋊C4C2212yes—
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2212yes—
32C2×C4×Dic131C2212yes—
33C131⋊#95C2211yescharacteristic
34C2×C4⋊C4×C131C2212yes—
35C2×C4⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2211yescharacteristic
38#95×C131C2211yescharacteristic
39C23×D131C2311yescharacteristic
40C23×D131C2311yescharacteristic
41C2×D524—22no—
42C2×D524—24no—
43C2×C22⋊D131—22no—
44C2×C22⋊D131—24no—
45C2×C22⋊D131—22no—
46C2×C22⋊D131—24no—
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2312yes—
50C22×Dic131C2311yescharacteristic
51C2×C4×D131—24no—
52C2×C4×D131—24no—
53C2×C4×D131—24no—
54C2×C4×D131—24no—
55C2×C4×D131D414yes—
56C4⋊Dic131—24no—
57C4⋊Dic131—22no—
58Dic131⋊C4—22no—
59Dic131⋊C4—24no—
60Dic131⋊C4—24no—
61Dic131⋊C4—24no—
62C22⋊Dic131—24no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—22no—
65(C2×D131)⋊C4—24no—
66(C2×D131)⋊C4D418yescomplemented
67C4×Dic131—24no—
68C24×C131C2311yescharacteristic
69C22×C4×C131C2312yes—
70C22×C4×C131C2311yescharacteristic
71C22×C4×C131C2311yescharacteristic
72C22×C4×C131C2312yes—
73C4⋊C4×C131—24no—
74C4⋊C4×C131—22no—
75C22⋊C4×C131—22no—
76C22⋊C4×C131—24no—
77C22×D131—22no—
78C22×D131—21no—
79C22×D131—21no—
80C22×D131—22no—
81C22×D131—21no—
82C22×D131—22no—
83C22×D131—21no—
84C22×D131—21no—
85C22×D131C2×D414yes—
86C22×D131C2×D412yes—
87D524—44no—
88C22⋊D131—44no—
89C22⋊D131—44no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—21no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—22no—
97C2×Dic131C2×D414yes—
98C2×Dic131C2×D412yes—
99C2×Dic131#4114yes—
100C4×D131—28no—
101C4×D131—48no—
102C23×C131—22no—
103C23×C131—21no—
104C23×C131—22no—
105C23×C131—21no—
106C23×C131—21no—
107C23×C131C2411yescharacteristic
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—21no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—22no—
116C2×C4×C131C2×D414yes—
117C2×C4×C131C2×D412yes—
118C22×C131—41no—
119C22×C131—41no—
120C22×C131—41no—
121C22×C131—42no—
122C22×C131—42no—
123C22×C131C22×D411yescharacteristic
124C22×C131C22×D412yes—
125C22×C131C2×#4112yes—
126C22×C1312−1+411yescharacteristic
127C22×C1312−1+411yescharacteristic
128C2×D131—22no—
129C2×D131—22no—
130C2×D131—24no—
131C2×D131—24no—
132C2×D131—41no—
133C2×D131—41no—
134C2×D131—41no—
135C2×D131—42no—
136C2×D131—42no—
137C2×D131#12714yes—
138C4×C131—22no—
139C4×C131—24no—
140C4×C131—42no—
141C4×C131—41no—
142Dic131—24no—
143Dic131—42no—
144Dic131—41no—
145Dic131#12314yescomplemented
146C2×C131—81no—
147C2×C131C2×#12311yescharacteristic
148C2×C131C2×#12711yescharacteristic
149C2×C131#53611yescharacteristic
150C2×C131#54712yes—
151C2×C131#54712yes—
152D131—24no—
153D131—81no—
154C131#250111yescharacteristic, complemented, Sylow
155#2501—1311nomaximal, Sylow
156C2×#123—1312no—
157C2×#123—1312no—
158C2×#121—1311no—
159C22×C4⋊C4—1311no—
160#399—1311no—
161#399—1311no—
162#399—1311no—
163#399D13111yescharacteristic, complemented
164#390—1312no—
165#384—1312no—
166C4×C22⋊C4—1311no—
167C22×D4—1311no—
168C22×D4—1311no—
169C22×D4—1311no—
170C23×C4—1312no—
171C23×C4—1311no—
172#123—2628no—
173#123—2628no—
174#121—2624no—
175C2×C4⋊C4—2624no—
176C2×C4⋊C4—1312no—
177C2×C4⋊C4—1311no—
178C2×C4⋊C4—1311no—
179C2×C4⋊C4—1312no—
180C2×C4⋊C4—1312no—
181C2×C4⋊C4—1312no—
182C2×C4⋊C4C2×D13111yescharacteristic
183C2×C4⋊C4C2×D13112yes—
184C2×C22⋊C4—1312no—
185C2×C22⋊C4—1311no—
186C2×C22⋊C4—1312no—
187C2×C22⋊C4—1312no—
188C2×C22⋊C4—1312no—
189C2×C22⋊C4C2×D13111yescharacteristic
190C2×C22⋊C4C2×D13112yes—
191C2×C42—1312no—
192#95—1311no—
193#95C2×D13111yescharacteristic
194C24—1311no—
195C24—1311no—
196C24C22×D13111yescharacteristic
197C2×D4—2622no—
198C2×D4—2624no—
199C2×D4—2622no—
200C2×D4—2624no—
201C2×D4—2622no—
202C2×D4—2624no—
203C22×C4—2624no—
204C22×C4—2624no—
205C22×C4—2624no—
206C22×C4—2624no—
207C22×C4—1312no—
208C22×C4—1311no—
209C22×C4—1312no—
210C22×C4—1314no—
211C22×C4—1311no—
212C22×C4C22×D13112yes—
213C22×C4C22×D13111yescharacteristic
214C22×C4C22×D13111yescharacteristic
215C22×C4C22×D13112yes—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—2624no—
219C4⋊C4—2622no—
220C4⋊C4—2624no—
221C4⋊C4—2624no—
222C4⋊C4—2622no—
223C4⋊C4—2624no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—2624no—
227C22⋊C4—2624no—
228C22⋊C4—2622no—
229C22⋊C4—2624no—
230C22⋊C4—1318no—
231C42—2624no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—2622no—
238C23—2621no—
239C23—2621no—
240C23—2622no—
241C23—2621no—
242C23—2622no—
243C23—2621no—
244C23—2621no—
245C23—1314no—
246C23—1312no—
247C23C23×D13111yescharacteristic
248D4—5244no—
249D4—5244no—
250D4—5244no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—2628no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—2622no—
264C2×C4—2621no—
265C2×C4—2622no—
266C2×C4—2622no—
267C2×C4—5248no—
268C2×C4—1314no—
269C2×C4—1312no—
270C2×C4—1314no—
271C2×C4D4×D13114yes—
272C2×C4D4×D13112yes—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—2622no—
279C22—2624no—
280C22—2624no—
281C22—2622no—
282C22—5241no—
283C22—5242no—
284C22—5242no—
285C22—5241no—
286C22—5241no—
287C22—1314no—
288C22C2×D4×D13111yescharacteristic
289C22C2×D4×D13112yes—
290C22#41×D13112yes—
291C22#1686229311yescharacteristic
292C22#1686229411yescharacteristic
293C4—24no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—2624no—
298C4—5242no—
299C4—5241no—
300C4—1314no—
301C2—2624no—
302C2—81no—
303C2—10481no—
304C2#123×D13111yescharacteristic
305C2#127×D13111yescharacteristic
306C2#2255388511yescharacteristic
307C2#2255396012yes—
308C2#2255396412yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
prime radicalC2×#16862219C2
hypercenter#399D131
hypocenterC131#2501
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16862231C2semidirect
2. C2×#16862228C2semidirect
4. C2×C4⋊C4×D131C2semidirect
5. #22553249C2semidirect
6. #22553248C2semidirect
7. #22553247C2semidirect
8. #22553212C2semidirect
9. #22553181C2semidirect
10. #22553159C2semidirect
11. #399×C131C2semidirect
66. (C2×D131)⋊C4D4semidirect
145. Dic131#123semidirect
154. C131#2501semidirect
163. #399D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622312
C2×#168622282
C2×#168622192
C2×C4⋊C4×D1312
#225532492
#225532482
#225532472
#225532122
#225531812
#225531592
#399×C1312
#2501131