Finite Group Databasebeta

← #30675592

Subgroups — #30675592

Order 16768.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16862279C211yesmaximal, characteristic, complemented
2C2×M16×D131C211yesmaximal, characteristic, complemented
3#22553824C211yesmaximal, characteristic
4C2×#16862226C211yesmaximal, characteristic, complemented
5#22553571C212yesmaximal, complemented
6#22553559C212yesmaximal, complemented
7#22553554C212yesmaximal, complemented
8#22553542C212yesmaximal, complemented
9#22553320C211yesmaximal, characteristic, complemented
10#22553311C211yesmaximal, characteristic, complemented
11#428×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×Dic262C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15#16862279—24no—
16C2×Dic524C2212yescomplemented
17C2×#14523502C2212yescomplemented
18C2×#14523497C2211yescharacteristic
19C2×#14523496C2212yes—
20M16×D131—24no—
21C2×C8×D131C2212yes—
22#16862230—21no—
23#16862226—21no—
24#16862226—21no—
25#16862226C2212yes—
26#16862226C2211yescharacteristic
27#16862226C2212yes—
28#16862226C2211yescharacteristic
29#16862221C2211yescharacteristic
30#16862221C2211yescharacteristic
31#16862216—21no—
32C4×D524—21no—
33C4×D524—21no—
34C2×C4⋊Dic131—21no—
35C2×(C2×D131)⋊C4—21no—
36#16862169C2212yescomplemented
37#16862169C2212yescomplemented
38#16862169C2212yescomplemented
39#16862166C2212yescomplemented
40C2×M16×C131C2211yescharacteristic
41#117×C131C2211yescharacteristic
42C2×C4⋊C4×C131C2211yescharacteristic
43#107×C131C2212yescomplemented
44#106×C131C2212yescomplemented
45C23×D131D411yescharacteristic
46#14523513—22no—
47#14523513—22no—
48C2×Dic262—21no—
49C2×Dic262—22no—
50C2×Dic262—22no—
51C2×Dic262C2312yes—
52C2×Dic262C2311yescharacteristic
53C2×D524—21no—
54C2×D524—21no—
55C2×D524C2311yescharacteristic
56C2×C22⋊D131—21no—
57C22×Dic131—21no—
58C22×Dic131D411yescharacteristic
59C2×C4×D131—22no—
60C2×C4×D131—21no—
61C2×C4×D131—22no—
62C2×C4×D131D411yescharacteristic
63C2×C4×D131D411yescharacteristic
64C2×C4×D131C2311yescharacteristic
65C2×C4×D131C2311yescharacteristic
66Dic524—24no—
67#14523502—24no—
68#14523497—22no—
69#14523496—24no—
70C2×C131⋊C8C2312yes—
71C8×D131—24no—
72C4⋊Dic131—21no—
73C4⋊Dic131—21no—
74C4⋊Dic131—22no—
75C4⋊Dic131—22no—
76(C2×D131)⋊C4—21no—
77(C2×D131)⋊C4—21no—
78(C2×D131)⋊C4—22no—
79(C2×D131)⋊C4—22no—
80(C2×D131)⋊C4—21no—
81(C2×D131)⋊C4—21no—
82C22×C4×C131—21no—
83C22×C4×C131C2311yescharacteristic
84M16×C131D414yescomplemented
85C2×C8×C131C2312yes—
86C4⋊C4×C131—21no—
87C4⋊C4×C131C2311yescharacteristic
88C4⋊C4×C131C2311yescharacteristic
89C4⋊C4×C131C2312yes—
90C22⋊C4×C131—21no—
91C42×C131—21no—
92C22×D131—22no—
93C22×D131—22no—
94C22×D131—21no—
95C22×D131—41no—
96C22×D131C2×D411yescharacteristic
97C22×D131C2×D411yescharacteristic
98Dic262—22no—
99Dic262—21no—
100Dic262—22no—
101Dic262—21no—
102Dic262—42no—
103D524—21no—
104D524—21no—
105D524—42no—
106D524—42no—
107C22⋊D131—42no—
108C2×Dic131—22no—
109C2×Dic131—21no—
110C2×Dic131—22no—
111C2×Dic131—42no—
112C2×Dic131C2×D411yescharacteristic
113C2×Dic131C2×D411yescharacteristic
114C4×D131—22no—
115C4×D131—22no—
116C4×D131—21no—
117C4×D131—21no—
118C4×D131—21no—
119C4×D131—21no—
120C4×D131—42no—
121C131⋊C8—22no—
122C23×C131C2×D411yescharacteristic
123C2×C4×C131—21no—
124C2×C4×C131—21no—
125C2×C4×C131—22no—
126C2×C4×C131—42no—
127C2×C4×C131C2×D411yescharacteristic
128C2×C4×C131C2×D412yes—
129C2×C4×C131#4112yes—
130C2×C4×C131C2411yescharacteristic
131C8×C131C2×D414yes—
132C22×C131—21no—
133C22×C131—21no—
134C22×C131#12112yes—
135C22×C131C22×D411yescharacteristic
136C2×D131—21no—
137C2×D131—21no—
138C2×D131—21no—
139C2×D131—21no—
140C2×D131—21no—
141C2×D131—44no—
142C2×D131—41no—
143C2×D131—41no—
144C2×D131—41no—
145C2×D131#13712yescomplemented
146C4×C131—41no—
147C4×C131—42no—
148C4×C131—41no—
149C4×C131#12112yes—
150C4×C131C22×D411yescharacteristic
151C4×C131C2×#4111yescharacteristic
152Dic131—21no—
153Dic131—21no—
154Dic131—41no—
155Dic131—42no—
156C2×C131—22no—
157C2×C131C2×#12111yescharacteristic
158C2×C131C2×#13711yescharacteristic
159C2×C131#57511yescharacteristic
160D131—22no—
161D131—41no—
162D131—81no—
163C131#305611yescharacteristic, complemented, Sylow
164#3056—1311nomaximal, Sylow
165C2×#137—1311no—
166C22×M16—1311no—
167#525—1311no—
168C2×C22⋊Q8—1311no—
169#470—1312no—
170#468—1312no—
171#467—1312no—
172#465—1312no—
173#428D13111yescharacteristic, complemented
174#419—1311no—
175#417—1311no—
176C2×#41—1311no—
177C22×Q8—1311no—
178C23×C4—1311no—
179#137—2624no—
180C2×Q16—1312no—
181C2×SD16—1312no—
182C2×M16—2624no—
183C2×M16—1311no—
184C2×M16—1312no—
185C2×M16C2×D13111yescharacteristic
186C22×C8—1312no—
187#123—2621no—
188C22⋊Q8—2621no—
189C22⋊Q8—2621no—
190C22⋊Q8—1312no—
191C22⋊Q8—1312no—
192C22⋊Q8—1311no—
193C22⋊Q8—1311no—
194#121—1311no—
195#121—1311no—
196#120—2621no—
197C4×D4—2621no—
198C4×D4—2621no—
199#117C2×D13111yescharacteristic
200C2×C4⋊C4—2621no—
201C2×C4⋊C4C2×D13111yescharacteristic
202C2×C22⋊C4—2621no—
203#107C2×D13112yescomplemented
204#106C2×D13112yescomplemented
205#103—1312no—
206#103—1312no—
207#103—1312no—
208#102—1312no—
209C24—1311no—
210#41—2622no—
211#41—2622no—
212C2×Q8—2621no—
213C2×Q8—2622no—
214C2×Q8—2622no—
215C2×Q8—1312no—
216C2×Q8—1311no—
217C2×D4—2621no—
218C2×D4—2621no—
219C2×D4—2621no—
220C2×D4—1311no—
221C22×C4—21no—
222C22×C4—2622no—
223C22×C4—2622no—
224C22×C4—2621no—
225C22×C4—2621no—
226C22×C4—1311no—
227C22×C4—1311no—
228C22×C4—1311no—
229C22×C4—1311no—
230C22×C4—1311no—
231C22×C4C22×D13111yescharacteristic
232Q16—2624no—
233SD16—2624no—
234M16—2622no—
235M16—2624no—
236M16D52414yescomplemented
237C2×C8—2624no—
238C2×C8—1312no—
239C2×C8C22×D13112yes—
240C4⋊C4—21no—
241C4⋊C4—2622no—
242C4⋊C4—2622no—
243C4⋊C4—2621no—
244C4⋊C4—2621no—
245C4⋊C4C22×D13112yes—
246C4⋊C4C22×D13111yescharacteristic
247C4⋊C4C22×D13111yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2622no—
250C22⋊C4—2622no—
251C22⋊C4—2621no—
252C22⋊C4—2621no—
253C22⋊C4—2621no—
254C22⋊C4—2621no—
255C42—21no—
256C23—2622no—
257C23—2622no—
258C23—2621no—
259C23—5241no—
260C23—1311no—
261C23—1311no—
262C23D4×D13111yescharacteristic
263Q8—2622no—
264Q8—2621no—
265Q8—2622no—
266Q8—2621no—
267Q8—5242no—
268D4—2621no—
269D4—2621no—
270D4—5242no—
271D4—5242no—
272D4—5242no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—2622no—
278C2×C4—2622no—
279C2×C4—2622no—
280C2×C4—2621no—
281C2×C4—2621no—
282C2×C4—2621no—
283C2×C4—2621no—
284C2×C4—2621no—
285C2×C4—2622no—
286C2×C4—5242no—
287C2×C4—5242no—
288C2×C4—1311no—
289C2×C4—1311no—
290C2×C4D4×D13111yescharacteristic
291C2×C4C2×D52412yes—
292C2×C4#1452351412yes—
293C2×C4C23×D13111yescharacteristic
294C8—2622no—
295C8C2×D52414yes—
296C22—21no—
297C22—21no—
298C22—2621no—
299C22—2621no—
300C22—2621no—
301C22—2621no—
302C22—2621no—
303C22—5244no—
304C22—5241no—
305C22—5241no—
306C22—5241no—
307C22—1312no—
308C22#1686222112yes—
309C22C2×D4×D13111yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—2621no—
314C4—2621no—
315C4—5241no—
316C4—5242no—
317C4#1686222112yes—
318C4C22×D52411yescharacteristic
319C4C2×#1452351411yescharacteristic
320C2—22no—
321C2—2622no—
322C2—5241no—
323C2—10481no—
324C2C2×#1686222111yescharacteristic
325C2#137×D13111yescharacteristic
326C2#2255410811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#428×C131C2
socleC22×C131C22×D4
prime radical#22553824C2
hypercenter#428D131
hypocenterC131#3056
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16862279C2semidirect
2. C2×M16×D131C2semidirect
4. C2×#16862226C2semidirect
5. #22553571C2semidirect
6. #22553559C2semidirect
7. #22553554C2semidirect
8. #22553542C2semidirect
9. #22553320C2semidirect
10. #22553311C2semidirect
11. #428×C131C2semidirect
16. C2×Dic524C22semidirect
17. C2×#14523502C22semidirect
36. #16862169C22semidirect
37. #16862169C22semidirect
38. #16862169C22semidirect
39. #16862166C22semidirect
43. #107×C131C22semidirect
44. #106×C131C22semidirect
84. M16×C131D4semidirect
145. C2×D131#137semidirect
163. C131#3056semidirect
173. #428D131semidirect
203. #107C2×D131semidirect
204. #106C2×D131semidirect
236. M16D524semidirect

Maximal subgroups

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

subgroupindex
C2×#168622792
C2×M16×D1312
#225538242
C2×#168622262
#225535712
#225535592
#225535542
#225535422
#225533202
#225533112
#428×C1312
#3056131