Finite Group Databasebeta

← #30669175

Subgroups — #30669175

Order 16768.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C22×#14523512C211yesmaximal, characteristic, complemented
2C2×#16862208C211yesmaximal, characteristic, complemented
3C2×(C22⋊D131)⋊C4C212yesmaximal, complemented
4#22553232C211yesmaximal, characteristic, complemented
5#22553227C212yesmaximal, complemented
6#22553225C211yesmaximal, characteristic, complemented
7#22553221C212yesmaximal, complemented
8#22553220C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic131C212yesmaximal, complemented
10#22553159C211yesmaximal, characteristic, complemented
11#394×C131C211yesmaximal, characteristic, complemented
12C2×#14523512—28no—
13C2×#14523512—24no—
14C22×Dic262C2211yescharacteristic
15C22×D524C2211yescharacteristic
16C22×C22⋊D131C2212yes—
17C22×C22⋊D131C2212yes—
18C23×Dic131C2212yes—
19C22×C4×D131C2211yescharacteristic
20C22×C4×D131C2211yescharacteristic
21#16862208—24no—
22(C22⋊D131)⋊C4—28no—
23C2×Dic131⋊C4C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2211yescharacteristic
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×C4×Dic131C2212yes—
30C2×C4×Dic131C2211yescharacteristic
31C131⋊#95C2211yescharacteristic
32C131⋊#95C2212yes—
33C131⋊#95C2212yes—
34C22×D4×C131C2211yescharacteristic
35C2×C4⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2212yes—
38#95×C131C2211yescharacteristic
39C23×D131C2311yescharacteristic
40C23×D131C2311yescharacteristic
41#14523512—232no—
42C2×Dic262—24no—
43C2×Dic262—22no—
44C2×D524—22no—
45C2×D524—22no—
46C2×D524—22no—
47C2×C22⋊D131—24no—
48C2×C22⋊D131—24no—
49C2×C22⋊D131—24no—
50C2×C22⋊D131—24no—
51C2×C22⋊D131—24no—
52C2×C22⋊D131—24no—
53C22×Dic131—24no—
54C22×Dic131—24no—
55C22×Dic131D418yescomplemented
56C22×Dic131C2311yescharacteristic
57C22×Dic131C2312yes—
58C22×Dic131C2312yes—
59C22×Dic131C2311yescharacteristic
60C2×C4×D131—24no—
61C2×C4×D131—22no—
62C2×C4×D131—24no—
63C2×C4×D131D414yescomplemented
64Dic131⋊C4—24no—
65C22⋊Dic131—24no—
66(C2×D131)⋊C4—22no—
67(C2×D131)⋊C4—24no—
68(C2×D131)⋊C4—24no—
69(C2×D131)⋊C4—22no—
70C4×Dic131—22no—
71C4×Dic131—24no—
72C24×C131C2312yes—
73C2×D4×C131—24no—
74C2×D4×C131—22no—
75C22×C4×C131C2311yescharacteristic
76C22×C4×C131C2312yes—
77C22×C4×C131C2311yescharacteristic
78C22×C4×C131C2311yescharacteristic
79C4⋊C4×C131—22no—
80C22⋊C4×C131—24no—
81C22⋊C4×C131—24no—
82C22×D131—22no—
83C22×D131—21no—
84C22×D131—22no—
85C22×D131—22no—
86C22×D131—22no—
87C22×D131—21no—
88C22×D131—21no—
89C22×D131—21no—
90C22×D131C2×D412yes—
91C22×D131#4112yescomplemented
92Dic262—28no—
93D524—28no—
94C22⋊D131—216no—
95C22⋊D131—216no—
96C2×Dic131—22no—
97C2×Dic131—22no—
98C2×Dic131—22no—
99C2×Dic131—22no—
100C2×Dic131—216no—
101C2×Dic131—21no—
102C2×Dic131—22no—
103C2×Dic131C2×D414yes—
104C2×Dic131C2×D414yes—
105C2×Dic131C2×D412yes—
106C2×Dic131#4114yescomplemented
107C4×D131—28no—
108C4×D131—44no—
109C23×C131—22no—
110C23×C131—22no—
111C23×C131—22no—
112C23×C131—22no—
113C23×C131—22no—
114C23×C131—22no—
115C23×C131C2×D414yes—
116C23×C131C2411yescharacteristic
117D4×C131—28no—
118C2×C4×C131—21no—
119C2×C4×C131—22no—
120C2×C4×C131—21no—
121C2×C4×C131—22no—
122C2×C4×C131—22no—
123C2×C4×C131—21no—
124C2×C4×C131—22no—
125C2×C4×C131—22no—
126C2×C4×C131C2×D412yes—
127C2×C4×C131#4112yescomplemented
128C22×C131—24no—
129C22×C131—24no—
130C22×C131—24no—
131C22×C131—42no—
132C22×C131—42no—
133C22×C131—42no—
134C22×C131—42no—
135C22×C131C22×D412yes—
136C22×C131C22×D411yescharacteristic
137C22×C131C2×#4111yescharacteristic
138C22×C131C2×#4112yes—
139C22×C131C2×#4111yescharacteristic
140C2×D131—22no—
141C2×D131—24no—
142C2×D131—22no—
143C2×D131—22no—
144C2×D131—22no—
145C2×D131—41no—
146C2×D131—42no—
147C2×D131—42no—
148C2×D131—42no—
149C2×D131—41no—
150C4×C131—22no—
151C4×C131—22no—
152C4×C131—41no—
153C4×C131—42no—
154Dic131—24no—
155Dic131—24no—
156Dic131—41no—
157Dic131#12014yes—
158C2×C131—44no—
159C2×C131C2×#12011yescharacteristic
160C2×C131#52512yes—
161C2×C131#52511yescharacteristic
162C2×C131#53212yes—
163C2×C131#53211yescharacteristic
164D131—42no—
165D131—42no—
166C131#238811yescharacteristic, complemented, Sylow
167#2388—1311nomaximal, Sylow
168C22×#41—1311no—
169C2×C4×D4—1311no—
170C2×C4×D4—1312no—
171#395—1311no—
172#394—1312no—
173#394D13111yescharacteristic, complemented
174#393—1311no—
175#392—1311no—
176#392—1312no—
177C4×C22⋊C4—1312no—
178C4×C22⋊C4—1311no—
179C2×#41—2628no—
180C2×#41—2624no—
181C22×Q8—1311no—
182C22×D4—1312no—
183C22×D4—1312no—
184C22×D4—1311no—
185C22×D4C2×D13111yescharacteristic
186C23×C4—1311no—
187C23×C4—1312no—
188C23×C4—1311no—
189C4×D4—2628no—
190C4×D4—2624no—
191C2×C4⋊C4—1312no—
192C2×C4⋊C4C2×D13111yescharacteristic
193C2×C22⋊C4—1312no—
194C2×C22⋊C4—1312no—
195C2×C22⋊C4—1311no—
196C2×C22⋊C4—1311no—
197C2×C22⋊C4—1312no—
198C2×C22⋊C4C2×D13112yes—
199C2×C22⋊C4C2×D13112yes—
200C2×C42—1311no—
201C2×C42—1312no—
202#95—1311no—
203#95—1312no—
204#95—1312no—
205#95C2×D13111yescharacteristic
206C24—1311no—
207C24—1311no—
208C24C22×D13112yes—
209#41—26232no—
210C2×Q8—2624no—
211C2×Q8—2622no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—2622no—
215C2×D4—2622no—
216C2×D4—2624no—
217C2×D4—2624no—
218C2×D4—2624no—
219C2×D4—2624no—
220C2×D4—2624no—
221C2×D4—2624no—
222C2×D4—2622no—
223C22×C4—2624no—
224C22×C4—2624no—
225C22×C4—2622no—
226C22×C4—2624no—
227C22×C4—2624no—
228C22×C4—1311no—
229C22×C4—1318no—
230C22×C4—1312no—
231C22×C4—1312no—
232C22×C4—1314no—
233C22×C4—1311no—
234C22×C4C22×D13112yes—
235C22×C4C22×D13111yescharacteristic
236C22×C4C22×D13111yescharacteristic
237C22×C4C22×D13111yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—2624no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—2622no—
243C22⋊C4—2624no—
244C22⋊C4—2624no—
245C22⋊C4—2624no—
246C22⋊C4—2622no—
247C42—2622no—
248C42—2624no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—2622no—
256C23—2621no—
257C23—2622no—
258C23—2622no—
259C23—2622no—
260C23—2621no—
261C23—2621no—
262C23—2621no—
263C23—1312no—
264C23—1312no—
265C23D4×D13114yes—
266C23C23×D13111yescharacteristic
267Q8—2628no—
268D4—28no—
269D4—2628no—
270D4—26216no—
271D4—26216no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—2628no—
281C2×C4—2622no—
282C2×C4—2622no—
283C2×C4—2622no—
284C2×C4—2622no—
285C2×C4—26216no—
286C2×C4—2621no—
287C2×C4—2622no—
288C2×C4—5244no—
289C2×C4—1314no—
290C2×C4—1314no—
291C2×C4—1314no—
292C2×C4—1312no—
293C2×C4D4×D13112yes—
294C2×C4#1452351212yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—2622no—
303C22—2622no—
304C22—2624no—
305C22—2622no—
306C22—2622no—
307C22—5242no—
308C22—5242no—
309C22—5241no—
310C22—5241no—
311C22—5242no—
312C22C2×D4×D13112yes—
313C22C2×D4×D13111yescharacteristic
314C22#41×D13112yes—
315C22#41×D13111yescharacteristic
316C22C2×#1452351211yescharacteristic
317C4—22no—
318C4—22no—
319C4—42no—
320C4—41no—
321C4—2624no—
322C4—2624no—
323C4—5241no—
324C4—1314no—
325C2—44no—
326C2—5242no—
327C2—5242no—
328C2#120×D13111yescharacteristic
329C2#2255382012yes—
330C2#2255382211yescharacteristic
331C2#2255385712yes—
332C2#2255386111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#394×C131C2
hypercenter#394D131
hypocenterC131#2388
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#14523512C2semidirect
2. C2×#16862208C2semidirect
3. C2×(C22⋊D131)⋊C4C2semidirect
4. #22553232C2semidirect
5. #22553227C2semidirect
6. #22553225C2semidirect
7. #22553221C2semidirect
8. #22553220C2semidirect
9. C22⋊C4×Dic131C2semidirect
10. #22553159C2semidirect
11. #394×C131C2semidirect
55. C22×Dic131D4semidirect
63. C2×C4×D131D4semidirect
91. C22×D131#41semidirect
106. C2×Dic131#41semidirect
127. C2×C4×C131#41semidirect
166. C131#2388semidirect
173. #394D131semidirect
294. C2×C4#14523512semidirect

Maximal subgroups

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

subgroupindex
C22×#145235122
C2×#168622082
C2×(C22⋊D131)⋊C42
#225532322
#225532272
#225532252
#225532212
#225532202
C22⋊C4×Dic1312
#225531592
#394×C1312
#2388131