Finite Group Databasebeta

← #29577478

Subgroups — #29577478

Order 15616.

720 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#21450750C211yesmaximal, characteristic
2#21444334C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21443705C212yesmaximal, complemented
5C61×#2312C211yesmaximal, characteristic, complemented
6C2×#14519760—21no—
7C2×#14519751C2211yescharacteristic
8C2×#14519746—21no—
9#16839726C2212yes—
10#16839698C412yescomplemented
11#16839671C412yescomplemented
12#16839662—22no—
13C4×C22⋊Dic61C2211yescharacteristic
14#16839495—24no—
15#16839486—22no—
16C22×C22⋊C4×C61C2211yescharacteristic
17C61×#388—24no—
18C61×#380—21no—
19C61×#379—21no—
20C4×C22⋊C4×C61C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23C22×C22⋊D61—22no—
24C2×#11758194—22no—
25#14519760—44no—
26#14519751—22no—
27#14519751—22no—
28#14519746—44no—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758196—22no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×#11758198—22no—
34C2×#11758198—21no—
35C2×C4×Dic61—21no—
36C22⋊#92418—22no—
37C22⋊#92418—22no—
38#14519684C2×C412yes—
39C25×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C2×C4⋊C4×C61—22no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61C2×C412yes—
49C2×C22⋊C4×C61D412yes—
50C2×C42×C61—22no—
51#95×C61—22no—
52#95×C61D414yescomplemented
53C24×C61—21no—
54C24×C61—21no—
55C24×C61—42no—
56C24×C61—42no—
57C24×C61—81no—
58C24×C61C22⋊C412yes—
59C24×C61C2×D411yescharacteristic
60C23×D61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—24no—
64C22×C4×C61—42no—
65C22×C4×C61—81no—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4112yes—
71C2×C22⋊D61—44no—
72C2×C22⋊D61—44no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C2×C22⋊D61—82no—
78C22×Dic61—22no—
79C22×Dic61—21no—
80C2×D244—22no—
81C2×D244—81no—
82C4⋊C4×C61—82no—
83C22⋊C4×C61—24no—
84C22⋊C4×C61—24no—
85C22⋊C4×C61—44no—
86C22⋊C4×C61—82no—
87C22⋊C4×C61—82no—
88C42×C61—82no—
89C22⋊Dic61—24no—
90C22⋊Dic61—44no—
91#11758198—44no—
92#11758198—42no—
93C4⋊Dic61—22no—
94#11758196—44no—
95C4×Dic61—81no—
96#11758194—82no—
97C2×#92418—42no—
98C23×C61—22no—
99C23×C61—22no—
100C23×C61—22no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—41no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—42no—
109C23×C61—42no—
110C23×C61—81no—
111C23×C61—81no—
112C23×C61—88no—
113C23×C61C2×C22⋊C411yescharacteristic
114C23×C61#12011yescharacteristic
115C23×C61#12311yescharacteristic
116C22×D61—41no—
117C22×D61—42no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—42no—
121C2×C4×C61—22no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—44no—
129C2×C4×C61—44no—
130C2×C4×C61—44no—
131C2×C4×C61—81no—
132C2×C4×C61C4×D412yes—
133C2×C4×C61#12112yes—
134C22⋊D61—88no—
135C22⋊D61—82no—
136C22⋊D61—82no—
137C2×Dic61—24no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143D244—82no—
144#92418—82no—
145#9413—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—44no—
154C22×C61—44no—
155C22×C61—42no—
156C22×C61—42no—
157C22×C61—42no—
158C22×C61—82no—
159C22×C61—84no—
160C22×C61—84no—
161C22×C61#38611yescharacteristic
162C22×C61#42112yes—
163C22×C61#45312yes—
164C2×D61—82no—
165C2×D61—82no—
166C2×D61—81no—
167C2×D61—81no—
168C2×D61—81no—
169C4×C61—22no—
170C4×C61—22no—
171C4×C61—42no—
172C4×C61—84no—
173Dic61—44no—
174Dic61—81no—
175#2539—611no—
176#2312D6111yescharacteristic, complemented
177#1507—612no—
178#1462—611no—
179#1297—612no—
180C2×C61—22no—
181C2×C61—22no—
182C2×C61—41no—
183C2×C61—82no—
184C2×C61#188711yescharacteristic
185C2×C61#189711yescharacteristic
186C2×C61#191311yescharacteristic
187D61—161no—
188C2×#123—1221no—
189C2×#121—611no—
190C2×#120—1221no—
191C22×C22⋊C4C2×D6111yescharacteristic
192#407—612no—
193#399—612no—
194#392—612no—
195#390—1222no—
196#388—24no—
197#380—21no—
198#379—21no—
199C4×C22⋊C4—611no—
200C4×C22⋊C4C2×D6112yes—
201#327—1224no—
202#323—1222no—
203C61#941311yescharacteristic, complemented, Sylow
204C25C22⋊D6111yescharacteristic
205C22×D4—611no—
206C22×D4—1222no—
207C22×D4—1221no—
208C23×C4C22⋊D6111yescharacteristic
209C23×C4C22×D6111yescharacteristic
210C2×M16—1222no—
211#123—2444no—
212#121—1222no—
213#121—1222no—
214#120—2444no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—1222no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—1221no—
224C2×C22⋊C4—1222no—
225C2×C22⋊C4—1222no—
226C2×C22⋊C4—1221no—
227C2×C22⋊C4C4×D6112yes—
228C2×C22⋊C4D24412yes—
229C2×C42—22no—
230C2×C42—1221no—
231C22⋊C8—1222no—
232C22⋊C8—1222no—
233#95—22no—
234#95—612no—
235#95C22⋊D6114yescomplemented
236C24—21no—
237C24—21no—
238C24—42no—
239C24—42no—
240C24—81no—
241C24—1221no—
242C24#1175819812yes—
243C24C2×C22⋊D6111yescharacteristic
244C2×D4—1222no—
245C2×D4—4881no—
246C2×D4—4882no—
247C2×D4—2444no—
248C2×D4—2442no—
249C2×D4—2442no—
250C2×D4—2444no—
251C2×D4—2441no—
252C2×D4—2441no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—1222no—
259C22×C4—1221no—
260C22×C4#1175819812yes—
261C22×C4C2×C4×D6111yescharacteristic
262C22×C4C2×D24411yescharacteristic
263C22×C4#1175822212yes—
264C22×C4C2×C22⋊D6112yes—
265M16—4882no—
266C2×C8—2442no—
267C4⋊C4—82no—
268C4⋊C4—1222no—
269C4⋊C4—2444no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—44no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—1224no—
276C22⋊C4—2444no—
277C22⋊C4—2444no—
278C22⋊C4—2442no—
279C42—82no—
280C42—4881no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—81no—
294C23—81no—
295C23—88no—
296C23—2441no—
297C23—2442no—
298C23—2441no—
299C23—2442no—
300C23—2441no—
301C23C2×#1175819811yescharacteristic
302C23#1451974611yescharacteristic
303C23#1451976011yescharacteristic
304D4—4882no—
305D4—4882no—
306D4—4882no—
307D4—4888no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—81no—
319C2×C4—1224no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4C4×C22⋊D6112yes—
326C2×C4#1451975112yes—
327C8—4882no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—84no—
343C22—4882no—
344C22—4882no—
345C22—4881no—
346C22—4881no—
347C22—4881no—
348C22#1683964111yescharacteristic
349C22#1683978012yes—
350C22#1683993012yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—84no—
355C4—4881no—
356C4—2444no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—9761no—
362C2#2144621711yescharacteristic
363C2#2144626711yescharacteristic
364C2#2144636111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2312C2
socleC22×C61#386
prime radical#21450750C2
cosocleC23×C4×C61C23
hypercenter#2312D61
hypocenterC61#9413
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #21444334C2semidirect
3. #21444146C2semidirect
4. #21443705C2semidirect
5. C61×#2312C2semidirect
10. #16839698C4semidirect
11. #16839671C4semidirect
52. #95×C61D4semidirect
176. #2312D61semidirect
203. C61#9413semidirect
235. #95C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214507502
#214443342
#214441462
#214437052
C61×#23122
#941361