Finite Group Databasebeta

← #29577459

Subgroups — #29577459

Order 15616.

840 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 115 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#21448172C211yesmaximal, characteristic, complemented
2#21444335C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21443705C212yesmaximal, complemented
5C61×#2539C211yesmaximal, characteristic
6C22×C22⋊Dic61—21no—
7#16839726C2212yes—
8C22⋊C4×Dic61—24no—
9C4×C22⋊Dic61—21no—
10C4×C22⋊Dic61C2212yes—
11#16839496—24no—
12#16839486—22no—
13C2×#123×C61—21no—
14C2×#121×C61C2211yescharacteristic
15C2×#120×C61—21no—
16C22×C22⋊C4×C61C2211yescharacteristic
17C61×#399C412yescomplemented
18C61×#392C412yescomplemented
19C61×#390—22no—
20C4×C22⋊C4×C61C2211yescharacteristic
21C23×Dic61—22no—
22C2×#11758194—22no—
23C2×C22⋊Dic61—28no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C4×Dic61—22no—
27C2×C4×Dic61—22no—
28C22⋊#92418—22no—
29C22⋊#92418—22no—
30#14519684—22no—
31#14519684D414yescomplemented
32C25×C61D411yescharacteristic
33C22×D4×C61—21no—
34C22×D4×C61—22no—
35C22×D4×C61C2×C411yescharacteristic, complemented
36C23×C4×C61D411yescharacteristic
37C23×C4×C61C2311yescharacteristic
38#123×C61—44no—
39#121×C61—22no—
40#121×C61—22no—
41#120×C61—44no—
42C2×C4⋊C4×C61—22no—
43C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61C2×C412yes—
51C2×C22⋊C4×C61D412yes—
52C2×C42×C61—21no—
53#95×C61C2×C412yescomplemented
54C24×C61—21no—
55C24×C61—24no—
56C24×C61—21no—
57C24×C61—21no—
58C24×C61—42no—
59C24×C61—81no—
60C24×C61C22⋊C412yes—
61C24×C61C2×D411yescharacteristic
62C2×D4×C61—22no—
63C2×D4×C61—44no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C2×D4×C61—44no—
67C2×D4×C61—41no—
68C2×D4×C61—41no—
69C2×D4×C61—82no—
70C2×D4×C61—81no—
71C22×C4×C61—22no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—42no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C412yes—
78C22×C4×C61C22×C411yescharacteristic
79C22×C4×C61C2×D412yes—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61#4112yes—
82C22×Dic61—24no—
83C22×Dic61—28no—
84C22×Dic61—82no—
85C22×Dic61—82no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—44no—
88C22⋊C4×C61—24no—
89C22⋊C4×C61—24no—
90C22⋊C4×C61—24no—
91C22⋊C4×C61—44no—
92C22⋊C4×C61—44no—
93C22⋊C4×C61—42no—
94C42×C61—81no—
95C22⋊Dic61—28no—
96C22⋊Dic61—28no—
97C4×Dic61—82no—
98C4×Dic61—82no—
99#11758194—82no—
100C2×#92418—42no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—24no—
104C23×C61—22no—
105C23×C61—22no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—41no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—41no—
117C23×C61—42no—
118C23×C61—81no—
119C23×C61—81no—
120C23×C61—88no—
121C23×C61#10414yes—
122C23×C61C2×C22⋊C411yescharacteristic
123C23×C61#12011yescharacteristic
124C23×C61#12311yescharacteristic
125D4×C61—82no—
126D4×C61—88no—
127D4×C61—82no—
128D4×C61—82no—
129C2×C4×C61—24no—
130C2×C4×C61—22no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—81no—
143C2×C4×C61C4×D412yes—
144C2×C4×C61#12112yes—
145C2×Dic61—28no—
146C2×Dic61—44no—
147C2×Dic61—44no—
148C2×Dic61—88no—
149#92418—82no—
150#9409—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—24no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—44no—
160C22×C61—44no—
161C22×C61—44no—
162C22×C61—44no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—82no—
166C22×C61—82no—
167C22×C61—82no—
168C22×C61—84no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61#38611yescharacteristic
173C22×C61C2×#10412yes—
174C22×C61#45312yes—
175C4×C61—22no—
176C4×C61—22no—
177C4×C61—44no—
178C4×C61—42no—
179C4×C61—81no—
180Dic61—48no—
181#2539D6111yescharacteristic
182#2273—611no—
183#1507—612no—
184#1462—611no—
185#1297—612no—
186C2×C61—22no—
187C2×C61—22no—
188C2×C61—41no—
189C2×C61—44no—
190C2×C61—161no—
191C2×C61#188711yescharacteristic
192C2×C61#189611yescharacteristic
193C2×C61#191311yescharacteristic
194C2×#123—21no—
195C2×#121C2×D6111yescharacteristic
196C2×#120—21no—
197C22×C22⋊C4—1221no—
198C22×C22⋊C4C2×D6111yescharacteristic
199#407—612no—
200#399Dic6112yescomplemented
201#392Dic6112yescomplemented
202#390—22no—
203C4×C22⋊C4—612no—
204C4×C22⋊C4—1224no—
205C4×C22⋊C4—1221no—
206C4×C22⋊C4C2×D6111yescharacteristic
207#327—1224no—
208#323—1222no—
209C61#940911yescharacteristic, complemented, Sylow
210C25C22⋊D6111yescharacteristic
211C22×D4—21no—
212C22×D4—22no—
213C22×D4C2×Dic6111yescharacteristic, complemented
214C23×C4—1222no—
215C23×C4C22⋊D6111yescharacteristic
216C23×C4C22×D6111yescharacteristic
217C2×M16—1222no—
218#123—44no—
219#121—22no—
220#121—22no—
221#120—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×Dic6111yescharacteristic, complemented
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—1222no—
231C2×C22⋊C4—1228no—
232C2×C22⋊C4—1222no—
233C2×C22⋊C4C2×Dic6112yes—
234C2×C22⋊C4C22⋊D6112yes—
235C2×C42—21no—
236C2×C42—1222no—
237C2×C42—1222no—
238C22⋊C8—1222no—
239C22⋊C8—1222no—
240#95—614no—
241#95—1222no—
242#95C2×Dic6112yescomplemented
243C24—21no—
244C24—24no—
245C24—21no—
246C24—21no—
247C24—42no—
248C24—81no—
249C24C22⋊Dic6112yes—
250C24C2×C22⋊D6111yescharacteristic
251C2×D4—22no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—82no—
259C2×D4—81no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—1228no—
267C22×C4—1224no—
268C22×C4—4882no—
269C22×C4—4882no—
270C22×C4C22⋊Dic6112yes—
271C22×C4D4×D6112yes—
272C22×C4#1175822412yes—
273C22×C4C22×Dic6111yescharacteristic
274C22×C4C2×C22⋊D6111yescharacteristic
275M16—4882no—
276C2×C8—2442no—
277C4⋊C4—22no—
278C4⋊C4—44no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—1228no—
286C22⋊C4—1228no—
287C42—81no—
288C42—4882no—
289C42—4882no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—41no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—41no—
307C23—81no—
308C23—81no—
309C23—88no—
310C23#1451970414yes—
311C23C2×C22⋊Dic6111yescharacteristic
312C23#1451974511yescharacteristic
313C23#1451975811yescharacteristic
314D4—82no—
315D4—88no—
316D4—82no—
317D4—82no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—1228no—
333C2×C4—4888no—
334C2×C4—2444no—
335C2×C4—2444no—
336C2×C4D4×Dic6112yes—
337C2×C4#1451974912yes—
338C8—4882no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1683964211yescharacteristic
361C22C2×#1451970412yes—
362C22#1683992912yes—
363C4—22no—
364C4—22no—
365C4—44no—
366C4—42no—
367C4—81no—
368C4—2448no—
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—161no—
374C2#2144621811yescharacteristic
375C2#2144626311yescharacteristic
376C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#2539C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#2539D61
hypocenterC61#9409
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
1. #21448172C2semidirect
2. #21444335C2semidirect
3. #21444146C2semidirect
4. #21443705C2semidirect
17. C61×#399C4semidirect
18. C61×#392C4semidirect
31. #14519684D4semidirect
35. C22×D4×C61C2×C4semidirect
43. C2×C4⋊C4×C61C2×C4semidirect
53. #95×C61C2×C4semidirect
200. #399Dic61semidirect
201. #392Dic61semidirect
209. C61#9409semidirect
213. C22×D4C2×Dic61semidirect
223. C2×C4⋊C4C2×Dic61semidirect
242. #95C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214481722
#214443352
#214441462
#214437052
C61×#25392
#940961