Finite Group Databasebeta

← #29577642

Subgroups — #29577642

Order 15616.

576 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456283C211yesmaximal, characteristic
2#21444443C211yesmaximal, characteristic, complemented
3#21444326C212yesmaximal, complemented
4#21443712C212yesmaximal, complemented
5C61×#2372C211yesmaximal, characteristic, complemented
6C2×#14519808—21no—
7C2×#14519803—21no—
8#16840312C2211yescharacteristic
9#16840157C412yescomplemented
10#16840152C412yescomplemented
11#16840144—22no—
12#16839819C2212yes—
13C4×#11758194C2211yescharacteristic
14#16839502—22no—
15#16839495—22no—
16#16839495—22no—
17C2×#124×C61C2211yescharacteristic
18C61×#391—21no—
19C61×#388—22no—
20C61×#386—21no—
21C4×C22⋊C4×C61—22no—
22C61×#376C2212yes—
23C2×#11758222—21no—
24#14519808—42no—
25#14519808—42no—
26#14519803—42no—
27#14519803—42no—
28C2×#11758202—22no—
29C2×#11758200—22no—
30C2×#11758201—22no—
31C2×#11758199—22no—
32C2×#11758194—21no—
33#14519777C2×C411yescharacteristic, complemented
34#14519774C2×C411yescharacteristic, complemented
35#14519765—21no—
36#14519751—22no—
37C4×D244—22no—
38C4⋊#92418—22no—
39C22⋊#92418—22no—
40#14519688—22no—
41#14519688C2×C412yescomplemented
42C4×#92418—21no—
43C22×Q8×C61D411yescharacteristic
44C22×D4×C61D411yescharacteristic
45C23×C4×C61—21no—
46#124×C61—22no—
47#124×C61C2×C412yes—
48#124×C61D412yes—
49C2×C4⋊C4×C61—21no—
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61—21no—
53C2×C42×C61—22no—
54C2×C42×C61—21no—
55C2×C42×C61C2311yescharacteristic
56#95×C61—22no—
57#95×C61—22no—
58#95×C61D412yes—
59#95×C61D412yes—
60C24×C61—21no—
61C2×Q8×C61—21no—
62C2×Q8×C61—41no—
63C2×Q8×C61—41no—
64C2×Q8×C61C22⋊C412yes—
65C2×D4×C61—21no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C2×D4×C61C22⋊C412yes—
69C22×C4×C61—22no—
70C22×C4×C61—21no—
71C22×C4×C61—22no—
72C22×C4×C61—44no—
73C22×C4×C61—81no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D412yes—
76C2×C22⋊D61—41no—
77#11758222—41no—
78#11758222—41no—
79#11758222—41no—
80#11758222—41no—
81C2×D244—21no—
82C2×D244—22no—
83C2×C4×D61—41no—
84C2×Dic122—21no—
85C4⋊C4×C61—81no—
86C22⋊C4×C61—24no—
87C22⋊C4×C61—24no—
88C22⋊C4×C61—44no—
89C22⋊C4×C61—81no—
90C42×C61—44no—
91C42×C61—81no—
92C42×C61C22×C411yescharacteristic
93C42×C61C2×D411yescharacteristic
94C42×C61#4112yes—
95#11758202—44no—
96#11758201—44no—
97#11758200—44no—
98#11758199—44no—
99#11758198—42no—
100C4⋊Dic61—22no—
101#11758194—22no—
102#11758194—22no—
103C2×#92418—22no—
104C2×#92418—42no—
105C23×C61—22no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61#12011yescharacteristic
111C22×D61—41no—
112Q8×C61—42no—
113Q8×C61—81no—
114D4×C61—42no—
115D4×C61—81no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—24no—
119C2×C4×C61—22no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—84no—
129C2×C4×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C4×D412yes—
131C2×C4×C61#12112yes—
132C2×C4×C61#12311yescharacteristic
133C22⋊D61—81no—
134C22⋊D61—81no—
135C2×Dic61—41no—
136D244—41no—
137D244—41no—
138D244—82no—
139C4×D61—81no—
140C4×D61—81no—
141Dic122—41no—
142Dic122—41no—
143#92418—24no—
144#92418—82no—
145#9441—611nomaximal, Sylow
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—44no—
149C22×C61—81no—
150C22×C61—81no—
151C22×C61—82no—
152C22×C61#38611yescharacteristic
153C22×C61#45311yescharacteristic
154C22×C61#45311yescharacteristic
155C2×D61—81no—
156C2×D61—82no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—44no—
162C4×C61—42no—
163C4×C61—82no—
164Dic61—81no—
165#3153—611no—
166#2372D6111yescharacteristic, complemented
167#1533—611no—
168#1505—612no—
169#1300—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61—82no—
173C2×C61#188711yescharacteristic
174C2×C61#190612yes—
175D61—161no—
176C2×#137—1221no—
177C2×AGL(1,ℤ/8)—1221no—
178#532—611no—
179C2×#124C2×D6111yescharacteristic
180#497—612no—
181#496—612no—
182#495—1222no—
183#432—612no—
184C4×M16—611no—
185#391—21no—
186#388—22no—
187#386—21no—
188C4×C22⋊C4—22no—
189#376C2×D6112yes—
190#329—1222no—
191#327—1222no—
192#327—1222no—
193C61#944111yescharacteristic, complemented, Sylow
194C2×#41—1221no—
195C22×Q8C22⋊D6111yescharacteristic
196C22×D4C22⋊D6111yescharacteristic
197C23×C4—21no—
198#137—2442no—
199#137—2442no—
200AGL(1,ℤ/8)—2442no—
201AGL(1,ℤ/8)—2442no—
202C2×Q16—1222no—
203C2×SD16—1222no—
204C2×SD16—1222no—
205C2×D8—1222no—
206C2×M16—1221no—
207C4⋊Q8—611no—
208#127—611no—
209#124—22no—
210#124—1221no—
211#124C4×D6112yes—
212#124D24412yes—
213#121—1222no—
214C4×D4—1222no—
215C2×C4⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42C22×D6111yescharacteristic
222C4⋊C8—1222no—
223C22⋊C8—1222no—
224#97—612no—
225#97—1222no—
226C4×C8—1221no—
227#95—22no—
228#95—22no—
229#95C22⋊D6112yes—
230#95C22⋊D6112yes—
231C24—21no—
232#41—2441no—
233#41—2441no—
234#41—2441no—
235#41—2441no—
236C2×Q8—21no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—1221no—
240C2×Q8#1175819812yes—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—1221no—
245C2×D4—1222no—
246C2×D4—2441no—
247C2×D4#1175819812yes—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—81no—
253C22×C4—2441no—
254C22×C4C2×C22⋊D6111yescharacteristic
255C22×C4C2×C22⋊D6112yes—
256Q16—2444no—
257SD16—2444no—
258SD16—2444no—
259D8—2444no—
260M16—1222no—
261M16—1222no—
262C2×C8—1222no—
263C2×C8—2442no—
264C4⋊C4—81no—
265C4⋊C4—1222no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—44no—
269C22⋊C4—81no—
270C22⋊C4—2442no—
271C42—44no—
272C42—81no—
273C42C2×C4×D6111yescharacteristic
274C42C2×D24411yescharacteristic
275C42#1175822212yes—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—2441no—
282C23#1451974611yescharacteristic
283Q8—42no—
284Q8—81no—
285Q8—2441no—
286Q8—2441no—
287D4—42no—
288D4—81no—
289D4—4882no—
290D4—4881no—
291D4—4881no—
292D4—2441no—
293D4—2441no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—84no—
307C2×C4—4881no—
308C2×C4—4881no—
309C2×C4—2441no—
310C2×C4C2×#1175819811yescharacteristic
311C2×C4C4×C22⋊D6112yes—
312C2×C4#1451975112yes—
313C2×C4#1451976011yescharacteristic
314C8—1224no—
315C8—4882no—
316C22—22no—
317C22—42no—
318C22—44no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—4881no—
323C22—4882no—
324C22#1683964111yescharacteristic
325C22#1683993011yescharacteristic
326C22#1683993011yescharacteristic
327C4—21no—
328C4—21no—
329C4—22no—
330C4—22no—
331C4—44no—
332C4—42no—
333C4—82no—
334C4—4881no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2—9761no—
339C2#2144621711yescharacteristic
340C2#2144632012yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2372C2
socleC22×C61#386
prime radical#21456283C2
cosocleC2×C42×C61C23
hypercenter#2372D61
hypocenterC61#9441
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2372

Complements

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

NH ≅ G/Ntype
2. #21444443C2semidirect
3. #21444326C2semidirect
4. #21443712C2semidirect
5. C61×#2372C2semidirect
9. #16840157C4semidirect
10. #16840152C4semidirect
33. #14519777C2×C4semidirect
34. #14519774C2×C4semidirect
41. #14519688C2×C4semidirect
166. #2372D61semidirect
193. C61#9441semidirect

Maximal subgroups

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

subgroupindex
#214562832
#214444432
#214443262
#214437122
C61×#23722
#944161