Finite Group Databasebeta

← #29592183

Subgroups — #29592183

Order 15616.

422 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 59 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#21447769C212yesmaximal, complemented
2#21446886C211yesmaximal, characteristic
3#21443807C212yesmaximal, complemented
4#21443755C211yesmaximal, characteristic, complemented
5C61×#3195C211yesmaximal, characteristic, complemented
6#16840562C2211yescharacteristic
7#16840273—21no—
8#16840203—22no—
9#16839778—21no—
10#16839571—22no—
11#16839544C2212yes—
12#16839519C2211yescharacteristic
13C2×C22⋊Q8×C61—21no—
14C61×#483—22no—
15C61×#470—21no—
16C61×#470C2212yes—
17C61×#451—22no—
18C61×#407C2211yescharacteristic
19C2×#11758222—21no—
20#14519810—41no—
21#14519805—41no—
22#14519799—22no—
23#14519760—41no—
24#14519756—41no—
25#14519751—41no—
26#14519746—41no—
27C4×C22⋊D61—42no—
28#14519736—21no—
29#14519723—42no—
30#14519720—42no—
31#14519717—22no—
32#14519703—42no—
33C22×Q8×C61—21no—
34C23×C4×C61D411yescharacteristic
35C2×Q16×C61—21no—
36C2×Q16×C61D412yescomplemented
37C2×M16×C61D411yescharacteristic
38C2×M16×C61C2311yescharacteristic
39C22⋊Q8×C61—22no—
40C22⋊Q8×C61—22no—
41C22⋊Q8×C61—41no—
42C22⋊Q8×C61—41no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—41no—
45C2×C22⋊C4×C61—41no—
46#106×C61—21no—
47#106×C61D412yescomplemented
48#103×C61—22no—
49#103×C61—22no—
50C22⋊C8×C61—22no—
51C24×C61D811yescharacteristic
52C2×Q8×C61—22no—
53C2×Q8×C61—41no—
54C2×Q8×C61—81no—
55C2×Q8×C61—81no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61D811yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C2×C22⋊D61—42no—
65C2×C22⋊D61—41no—
66#11758222—41no—
67#11758222—41no—
68#11758222—41no—
69#11758222—41no—
70C2×D244—21no—
71C2×C4×D61—41no—
72C2×Dic122—21no—
73Q16×C61—42no—
74Q16×C61—81no—
75M16×C61—41no—
76M16×C61—41no—
77C2×C8×C61—21no—
78C2×C8×C61C2×D412yes—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—42no—
82C4⋊C4×C61—42no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—42no—
85C22⋊Dic61—42no—
86C22⋊Dic61—41no—
87#11758198—42no—
88C4⋊Dic61—41no—
89#11758196—42no—
90C4×Dic61—22no—
91Dic244—22no—
92D488—22no—
93#11758190—42no—
94#11758185—42no—
95C23×C61—21no—
96C23×C61—41no—
97C23×C61—42no—
98C23×C61C2×D811yescharacteristic
99C22×D61—41no—
100Q8×C61—42no—
101Q8×C61—42no—
102Q8×C61—82no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—81no—
112C2×C4×C61—82no—
113C2×C4×C61—82no—
114C2×C4×C61#12011yescharacteristic
115C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
116C22⋊D61—81no—
117C22⋊D61—84no—
118C22⋊D61—81no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121D244—41no—
122D244—41no—
123C4×D61—81no—
124C4×D61—81no—
125Dic122—41no—
126Dic122—41no—
127C8×C61—22no—
128C8×C61—41no—
129#12115—611nomaximal, Sylow
130C22×C61—22no—
131C22×C61—21no—
132C22×C61—41no—
133C22×C61—41no—
134C22×C61—42no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61#44711yescharacteristic
138C2×D61—82no—
139C2×D61—81no—
140C4×C61—21no—
141C4×C61—41no—
142C4×C61—82no—
143C4×C61—82no—
144C4×C61#45311yescharacteristic
145C4×C61B(4,𝔽2)11yescharacteristic
146Dic61—82no—
147Dic61—81no—
148#3195D6111yescharacteristic, complemented
149#2191—612no—
150#2010—611no—
151#1340—612no—
152#1320—611no—
153C2×C61—21no—
154C2×C61—41no—
155C2×C61—42no—
156C2×C61#159511yescharacteristic
157D61—161no—
158#575—611no—
159#525—1221no—
160C2×C22⋊Q8—21no—
161#510—1222no—
162#483—22no—
163#470—21no—
164#470C2×D6112yes—
165#451—22no—
166#421—1221no—
167#407C2×D6111yescharacteristic
168#360—1222no—
169#349—612no—
170H(3,ℤ/4)—611no—
171C61#1211511yescharacteristic, complemented, Sylow
172C2×#41—1221no—
173C22×Q8—21no—
174C23×C4C22⋊D6111yescharacteristic
175#137—2441no—
176AGL(1,ℤ/8)—2441no—
177#135—1222no—
178C2×Q16—21no—
179C2×Q16C22⋊D6112yescomplemented
180C2×M16C22⋊D6111yescharacteristic
181C2×M16C22×D6111yescharacteristic
182#123—2441no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—2441no—
188#121—2441no—
189#120—2441no—
190C4×D4—2442no—
191#117—1221no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—41no—
194C2×C22⋊C4—41no—
195Q32—2442no—
196SD32—2442no—
197M32—1222no—
198#106—21no—
199#106C22⋊D6112yescomplemented
200#104—2442no—
201#103—22no—
202#103—22no—
203C22⋊C8—22no—
204C24#1175819911yescharacteristic
205#41—2441no—
206#41—2441no—
207#41—2441no—
208#41—2441no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—81no—
212C2×Q8—81no—
213C2×Q8—1221no—
214C2×D4—1221no—
215C2×D4—2442no—
216C2×D4—2441no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—2441no—
224C22×C4#1175819911yescharacteristic
225C22×C4C2×C22⋊D6111yescharacteristic
226Q16—42no—
227Q16—81no—
228Q16—1222no—
229SD16—2442no—
230D8—1222no—
231M16—41no—
232M16—41no—
233C2×C8—21no—
234C2×C8C2×C22⋊D6112yes—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—2441no—
240C4⋊C4—2442no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C22⋊C4—2441no—
246C42—1222no—
247C16—2442no—
248C23—21no—
249C23—41no—
250C23—42no—
251C23—2441no—
252C23C2×#1175819911yescharacteristic
253Q8—42no—
254Q8—42no—
255Q8—82no—
256Q8—2441no—
257Q8—2441no—
258D4—4881no—
259D4—4884no—
260D4—4881no—
261D4—2441no—
262D4—2441no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—4881no—
275C2×C4—4881no—
276C2×C4—2442no—
277C2×C4—2441no—
278C2×C4#1451974611yescharacteristic
279C2×C4#1451980311yescharacteristic
280C8—22no—
281C8—41no—
282C22—22no—
283C22—21no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—81no—
289C22—4882no—
290C22—4881no—
291C22#1683989611yescharacteristic
292C4—21no—
293C4—41no—
294C4—82no—
295C4—82no—
296C4—4882no—
297C4—4881no—
298C4#1683993011yescharacteristic
299C4#1683994911yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—9761no—
304C2#2144473711yescharacteristic
subgroupG/H
centerC2#21444737
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#3195C2
socleC2×C61#1595
prime radical#21446886C2
hypercenter#3195D61
hypocenterC61#12115
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. #21447769C2semidirect
3. #21443807C2semidirect
4. #21443755C2semidirect
5. C61×#3195C2semidirect
36. C2×Q16×C61D4semidirect
47. #106×C61D4semidirect
148. #3195D61semidirect
171. C61#12115semidirect
179. C2×Q16C22⋊D61semidirect
199. #106C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214477692
#214468862
#214438072
#214437552
C61×#31952
#1211561