Finite Group Databasebeta

← #29592814

Subgroups — #29592814

Order 15616.

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

normal subgroups only

This group has 484 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#21454072C211yesmaximal, characteristic
2#21444776C212yesmaximal, complemented
3#21443839C212yesmaximal, complemented
4#21443747C211yesmaximal, characteristic
5C61×#2029C211yesmaximal, characteristic
6#16839917—22no—
7#16839674—22no—
8#16839674—21no—
9#16839674—22no—
10#16839674C2212yes—
11#16839618—21no—
12#16839499—22no—
13#16839486C2212yes—
14C61×#543C2211yescharacteristic
15C61×#379—21no—
16C61×#379C2211yescharacteristic
17C61×#342—21no—
18C61×#342C2211yescharacteristic
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×#11758202—42no—
22C2×C22⋊Dic61—21no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—21no—
25C2×C22⋊Dic61—22no—
26#14519700—42no—
27C22⋊#92418—22no—
28#14519684—22no—
29#14519684—21no—
30#14519684—22no—
31#14519684D412yes—
32C25×C61D411yescharacteristic
33C22⋊Q8×C61—22no—
34C22⋊Q8×C61—42no—
35C2×C22⋊C4×C61—21no—
36C2×C22⋊C4×C61—21no—
37C2×C22⋊C4×C61—22no—
38C2×C22⋊C4×C61—42no—
39C2×C22⋊C4×C61D411yescharacteristic
40C2×C22⋊C4×C61C2311yescharacteristic
41C24×C61—21no—
42C24×C61—21no—
43C24×C61—41no—
44C24×C61—41no—
45C24×C61—82no—
46C24×C61Q1612yescomplemented
47C24×C61C2×D411yescharacteristic
48C2×Q8×C61—22no—
49C22×C4×C61—21no—
50C22×C4×C61—22no—
51C22×C4×C61—41no—
52C22×C4×C61C2×D412yes—
53C22×Dic61—22no—
54C22×Dic61—22no—
55C2×Dic122—42no—
56C2×Dic122—82no—
57C2×Dic122—82no—
58C2×Dic122—81no—
59C4⋊C4×C61—22no—
60C4⋊C4×C61—42no—
61C22⋊C4×C61—41no—
62C22⋊C4×C61—41no—
63C22⋊C4×C61—41no—
64C22⋊C4×C61—41no—
65C22⋊C4×C61—41no—
66C22⋊C4×C61—41no—
67C22⋊C4×C61—82no—
68C22⋊C4×C61—84no—
69C22⋊Dic61—82no—
70C22⋊Dic61—82no—
71C22⋊Dic61—81no—
72C22⋊Dic61—81no—
73#11758202—84no—
74C2×#92418—42no—
75C23×C61—21no—
76C23×C61—21no—
77C23×C61—22no—
78C23×C61—42no—
79C23×C61—42no—
80C23×C61—42no—
81C23×C61—42no—
82C23×C61—44no—
83C23×C61—41no—
84C23×C61—41no—
85C23×C61—81no—
86C23×C61—82no—
87C23×C61—82no—
88C23×C61—84no—
89C23×C61—82no—
90C23×C61#12011yescharacteristic
91C23×C61C2×Q1611yescharacteristic
92C23×C61#13711yescharacteristic
93Q8×C61—82no—
94C2×C4×C61—22no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61—41no—
98C2×C4×C61—42no—
99C2×C4×C61—82no—
100C2×C4×C61—81no—
101C2×C4×C61—82no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61—42no—
108Dic122—44no—
109Dic122—84no—
110Dic122—82no—
111#92418—82no—
112#12217—611nomaximal, Sylow
113C22×C61—22no—
114C22×C61—21no—
115C22×C61—21no—
116C22×C61—41no—
117C22×C61—41no—
118C22×C61—42no—
119C22×C61—42no—
120C22×C61—42no—
121C22×C61—44no—
122C22×C61—42no—
123C22×C61—81no—
124C22×C61—82no—
125C22×C61—82no—
126C22×C61—84no—
127C22×C61—82no—
128C22×C61#45111yescharacteristic
129C22×C61#45511yescharacteristic
130C22×C61B(4,𝔽2)11yescharacteristic
131C4×C61—42no—
132C4×C61—81no—
133C4×C61—82no—
134C4×C61—161no—
135Dic61—82no—
136Dic61—82no—
137#2840—611no—
138#2029D6111yescharacteristic
139#1602—612no—
140#1351—612no—
141#1316—611no—
142C2×C61—21no—
143C2×C61—21no—
144C2×C61—41no—
145C2×C61—41no—
146C2×C61—82no—
147C2×C61#160211yescharacteristic
148C2×C61#219912yes—
149#543C2×D6111yescharacteristic
150#451—1222no—
151#393—612no—
152#393—1222no—
153#393—1222no—
154#393—1221no—
155#379—21no—
156#379—1221no—
157#379C2×D6111yescharacteristic
158#342—21no—
159#342C2×D6111yescharacteristic
160#328—1222no—
161#323—612no—
162C61#1221711yescharacteristic, complemented, Sylow
163C25C22⋊D6111yescharacteristic
164C22×Q8—612no—
165C22×Q8—1221no—
166C2×Q16—2442no—
167C22⋊Q8—22no—
168C22⋊Q8—42no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—1221no—
174C2×C22⋊C4—1222no—
175C2×C22⋊C4—1221no—
176C2×C22⋊C4—1222no—
177C2×C22⋊C4C22⋊D6111yescharacteristic
178C2×C22⋊C4C22×D6111yescharacteristic
179#103—2442no—
180C22⋊C8—1222no—
181#95—612no—
182#95—1222no—
183#95—1221no—
184#95—1222no—
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24—82no—
190C24#1175820212yescomplemented
191C24C2×C22⋊D6111yescharacteristic
192C2×Q8—22no—
193C2×Q8—4882no—
194C2×Q8—4881no—
195C2×Q8—4882no—
196C2×Q8—2442no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—41no—
200C22×C4—1222no—
201C22×C4—1222no—
202C22×C4D4×D6112yes—
203Q16—4884no—
204C2×C8—2442no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C22⋊C4—84no—
215C22⋊C4—4882no—
216C22⋊C4—4882no—
217C22⋊C4—4881no—
218C22⋊C4—4881no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23#1451974511yescharacteristic
235C23C2×#1175820211yescharacteristic
236C23#1451980711yescharacteristic
237Q8—82no—
238Q8—4882no—
239Q8—4884no—
240Q8—2444no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—2442no—
250C2×C4—2442no—
251C2×C4—2442no—
252C2×C4—2442no—
253C2×C4—2442no—
254C2×C4—2442no—
255C8—4882no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22#1683991711yescharacteristic
272C22#1683993911yescharacteristic
273C22#1683994811yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—161no—
278C4—4882no—
279C4—4882no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2#2144477511yescharacteristic
286C2#2144781812yes—
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2029C2
socleC22×C61B(4,𝔽2)
prime radicalC25×C61D4
hypercenter#2029D61
hypocenterC61#12217
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2029

Complements

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

NH ≅ G/Ntype
2. #21444776C2semidirect
3. #21443839C2semidirect
46. C24×C61Q16semidirect
162. C61#12217semidirect
190. C24#11758202semidirect

Maximal subgroups

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

subgroupindex
#214540722
#214447762
#214438392
#214437472
C61×#20292
#1221761