Finite Group Databasebeta

← #29580967

Subgroups — #29580967

Order 15616.

520 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 85 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#21455914C212yesmaximal
2#21446616C211yesmaximal, characteristic
3#21444434C212yesmaximal, complemented
4#21443689C211yesmaximal, characteristic, complemented
5C61×#3249C211yesmaximal, characteristic, complemented
6C2×C4×D244C2211yescharacteristic
7#16840095—22no—
8#16840039—22no—
9#16839964—22no—
10#16839912—22no—
11#16839860—24no—
12#16839819C2212yes—
13C2×#14519689—22no—
14C2×#14519688C2211yescharacteristic
15#16839504—24no—
16C61×#533C2212yes—
17C61×#477—22no—
18C61×#467—22no—
19C61×#462—22no—
20C2×C4⋊C8×C61C2211yescharacteristic
21C22×D244D411yescharacteristic
22C22×C4×D61—21no—
23C2×#11758201—24no—
24C22×#92418—21no—
25C4×D244—22no—
26C4×D244—22no—
27C2×C4⋊Dic61D411yescharacteristic
28C2×#11758198—21no—
29#14519710—24no—
30C4⋊#92418—22no—
31#14519701—24no—
32#14519697—24no—
33C22⋊#92418—22no—
34#14519689—44no—
35#14519688—22no—
36#14519688D414yes—
37C2×Q16×C61—22no—
38C22×C8×C61D412yes—
39C4⋊Q8×C61D412yes—
40C4⋊Q8×C61D412yes—
41#125×C61—22no—
42C22⋊Q8×C61—22no—
43C22⋊Q8×C61—22no—
44C4×Q8×C61—22no—
45C4×Q8×C61—22no—
46#117×C61—22no—
47C2×C42×C61C2311yescharacteristic
48#107×C61—22no—
49C4⋊C8×C61—22no—
50#103×C61—24no—
51C23×D61—21no—
52C2×Q8×C61—22no—
53C2×Q8×C61—22no—
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61#4111yescharacteristic
57C22×Dic61—21no—
58C2×D244—22no—
59C2×D244—42no—
60C2×C4×D61—42no—
61C2×C4×D61—42no—
62C2×C4×D61—41no—
63C2×C4×D61—41no—
64Q16×C61—82no—
65C2×C8×C61—22no—
66C2×C8×C61—42no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—22no—
69C4⋊C4×C61—22no—
70C4⋊C4×C61—22no—
71C4⋊C4×C61—42no—
72C4⋊C4×C61—42no—
73C22⋊C4×C61—42no—
74C42×C61—42no—
75C42×C61C2×D412yes—
76C42×C61C2×D412yes—
77#11758201—48no—
78#11758198—42no—
79C4⋊Dic61—22no—
80C2×#92418—22no—
81C2×#92418—24no—
82C2×#92418—41no—
83C2×#92418—41no—
84C23×C61#12111yescharacteristic
85C22×D61—22no—
86C22×D61—42no—
87C22×D61—41no—
88C22×D61—41no—
89C22×D61—42no—
90Q8×C61—44no—
91Q8×C61—42no—
92Q8×C61—42no—
93C2×C4×C61—22no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—41no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61#12012yes—
100C2×C4×C61#12011yescharacteristic
101C2×C4×C61#12112yes—
102C2×C4×C61#12711yescharacteristic
103C2×Dic61—42no—
104C2×Dic61—41no—
105D244—24no—
106D244—42no—
107C4×D61—82no—
108C4×D61—81no—
109C4×D61—81no—
110C8×C61—42no—
111#92418—42no—
112#92418—44no—
113#10030—611nomaximal, Sylow
114C22×C61—21no—
115C22×C61—21no—
116C22×C61#39211yescharacteristic
117C22×C61#47312yes—
118C2×D61—44no—
119C2×D61—42no—
120C2×D61—82no—
121C2×D61—81no—
122C2×D61—81no—
123C4×C61—21no—
124C4×C61—22no—
125C4×C61—22no—
126C4×C61—82no—
127C4×C61—82no—
128C4×C61#45311yescharacteristic
129C4×C61#45411yescharacteristic
130Dic61—81no—
131#3249D6111yescharacteristic, complemented
132#3110—612no—
133#1955—611no—
134#1531—612no—
135#1290—611no—
136C2×C61—22no—
137C2×C61#200611yescharacteristic
138C2×C61#201011yescharacteristic
139C2×C61#202011yescharacteristic
140D61—82no—
141#533C2×D6112yes—
142C2×C4×D4—611no—
143#484—1222no—
144#477—22no—
145#474—1222no—
146#467—22no—
147#462—22no—
148#461—1222no—
149#450—1222no—
150#440—1224no—
151#432—612no—
152C2×C4⋊C8C2×D6111yescharacteristic
153C2×C22⋊C8—1222no—
154C2×#97—611no—
155#330—1224no—
156C61#1003011yescharacteristic, complemented, Sylow
157C22×D4—611no—
158C23×C4—1221no—
159C2×Q16—22no—
160C2×SD16—1224no—
161C22×C8—1221no—
162C22×C8C22⋊D6112yes—
163C4⋊Q8C22⋊D6112yes—
164C4⋊Q8C22⋊D6112yes—
165#125—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×D4—1222no—
171C4×D4—1222no—
172#117—22no—
173C2×C4⋊C4—611no—
174C2×C22⋊C4—1221no—
175C2×C42C22×D6111yescharacteristic
176#107—22no—
177#107—1224no—
178C4⋊C8—22no—
179C4⋊C8—1222no—
180#103—24no—
181#103—1224no—
182#102—1224no—
183C22⋊C8—1222no—
184C22⋊C8—2444no—
185#97—614no—
186#97—1222no—
187C24—1221no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×D4—1222no—
191C2×D4—2442no—
192C22×C4—1221no—
193C22×C4—2442no—
194C22×C4—2442no—
195C22×C4—2441no—
196C22×C4—2441no—
197C22×C4D4×D6111yescharacteristic
198C22×C4#1175822411yescharacteristic
199C22×C4C2×C22⋊D6111yescharacteristic
200Q16—82no—
201SD16—2448no—
202C2×C8—22no—
203C2×C8—42no—
204C2×C8—1222no—
205C2×C8—1224no—
206C2×C8—2441no—
207C2×C8—2441no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—1222no—
215C22⋊C4—42no—
216C22⋊C4—2442no—
217C42—42no—
218C42D4×D6112yes—
219C42C2×C22⋊D6112yes—
220C23—1222no—
221C23—2442no—
222C23—2441no—
223C23—2442no—
224C23—2441no—
225C23#1451974811yescharacteristic
226Q8—44no—
227Q8—42no—
228Q8—42no—
229D4—1224no—
230D4—2442no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—4882no—
238C2×C4—4881no—
239C2×C4—4881no—
240C2×C4—2442no—
241C2×C4—2441no—
242C2×C4#1451974512yes—
243C2×C4#1451974611yescharacteristic
244C2×C4#1451974912yes—
245C2×C4#1451977311yescharacteristic
246C8—42no—
247C8—2442no—
248C8—2444no—
249C22—21no—
250C22—21no—
251C22—4882no—
252C22—4881no—
253C22—4881no—
254C22—2444no—
255C22—2442no—
256C22#1683967011yescharacteristic
257C22#1684003412yes—
258C4—22no—
259C4—21no—
260C4—22no—
261C4—82no—
262C4—82no—
263C4—4881no—
264C4#1683993011yescharacteristic
265C4#1683993511yescharacteristic
266C2—22no—
267C2—4882no—
268C2#2144686111yescharacteristic
269C2#2144688311yescharacteristic
270C2#2144693811yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3249C2
socleC22×C61#392
prime radicalC22×D244D4
hypercenter#3249D61
hypocenterC61#10030
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
3. #21444434C2semidirect
4. #21443689C2semidirect
5. C61×#3249C2semidirect
131. #3249D61semidirect
156. C61#10030semidirect

Maximal subgroups

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

subgroupindex
#214559142
#214466162
#214444342
#214436892
C61×#32492
#1003061