Finite Group Databasebeta

← #29592134

Subgroups — #29592134

Order 15616.

424 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456594C211yesmaximal, characteristic
2#21447755C212yesmaximal, complemented
3#21446829C211yesmaximal, characteristic
4#21443832C212yesmaximal, complemented
5C61×#1314C211yesmaximal, characteristic
6C22×Dic244C2211yescharacteristic
7C2×#14519756—21no—
8C2×#14519756—21no—
9C2×Dic488—22no—
10#16840098—22no—
11#16840003—21no—
12#16840003C2212yes—
13#16839920—22no—
14C2×#14519702—21no—
15#16839567—22no—
16#16839525C2211yescharacteristic
17C2×C22⋊C8×C61C2211yescharacteristic
18C61×C22⋊C16C2212yes—
19C22×Dic122—21no—
20C22×Dic122—21no—
21C2×Dic244—22no—
22C2×Dic244—21no—
23C2×Dic244—42no—
24C2×Dic244D412yes—
25#14519756—22no—
26#14519756—22no—
27#14519756—42no—
28#14519756—42no—
29#14519756—41no—
30#14519756—41no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×#11758196—41no—
34C2×#11758196—41no—
35C2×C22⋊Dic61—41no—
36C2×C22⋊Dic61—41no—
37Dic488—44no—
38#14519711—21no—
39#14519711D412yes—
40#14519702—22no—
41#14519702—22no—
42#14519702—42no—
43C23×C4×C61D411yescharacteristic
44C22×C8×C61D411yescharacteristic
45C22×C8×C61C2311yescharacteristic
46C2×C16×C61—22no—
47C22⋊C8×C61—22no—
48C24×C61D811yescharacteristic
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61D811yescharacteristic
54C22×C4×C61C2×D411yescharacteristic
55C22×Dic61—41no—
56C22×Dic61—41no—
57C22×Dic61—41no—
58C22×Dic61—41no—
59C2×Dic122—22no—
60C2×Dic122—22no—
61C2×Dic122—41no—
62C2×Dic122—41no—
63C2×Dic122—41no—
64C2×Dic122—41no—
65C2×Dic122—41no—
66C2×Dic122—41no—
67C2×Dic122—81no—
68C2×Dic122—81no—
69C2×C8×C61—21no—
70C2×C8×C61—41no—
71C2×C8×C61—41no—
72C2×C8×C61C2×D412yes—
73C22⋊Dic61—42no—
74C22⋊Dic61—44no—
75C22⋊Dic61—42no—
76C4⋊Dic61—22no—
77C4⋊Dic61—22no—
78C4⋊Dic61—41no—
79C4⋊Dic61—41no—
80#11758196—42no—
81#11758196—44no—
82#11758196—42no—
83Dic244—24no—
84Dic244—42no—
85Dic244—42no—
86Dic244—81no—
87C16×C61—42no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61Q3212yescomplemented
93C23×C61C2×D811yescharacteristic
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—41no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—81no—
101C2×C4×C61#12011yescharacteristic
102C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
103C2×Dic61—42no—
104C2×Dic61—42no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61—81no—
108C2×Dic61—81no—
109C2×Dic61—82no—
110C2×Dic61—82no—
111C2×Dic61—81no—
112C2×Dic61—81no—
113Dic122—42no—
114Dic122—42no—
115Dic122—42no—
116Dic122—42no—
117Dic122—81no—
118Dic122—82no—
119Dic122—81no—
120C8×C61—22no—
121C8×C61—41no—
122#12106—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—22no—
126C22×C61—42no—
127C22×C61—41no—
128C22×C61—41no—
129C22×C61—81no—
130C22×C61#44711yescharacteristic
131C22×C61C2×Q3211yescharacteristic
132C22×C61#51011yescharacteristic
133C4×C61—21no—
134C4×C61—41no—
135C4×C61#45311yescharacteristic
136C4×C61B(4,𝔽2)11yescharacteristic
137Dic61—82no—
138Dic61—82no—
139Dic61—82no—
140Dic61—82no—
141#3189—611no—
142#2188—612no—
143#2001—611no—
144#1348—612no—
145#1314D6111yescharacteristic
146C2×C61—21no—
147C2×C61—21no—
148C2×C61—42no—
149C2×C61#159511yescharacteristic
150C2×C61#218811yescharacteristic
151C2×C61#219111yescharacteristic
152C22×Q16—611no—
153C2×C22⋊Q8—1221no—
154C2×C22⋊Q8—1221no—
155C2×Q32—1222no—
156#484—1222no—
157#467—612no—
158#467—1221no—
159#451—1222no—
160C2×#103—1221no—
161C2×C22⋊C8C2×D6111yescharacteristic
162#358—1222no—
163C22⋊C16C2×D6112yes—
164#340—611no—
165C61#1210611yescharacteristic, complemented, Sylow
166C22×Q8—1221no—
167C22×Q8—1221no—
168C23×C4D24411yescharacteristic
169C2×Q16—612no—
170C2×Q16—1221no—
171C2×Q16—1222no—
172C2×Q16—2442no—
173C22×C8D24411yescharacteristic
174C22×C8C22×D6111yescharacteristic
175C22⋊Q8—1222no—
176C22⋊Q8—1222no—
177C22⋊Q8—2442no—
178C22⋊Q8—2441no—
179C22⋊Q8—2442no—
180C22⋊Q8—2441no—
181C2×C4⋊C4—1221no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—2441no—
184C2×C4⋊C4—2441no—
185C2×C22⋊C4—2441no—
186C2×C22⋊C4—2441no—
187Q32—2444no—
188C2×C16—22no—
189#107—612no—
190#107—1221no—
191#103—1222no—
192#103—1222no—
193#103—2442no—
194C22⋊C8—22no—
195C24D48811yescharacteristic
196C2×Q8—1222no—
197C2×Q8—1222no—
198C2×Q8—4881no—
199C2×Q8—4881no—
200C2×Q8—2441no—
201C2×Q8—2441no—
202C2×Q8—2441no—
203C2×Q8—2441no—
204C2×Q8—2441no—
205C2×Q8—2441no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—2441no—
211C22×C4—2441no—
212C22×C4—2441no—
213C22×C4—2441no—
214C22×C4D48811yescharacteristic
215C22×C4C2×D24411yescharacteristic
216Q16—1224no—
217Q16—4881no—
218Q16—2442no—
219Q16—2442no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8D4×D6112yes—
224C4⋊C4—1222no—
225C4⋊C4—1222no—
226C4⋊C4—2441no—
227C4⋊C4—2441no—
228C4⋊C4—2442no—
229C4⋊C4—2444no—
230C4⋊C4—2442no—
231C22⋊C4—2442no—
232C22⋊C4—2444no—
233C22⋊C4—2442no—
234C16—42no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23Dic48812yescomplemented
240C23C2×D48811yescharacteristic
241Q8—4881no—
242Q8—4882no—
243Q8—4881no—
244Q8—2442no—
245Q8—2442no—
246Q8—2442no—
247Q8—2442no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—81no—
255C2×C4—4881no—
256C2×C4—4881no—
257C2×C4—4882no—
258C2×C4—4882no—
259C2×C4—4881no—
260C2×C4—4881no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2442no—
264C2×C4—2442no—
265C2×C4#1451974711yescharacteristic
266C2×C4#1451980511yescharacteristic
267C8—22no—
268C8—41no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22#1683989811yescharacteristic
277C22C2×Dic48811yescharacteristic
278C22#1684020511yescharacteristic
279C4—21no—
280C4—41no—
281C4—4882no—
282C4—4882no—
283C4—4882no—
284C4—4882no—
285C4#1683993211yescharacteristic
286C4#1683995011yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2#2144473911yescharacteristic
291C2#2144775511yescharacteristic
292C2#2144777211yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1314C2
socleC22×C61#447
prime radicalC24×C61D8
hypercenter#1314D61
hypocenterC61#12106
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21447755C2semidirect
4. #21443832C2semidirect
92. C23×C61Q32semidirect
165. C61#12106semidirect
239. C23Dic488semidirect

Maximal subgroups

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

subgroupindex
#214565942
#214477552
#214468292
#214438322
C61×#13142
#1210661