Finite Group Databasebeta

← #30595652

Subgroups — #30595652

Order 15616.

624 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 122 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#21461736C212yesmaximal, complemented
2#21460636C212yesmaximal, complemented
3#21460635C212yesmaximal, complemented
4C4×#14519773C211yesmaximal, characteristic
5C2×C4⋊#92420—22no—
6#16840628—22no—
7#16840628C2214yes—
8C2×#14519773C411yescharacteristic
9C2×C4×D244C411yescharacteristic
10C2×C4×C22⋊D61—22no—
11#16839665C2211yescharacteristic
12#16839663C2211yescharacteristic
13#16839662—22no—
14#16839631C411yescharacteristic
15C42×Dic61C2211yescharacteristic
16C2×C4×D4×C61C411yescharacteristic
17C22×#92420—24no—
18C4⋊#92420—44no—
19C22×D244C2×C411yescharacteristic
20C22×C22⋊D61—22no—
21C22×C4×D61—21no—
22#14519773—22no—
23#14519773—22no—
24C4×D244—42no—
25C4×C22⋊D61—44no—
26C4×C22⋊D61—44no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—22no—
29C2×C22⋊Dic61—22no—
30C2×#11758198—21no—
31C2×#11758198—22no—
32C2×C4×Dic61C2×C411yescharacteristic
33C2×C4×Dic61D412yes—
34C2×C4×Dic61D412yes—
35C2×C4×Dic61C2311yescharacteristic
36C22×D4×C61C2×C411yescharacteristic
37C23×C4×C61—21no—
38C4×D4×C61—42no—
39C2×C4⋊C4×C61C2×C411yescharacteristic
40C2×C22⋊C4×C61—21no—
41C2×C42×C61C2×C411yescharacteristic
42C24×C61—21no—
43C23×D61—21no—
44C2×D4×C61—21no—
45C2×D4×C61—41no—
46C2×D4×C61—41no—
47C2×D4×C61M1612yes—
48C22×C4×C61—21no—
49C22×C4×C61—42no—
50C22×C4×C61—44no—
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22⋊C411yescharacteristic
53C22×C4×C61C22×C411yescharacteristic
54C2×C22⋊D61—28no—
55C2×C22⋊D61—44no—
56C2×C22⋊D61—44no—
57C22×Dic61—21no—
58C22×Dic61C22⋊C412yes—
59C22×Dic61C2×D412yes—
60C2×D244—21no—
61C2×D244—41no—
62C2×D244—41no—
63C2×D244M1612yes—
64C2×C4×D61—42no—
65C2×C4×D61—44no—
66C2×#92420—44no—
67C2×#92420—44no—
68C2×#92420—44no—
69C4⋊C4×C61—41no—
70C22⋊C4×C61—42no—
71C42×C61—22no—
72C22⋊Dic61—42no—
73C22⋊Dic61—42no—
74#11758198—42no—
75#11758198—42no—
76#11758198—42no—
77C4⋊Dic61—41no—
78#11758196—44no—
79C4×Dic61—21no—
80C4×Dic61—21no—
81C4×Dic61—42no—
82C4×Dic61—41no—
83C4×Dic61D814yes—
84C4×Dic61SD1614yes—
85C23×C61—22no—
86C23×C61—41no—
87C23×C61—41no—
88C23×C61—42no—
89C23×C61—42no—
90C23×C61C2×C22⋊C411yescharacteristic
91C22×D61—22no—
92C22×D61—41no—
93C22×D61—41no—
94C22×D61—42no—
95C22×D61—42no—
96D4×C61—42no—
97D4×C61—42no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61—84no—
105C2×C4×C61#10012yes—
106C2×C4×C61#10214yes—
107C2×C4×C61C2×M1611yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C22⋊D61—416no—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—22no—
113C2×Dic61—22no—
114C2×Dic61—41no—
115C2×Dic61—42no—
116C2×Dic61#10214yes—
117C2×Dic61C2×D812yes—
118C2×Dic61C2×SD1612yes—
119D244—42no—
120D244—42no—
121C4×D61—84no—
122#92420—84no—
123#7651—611nomaximal, Sylow
124C22×C61—42no—
125C22×C61—44no—
126C22×C61—81no—
127C22×C61—81no—
128C22×C61—82no—
129C22×C61#40711yescharacteristic
130C22×C61#41011yescharacteristic
131C22×C61C2×#10011yescharacteristic
132C22×C61C2×#10212yes—
133C22×C61#42112yes—
134C2×D61—42no—
135C2×D61—44no—
136C2×D61—81no—
137C2×D61—81no—
138C2×D61—82no—
139C4×C61—21no—
140C4×C61—21no—
141C4×C61—41no—
142C4×C61—81no—
143C4×C61#33114yes—
144Dic61—24no—
145Dic61—22no—
146Dic61—22no—
147Dic61—81no—
148C4×#127—611no—
149#1831—612no—
150#1270—612no—
151#1270—612no—
152C2×C61—82no—
153C2×C61#146611yescharacteristic
154C2×C61#148612yes—
155C2×C61#151712yes—
156C2×C61#153611yescharacteristic
157C2×C61C2×#33111yescharacteristic
158D61—82no—
159C2×#127—611no—
160C2×C4×D4—611no—
161C2×C4×D4—1222no—
162C2×C4×D4#692311yescharacteristic
163C2×C4⋊C8—1222no—
164#390—611no—
165#390—611no—
166#390—1222no—
167#384—611no—
168C43—611no—
169#336—614no—
170#336—1222no—
171C61#765111yescharacteristic, complemented, Sylow
172C22×D4—611no—
173C22×D4—1222no—
174C22×D4C2×#692311yescharacteristic
175C23×C4—21no—
176C23×C4—1221no—
177C22×C8—1224no—
178#127—1222no—
179#127—1222no—
180C4×D4—42no—
181C4×D4—2442no—
182C4×D4—2444no—
183C4×D4—2444no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—1222no—
186C2×C4⋊C4C2×#692311yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1221no—
191C2×C42—612no—
192C2×C42—611no—
193C2×C42—612no—
194C2×C42—611no—
195C2×C42C2×#692311yescharacteristic
196C4⋊C8—2444no—
197C24—21no—
198C24—1221no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—612no—
203C2×D4—1228no—
204C2×D4—1221no—
205C2×D4—2441no—
206C2×D4—2441no—
207C2×D4—2444no—
208C2×D4—2444no—
209C2×D4#1175821312yes—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—612no—
214C22×C4—612no—
215C22×C4—1221no—
216C22×C4—2442no—
217C22×C4—2444no—
218C22×C4C22⋊#692311yescharacteristic
219C22×C4C22⋊#692311yescharacteristic
220C22×C4C22×#692311yescharacteristic
221C2×C8—2444no—
222C2×C8—2444no—
223C2×C8—2444no—
224C4⋊C4—41no—
225C4⋊C4—2441no—
226C4⋊C4—2444no—
227C22⋊C4—42no—
228C22⋊C4—2442no—
229C22⋊C4—2442no—
230C22⋊C4—2442no—
231C22⋊C4—2442no—
232C22⋊C4—2442no—
233C42—22no—
234C42—614no—
235C42—614no—
236C42—1221no—
237C42—1221no—
238C42—2441no—
239C42—2442no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—1222no—
246C23—2441no—
247C23—2441no—
248C23—2442no—
249C23—2442no—
250C23C2×C22⋊#692311yescharacteristic
251D4—42no—
252D4—42no—
253D4—2442no—
254D4—2442no—
255D4—24416no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—614no—
264C2×C4—612no—
265C2×C4—612no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—1222no—
269C2×C4—1222no—
270C2×C4—4884no—
271C2×C4—2441no—
272C2×C4—2442no—
273C2×C4#1451983712yes—
274C2×C4#1451983914yes—
275C2×C4D61⋊M1611yescharacteristic
276C2×C4C2×#1175821311yescharacteristic
277C8—4884no—
278C22—42no—
279C22—44no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22—4882no—
284C22—4881no—
285C22—4881no—
286C22—2442no—
287C22—2444no—
288C22#1684073311yescharacteristic
289C22#1684074111yescharacteristic
290C22C2×#1451983711yescharacteristic
291C22C2×#1451983912yes—
292C22#1684075912yes—
293C4—21no—
294C4—21no—
295C4—41no—
296C4—81no—
297C4—1222no—
298C4—1222no—
299C4—1224no—
300C4—4881no—
301C4#1684062014yes—
302C2—82no—
303C2—4882no—
304C2#2146107811yescharacteristic
305C2#2146114512yes—
306C2#2146124912yes—
307C2#2146130711yescharacteristic
308C2C2×#1684062011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519773C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×D4#6923
hypocenterC61#7651
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461736C2semidirect
2. #21460636C2semidirect
3. #21460635C2semidirect
171. C61#7651semidirect

Maximal subgroups

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

subgroupindex
#214617362
#214606362
#214606352
C4×#145197732
#765161