Finite Group Databasebeta

← #30598193

Subgroups — #30598193

Order 15616.

1146 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 144 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#21461120C214yesmaximal, complemented
2#21461050C212yesmaximal, complemented
3C2×#16840236C211yesmaximal, characteristic
4#16840778C2214yescomplemented
5#16840610—28no—
6#16840607—22no—
7#16840605—22no—
8C23×D244C411yescharacteristic
9#16840236—28no—
10C2×C4×D244—22no—
11C2×#14519739C412yes—
12C2×#14519739C2212yes—
13C2×#14519736C2211yescharacteristic
14C2×C22⋊C4×D61—22no—
15C2×#117×C61C411yescharacteristic
16C4⋊#92420—24no—
17C22⋊#92420—24no—
18C4×#92420—24no—
19C24×D61—21no—
20C22×D244—21no—
21C22×D244—22no—
22C22×D244—24no—
23C22×D244—24no—
24C22×D244C814yescomplemented
25C22×D244C2×C412yes—
26C22×C4×D61—22no—
27C22×C4×D61—22no—
28C4×D244—216no—
29#14519739—24no—
30#14519739—24no—
31#14519739—24no—
32#14519739D418yes—
33#14519736—24no—
34C2×C4⋊Dic61D412yes—
35C2×C22⋊Dic61D412yes—
36C2×#11758198—22no—
37C2×#11758198—22no—
38C22⋊C4×D61—216no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61C2311yescharacteristic
41C23×C4×C61C2×C411yescharacteristic
42#117×C61—24no—
43C2×C4⋊C4×C61C2×C412yes—
44C2×C22⋊C4×C61—21no—
45C2×C42×C61—21no—
46C24×C61C22⋊C411yescharacteristic
47C23×D61—22no—
48C23×D61—24no—
49C23×D61—24no—
50C23×D61—24no—
51C23×D61—84no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C2×C812yes—
57C22×C4×C61M1612yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×Dic61C22⋊C412yes—
60C22×Dic61C2×D412yes—
61C2×D244—24no—
62C2×D244—216no—
63C2×D244—24no—
64C2×D244—24no—
65C2×D244—22no—
66C2×D244—22no—
67C2×D244—22no—
68C2×D244—216no—
69C2×D244C22⋊C414yes—
70C2×D244C2×C814yes—
71C2×D244M1614yes—
72C2×C4×D61—28no—
73C2×C4×D61—28no—
74C2×C4×D61—84no—
75C2×C4×D61—84no—
76C2×#92420—44no—
77C4⋊C4×C61—22no—
78C4⋊C4×C61C22⋊C414yes—
79C22⋊C4×C61—24no—
80C42×C61—24no—
81C22⋊Dic61—24no—
82#11758198—28no—
83#11758198—28no—
84C4⋊Dic61—24no—
85C4×Dic61—22no—
86C4×Dic61C2×D414yes—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—42no—
90C23×C61C22⋊C812yes—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—24no—
93C22×D61—24no—
94C22×D61—24no—
95C22×D61—216no—
96C22×D61—44no—
97C22×D61—44no—
98C22×D61—88no—
99C22×D61—88no—
100C22×D61—88no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—24no—
104C2×C4×C61—24no—
105C2×C4×C61—24no—
106C2×C4×C61—44no—
107C2×C4×C61C22⋊C812yes—
108C2×C4×C61C2×C22⋊C412yes—
109C2×C4×C61C22×C811yescharacteristic
110C2×C4×C61C2×M1612yes—
111C2×C4×C61C2×M1611yescharacteristic
112C2×Dic61—22no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61#12014yes—
116D244—216no—
117D244—24no—
118D244—24no—
119D244—24no—
120D244C22⋊C818yes—
121C4×D61—88no—
122C4×D61—88no—
123#92420—84no—
124#8322—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61#37911yescharacteristic
131C22×C61C2×C22⋊C811yescharacteristic
132C22×C61#40711yescharacteristic
133C2×D61—28no—
134C2×D61—48no—
135C2×D61—48no—
136C2×D61—44no—
137C2×D61—44no—
138C2×D61—816no—
139C4×C61—22no—
140C4×C61—44no—
141C4×C61C2×C22⋊C812yes—
142C4×C61#40712yes—
143Dic61—24no—
144Dic61—42no—
145C2×#518—611no—
146#1478—614no—
147#1458—612no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61—42no—
151C2×C61#177911yescharacteristic
152C2×C61#179512yes—
153D61—48no—
154C23×D4—611no—
155#518—1228no—
156C2×C4×D4—612no—
157C2×C4×D4—612no—
158C2×C4×D4—1222no—
159C2×#117—611no—
160C2×#117#692311yescharacteristic
161C22×C22⋊C4—1222no—
162#431—614no—
163D4⋊C8—1228no—
164#324—1222no—
165#323—1222no—
166C61#832211yescharacteristic, complemented, Sylow
167C25—1221no—
168C22×D4—614no—
169C22×D4—612no—
170C22×D4—1224no—
171C22×D4—1224no—
172C22×D4—1222no—
173C22×D4—1221no—
174C23×C4—1222no—
175C23×C4—1222no—
176C23×C4C2×#692311yescharacteristic
177C4×D4—618no—
178C4×D4—12216no—
179C4×D4—1224no—
180C4×D4—1224no—
181C4×D4—1224no—
182#117—24no—
183#117—1224no—
184C2×C4⋊C4—612no—
185C2×C4⋊C4C2×#692312yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—612no—
188C2×C22⋊C4—12216no—
189C2×C22⋊C4—1222no—
190C2×C22⋊C4—1222no—
191C2×C42—21no—
192C2×C42—611no—
193C2×C42—611no—
194C4⋊C8—1224no—
195C22⋊C8—1224no—
196C4×C8—1224no—
197C24—1224no—
198C24—1224no—
199C24—1224no—
200C24—1222no—
201C24—4884no—
202C24C22⋊#692311yescharacteristic
203C2×D4—614no—
204C2×D4—614no—
205C2×D4—614no—
206C2×D4—12216no—
207C2×D4—1224no—
208C2×D4—1224no—
209C2×D4—12216no—
210C2×D4—1224no—
211C2×D4—1222no—
212C2×D4—1222no—
213C2×D4—1222no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—612no—
218C22×C4—612no—
219C22×C4—1228no—
220C22×C4—1228no—
221C22×C4—4884no—
222C22×C4—4884no—
223C22×C4D61⋊C812yes—
224C22×C4#1175821312yes—
225C22×C4C22⋊#692311yescharacteristic
226C22×C4C22×#692311yescharacteristic
227C2×C8—2444no—
228C4⋊C4—22no—
229C4⋊C4—1224no—
230C4⋊C4C22⋊#692314yes—
231C22⋊C4—24no—
232C22⋊C4—1228no—
233C22⋊C4—1228no—
234C22⋊C4—1224no—
235C42—24no—
236C42—614no—
237C42—1222no—
238C23—21no—
239C23—21no—
240C23—42no—
241C23—12216no—
242C23—1224no—
243C23—1224no—
244C23—1224no—
245C23—4888no—
246C23—4888no—
247C23—4888no—
248C23—2444no—
249C23—2444no—
250C23#1451983412yes—
251C23C2×C22⋊#692311yescharacteristic
252D4—618no—
253D4—12216no—
254D4—1224no—
255D4—1224no—
256D4—1224no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—44no—
263C2×C4—614no—
264C2×C4—1222no—
265C2×C4—4888no—
266C2×C4—4888no—
267C2×C4—2442no—
268C2×C4—2442no—
269C2×C4#1451983412yes—
270C2×C4C2×C22⋊#692312yes—
271C2×C4C2×D61⋊C811yescharacteristic
272C2×C4D61⋊M1612yes—
273C2×C4C2×#1175821311yescharacteristic
274C8—4884no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—1228no—
281C22—48816no—
282C22—2448no—
283C22—2448no—
284C22—2444no—
285C22—2444no—
286C22#1684070311yescharacteristic
287C22C2×#1451983411yescharacteristic
288C22#1684073411yescharacteristic
289C4—22no—
290C4—44no—
291C4—1224no—
292C4—2442no—
293C4D61⋊(C22⋊C8)12yes—
294C4#1684073312yes—
295C2—21no—
296C2—21no—
297C2—42no—
298C2—2448no—
299C2#2146164211yescharacteristic
300C2#2146167012yes—
subgroupG/H
centerC22#16840703
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61#379
prime radicalC23×D244C4
cosocleC2×C4×Dic61C23
hypercenterC2×#117#6923
hypocenterC61#8322
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21461120C2semidirect
2. #21461050C2semidirect
4. #16840778C22semidirect
24. C22×D244C8semidirect
166. C61#8322semidirect

Maximal subgroups

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

subgroupindex
#214611202
#214610502
C2×#168402362
#832261