Finite Group Databasebeta

← #29571372

Subgroups — #29571372

Order 15616.

1174 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×#14519756C211yesmaximal, characteristic
2#21445729C211yesmaximal, characteristic
3C2×#16839487C214yesmaximal, complemented
4C61×#1779C211yesmaximal, characteristic
5C23×Dic122C411yescharacteristic
6C2×#14519756—28no—
7C2×#14519756—22no—
8C2×#14519756—22no—
9C22×C4⋊Dic61C411yescharacteristic
10C22×#11758196C2212yes—
11C22×C22⋊Dic61C412yes—
12C2×C22⋊#92418—22no—
13C2×C22⋊#92418C2212yes—
14#16839487—28no—
15C24×C4×C61C2211yescharacteristic
16C2×C22⋊C8×C61—22no—
17C2×C22⋊C8×C61C2212yes—
18C22×Dic122—22no—
19C22×Dic122—24no—
20C22×Dic122—44no—
21C23×Dic61C2×C412yes—
22C23×Dic61C2×C412yes—
23C22×#92418—22no—
24#14519756—216no—
25#14519756—216no—
26C2×C4⋊Dic61—22no—
27C2×C4⋊Dic61—24no—
28C2×#11758196—22no—
29C2×#11758196—24no—
30C2×#11758196—22no—
31C2×#11758196D418yes—
32C2×C22⋊Dic61—24no—
33C2×C22⋊Dic61—24no—
34C2×C22⋊Dic61—24no—
35C22⋊#92418—24no—
36C22⋊#92418—44no—
37C25×C61C2×C411yescharacteristic
38C23×C4×C61—24no—
39C23×C4×C61—44no—
40C23×C4×C61C2×C411yescharacteristic
41C23×C4×C61D412yes—
42C23×C4×C61D412yes—
43C23×C4×C61C2311yescharacteristic
44C22×C8×C61—22no—
45C22⋊C8×C61—24no—
46C22⋊C8×C61—44no—
47C24×C61—24no—
48C24×C61—22no—
49C24×C61—22no—
50C24×C61—22no—
51C24×C61—42no—
52C24×C61C22⋊C412yes—
53C24×C61C22×C411yescharacteristic
54C22×C4×C61—22no—
55C22×C4×C61—24no—
56C22×C4×C61—24no—
57C22×C4×C61—24no—
58C22×C4×C61—24no—
59C22×C4×C61—24no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61—416no—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D414yes—
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—24no—
73C22×Dic61—24no—
74C22×Dic61—44no—
75C22×Dic61C22⋊C414yes—
76C22×Dic61C22⋊C414yes—
77C2×Dic122—28no—
78C2×Dic122—416no—
79C2×Dic122—44no—
80C2×Dic122—44no—
81C2×C8×C61—44no—
82C2×C8×C61—42no—
83C22⋊Dic61—216no—
84C4⋊Dic61—28no—
85#11758196—28no—
86#11758196—28no—
87C2×#92418—44no—
88C2×#92418—42no—
89C23×C61—22no—
90C23×C61—22no—
91C23×C61—28no—
92C23×C61—22no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61—48no—
99C23×C61—48no—
100C23×C61—44no—
101C23×C61—44no—
102C23×C61#10114yes—
103C23×C61C2×C22⋊C412yes—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61#13112yes—
106C23×C61#13112yes—
107C2×C4×C61—28no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—44no—
113C2×C4×C61—48no—
114C2×C4×C61—48no—
115C2×C4×C61#9914yes—
116C2×C4×C61#12012yes—
117C2×C4×C61#12012yes—
118C2×Dic61—24no—
119C2×Dic61—24no—
120C2×Dic61—28no—
121C2×Dic61—44no—
122C2×Dic61—44no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—44no—
126C2×Dic61—44no—
127Dic122—416no—
128C8×C61—82no—
129#92418—82no—
130#8308—611nomaximal, Sylow
131C22×C61—24no—
132C22×C61—24no—
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—24no—
136C22×C61—24no—
137C22×C61—24no—
138C22×C61—416no—
139C22×C61—42no—
140C22×C61—42no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61#37911yescharacteristic
144C22×C61#40814yes—
145C22×C61#40811yescharacteristic
146C22×C61#40811yescharacteristic
147C22×C61C2×#9912yes—
148C22×C61C2×#10112yes—
149C4×C61—22no—
150C4×C61—22no—
151C4×C61—24no—
152Dic61—44no—
153Dic61—44no—
154C22×C22⋊Q8—611no—
155#1779—611no—
156#1779D6111yescharacteristic
157C2×#324—614no—
158C2×C61—24no—
159C2×C61—44no—
160C2×C61#147212yes—
161C2×C61#147212yes—
162C2×C61#178011yescharacteristic
163C2×C61#178111yescharacteristic
164C2×C61#178411yescharacteristic
165C23×Q8—611no—
166C24×C4C2×D6111yescharacteristic
167C2×C22⋊Q8—1228no—
168C2×C22⋊Q8—1222no—
169C2×C22⋊Q8—1222no—
170C22×C4⋊C4—612no—
171C22×C4⋊C4—611no—
172C22×C22⋊C4—612no—
173C2×C22⋊C8—22no—
174C2×C22⋊C8—612no—
175C2×C22⋊C8—1222no—
176C2×C22⋊C8C2×D6112yes—
177#324—1228no—
178C61#830811yescharacteristic, complemented, Sylow
179C25C4×D6111yescharacteristic
180C22×Q8—1224no—
181C22×Q8—1222no—
182C22×Q8—2444no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—612no—
186C23×C4—612no—
187C23×C4C4×D6111yescharacteristic
188C23×C4D24412yes—
189C23×C4C22⋊D6112yes—
190C23×C4C22×D6111yescharacteristic
191C22×C8—22no—
192C22×C8—1222no—
193C22⋊Q8—12216no—
194C22⋊Q8—12216no—
195C2×C4⋊C4—618no—
196C2×C4⋊C4—1224no—
197C2×C4⋊C4—1222no—
198C2×C4⋊C4—1222no—
199C2×C4⋊C4—1224no—
200C2×C4⋊C4—1222no—
201C2×C22⋊C4—1224no—
202C2×C22⋊C4—1224no—
203C2×C22⋊C4—1224no—
204C22⋊C8—24no—
205C22⋊C8—44no—
206C22⋊C8—1224no—
207C22⋊C8—2444no—
208C24—24no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—42no—
213C24#1175819812yes—
214C24C2×C4×D6111yescharacteristic
215C2×Q8—1228no—
216C2×Q8—24416no—
217C2×Q8—2444no—
218C2×Q8—2444no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—416no—
229C22×C4—614no—
230C22×C4—614no—
231C22×C4—1222no—
232C22×C4—1222no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4—1224no—
236C22×C4—1224no—
237C22×C4—2444no—
238C22×C4#1175819812yes—
239C22×C4C2×D24411yescharacteristic
240C22×C4D4×D6114yes—
241C22×C4C2×C22⋊D6111yescharacteristic
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—2444no—
245C2×C8—2442no—
246C4⋊C4—1228no—
247C4⋊C4—1228no—
248C4⋊C4—1228no—
249C22⋊C4—12216no—
250C23—22no—
251C23—22no—
252C23—28no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—48no—
260C23—48no—
261C23—44no—
262C23—44no—
263C23#1451969514yes—
264C23C22⋊C4×D6112yes—
265C23C2×#1175819811yescharacteristic
266C23#1451978312yes—
267C23#1451978512yes—
268Q8—24416no—
269C2×C4—28no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—44no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—1228no—
278C2×C4—1224no—
279C2×C4—1224no—
280C2×C4—2444no—
281C2×C4—2444no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2444no—
285C2×C4—2444no—
286C2×C4#1451969114yes—
287C2×C4#1451974512yes—
288C2×C4#1451974712yes—
289C8—82no—
290C8—4882no—
291C22—24no—
292C22—24no—
293C22—22no—
294C22—22no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—416no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22#1683961711yescharacteristic
304C22#1683972714yes—
305C22#1683972811yescharacteristic
306C22#1683973011yescharacteristic
307C22C2×#1451969112yes—
308C22C2×#1451969512yes—
309C4—22no—
310C4—22no—
311C4—24no—
312C4—2444no—
313C4—2444no—
314C2—24no—
315C2—44no—
316C2#2144418212yes—
317C2#2144418412yes—
318C2#2144573011yescharacteristic
319C2#2144573311yescharacteristic
320C2#2144574211yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1779C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#1779D61
hypocenterC61#8308
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. C2×#16839487C2semidirect
178. C61#8308semidirect

Maximal subgroups

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

subgroupindex
C22×#145197562
#214457292
C2×#168394872
C61×#17792
#830861