Finite Group Databasebeta

← #30598792

Subgroups — #30598792

Order 15616.

1134 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461653C212yesmaximal, complemented
2#21461069C212yesmaximal, complemented
3#21461066C212yesmaximal
4C22×#14519755C211yesmaximal, characteristic
5C2×#14519838—24no—
6#16840732C2214yes—
7#16840608—24no—
8#16840606—24no—
9C23×Dic122C2211yescharacteristic
10C23×C4×D61C411yescharacteristic
11C2×#14519755—24no—
12C2×#14519755—24no—
13C2×#14519755—24no—
14C22×#11758196C411yescharacteristic
15C22×#11758196C2211yescharacteristic
16C22×#11758198C411yescharacteristic
17C22×#11758198C2211yescharacteristic
18C22×C4⋊C4×C61C411yescharacteristic
19C2×#11758217—24no—
20#14519838—48no—
21C22⋊#92420—24no—
22C22⋊#92420—24no—
23C24×D61C2×C411yescharacteristic
24C22×Dic122—22no—
25C22×Dic122—22no—
26C22×Dic122—44no—
27C22×Dic122D414yes—
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2311yescharacteristic
31C22×C4×D61—22no—
32C22×C4×D61—22no—
33C22×C4×D61—22no—
34C22×C4×D61—44no—
35#14519755—232no—
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×#11758196—22no—
39C2×#11758196—22no—
40C2×#11758196—22no—
41C2×#11758196D414yes—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C2×#11758198D414yes—
48C23×C4×C61C2×C411yescharacteristic
49C23×C4×C61C2×C411yescharacteristic
50C23×C4×C61C2×C411yescharacteristic
51C2×C4⋊C4×C61—22no—
52C2×C4⋊C4×C61—22no—
53C2×C4⋊C4×C61—22no—
54C24×C61C22×C411yescharacteristic
55C23×D61—21no—
56C23×D61—21no—
57C23×D61—24no—
58C23×D61—42no—
59C23×D61—42no—
60C23×D61C22⋊C412yes—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—42no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61C22⋊C412yes—
71C22×C4×C61C22⋊C412yes—
72C22×C4×C61C22⋊C412yes—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—24no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—42no—
79C22×Dic61—42no—
80C22×Dic61C22⋊C412yes—
81C22×Dic61C22⋊C412yes—
82C22×Dic61C2×D412yes—
83C22×Dic61C2×D412yes—
84C22×Dic61C2×D412yes—
85C2×C4×D61—28no—
86C2×C4×D61—48no—
87C2×C4×D61—48no—
88C2×C4×D61—44no—
89C2×C4×D61—44no—
90C2×Dic122—28no—
91C2×Dic122—28no—
92C2×Dic122—44no—
93C2×Dic122—48no—
94C2×Dic122—48no—
95#11758217—48no—
96C2×#92420—44no—
97C4⋊C4×C61—28no—
98#11758198—28no—
99#11758198—28no—
100#11758196—28no—
101#11758196—28no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—22no—
105C23×C61C2×C22⋊C411yescharacteristic
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61C2×C22⋊C411yescharacteristic
108C22×D61—24no—
109C22×D61—24no—
110C22×D61—22no—
111C22×D61—22no—
112C22×D61—42no—
113C22×D61—416no—
114C22×D61—42no—
115C22×D61—42no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61#10114yescomplemented
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—24no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×Dic61—24no—
133C2×Dic61—24no—
134C2×Dic61—22no—
135C2×Dic61—24no—
136C2×Dic61—22no—
137C2×Dic61—22no—
138C2×Dic61—24no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61#9914yes—
144C2×Dic61#12014yes—
145C4×D61—416no—
146Dic122—416no—
147#92420—84no—
148#8550—611nomaximal, Sylow
149C22×C61—28no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61#37911yescharacteristic
157C22×C61C2×#9912yes—
158C22×C61C2×#10112yes—
159C22×C61#41312yes—
160C2×D61—24no—
161C2×D61—48no—
162C2×D61—44no—
163C2×D61—48no—
164C2×D61—42no—
165C2×D61—42no—
166C2×D61—44no—
167C4×C61—42no—
168C4×C61—42no—
169C4×C61—42no—
170Dic61—24no—
171Dic61—24no—
172Dic61—42no—
173C22×C22⋊Q8—611no—
174#1784—612no—
175#1463—612no—
176#1462—612no—
177C2×C61—22no—
178C2×C61—22no—
179C2×C61—22no—
180C2×C61#178111yescharacteristic
181C2×C61#178411yescharacteristic
182C2×C61#178511yescharacteristic
183D61—44no—
184C23×Q8—611no—
185C24×C4—611no—
186C2×C22⋊Q8—1224no—
187C2×C22⋊Q8—1224no—
188C2×C22⋊Q8—1224no—
189C22×C4⋊C4—611no—
190C22×C4⋊C4—611no—
191C22×C4⋊C4#692311yescharacteristic
192C22×C22⋊C4—611no—
193C22×C22⋊C4—611no—
194C2×#101—1224no—
195#407—614no—
196#324—1224no—
197#323—1224no—
198C61#855011yescharacteristic, complemented, Sylow
199C25—611no—
200C22×Q8—614no—
201C22×Q8—1222no—
202C22×Q8—1222no—
203C22×Q8—2444no—
204C23×C4—611no—
205C23×C4—611no—
206C23×C4—611no—
207C23×C4—1222no—
208C23×C4—1222no—
209C23×C4—1222no—
210C23×C4—2444no—
211C23×C4C2×#692311yescharacteristic
212C23×C4C2×#692311yescharacteristic
213C23×C4C2×#692311yescharacteristic
214C2×M16—1224no—
215C22⋊Q8—12232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—614no—
220C2×C4⋊C4—1222no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4—1222no—
223C2×C4⋊C4—1222no—
224C2×C4⋊C4—1222no—
225C2×C22⋊C4—614no—
226C2×C22⋊C4—1222no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4—1222no—
229C2×C22⋊C4—1222no—
230C2×C22⋊C4—1222no—
231#101—2448no—
232C22⋊C8—1224no—
233C22⋊C8—1224no—
234C24—612no—
235C24—1224no—
236C24—1221no—
237C24—1221no—
238C24—2442no—
239C24—2442no—
240C24C22×#692311yescharacteristic
241C2×Q8—1228no—
242C2×Q8—1228no—
243C2×Q8—2444no—
244C2×Q8—2448no—
245C2×Q8—2448no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—612no—
259C22×C4—612no—
260C22×C4—1228no—
261C22×C4—1224no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—2444no—
267C22×C4—2448no—
268C22×C4—2444no—
269C22×C4—2448no—
270C22×C4—2442no—
271C22×C4—2442no—
272C22×C4C22⋊#692312yes—
273C22×C4C22⋊#692312yes—
274C22×C4C22⋊#692312yes—
275M16—2448no—
276C2×C8—2444no—
277C4⋊C4—28no—
278C4⋊C4—1228no—
279C4⋊C4—1228no—
280C22⋊C4—1228no—
281C22⋊C4—1228no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—614no—
286C23—1224no—
287C23—1224no—
288C23—1222no—
289C23—1222no—
290C23—2442no—
291C23—2442no—
292C23—2442no—
293C23—24416no—
294C23—2442no—
295C23—2442no—
296C23—2442no—
297C23C2×C22⋊#692311yescharacteristic
298C23C2×C22⋊#692311yescharacteristic
299C23C2×C22⋊#692311yescharacteristic
300Q8—24416no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—614no—
314C2×C4—614no—
315C2×C4—1224no—
316C2×C4—1224no—
317C2×C4—1224no—
318C2×C4—1222no—
319C2×C4—1222no—
320C2×C4—1224no—
321C2×C4—1222no—
322C2×C4—24416no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2442no—
327C8—4884no—
328C22—28no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—1224no—
336C22—2444no—
337C22—2448no—
338C22—2444no—
339C22—2448no—
340C22—2442no—
341C22—2442no—
342C22#1684070311yescharacteristic
343C22D61⋊#9912yes—
344C22D61⋊#10112yes—
345C22#1684074812yes—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—1224no—
350C4—1224no—
351C4—2442no—
352C2—22no—
353C2—22no—
354C2—22no—
355C2—2444no—
356C2#2146164611yescharacteristic
357C2#2146165411yescharacteristic
358C2#2146165511yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61#379
prime radicalC24×D61C2×C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8550
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #21461653C2semidirect
2. #21461069C2semidirect
119. C22×D61#101semidirect
198. C61#8550semidirect

Maximal subgroups

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

subgroupindex
#214616532
#214610692
#214610662
C22×#145197552
#855061