Finite Group Databasebeta

← #30607244

Subgroups — #30607244

Order 15616.

914 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461647C212yesmaximal
2#21460914C214yesmaximal, complemented
3C2×#16839663C211yesmaximal, characteristic
4C2×#14519836—22no—
5C2×#14519836C2214yes—
6#16840703—22no—
7C23×D244C2211yescharacteristic
8C22×#11758198—22no—
9C22×C4×Dic61C411yescharacteristic, complemented
10#16839663—22no—
11#16839663C412yescomplemented
12#16839663C2212yes—
13C22×C4⋊C4×C61C411yescharacteristic, complemented
14C2×C22⋊#6923—24no—
15C2×C22⋊#6923—42no—
16C2×C22⋊#6923—42no—
17#14519836—28no—
18#14519836—44no—
19C24×D61D412yes—
20C22×D244—24no—
21C22×D244—42no—
22C22×D244—41no—
23C22×D244—41no—
24C22×D244C2×C411yescharacteristic
25C22×D244D412yes—
26C22×D244C2311yescharacteristic
27C23×Dic61—21no—
28C2×#11758198—24no—
29C2×#11758198—24no—
30C2×#11758198—48no—
31C2×C4×Dic61—21no—
32C2×C4×Dic61—42no—
33C2×C4×Dic61C2×C412yes—
34C23×C4×C61—21no—
35C23×C4×C61C2×C411yescharacteristic
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61—41no—
38C2×C4⋊C4×C61—41no—
39C2×C4⋊C4×C61C2×C412yes—
40C24×C61C22⋊C411yescharacteristic
41C23×D61—22no—
42C23×D61—22no—
43C23×D61—24no—
44C23×D61—42no—
45C23×D61—44no—
46C23×D61—44no—
47C23×D61C22⋊C412yes—
48C23×D61C2×D412yes—
49C22×#6923—44no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—44no—
56C22×C4×C61—41no—
57C22×C4×C61—41no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—24no—
61C22×Dic61—22no—
62C22×Dic61—42no—
63C22×Dic61—42no—
64C2×D244—24no—
65C2×D244—24no—
66C2×D244—22no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244—42no—
70C2×D244—41no—
71C2×D244—41no—
72C2×D244—48no—
73C2×D244—48no—
74C2×D244#4114yes—
75C22⋊#6923—44no—
76C22⋊#6923—44no—
77C22⋊#6923—84no—
78C22⋊#6923—84no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—42no—
81#11758198—48no—
82#11758198—48no—
83C4×Dic61—42no—
84C4×Dic61—42no—
85C23×C61—22no—
86C23×C61—22no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61#9911yescharacteristic
90C23×C61#9911yescharacteristic
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—22no—
93C22×D61—22no—
94C22×D61—22no—
95C22×D61—22no—
96C22×D61—22no—
97C22×D61—44no—
98C22×D61—44no—
99C22×D61—42no—
100C22×D61—42no—
101C22×D61—42no—
102C22×D61—42no—
103C22×D61—416no—
104C22×D61—44no—
105C22×D61—44no—
106C22×D61—82no—
107C22×D61—82no—
108C22×D61—88no—
109C22×D61#12314yes—
110C2×#6923—84no—
111C2×#6923—84no—
112C2×#6923—84no—
113C2×C4×C61—24no—
114C2×C4×C61—24no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—44no—
120C2×C4×C61—42no—
121C2×C4×C61—82no—
122C2×C4×C61—82no—
123C2×C4×C61#9912yes—
124C2×C4×C61#11712yes—
125C2×Dic61—24no—
126C2×Dic61—48no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129D244—44no—
130D244—48no—
131D244—44no—
132#12856—611nomaximal, Sylow
133C22×C61—24no—
134C22×C61—24no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61#35312yes—
142C22×C61#35412yes—
143C22×C61#38111yescharacteristic
144C22×C61C2×#9911yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C2×D61—24no—
147C2×D61—42no—
148C2×D61—42no—
149C2×D61—44no—
150C2×D61—44no—
151C2×D61—42no—
152C2×D61—42no—
153C2×D61—42no—
154C2×D61—82no—
155C2×D61—88no—
156C2×D61—88no—
157C2×D61—82no—
158#6923—164no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—82no—
162Dic61—44no—
163C2×#390—611no—
164#1781—612no—
165#1393—614no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—22no—
169C2×C61—22no—
170C2×C61#181911yescharacteristic
171C2×C61C2×#35311yescharacteristic
172C2×C61C2×#35411yescharacteristic
173D61—44no—
174D61—82no—
175C23×D4—611no—
176C22×C4⋊C4#692311yescharacteristic, complemented
177C22×C22⋊C4—1222no—
178C22×C42—611no—
179C2×#99—614no—
180C2×#99—1222no—
181#390—612no—
182#390—612no—
183#390—1222no—
184#379—1222no—
185C61#1285611yescharacteristic, complemented, Sylow
186C25—612no—
187C22×D4—612no—
188C22×D4—611no—
189C22×D4—611no—
190C22×D4—1224no—
191C22×D4—2442no—
192C22×D4—2441no—
193C22×D4—2441no—
194C23×C4—21no—
195C23×C4—1221no—
196C23×C4C2×#692311yescharacteristic
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×#692312yes—
201C2×C22⋊C4—1224no—
202C2×C22⋊C4—1224no—
203C2×C22⋊C4—1224no—
204C2×C22⋊C4—2442no—
205C2×C22⋊C4—2442no—
206C2×C22⋊C4—2448no—
207C2×C42—612no—
208C2×C42—1221no—
209C2×C42—2442no—
210#99—1228no—
211#99—2444no—
212C24—612no—
213C24—612no—
214C24—1224no—
215C24—1222no—
216C24—1222no—
217C24—2442no—
218C24—2444no—
219C24—2444no—
220C24C22⋊#692311yescharacteristic
221C2×D4—614no—
222C2×D4—1224no—
223C2×D4—1224no—
224C2×D4—1222no—
225C2×D4—2442no—
226C2×D4—2441no—
227C2×D4—2441no—
228C2×D4—2441no—
229C2×D4—2441no—
230C2×D4—2448no—
231C2×D4—2448no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—1224no—
241C22×C4—1222no—
242C22×C4—2444no—
243C22×C4—2442no—
244C22×C4—2442no—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22×#692311yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—4884no—
250C22⋊C4—4884no—
251C22⋊C4—2448no—
252C22⋊C4—2448no—
253C22⋊C4—2444no—
254C22⋊C4—2444no—
255C42—2442no—
256C42—2442no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—614no—
262C23—1222no—
263C23—1222no—
264C23—1222no—
265C23—1222no—
266C23—1222no—
267C23—4888no—
268C23—4882no—
269C23—4882no—
270C23—2444no—
271C23—2444no—
272C23—24416no—
273C23—2442no—
274C23—2442no—
275C23—2442no—
276C23—2442no—
277C23—2444no—
278C23—2444no—
279C23#1451983511yescharacteristic
280C23#1451983511yescharacteristic
281C23C2×C22⋊#692311yescharacteristic
282D4—2444no—
283D4—2448no—
284D4—2444no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—1224no—
296C2×C4—4884no—
297C2×C4—4884no—
298C2×C4—4884no—
299C2×C4—2448no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4#1451983612yes—
303C2×C4#1451985612yes—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—1224no—
313C22—4882no—
314C22—4888no—
315C22—4888no—
316C22—4882no—
317C22—2444no—
318C22—2444no—
319C22—2442no—
320C22—2442no—
321C22—2442no—
322C22—2442no—
323C22—2442no—
324C22#1684066612yes—
325C22#1684066812yes—
326C22#1684070511yescharacteristic
327C22C2×#1451983511yescharacteristic
328C22C2×#1451983611yescharacteristic
329C4—22no—
330C4—22no—
331C4—82no—
332C4—9764no—
333C4—2444no—
334C2—21no—
335C2—22no—
336C2—22no—
337C2—21no—
338C2—4882no—
339C2—2444no—
340C2#2146171411yescharacteristic
341C2C2×#1684066611yescharacteristic
342C2C2×#1684066811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC23×D244C22
cosocleC22×D244C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#12856
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
9. C22×C4×Dic61C4semidirect
11. #16839663C4semidirect
13. C22×C4⋊C4×C61C4semidirect
176. C22×C4⋊C4#6923semidirect
185. C61#12856semidirect

Maximal subgroups

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

subgroupindex
#214616472
#214609142
C2×#168396632
#1285661