Finite Group Databasebeta

← #30603845

Subgroups — #30603845

Order 15616.

686 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461053C212yesmaximal, complemented
2#21460748C212yesmaximal, complemented
3#21460747C212yesmaximal, complemented
4#21448398C211yesmaximal, characteristic
5#16840779—22no—
6#16840607C2214yescomplemented
7#16840606—22no—
8C22×D4×D61C411yescharacteristic
9C2×#14519735C2211yescharacteristic
10C2×#14519734C411yescharacteristic
11#16839665C2211yescharacteristic
12#16839664C2211yescharacteristic
13#16839663C411yescharacteristic
14#16839662—22no—
15#16839617—22no—
16C61×#390C411yescharacteristic
17#14519844—42no—
18C22⋊#92420—24no—
19C4×#92420—42no—
20C24×D61—21no—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23C2×D4×D61—22no—
24C2×D4×D61—42no—
25C2×D4×D61—42no—
26C2×D4×D61C814yescomplemented
27C22×C4×D61C2×C411yescharacteristic
28#14519735—42no—
29#14519734—42no—
30C2×C4⋊Dic61C2311yescharacteristic
31C2×#11758196—21no—
32C2×C22⋊Dic61—22no—
33C2×#11758198—21no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×#11758198D412yes—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61D412yes—
39C22×D4×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C22⋊C4×C61—22no—
42C2×C42×C61C2×C411yescharacteristic
43C24×C61—21no—
44C23×D61—21no—
45C23×D61—24no—
46C23×D61—42no—
47C23×D61—42no—
48C23×D61—82no—
49C23×D61—82no—
50C23×D61C22⋊C411yescharacteristic
51C2×D4×C61—21no—
52C2×D4×C61—41no—
53C2×D4×C61—41no—
54C2×D4×C61C2×C812yes—
55C22×C4×C61—21no—
56C22×C4×C61C22⋊C412yes—
57C22×C4×C61C22×C411yescharacteristic
58C2×C22⋊D61—24no—
59C2×C22⋊D61—42no—
60C2×C22⋊D61—42no—
61C22×Dic61—21no—
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C2×D412yes—
64D4×D61—48no—
65D4×D61—48no—
66C2×D244—21no—
67C2×D244—41no—
68C2×D244—41no—
69C2×D244C2×C812yes—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61C2×C812yes—
73C2×C4×D61M1612yes—
74C2×#92420—44no—
75C4⋊C4×C61—41no—
76C22⋊C4×C61—82no—
77C42×C61—41no—
78C22⋊Dic61—82no—
79#11758198—42no—
80#11758198—42no—
81#11758198—82no—
82#11758198—82no—
83C4⋊Dic61—41no—
84#11758196—42no—
85C4×Dic61—22no—
86C4×Dic61—22no—
87C4×Dic61—41no—
88C23×C61—22no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61C2×C22⋊C411yescharacteristic
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—22no—
97C22×D61—48no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—42no—
101C22×D61—42no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—88no—
105C22×D61—88no—
106C22×D61—84no—
107C22×D61—84no—
108C22×D61C22⋊C812yes—
109D4×C61—42no—
110D4×C61—42no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—22no—
114C2×C4×C61—42no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61C22×C811yescharacteristic
119C2×C4×C61C2×M1611yescharacteristic
120C22⋊D61—48no—
121C2×Dic61—22no—
122C2×Dic61—42no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61C22⋊C812yes—
127C2×Dic61#9912yes—
128C2×Dic61#10012yes—
129D244—42no—
130D244—42no—
131C4×D61—24no—
132C4×D61—24no—
133#92420—84no—
134#11433—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—44no—
138C22×C61—42no—
139C22×C61—82no—
140C22×C61—81no—
141C22×C61—81no—
142C22×C61C2×C22⋊C811yescharacteristic
143C22×C61C2×#9911yescharacteristic
144C22×C61C2×#10011yescharacteristic
145C2×D61—22no—
146C2×D61—42no—
147C2×D61—41no—
148C2×D61—41no—
149C2×D61—42no—
150C2×D61—44no—
151C2×D61—42no—
152C2×D61—816no—
153C2×D61—82no—
154C2×D61—81no—
155C2×D61—81no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—42no—
159C4×C61—81no—
160Dic61—42no—
161Dic61—81no—
162Dic61#32314yes—
163#2309—611no—
164#1458—612no—
165#1318—612no—
166#1318—612no—
167C2×C61—21no—
168C2×C61—21no—
169C2×C61—82no—
170C2×C61C2×#32311yescharacteristic
171C2×C61#212811yescharacteristic
172C2×C61#213411yescharacteristic
173D61—42no—
174D61—82no—
175C23×D4—611no—
176C2×#117—611no—
177C2×#117—611no—
178#431—1222no—
179#390—611no—
180#390—611no—
181#390—611no—
182#390—1222no—
183#390#692311yescharacteristic
184#379—1222no—
185#324—614no—
186#323—1222no—
187C61#1143311yescharacteristic, complemented, Sylow
188C25—1221no—
189C22×D4—614no—
190C22×D4—611no—
191C22×D4—1222no—
192C22×D4—1221no—
193C22×D4—2442no—
194C22×D4—2442no—
195C22×D4C2×#692311yescharacteristic
196C23×C4—611no—
197#117—2442no—
198#117—2442no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4C2×#692311yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—612no—
204C2×C22⋊C4—1222no—
205C2×C22⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4—1221no—
208C2×C42—612no—
209C2×C42—611no—
210C2×C42C2×#692311yescharacteristic
211C4⋊C8—2442no—
212C22⋊C8—1224no—
213C4×C8—2442no—
214C24—21no—
215C24—611no—
216C24—1224no—
217C24—1221no—
218C24—4882no—
219C24—4882no—
220C24—2442no—
221C24—2442no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—612no—
226C2×D4—1224no—
227C2×D4—1221no—
228C2×D4—2441no—
229C2×D4—2441no—
230C2×D4—2442no—
231C2×D4—2442no—
232C2×D4—2448no—
233C2×D4—2448no—
234C2×D4D61⋊C812yes—
235C22×C4—21no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—612no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—1221no—
243C22×C4C22⋊#692312yes—
244C22×C4C22×#692311yescharacteristic
245C2×C8—2444no—
246C4⋊C4—41no—
247C4⋊C4—2441no—
248C4⋊C4—2442no—
249C22⋊C4—82no—
250C22⋊C4—4882no—
251C22⋊C4—4882no—
252C22⋊C4—4882no—
253C22⋊C4—2442no—
254C22⋊C4—2442no—
255C42—41no—
256C42—1222no—
257C42—1222no—
258C42—2441no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—41no—
264C23—612no—
265C23—1221no—
266C23—1221no—
267C23—1222no—
268C23—4888no—
269C23—4888no—
270C23—4884no—
271C23—4884no—
272C23—2442no—
273C23—2442no—
274C23—2441no—
275C23—2441no—
276C23—2448no—
277C23—2441no—
278C23—2441no—
279C23C2×C22⋊#692311yescharacteristic
280D4—42no—
281D4—42no—
282D4—2442no—
283D4—2442no—
284D4—2448no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—612no—
293C2×C4—612no—
294C2×C4—612no—
295C2×C4—1224no—
296C2×C4—1224no—
297C2×C4—1222no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4C2×D61⋊C811yescharacteristic
303C2×C4D61⋊M1611yescharacteristic
304C8—4884no—
305C22—21no—
306C22—21no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—81no—
311C22—81no—
312C22—1222no—
313C22—4882no—
314C22—48816no—
315C22—4881no—
316C22—4881no—
317C22—2444no—
318C22—2442no—
319C22—2442no—
320C22—2442no—
321C22—2441no—
322C22—2441no—
323C22D61⋊(C22⋊C8)11yescharacteristic
324C22D61⋊#9911yescharacteristic
325C22D61⋊#10011yescharacteristic
326C4—21no—
327C4—21no—
328C4—42no—
329C4—81no—
330C4—614no—
331C4—4881no—
332C4—2442no—
333C2—21no—
334C2—21no—
335C2—82no—
336C2—4882no—
337C2—2442no—
338C2D61⋊#32311yescharacteristic
339C2#2146209911yescharacteristic
340C2#2146211911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×D4×D61C4
cosocleC2×C4⋊Dic61C23
hypercenter#390#6923
hypocenterC61#11433
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461053C2semidirect
2. #21460748C2semidirect
3. #21460747C2semidirect
6. #16840607C22semidirect
26. C2×D4×D61C8semidirect
187. C61#11433semidirect

Maximal subgroups

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

subgroupindex
#214610532
#214607482
#214607472
#214483982
#1143361