Finite Group Databasebeta

← #30606303

Subgroups — #30606303

Order 15616.

1026 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1026 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
1C2×D61⋊#99C212yesmaximal
2#21460914C212yesmaximal, complemented
3#21460913C212yesmaximal, complemented
4#21448398C211yesmaximal, characteristic
5C22×C22⋊#6923—22no—
6C2×#14519836C2214yes—
7D61⋊#99—24no—
8D61⋊#99C418yescomplemented
9C2×#14519835—22no—
10C22×D4×D61C2211yescharacteristic
11C2×#14519735C411yescharacteristic, complemented
12C2×#14519734C411yescharacteristic, complemented
13#16839665C411yescharacteristic, complemented
14#16839664C2211yescharacteristic
15#16839663C2211yescharacteristic
16#16839662—22no—
17#16839617—22no—
18C61×#390C411yescharacteristic, complemented
19C23×#6923—42no—
20C2×C22⋊#6923—28no—
21C2×C22⋊#6923—24no—
22C2×C22⋊#6923—44no—
23C2×C22⋊#6923—42no—
24#14519836—24no—
25#14519836C2×C418yescomplemented
26#14519835—28no—
27C24×D61D412yes—
28C22×D244C2311yescharacteristic
29C22×C22⋊D61D412yes—
30C2×D4×D61—42no—
31C2×D4×D61—42no—
32C2×D4×D61C2×C414yes—
33C2×D4×D61D412yes—
34C2×D4×D61Q812yes—
35C22×C4×D61C2×C411yescharacteristic
36#14519735—42no—
37#14519734—42no—
38C2×C4⋊Dic61C2×C411yescharacteristic, complemented
39C2×#11758196—21no—
40C2×C22⋊Dic61—22no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61C2×C411yescharacteristic, complemented
47C22×D4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
49C2×C22⋊C4×C61—22no—
50C2×C42×C61C2×C411yescharacteristic, complemented
51C24×C61C22⋊C412yes—
52C23×D61—22no—
53C23×D61—22no—
54C23×D61—24no—
55C23×D61—82no—
56C23×D61—82no—
57C23×D61C22⋊C414yes—
58C23×D61C22⋊C411yescharacteristic
59C23×D61C4⋊C414yes—
60C23×D61C2×D412yes—
61C22×#6923—44no—
62C22×#6923—48no—
63C22×#6923—84no—
64C22×#6923—84no—
65C2×D4×C61—41no—
66C2×D4×C61—41no—
67C2×D4×C61C4212yes—
68C2×D4×C61C4⋊C412yes—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C22×C411yescharacteristic
72C2×C22⋊D61—42no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61C22⋊C414yes—
75C2×C22⋊D61C4⋊C414yes—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C411yescharacteristic
79D4×D61—44no—
80D4×D61—44no—
81D4×D61—44no—
82D4×D61—44no—
83C2×D244—41no—
84C2×D244—41no—
85C2×D244C22×C412yes—
86C2×D244C2×D411yescharacteristic
87C2×D244C2×Q811yescharacteristic
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61C4212yes—
91C2×C4×D61C4⋊C412yes—
92C22⋊#6923—28no—
93C22⋊#6923—44no—
94C22⋊#6923—48no—
95C4⋊C4×C61—41no—
96C22⋊C4×C61—82no—
97C42×C61—41no—
98C22⋊Dic61—82no—
99#11758198—42no—
100#11758198—42no—
101#11758198—82no—
102#11758198—82no—
103C4⋊Dic61—41no—
104#11758196—42no—
105C4×Dic61—42no—
106C4×Dic61—41no—
107C23×C61—22no—
108C23×C61—22no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61#9514yes—
112C23×C61C2×C22⋊C411yescharacteristic
113C22×D61—22no—
114C22×D61—22no—
115C22×D61—24no—
116C22×D61—24no—
117C22×D61—24no—
118C22×D61—24no—
119C22×D61—41no—
120C22×D61—41no—
121C22×D61—44no—
122C22×D61—44no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—82no—
126C22×D61—82no—
127C22×D61—84no—
128C22×D61—84no—
129C22×D61—88no—
130C22×D61#9512yes—
131C22×D61#9912yes—
132C22×D61#9912yes—
133C22×D61C2×C22⋊C412yes—
134C22×D61C2×C4⋊C412yes—
135C2×#6923—48no—
136C2×#6923—816no—
137C2×#6923—84no—
138C2×#6923—84no—
139D4×C61—42no—
140D4×C61—42no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61C2×C4211yescharacteristic
149C2×C4×C61C2×C4⋊C411yescharacteristic
150C22⋊D61—44no—
151C22⋊D61—44no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61#9512yes—
160D244—42no—
161D244—41no—
162D244—41no—
163C4×D61—24no—
164C4×D61—24no—
165#12582—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—24no—
169C22×C61—44no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—82no—
173C22×C61C2×#9511yescharacteristic
174C22×C61C2×#9911yescharacteristic
175C22×C61C2×#9911yescharacteristic
176C2×D61—22no—
177C2×D61—22no—
178C2×D61—22no—
179C2×D61—22no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—48no—
183C2×D61—48no—
184C2×D61—42no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—42no—
188C2×D61—84no—
189C2×D61—84no—
190C2×D61—82no—
191C2×D61#34214yes—
192#6923—88no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—81no—
196C4×C61—81no—
197Dic61—22no—
198Dic61—81no—
199Dic61—81no—
200C22×#99—612no—
201#2309—611no—
202#1393—612no—
203#1393—612no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—44no—
207C2×C61C2×#34211yescharacteristic
208C2×C61#212711yescharacteristic
209C2×C61#212711yescharacteristic
210D61—24no—
211D61—42no—
212D61—42no—
213C23×D4—611no—
214C2×#117—611no—
215C2×#117—611no—
216C22×C22⋊C4—1222no—
217C2×#99—618no—
218C2×#99—614no—
219C2×#99—1222no—
220C2×#99—1224no—
221#390—611no—
222#390—611no—
223#390—611no—
224#390—1222no—
225#390#692311yescharacteristic, complemented
226#379—1222no—
227C61#1258211yescharacteristic, complemented, Sylow
228C25—612no—
229C22×D4—612no—
230C22×D4—614no—
231C22×D4—612no—
232C22×D4—612no—
233C22×D4—611no—
234C22×D4—2442no—
235C22×D4—2442no—
236C22×D4C2×#692311yescharacteristic
237C23×C4—611no—
238C23×C4—2442no—
239#117—2442no—
240#117—2442no—
241C2×C4⋊C4—611no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4C2×#692311yescharacteristic, complemented
244C2×C22⋊C4—22no—
245C2×C22⋊C4—1228no—
246C2×C22⋊C4—1224no—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4—1222no—
249C2×C22⋊C4—1222no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—2442no—
253C2×C22⋊C4—2444no—
254C2×C42—611no—
255C2×C42—1221no—
256C2×C42C2×#692311yescharacteristic, complemented
257#99—618no—
258#99—1228no—
259#99—1224no—
260C24—611no—
261C24—614no—
262C24—614no—
263C24—612no—
264C24—1222no—
265C24—1224no—
266C24—1222no—
267C24—4882no—
268C24—4882no—
269C24C22⋊#692312yes—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—614no—
273C2×D4—614no—
274C2×D4—612no—
275C2×D4—611no—
276C2×D4—611no—
277C2×D4—2441no—
278C2×D4—2441no—
279C2×D4—2444no—
280C2×D4—2444no—
281C2×D4—2444no—
282C2×D4—2444no—
283C2×D4—2442no—
284C2×D4—2442no—
285C2×D4C4×#692312yes—
286C2×D4C4⋊#692312yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—611no—
290C22×C4—612no—
291C22×C4—612no—
292C22×C4—1222no—
293C22×C4—1222no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—4884no—
297C22×C4—4884no—
298C22×C4—2448no—
299C22×C4—2444no—
300C22×C4C22×#692311yescharacteristic
301C4⋊C4—41no—
302C4⋊C4—2441no—
303C4⋊C4—2442no—
304C22⋊C4—82no—
305C22⋊C4—1228no—
306C22⋊C4—4882no—
307C22⋊C4—4882no—
308C22⋊C4—4882no—
309C22⋊C4—2442no—
310C22⋊C4—2442no—
311C22⋊C4—2448no—
312C22⋊C4—2444no—
313C42—41no—
314C42—2441no—
315C42—2442no—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—612no—
321C23—612no—
322C23—612no—
323C23—612no—
324C23—612no—
325C23—1224no—
326C23—1224no—
327C23—1224no—
328C23—1224no—
329C23—1222no—
330C23—1222no—
331C23—4882no—
332C23—4882no—
333C23—4884no—
334C23—4884no—
335C23—4888no—
336C23—2441no—
337C23—2441no—
338C23—2442no—
339C23—2442no—
340C23—2444no—
341C23—2444no—
342C23(C2×#6923)⋊C414yes—
343C23C2×C22⋊#692311yescharacteristic
344D4—42no—
345D4—42no—
346D4—2442no—
347D4—2441no—
348D4—2441no—
349D4—2444no—
350D4—2444no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—612no—
359C2×C4—1224no—
360C2×C4—1224no—
361C2×C4—1221no—
362C2×C4—1221no—
363C2×C4—48816no—
364C2×C4—4884no—
365C2×C4—4884no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2442no—
369C2×C4—2448no—
370C2×C4—2441no—
371C2×C4—2441no—
372C2×C4C2×C4×#692311yescharacteristic
373C2×C4C2×C4⋊#692311yescharacteristic
374C22—21no—
375C22—21no—
376C22—24no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—82no—
381C22—614no—
382C22—1222no—
383C22—1222no—
384C22—1222no—
385C22—1222no—
386C22—4882no—
387C22—4884no—
388C22—4884no—
389C22—2442no—
390C22—2442no—
391C22—2442no—
392C22—2442no—
393C22—2448no—
394C22—2448no—
395C22—2442no—
396C22—2442no—
397C22C2×(C2×#6923)⋊C411yescharacteristic
398C22D61⋊#9911yescharacteristic
399C22D61⋊#9911yescharacteristic
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—1222no—
405C4—4881no—
406C4—4888no—
407C4—4881no—
408C2—21no—
409C2—21no—
410C2—44no—
411C2—1224no—
412C2—2442no—
413C2—2442no—
414C2#2146151611yescharacteristic
415C2#2146209511yescharacteristic
416C2#2146209711yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC22×C61C2×#99
prime radicalC22×D4×D61C22
cosocleC22×D244C23
hypercenter#390#6923
hypocenterC61#12582
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
3. #21460913C2semidirect
8. D61⋊#99C4semidirect
11. C2×#14519735C4semidirect
12. C2×#14519734C4semidirect
13. #16839665C4semidirect
18. C61×#390C4semidirect
25. #14519836C2×C4semidirect
38. C2×C4⋊Dic61C2×C4semidirect
46. C2×C4×Dic61C2×C4semidirect
48. C2×C4⋊C4×C61C2×C4semidirect
50. C2×C42×C61C2×C4semidirect
225. #390#6923semidirect
227. C61#12582semidirect
243. C2×C4⋊C4C2×#6923semidirect
256. C2×C42C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×D61⋊#992
#214609142
#214609132
#214483982
#1258261