Finite Group Databasebeta

← #30605383

Subgroups — #30605383

Order 15616.

776 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461679C211yesmaximal, characteristic, complemented
2#21461058C211yesmaximal, characteristic, complemented
3#21460797C214yesmaximal, complemented
4#21449169C211yesmaximal, characteristic
5#16840779—21no—
6C2×#14519835—21no—
7#16840704—21no—
8#16840640C2212yescomplemented
9#16840606C2212yescomplemented
10#16840605—21no—
11C2×#14519748—21no—
12C2×#14519746—21no—
13C2×C4×C22⋊D61C2211yescharacteristic
14#16839676C412yescomplemented
15#16839670C2212yes—
16#16839665—21no—
17#16839622C411yescharacteristic, complemented
18#16839618—21no—
19C23×D4×C61C411yescharacteristic, complemented
20C2×C4⋊#6923—41no—
21C2×C22⋊#6923—22no—
22#14519844—41no—
23#14519835—42no—
24C22⋊#92420—22no—
25C4×#92420—41no—
26(C2×#6923)⋊C4—41no—
27C22×C22⋊D61—22no—
28C22×C22⋊D61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30#14519748—44no—
31#14519746—42no—
32#14519746—42no—
33C4×C22⋊D61—41no—
34C4×C22⋊D61—41no—
35C2×C4⋊Dic61—21no—
36C2×#11758196C2×C411yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61D411yescharacteristic
42C2×#11758198C2311yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44#14519684C2×C412yes—
45C25×C61—21no—
46C22×D4×C61—21no—
47C22×D4×C61—41no—
48C22×D4×C61—41no—
49C22×D4×C61—44no—
50C22×D4×C61C2×C412yes—
51C23×C4×C61C2×C411yescharacteristic
52C24×C61—22no—
53C24×C61—42no—
54C24×C61—44no—
55C24×C61—42no—
56C24×C61—42no—
57C24×C61—82no—
58C24×C61C22⋊C411yescharacteristic
59C23×D61C2×D411yescharacteristic
60C22×#6923—42no—
61C2×D4×C61—24no—
62C2×D4×C61—42no—
63C2×D4×C61—42no—
64C2×D4×C61—48no—
65C2×D4×C61—44no—
66C2×D4×C61—44no—
67C2×D4×C61—44no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C2×C22⋊D61—44no—
78C2×C22⋊D61—42no—
79C2×C22⋊D61—42no—
80C2×C22⋊D61—82no—
81C2×C22⋊D61D812yes—
82C2×C22⋊D61SD1612yes—
83C22×Dic61—22no—
84C22×Dic61C22⋊C411yescharacteristic
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C2×D411yescharacteristic
87C2×C4×D61—21no—
88C2×C4×D61—21no—
89C2×C4×D61—42no—
90C22⋊#6923—42no—
91C22⋊#6923—42no—
92C2×#92420—42no—
93C4⋊#6923—82no—
94C22⋊Dic61—41no—
95C22⋊Dic61—44no—
96C22⋊Dic61—42no—
97C22⋊Dic61—44no—
98C22⋊Dic61—82no—
99#11758198—41no—
100C4⋊Dic61—42no—
101#11758196—41no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—24no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—48no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—84no—
117C23×C61—88no—
118C23×C61—88no—
119C23×C61—84no—
120C23×C61#10212yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61C2×D811yescharacteristic
127C22×D61C2×SD1611yescharacteristic
128C2×#6923—82no—
129C2×#6923—82no—
130C2×#6923—82no—
131D4×C61—44no—
132D4×C61—44no—
133D4×C61—44no—
134D4×C61—44no—
135C2×C4×C61—22no—
136C2×C4×C61—21no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61#9911yescharacteristic
141C2×C4×C61#9911yescharacteristic
142C22⋊D61—42no—
143C22⋊D61—42no—
144C22⋊D61—84no—
145C22⋊D61—84no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—42no—
153C2×Dic61—42no—
154C2×Dic61—42no—
155C2×Dic61#10212yes—
156C4×D61—42no—
157C4×D61—81no—
158#92420—82no—
159#11873—611nomaximal, Sylow
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—22no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—44no—
166C22×C61—44no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—44no—
170C22×C61—82no—
171C22×C61—82no—
172C22×C61—816no—
173C22×C61—82no—
174C22×C61—82no—
175C22×C61C2×#9911yescharacteristic
176C22×C61C2×#10211yescharacteristic
177C22×C61#42111yescharacteristic
178C2×D61—21no—
179C2×D61—21no—
180C2×D61—41no—
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—81no—
185#6923—162no—
186C4×C61—21no—
187C4×C61—41no—
188C4×C61AΣL(1,𝔽16)12yes—
189Dic61—22no—
190Dic61—41no—
191Dic61—41no—
192Dic61—82no—
193#2408—611no—
194#1799—611no—
195#1460—611no—
196#1343—614no—
197C2×C61—21no—
198C2×C61—21no—
199C2×C61—42no—
200C2×C61—84no—
201C2×C61#151911yescharacteristic
202C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
203C2×C61#212811yescharacteristic
204D61—41no—
205D61—41no—
206C23×D4#692311yescharacteristic, complemented
207C2×#121—1221no—
208C2×#120—1221no—
209C2×C4×D4—611no—
210#431—1221no—
211C2×#99—1221no—
212#394—612no—
213#392—612no—
214#390—1221no—
215#380—611no—
216#380—1221no—
217#379—1221no—
218#342—612no—
219#323—612no—
220#323—1221no—
221C61#1187311yescharacteristic, complemented, Sylow
222C25—21no—
223C22×D4—21no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—44no—
227C22×D4—611no—
228C22×D4—1222no—
229C22×D4C2×#692312yes—
230C23×C4—611no—
231C23×C4C2×#692311yescharacteristic
232#121—2444no—
233#120—2442no—
234#120—2442no—
235C4×D4—2441no—
236C4×D4—2441no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—2441no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—611no—
242C2×C22⋊C4—1222no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4—1222no—
247C2×C42—611no—
248C4⋊C8—2441no—
249#99—2442no—
250C22⋊C8—1222no—
251C4×C8—2441no—
252#95—612no—
253#95—2441no—
254C24—22no—
255C24—42no—
256C24—44no—
257C24—42no—
258C24—42no—
259C24—82no—
260C24—611no—
261C24C22⋊#692311yescharacteristic
262C2×D4—24no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—48no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—612no—
272C2×D4—612no—
273C2×D4—4882no—
274C2×D4—2441no—
275C2×D4—2441no—
276C2×D4—2444no—
277C2×D4—2442no—
278C2×D4—2442no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4C22⋊#692311yescharacteristic
291C22×C4C22×#692311yescharacteristic
292C2×C8—2442no—
293C4⋊C4—4882no—
294C4⋊C4—2441no—
295C4⋊C4—2442no—
296C22⋊C4—4882no—
297C22⋊C4—2441no—
298C22⋊C4—2444no—
299C22⋊C4—2441no—
300C22⋊C4—2442no—
301C22⋊C4—2442no—
302C22⋊C4—2444no—
303C22⋊C4—2442no—
304C42—1221no—
305C42—1221no—
306C23—21no—
307C23—21no—
308C23—24no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—88no—
320C23—88no—
321C23—84no—
322C23—611no—
323C23—611no—
324C23—1221no—
325C23—1221no—
326C23—2441no—
327C23—2441no—
328C23#1451983912yes—
329C23C2×C22⋊#692311yescharacteristic
330D4—44no—
331D4—44no—
332D4—44no—
333D4—44no—
334D4—4884no—
335D4—4884no—
336D4—2442no—
337D4—2442no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—612no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—4881no—
349C2×C4—4882no—
350C2×C4—4882no—
351C2×C4—4882no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4#1451983511yescharacteristic
359C2×C4#1451983511yescharacteristic
360C8—4882no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22—44no—
368C22—41no—
369C22—41no—
370C22—44no—
371C22—82no—
372C22—82no—
373C22—816no—
374C22—82no—
375C22—82no—
376C22—1221no—
377C22—1221no—
378C22—4881no—
379C22—2441no—
380C22—2441no—
381C22—2441no—
382C22—2441no—
383C22C2×#1451983511yescharacteristic
384C22C2×#1451983911yescharacteristic
385C22#1684076011yescharacteristic
386C4—21no—
387C4—41no—
388C4—1222no—
389C4—9762no—
390C4—4882no—
391C4—2441no—
392C4—2441no—
393C4#1684066212yes—
394C2—21no—
395C2—21no—
396C2—42no—
397C2—84no—
398C2—2441no—
399C2—2441no—
400C2#2146125511yescharacteristic
401C2C2×#1684066211yescharacteristic
402C2#2146209811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×D4×C61C4
socleC22×C61C2×#99
prime radical#21449169C2
cosocleC2×#11758198C23
hypercenterC23×D4#6923
hypocenterC61#11873
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461679C2semidirect
2. #21461058C2semidirect
3. #21460797C2semidirect
8. #16840640C22semidirect
9. #16840606C22semidirect
14. #16839676C4semidirect
17. #16839622C4semidirect
19. C23×D4×C61C4semidirect
206. C23×D4#6923semidirect
221. C61#11873semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610582
#214607972
#214491692
#1187361