Finite Group Databasebeta

← #29661245

Subgroups — #29661245

Order 15616.

936 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 171 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#21457009C212yesmaximal, complemented
2#21456283C211yesmaximal, characteristic
3#21445130C214yesmaximal, complemented
4#21445117C214yesmaximal, complemented
5#21444443C211yesmaximal, characteristic, complemented
6#21444421C212yesmaximal, complemented
7C2×C61×#563C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9C2×#14519803—21no—
10#16840312C2211yescharacteristic
11#16840172C2214yescomplemented
12#16840157C2212yes—
13#16840152C2212yes—
14#16840144—22no—
15#16840128C2214yescomplemented
16#16840080—22no—
17#16840075—22no—
18#16840059—24no—
19#16839819C2212yes—
20C4×#11758194C2211yescharacteristic
21#16839504C2218yescomplemented
22#16839502C2214yes—
23C61×#563—24no—
24C61×#563—22no—
25C61×#563C2214yes—
26C2×C4⋊Q8×C61—21no—
27C2×#126×C61—22no—
28C2×#125×C61C2212yes—
29C2×#124×C61C2211yescharacteristic
30C2×#123×C61—21no—
31C2×C22⋊Q8×C61—22no—
32C2×#11758222—21no—
33#14519808—44no—
34#14519803—44no—
35C2×#11758202—22no—
36C2×#11758200—22no—
37C2×#11758201—22no—
38C2×#11758199—22no—
39C2×#11758194—21no—
40#14519777C2311yescharacteristic
41#14519774C2311yescharacteristic
42#14519765—21no—
43#14519751—22no—
44C4×D244—22no—
45#14519710—24no—
46#14519708—24no—
47C4⋊#92418—22no—
48#14519697—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51#14519688C2312yes—
52#14519688C2314yes—
53C4×#92418—21no—
54C22×Q8×C61—21no—
55C22×Q8×C61D411yescharacteristic
56C22×D4×C61D411yescharacteristic
57C23×C4×C61—21no—
58C4⋊Q8×C61—24no—
59C4⋊Q8×C61—24no—
60#126×C61—28no—
61#126×C61—28no—
62#125×C61—22no—
63#125×C61—22no—
64#125×C61D414yes—
65#125×C61C2314yes—
66#124×C61—22no—
67#124×C61D412yes—
68#124×C61C2312yes—
69#123×C61—24no—
70#123×C61—24no—
71C22⋊Q8×C61—28no—
72C22⋊Q8×C61—28no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61D412yes—
77C2×C4⋊C4×C61D412yes—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—22no—
81C2×C42×C61—21no—
82C2×C42×C61C2311yescharacteristic
83C24×C61—21no—
84C2×Q8×C61—24no—
85C2×Q8×C61—21no—
86C2×Q8×C61—42no—
87C2×Q8×C61—82no—
88C2×Q8×C61C2×D412yes—
89C2×D4×C61—21no—
90C2×D4×C61—42no—
91C2×D4×C61C2×D412yes—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—24no—
96C22×C4×C61—82no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D412yes—
99C2×C22⋊D61—41no—
100#11758222—42no—
101#11758222—42no—
102C2×D244—21no—
103C2×D244—22no—
104C2×C4×D61—41no—
105C2×Dic122—21no—
106C4⋊C4×C61—28no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61C2×D414yes—
114C4⋊C4×C61C2×D414yes—
115C22⋊C4×C61—28no—
116C22⋊C4×C61—24no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—24no—
119C42×C61—24no—
120C42×C61C2×D412yes—
121C42×C61C2×D411yescharacteristic
122C42×C61C2411yescharacteristic
123#11758202—44no—
124#11758201—44no—
125#11758200—44no—
126#11758199—44no—
127#11758198—42no—
128C4⋊Dic61—22no—
129#11758194—24no—
130C2×#92418—22no—
131C2×#92418—24no—
132C23×C61—22no—
133C23×C61—44no—
134C23×C61—42no—
135C23×C61#12011yescharacteristic
136C22×D61—41no—
137Q8×C61—42no—
138Q8×C61—81no—
139Q8×C61—84no—
140D4×C61—42no—
141D4×C61—81no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—24no—
146C2×C4×C61—42no—
147C2×C4×C61—44no—
148C2×C4×C61—44no—
149C2×C4×C61—42no—
150C2×C4×C61—84no—
151C2×C4×C61#12012yes—
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61C22×D412yes—
154C2×C4×C61C22×D411yescharacteristic
155C22⋊D61—82no—
156C2×Dic61—41no—
157D244—41no—
158D244—41no—
159D244—44no—
160C4×D61—82no—
161Dic122—41no—
162Dic122—41no—
163#92418—24no—
164#92418—44no—
165#18994—611nomaximal, Sylow
166C22×C61—22no—
167C22×C61—42no—
168C22×C61—44no—
169C22×C61—84no—
170C22×C61#45312yes—
171C22×C61C2×#12011yescharacteristic
172C2×D61—81no—
173C2×D61—82no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—22no—
177C4×C61—22no—
178C4×C61—44no—
179C4×C61—42no—
180C4×C61—44no—
181Dic61—81no—
182C2×#563D6111yescharacteristic, complemented
183#3234—612no—
184#3153—611no—
185#1660—614no—
186#1657—614no—
187#1533—611no—
188#1528—612no—
189C2×C61—22no—
190C2×C61—82no—
191C2×C61C2×#45311yescharacteristic
192C2×C61#302012yes—
193D61—161no—
194C2×#137—1221no—
195C2×AGL(1,ℤ/8)—1221no—
196#563—24no—
197#563—22no—
198#563C2×D6114yes—
199#532—611no—
200C2×C4⋊Q8—21no—
201C2×#126—22no—
202C2×#125C2×D6112yes—
203C2×#124C2×D6111yescharacteristic
204C2×#123—21no—
205C2×C22⋊Q8—22no—
206#501—614no—
207#497—612no—
208#496—612no—
209#495—1222no—
210#490—614no—
211#481—1222no—
212#480—1222no—
213#478—1224no—
214#432—612no—
215C4×M16—611no—
216#330—618no—
217#329—614no—
218C61#1899411yescharacteristic, complemented, Sylow
219C2×#41—1221no—
220C22×Q8—21no—
221C22×Q8C22⋊D6111yescharacteristic
222C22×D4C22⋊D6111yescharacteristic
223C23×C4—21no—
224#137—2444no—
225AGL(1,ℤ/8)—2444no—
226C2×Q16—1222no—
227C2×SD16—1222no—
228C2×SD16—1222no—
229C2×D8—1222no—
230C2×M16—1221no—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—611no—
234#127—611no—
235#126—28no—
236#126—28no—
237#125—22no—
238#125—22no—
239#125C22⋊D6114yes—
240#125C22×D6114yes—
241#124—22no—
242#124—1221no—
243#124C22⋊D6112yes—
244#124C22×D6112yes—
245#123—24no—
246#123—24no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249#121—1222no—
250C4×D4—1222no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C22⋊D6112yes—
255C2×C4⋊C4C22⋊D6112yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C42—21no—
260C2×C42C22×D6111yescharacteristic
261#107—1224no—
262#106—1224no—
263C4⋊C8—1222no—
264#102—1224no—
265#102—1224no—
266C22⋊C8—1222no—
267#97—614no—
268#97—612no—
269C4×C8—1221no—
270C24—21no—
271#41—2442no—
272#41—2442no—
273C2×Q8—24no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—82no—
277C2×Q8—1221no—
278C2×Q8C2×C22⋊D6112yes—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—1222no—
282C2×D4—1221no—
283C2×D4—2441no—
284C2×D4C2×C22⋊D6112yes—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—82no—
290C22×C4—2441no—
291C22×C4C2×C22⋊D6111yescharacteristic
292C22×C4C2×C22⋊D6112yes—
293Q16—2444no—
294SD16—2444no—
295SD16—2444no—
296D8—2444no—
297M16—1224no—
298C2×C8—1222no—
299C2×C8—1224no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—1222no—
308C4⋊C4C2×C22⋊D6114yes—
309C4⋊C4C2×C22⋊D6114yes—
310C22⋊C4—28no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—2442no—
315C42—24no—
316C42C2×C22⋊D6111yescharacteristic
317C42C2×C22⋊D6112yes—
318C42C23×D6111yescharacteristic
319C23—22no—
320C23—44no—
321C23—42no—
322C23—2441no—
323C23#1451974611yescharacteristic
324Q8—42no—
325Q8—81no—
326Q8—84no—
327Q8—2441no—
328Q8—2441no—
329D4—42no—
330D4—81no—
331D4—4882no—
332D4—2441no—
333D4—2441no—
334D4—2444no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4—4882no—
345C2×C4—2441no—
346C2×C4#1451974612yes—
347C2×C4#1451974611yescharacteristic
348C2×C4C22×C22⋊D6112yes—
349C2×C4C22×C22⋊D6111yescharacteristic
350C8—1224no—
351C8—2444no—
352C22—22no—
353C22—42no—
354C22—44no—
355C22—84no—
356C22—4881no—
357C22—4882no—
358C22#1683993012yes—
359C22C2×#1451974611yescharacteristic
360C4—21no—
361C4—21no—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—42no—
366C4—44no—
367C4—4881no—
368C2—22no—
369C2—82no—
370C2—9761no—
371C2C2×#1683993011yescharacteristic
372C2#2145518312yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#563C2
socleC22×C61C2×#120
prime radical#21456283C2
hypercenterC2×#563D61
hypocenterC61#18994
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. #21457009C2semidirect
3. #21445130C2semidirect
4. #21445117C2semidirect
5. #21444443C2semidirect
6. #21444421C2semidirect
7. C2×C61×#563C2semidirect
11. #16840172C22semidirect
15. #16840128C22semidirect
21. #16839504C22semidirect
182. C2×#563D61semidirect
218. C61#18994semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214562832
#214451302
#214451172
#214444432
#214444212
C2×C61×#5632
#1899461