Finite Group Databasebeta

← #29653132

Subgroups — #29653132

Order 15616.

1154 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 203 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#21455846C212yesmaximal, complemented
2#21451181C211yesmaximal, characteristic, complemented
3#21444856C214yesmaximal, complemented
4#21444788C214yesmaximal, complemented
5#21444379C212yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7C61×#3497C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16840078C2214yes—
10#16840044—22no—
11#16839959—22no—
12#16839900C2214yes—
13C4×#11758200—24no—
14#16839815C2212yes—
15#16839701C2212yes—
16#16839682C2212yes—
17#16839669—21no—
18#16839658—21no—
19#16839646—22no—
20#16839628C2211yescharacteristic
21#16839496C2218yes—
22#16839488C2214yes—
23C2×2−1+4×C61—21no—
24C61×#550—22no—
25D4×Q8×C61—22no—
26C61×#548—24no—
27C61×#537—21no—
28C61×#537C2214yes—
29C61×#532—22no—
30C61×#532C2212yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758200—24no—
34C4×Dic122—22no—
35C4×Dic122—22no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61C2311yescharacteristic
38C2×#11758196—22no—
39C2×#11758196—21no—
40C2×C4×Dic61—21no—
41#14519708—24no—
42C4⋊#92418—22no—
43#14519700—24no—
44#14519698—24no—
45C22⋊#92418C2314yes—
46C4×#92418—22no—
47#14519684—22no—
48#14519684—22no—
49#14519684—22no—
50#14519684—21no—
51#14519684C2312yes—
522−1+4×C61—44no—
532−1+4×C61—44no—
54C2×#41×C61—24no—
55C2×#41×C61—24no—
56C2×#41×C61—22no—
57C22×Q8×C61—24no—
58C22×Q8×C61D411yescharacteristic
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61D412yes—
61#127×C61—22no—
62#127×C61D412yes—
63#124×C61—24no—
64#124×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—28no—
67#121×C61—24no—
68#121×C61—28no—
69#121×C61—22no—
70#121×C61—24no—
71#121×C61C2314yes—
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C4×D4×C61—24no—
75C4×D4×C61—24no—
76C4×D4×C61—22no—
77C4×D4×C61D414yes—
78C2×C4⋊C4×C61D411yescharacteristic
79C2×C4⋊C4×C61C2312yes—
80C2×C42×C61D412yes—
81C2×C42×C61C2311yescharacteristic
82#41×C61—48no—
83#41×C61—416no—
84#41×C61—48no—
85#41×C61—44no—
86#41×C61—44no—
87C2×Q8×C61—24no—
88C2×Q8×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—22no—
91C2×Q8×C61—416no—
92C2×Q8×C61SD1614yescomplemented
93C2×D4×C61—24no—
94C2×D4×C61—24no—
95C2×D4×C61—24no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61C2×D414yes—
99C22×C4×C61—28no—
100C22×C4×C61—24no—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—22no—
108C22×Dic61—21no—
109C2×Dic122—22no—
110C2×Dic122—42no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61C2×D414yes—
116C22⋊C4×C61—28no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—24no—
119C42×C61—22no—
120C42×C61—22no—
121C42×C61C2×D412yes—
122C42×C61C2×D412yes—
123#11758200—48no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126#11758196—44no—
127#11758196—41no—
128#11758196—41no—
129C4×Dic61—42no—
130C2×#92418—24no—
131C23×C61—24no—
132C23×C61C22×D411yescharacteristic
133Q8×C61—28no—
134Q8×C61—48no—
135Q8×C61—44no—
136D4×C61—44no—
137D4×C61—44no—
138D4×C61—48no—
139D4×C61—44no—
140D4×C61—44no—
141D4×C61—44no—
142D4×C61—84no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—24no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—416no—
150C2×C4×C61—48no—
151C2×C4×C61—42no—
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12012yes—
155C2×C4×C61C2×SD1614yes—
156C2×C4×C61C2×SD1612yes—
157C2×C4×C61AGL(1,ℤ/8)12yes—
158C2×C4×C61C22×D412yes—
159C2×Dic61—22no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167Dic122—24no—
168Dic122—42no—
169#92418—44no—
170#18219—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—44no—
174C22×C61—44no—
175C22×C61—44no—
176C22×C61C2×#12011yescharacteristic
177C22×C61C22×SD1611yescharacteristic
178C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
179C4×C61—24no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—22no—
183C4×C61—44no—
184C4×C61—42no—
185C4×C61#45014yes—
186Dic61—42no—
187Dic61—82no—
188Dic61—81no—
189#3497D6111yescharacteristic
190#3103—612no—
191#2581—611no—
192#1616—614no—
193#1604—614no—
194#1517—612no—
195#1461—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—84no—
199C2×C61C2×#45011yescharacteristic
200C2×C61#302011yescharacteristic
201C2×C61#302511yescharacteristic
202C2×2−1+4—21no—
203#550—22no—
204D4×Q8—22no—
205#548—24no—
206#537—21no—
207#537C2×D6114yes—
208#532—22no—
209#532C2×D6112yes—
210C2×C4×Q8—611no—
211C2×C4×Q8C2×D6111yescharacteristic
212#481—614no—
213#475—1222no—
214#460—1222no—
215#448—614no—
216C4×SD16—1224no—
217#431—612no—
218#400—612no—
219#395—612no—
220#391—1221no—
221#389—1221no—
222#387—1222no—
223#382—611no—
224#327—618no—
225#324—614no—
226C61#1821911yescharacteristic, complemented, Sylow
2272−1+4—44no—
2282−1+4—44no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—22no—
232C22×Q8—24no—
233C22×Q8—611no—
234C22×Q8C22⋊D6111yescharacteristic
235C2×SD16—1224no—
236C4⋊Q8—22no—
237C4⋊Q8C22⋊D6112yes—
238#127—22no—
239#127C22⋊D6112yes—
240#124—24no—
241#124—22no—
242C22⋊Q8—24no—
243C22⋊Q8—28no—
244#121—24no—
245#121—28no—
246#121—22no—
247#121—24no—
248#121C22×D6114yes—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—1222no—
252C4×Q8—1222no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—22no—
256C4×D4C22⋊D6114yes—
257C2×C4⋊C4—611no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1222no—
261C2×C4⋊C4C22⋊D6111yescharacteristic
262C2×C4⋊C4C22×D6112yes—
263C2×C42—1221no—
264C2×C42C22⋊D6112yes—
265C2×C42C22×D6111yescharacteristic
266#106—1224no—
267C4⋊C8—1222no—
268#103—1224no—
269#102—1224no—
270C22⋊C8—614no—
271C4×C8—1222no—
272#95—612no—
273#95—1221no—
274#95—1222no—
275#95—1222no—
276#95—1222no—
277#41—48no—
278#41—416no—
279#41—48no—
280#41—44no—
281#41—44no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—416no—
287C2×Q8—1222no—
288C2×Q8—2442no—
289C2×Q8#1175820014yescomplemented
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4C2×C22⋊D6114yes—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—1221no—
299C22×C4—1222no—
300C22×C4—1221no—
301C22×C4C2×C22⋊D6112yes—
302C22×C4C2×C22⋊D6112yes—
303C22×C4C2×C22⋊D6111yescharacteristic
304C22×C4C2×C22⋊D6111yescharacteristic
305C22×C4C23×D6111yescharacteristic
306SD16—2448no—
307C2×C8—1224no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—1222no—
313C4⋊C4—2444no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2442no—
317C4⋊C4C2×C22⋊D6114yes—
318C22⋊C4—28no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C42—22no—
322C42—22no—
323C42—2442no—
324C42C2×C22⋊D6112yes—
325C42C2×C22⋊D6112yes—
326C23—24no—
327C23C22×C22⋊D6111yescharacteristic
328Q8—28no—
329Q8—48no—
330Q8—44no—
331Q8—1224no—
332Q8—2442no—
333D4—44no—
334D4—44no—
335D4—48no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—84no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—416no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—1222no—
350C2×C4—2442no—
351C2×C4—2442no—
352C2×C4—2442no—
353C2×C4—2441no—
354C2×C4—2442no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4#1451974611yescharacteristic
358C2×C4#1451974611yescharacteristic
359C2×C4#1451974612yes—
360C2×C4C2×#1175820014yes—
361C2×C4C2×#1175820012yes—
362C2×C4#1451980312yes—
363C2×C4C22×C22⋊D6112yes—
364C8—2444no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22C2×#1451974611yescharacteristic
371C22C22×#1175820011yescharacteristic
372C22C2×#1451980311yescharacteristic
373C4—21no—
374C4—21no—
375C4—22no—
376C4—24no—
377C4—42no—
378C4—44no—
379C4—4882no—
380C4—4881no—
381C4—2442no—
382C4#1683991314yes—
383C2—21no—
384C2—21no—
385C2—84no—
386C2C2×#1683991311yescharacteristic
387C2#2145518311yescharacteristic
388C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18219
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21455846C2semidirect
2. #21451181C2semidirect
3. #21444856C2semidirect
4. #21444788C2semidirect
5. #21444379C2semidirect
6. #21444143C2semidirect
92. C2×Q8×C61SD16semidirect
226. C61#18219semidirect
289. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214558462
#214511812
#214448562
#214447882
#214443792
#214441432
C61×#34972
#1821961