Finite Group Databasebeta

← #29654856

Subgroups — #29654856

Order 15616.

1174 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 207 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#21456741C212yesmaximal, complemented
2#21451024C211yesmaximal, characteristic, complemented
3#21444788C218yesmaximal, complemented
4#21444314C212yesmaximal, complemented
5#21444143C211yesmaximal, characteristic, complemented
6C61×#3497C211yesmaximal, characteristic
7C2×#14519777C2211yescharacteristic
8#16840134—22no—
9#16840114—22no—
10#16839967—24no—
11#16839900C2218yes—
12#16839815C2212yes—
13#16839682C2214yes—
14#16839669—21no—
15#16839660—22no—
16#16839631—21no—
17C4×C4⋊Dic61C2211yescharacteristic
18#16839496C2218yes—
19#16839488C2214yes—
20C2×2−1+4×C61—21no—
21C61×#550—22no—
22D4×Q8×C61—22no—
23C61×#548—24no—
24C61×#537—21no—
25C61×#537C2214yes—
26C61×#532—22no—
27C61×#532C2212yes—
28C2×C4×Q8×C61C2211yescharacteristic
29C22×Dic122C2312yes—
30C2×#11758200—28no—
31#14519777—22no—
32#14519777—22no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61—21no—
36C2×#11758196—24no—
37C2×C4×Dic61—21no—
38C4⋊#92418—22no—
39#14519700—28no—
40C22⋊#92418C2314yes—
41C4×#92418—22no—
42#14519684—24no—
43#14519684C2312yes—
442−1+4×C61—44no—
452−1+4×C61—44no—
46C2×#41×C61—24no—
47C2×#41×C61—24no—
48C2×#41×C61—22no—
49C22×Q8×C61—24no—
50C22×Q8×C61D411yescharacteristic
51C4⋊Q8×C61—22no—
52C4⋊Q8×C61D412yes—
53#127×C61—22no—
54#127×C61D412yes—
55#124×C61—24no—
56#124×C61—22no—
57C22⋊Q8×C61—24no—
58C22⋊Q8×C61—28no—
59#121×C61—24no—
60#121×C61—28no—
61#121×C61—22no—
62#121×C61—24no—
63#121×C61C2314yes—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×D4×C61—24no—
67C4×D4×C61—24no—
68C4×D4×C61—22no—
69C4×D4×C61D414yes—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61C2312yes—
72C2×C42×C61D412yes—
73C2×C42×C61C2311yescharacteristic
74#41×C61—48no—
75#41×C61—416no—
76#41×C61—48no—
77#41×C61—44no—
78#41×C61—44no—
79C2×Q8×C61—24no—
80C2×Q8×C61—22no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—416no—
84C2×Q8×C61SD1614yescomplemented
85C2×D4×C61—24no—
86C2×D4×C61—24no—
87C2×D4×C61—24no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61C2×D414yes—
91C22×C4×C61—28no—
92C22×C4×C61—24no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61—22no—
100C2×Dic122—24no—
101C2×Dic122—82no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—24no—
106C4⋊C4×C61C2×D414yes—
107C22⋊C4×C61—28no—
108C22⋊C4×C61—24no—
109C22⋊C4×C61—24no—
110C42×C61—22no—
111C42×C61—22no—
112C42×C61C2×D412yes—
113C42×C61C2×D412yes—
114#11758200—88no—
115C4⋊Dic61—24no—
116C4⋊Dic61—41no—
117C4⋊Dic61—42no—
118C4⋊Dic61—41no—
119#11758196—84no—
120C4×Dic61—42no—
121C2×#92418—24no—
122C23×C61—24no—
123C23×C61C22×D411yescharacteristic
124Q8×C61—28no—
125Q8×C61—48no—
126Q8×C61—44no—
127D4×C61—44no—
128D4×C61—48no—
129D4×C61—44no—
130D4×C61—44no—
131D4×C61—44no—
132D4×C61—44no—
133D4×C61—84no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—41no—
140C2×C4×C61—416no—
141C2×C4×C61—48no—
142C2×C4×C61—42no—
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61#12012yes—
145C2×C4×C61#12011yescharacteristic
146C2×C4×C61C2×SD1614yes—
147C2×C4×C61C2×SD1612yes—
148C2×C4×C61AGL(1,ℤ/8)12yes—
149C2×C4×C61C22×D412yes—
150C2×Dic61—24no—
151C2×Dic61—42no—
152C2×Dic61—42no—
153C2×Dic61—44no—
154C2×Dic61—42no—
155Dic122—48no—
156#92418—44no—
157#18402—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—44no—
161C22×C61—44no—
162C22×C61—44no—
163C22×C61C2×#12011yescharacteristic
164C22×C61C22×SD1611yescharacteristic
165C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
166C4×C61—24no—
167C4×C61—21no—
168C4×C61—22no—
169C4×C61—44no—
170C4×C61—42no—
171C4×C61#45014yes—
172C4×C61#45612yes—
173Dic61—44no—
174Dic61—82no—
175#3497D6111yescharacteristic
176#3206—612no—
177#2567—611no—
178#1604—618no—
179#1502—612no—
180#1461—611no—
181C2×C61—21no—
182C2×C61—21no—
183C2×C61—84no—
184C2×C61C2×#45011yescharacteristic
185C2×C61C2×#45611yescharacteristic
186C2×C61#302711yescharacteristic
187C2×2−1+4—21no—
188#550—22no—
189D4×Q8—22no—
190#548—24no—
191#537—21no—
192#537C2×D6114yes—
193#532—22no—
194#532C2×D6112yes—
195C2×C4⋊Q8—611no—
196C2×C4×Q8C2×D6111yescharacteristic
197#492—1222no—
198#487—1222no—
199#461—1224no—
200#448—618no—
201#431—612no—
202#395—614no—
203#391—1221no—
204#389—1222no—
205#384—1221no—
206C4×C4⋊C4—611no—
207#327—618no—
208#324—614no—
209C61#1840211yescharacteristic, complemented, Sylow
2102−1+4—44no—
2112−1+4—44no—
212C2×#41—24no—
213C2×#41—24no—
214C2×#41—22no—
215C22×Q8—24no—
216C22×Q8—612no—
217C22×Q8C22⋊D6111yescharacteristic
218C2×SD16—1228no—
219C4⋊Q8—22no—
220C4⋊Q8—1222no—
221C4⋊Q8—1222no—
222C4⋊Q8C22⋊D6112yes—
223#127—22no—
224#127C22⋊D6112yes—
225#124—24no—
226#124—22no—
227C22⋊Q8—24no—
228C22⋊Q8—28no—
229#121—24no—
230#121—28no—
231#121—22no—
232#121—24no—
233#121C22×D6114yes—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—22no—
239C4×D4C22⋊D6114yes—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1224no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4C22⋊D6111yescharacteristic
245C2×C4⋊C4C22×D6112yes—
246C2×C42—1221no—
247C2×C42C22⋊D6112yes—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—1222no—
250#103—1228no—
251C22⋊C8—614no—
252C4×C8—1222no—
253#95—612no—
254#95—1224no—
255#41—48no—
256#41—416no—
257#41—48no—
258#41—44no—
259#41—44no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—416no—
265C2×Q8—1224no—
266C2×Q8—4882no—
267C2×Q8#1175820014yescomplemented
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4C2×C22⋊D6114yes—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—1222no—
277C22×C4—1222no—
278C22×C4C2×C22⋊D6111yescharacteristic
279C22×C4C2×C22⋊D6112yes—
280C22×C4C2×C22⋊D6112yes—
281C22×C4C2×C22⋊D6111yescharacteristic
282C22×C4C23×D6111yescharacteristic
283SD16—4888no—
284C2×C8—1224no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—1224no—
290C4⋊C4—4884no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C4⋊C4—2442no—
294C4⋊C4C2×C22⋊D6114yes—
295C22⋊C4—28no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C42—22no—
299C42—22no—
300C42—2442no—
301C42C2×C22⋊D6112yes—
302C42C2×C22⋊D6112yes—
303C23—24no—
304C23C22×C22⋊D6111yescharacteristic
305Q8—28no—
306Q8—48no—
307Q8—44no—
308Q8—2448no—
309D4—44no—
310D4—48no—
311D4—44no—
312D4—44no—
313D4—44no—
314D4—44no—
315D4—84no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—41no—
322C2×C4—416no—
323C2×C4—48no—
324C2×C4—42no—
325C2×C4—1224no—
326C2×C4—2444no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4#1451974611yescharacteristic
331C2×C4#1451974611yescharacteristic
332C2×C4#1451974612yes—
333C2×C4C2×#1175820014yes—
334C2×C4C2×#1175820012yes—
335C2×C4#1451980312yes—
336C2×C4C22×C22⋊D6112yes—
337C8—2444no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22—44no—
342C22—44no—
343C22C2×#1451974611yescharacteristic
344C22C22×#1175820011yescharacteristic
345C22C2×#1451980311yescharacteristic
346C4—21no—
347C4—22no—
348C4—24no—
349C4—42no—
350C4—44no—
351C4—4882no—
352C4—2444no—
353C4#1683991314yes—
354C4#1683994512yes—
355C2—21no—
356C2—21no—
357C2—84no—
358C2C2×#1683991311yescharacteristic
359C2C2×#1683994511yescharacteristic
360C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18402
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. #21456741C2semidirect
2. #21451024C2semidirect
3. #21444788C2semidirect
4. #21444314C2semidirect
5. #21444143C2semidirect
84. C2×Q8×C61SD16semidirect
209. C61#18402semidirect
267. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567412
#214510242
#214447882
#214443142
#214441432
C61×#34972
#1840261