Finite Group Databasebeta

← #29685186

Subgroups — #29685186

Order 15616.

1238 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840319C211yesmaximal, characteristic
2C22×C4×#92418C211yesmaximal, characteristic
3#21446779C212yesmaximal, complemented
4#21446620C212yesmaximal, complemented
5#21446240C214yesmaximal
6C4×#14519700C214yesmaximal, complemented
7C61×#2285C211yesmaximal, characteristic
8C23×#92418C2212yes—
9#16840319—24no—
10#16840319C418yescomplemented
11C2×#14519777C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×#14519769—21no—
14C2×#14519756—21no—
15C2×#14519756C2212yes—
16C2×C4×Dic122—21no—
17C2×C4×Dic122C2212yes—
18C2×#14519736—21no—
19C2×#14519700C2218yes—
20C2×C4×#92418—24no—
21C2×C4×#92418C2212yes—
22C2×C4×#92418C2212yes—
23#16839524C2218yes—
24C2×#117×C61—21no—
25C22×C42×C61C2211yescharacteristic
26C61×#384C2212yes—
27C61×#382—21no—
28C61×#380—21no—
29C61×#380C2212yes—
30C4×C4⋊C4×C61—21no—
31C4×C4⋊C4×C61C2212yes—
32C22×Dic122—21no—
33C22×Dic122C2312yes—
34C22×#92418—24no—
35C22×#92418—28no—
36C22×#92418C2314yes—
37#14519777—22no—
38#14519777—22no—
39#14519777C2×C414yes—
40#14519777C2×C414yes—
41#14519769—28no—
42#14519756—28no—
43#14519756—24no—
44#14519756C2×C418yes—
45C4×Dic122—28no—
46C4×Dic122—24no—
47C4×Dic122C2×C418yes—
48#14519736—28no—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×C22⋊Dic61—22no—
56C2×C4×Dic61—22no—
57#14519700—216no—
58C4×#92418—24no—
59C4×#92418—24no—
60C23×C4×C61D411yescharacteristic
61C23×C4×C61D411yescharacteristic
62C23×C4×C61C2311yescharacteristic
63#117×C61—44no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61C2314yes—
67C2×C22⋊C4×C61—22no—
68C2×C42×C61—22no—
69C2×C42×C61C2×C414yes—
70C2×C42×C61D414yes—
71C2×C42×C61D411yescharacteristic
72C2×C42×C61D411yescharacteristic
73C2×C42×C61C2311yescharacteristic
74C2×C42×C61C2311yescharacteristic
75#95×C61—22no—
76#95×C61—22no—
77C24×C61C2×D411yescharacteristic
78C22×C4×C61—24no—
79C22×C4×C61—24no—
80C22×C4×C61—24no—
81C22×C4×C61C22⋊C414yes—
82C22×C4×C61C22⋊C414yes—
83C22×C4×C61SD1614yes—
84C22×C4×C61Q1614yes—
85C22×C4×C61C22×C412yes—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C22×Dic61—22no—
94C22×Dic61—22no—
95C2×Dic122—28no—
96C2×Dic122—24no—
97C2×Dic122—22no—
98C2×Dic122—44no—
99C2×Dic122C22×C414yes—
100C4⋊C4×C61—28no—
101C4⋊C4×C61—44no—
102C4⋊C4×C61—44no—
103C22⋊C4×C61—44no—
104C42×C61—44no—
105C42×C61C22⋊C414yes—
106C42×C61C22⋊C414yes—
107C42×C61C22×C412yes—
108C42×C61C22×C412yes—
109C42×C61C2×D412yes—
110C42×C61C2×D412yes—
111C22⋊Dic61—28no—
112C4⋊Dic61—24no—
113C4⋊Dic61—24no—
114C4⋊Dic61—24no—
115C4⋊Dic61—22no—
116C4⋊Dic61C22×C414yes—
117#11758196—28no—
118#11758196—28no—
119C4×Dic61—28no—
120C2×#92418—216no—
121C2×#92418—24no—
122C2×#92418—24no—
123C2×#92418#4118yes—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61C2×C22⋊C412yes—
127C23×C61C2×SD1612yes—
128C23×C61C2×Q1612yes—
129C23×C61C22×D411yescharacteristic
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—24no—
133C2×C4×C61—24no—
134C2×C4×C61—28no—
135C2×C4×C61—44no—
136C2×C4×C61—44no—
137C2×C4×C61—44no—
138C2×C4×C61#103116yescomplemented
139C2×C4×C61C2×C22⋊C412yes—
140C2×C4×C61C2×C22⋊C412yes—
141C2×C4×C61C2×C22⋊C412yes—
142C2×C4×C61C2×C22⋊C412yes—
143C2×C4×C61C2×C22⋊C412yes—
144C2×C4×C61C2×SD1612yes—
145C2×C4×C61C2×SD1612yes—
146C2×C4×C61C2×Q1612yes—
147C2×C4×C61C2×Q1612yes—
148C2×C4×C61#13514yes—
149C2×C4×C61C23×C411yescharacteristic
150C2×C4×C61C22×D411yescharacteristic
151C2×C4×C61C2×#4114yes—
152C2×Dic61—24no—
153C2×Dic61—24no—
154C2×Dic61—44no—
155C2×Dic61—44no—
156Dic122—28no—
157Dic122—44no—
158Dic122—48no—
159#92418—28no—
160#21400—611nomaximal, Sylow
161C22×C61—22no—
162C22×C61—24no—
163C22×C61—24no—
164C22×C61—22no—
165C22×C61C2×#10314yes—
166C22×C61C22×C22⋊C411yescharacteristic
167C22×C61#53212yes—
168C22×C61C22×SD1611yescharacteristic
169C22×C61C22×Q1611yescharacteristic
170C22×C61C2×#13512yes—
171C4×C61—22no—
172C4×C61—22no—
173C4×C61—84no—
174C4×C61C2×#10314yes—
175C4×C61C2×#10314yes—
176C4×C61#41614yes—
177C4×C61C4×#4112yes—
178C4×C61#52512yes—
179Dic61—44no—
180Dic61—44no—
181C2×#533—611no—
182C22×C4×C8—611no—
183#2285D6111yescharacteristic
184#1990—612no—
185#1956—612no—
186#1892—614no—
187C4×#103—614no—
188C2×C61—24no—
189C2×C61#229711yescharacteristic
190C2×C61C22×#10311yescharacteristic
191C2×C61C2×#41611yescharacteristic
192C2×C61#316712yes—
193C2×C61#317012yes—
194C23×C8—612no—
195#533—618no—
196#533—1224no—
197C2×C4⋊Q8—611no—
198C2×C4⋊Q8—611no—
199C2×#125—1221no—
200C2×C22⋊Q8—612no—
201C2×C22⋊Q8—1221no—
202C2×C4×Q8—612no—
203C2×C4×Q8—1221no—
204C2×#117—21no—
205C2×#117—1221no—
206C22×C42C2×D6111yescharacteristic
207C2×#103—618no—
208C2×C4×C8—612no—
209C2×C4×C8—612no—
210C2×C4×C8—1224no—
211#384C2×D6112yes—
212#382—21no—
213#380—21no—
214#380C2×D6112yes—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2×D6112yes—
217#340—618no—
218C61#2140011yescharacteristic, complemented, Sylow
219C22×Q8—612no—
220C22×Q8—1221no—
221C23×C4C22⋊D6111yescharacteristic
222C23×C4C22⋊D6111yescharacteristic
223C23×C4C22×D6111yescharacteristic
224C22×C8—614no—
225C22×C8—1228no—
226C22×C8—1224no—
227C4⋊Q8—614no—
228C4⋊Q8—614no—
229C4⋊Q8—1222no—
230C4⋊Q8—1222no—
231#125—1228no—
232C22⋊Q8—618no—
233C22⋊Q8—1228no—
234C22⋊Q8—1224no—
235C4×Q8—618no—
236C4×Q8—1228no—
237C4×Q8—1224no—
238#117—44no—
239#117—1228no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—612no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4C22×D6114yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—1222no—
251C2×C42—22no—
252C2×C42—1222no—
253C2×C42C4×D6114yes—
254C2×C42D24414yes—
255C2×C42C22⋊D6111yescharacteristic
256C2×C42C22⋊D6111yescharacteristic
257C2×C42C22×D6111yescharacteristic
258C2×C42C22×D6111yescharacteristic
259#103—12216no—
260C4×C8—1224no—
261C4×C8—1224no—
262#95—22no—
263#95—22no—
264C24C2×C22⋊D6111yescharacteristic
265C2×Q8—614no—
266C2×Q8—1222no—
267C2×Q8—1224no—
268C2×Q8—1228no—
269C2×Q8—2444no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—1222no—
274C22×C4—1222no—
275C22×C4#1175819814yes—
276C22×C4#1175819814yes—
277C22×C4#1175820014yes—
278C22×C4#1175820214yes—
279C22×C4C2×C4×D6112yes—
280C22×C4C2×D24412yes—
281C22×C4C2×C22⋊D6111yescharacteristic
282C22×C4C2×C22⋊D6111yescharacteristic
283C22×C4C2×C22⋊D6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285C22×C4C2×C22⋊D6111yescharacteristic
286C22×C4C23×D6111yescharacteristic
287C2×C8—618no—
288C2×C8—12216no—
289C2×C8—1224no—
290C2×C8—1224no—
291C4⋊C4—28no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—614no—
295C4⋊C4—1222no—
296C4⋊C4—1228no—
297C4⋊C4—1228no—
298C4⋊C4—1224no—
299C4⋊C4—1224no—
300C4⋊C4—1224no—
301C22⋊C4—44no—
302C22⋊C4—1228no—
303C42—44no—
304C42—1228no—
305C42#1175819814yes—
306C42#1175819814yes—
307C42C2×C4×D6112yes—
308C42C2×C4×D6112yes—
309C42C2×D24412yes—
310C42C2×D24412yes—
311C23—22no—
312C23—22no—
313C23C2×#1175819812yes—
314C23C2×#1175820012yes—
315C23C2×#1175820212yes—
316C23C22×C22⋊D6111yescharacteristic
317Q8—1228no—
318Q8—2448no—
319Q8—2444no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—1224no—
329C2×C4—1224no—
330C2×C4—2444no—
331C2×C4—2444no—
332C2×C4#14519700116yescomplemented
333C2×C4C2×#1175819812yes—
334C2×C4C2×#1175819812yes—
335C2×C4C2×#1175819812yes—
336C2×C4C2×#1175819812yes—
337C2×C4C2×#1175819812yes—
338C2×C4C2×#1175820012yes—
339C2×C4C2×#1175820012yes—
340C2×C4C2×#1175820212yes—
341C2×C4C2×#1175820212yes—
342C2×C4#1451979714yes—
343C2×C4C22×C4×D6111yescharacteristic
344C2×C4C22×D24411yescharacteristic
345C2×C4#41×D6114yes—
346C8—1228no—
347C22—24no—
348C22—22no—
349C22—22no—
350C22—24no—
351C22C2×#1451970014yes—
352C22C22×#1175819811yescharacteristic
353C22#1684030812yes—
354C22C22×#1175820011yescharacteristic
355C22C22×#1175820211yescharacteristic
356C22C2×#1451979712yes—
357C4—22no—
358C4—22no—
359C4—84no—
360C4—2444no—
361C4—2444no—
362C4C2×#1451970014yes—
363C4C2×#1451970014yes—
364C4#1683975614yes—
365C4#1684023012yes—
366C4#1684027412yes—
367C2—24no—
368C2#2144830711yescharacteristic
369C2C22×#1451970011yescharacteristic
370C2C2×#1683975611yescharacteristic
371C2#2145637412yes—
372C2#2145640112yes—
subgroupG/H
centerC22×C4C2×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2285C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#2285D61
hypocenterC61#21400
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
3. #21446779C2semidirect
4. #21446620C2semidirect
6. C4×#14519700C2semidirect
10. #16840319C4semidirect
138. C2×C4×C61#103semidirect
218. C61#21400semidirect
332. C2×C4#14519700semidirect

Maximal subgroups

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

subgroupindex
C2×#168403192
C22×C4×#924182
#214467792
#214466202
#214462402
C4×#145197002
C61×#22852
#2140061