Finite Group Databasebeta

← #29594894

Subgroups — #29594894

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519716C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic
3C2×#107×D61C211yesmaximal, characteristic
4#21443911C211yesmaximal, characteristic, complemented
5#21443910C211yesmaximal, characteristic, complemented
6#21443910C211yesmaximal, characteristic, complemented
7C61×#1380C211yesmaximal, characteristic, complemented
8C22×C8×D61C2211yescharacteristic
9C2×#14519735—21no—
10C2×C4⋊C4×D61—21no—
11C2×#14519716—22no—
12C2×#14519716—22no—
13C2×#14519716C414yescomplemented
14C22×#11758185C2211yescharacteristic
15#16839799—22no—
16#16839799C414yescomplemented
17C2×#14519711C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710—21no—
20#107×D61—22no—
21#107×D61C414yescomplemented
22C2×#14519708—21no—
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C2×#107×C61C2211yescharacteristic
26C22×C4×D61—41no—
27C22×C4×D61D411yescharacteristic
28C22×#92418D411yescharacteristic
29C2×C8×D61C2×C412yes—
30C2×C8×D61C2×C412yes—
31C2×C8×D61C2×C412yes—
32C2×C8×D61D412yes—
33C2×C8×D61D412yes—
34C2×C8×D61Q812yes—
35#14519735—24no—
36#14519735—42no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×#11758196—41no—
40C4⋊C4×D61—24no—
41C4⋊C4×D61—42no—
42C2×#11758198—41no—
43C2×C4×Dic61—41no—
44#14519716—28no—
45C2×#11758185—21no—
46C2×#11758185—21no—
47C2×#11758185C2×C412yescomplemented
48#14519711—21no—
49#14519711—21no—
50#14519711C2×C412yes—
51#14519711C2×C412yescomplemented
52#14519710—24no—
53#14519708—24no—
54C22×C8×C61C2311yescharacteristic
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C16×C61—21no—
58C2×C16×C61—21no—
59C2×C16×C61C2×C412yescomplemented
60#107×C61—21no—
61#107×C61—21no—
62#107×C61C2×C412yes—
63#107×C61C2×C412yescomplemented
64C23×D61D811yescharacteristic
65C22×C4×C61—41no—
66C22×C4×C61—41no—
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—41no—
69C22×Dic61—41no—
70C22×Dic61D811yescharacteristic
71C2×C4×D61—44no—
72C2×C4×D61—82no—
73C2×C4×D61—82no—
74C2×C4×D61C22⋊C412yes—
75C2×C4×D61C22⋊C412yes—
76C2×C4×D61C4⋊C412yes—
77C2×C4×D61SD1612yes—
78C2×C4×D61SD1612yes—
79C2×C4×D61Q1612yes—
80C2×C8×C61C22×C411yescharacteristic
81C2×C8×C61C22×C411yescharacteristic
82C2×C8×C61C22×C411yescharacteristic
83C2×C8×C61C2×D411yescharacteristic
84C2×C8×C61C2×D411yescharacteristic
85C2×C8×C61C2×Q811yescharacteristic
86C4⋊C4×C61—22no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90#11758198—44no—
91C4⋊Dic61—22no—
92C4⋊Dic61—22no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95#11758196—44no—
96C4×Dic61—44no—
97C2×#92418C22⋊C412yes—
98C2×#92418C22⋊C412yes—
99C2×#92418C4⋊C412yes—
100C8×D61C4214yes—
101C8×D61C22⋊C414yes—
102C8×D61C4⋊C414yes—
103C8×D61C4⋊C414yes—
104C16×C61—22no—
105#11758185—22no—
106C23×C61C2×D811yescharacteristic
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—21no—
110C22×D61—21no—
111C22×D61#10212yes—
112C22×D61#10212yes—
113C22×D61#10712yes—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—81no—
117C2×C4×C61—81no—
118C2×C4×C61—81no—
119C2×C4×C61—81no—
120C2×C4×C61C2×C22⋊C411yescharacteristic
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×C4×C61C2×C4⋊C411yescharacteristic
123C2×C4×C61C2×SD1611yescharacteristic
124C2×C4×C61C2×SD1611yescharacteristic
125C2×C4×C61C2×Q1611yescharacteristic
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—81no—
129C2×Dic61—81no—
130C2×Dic61—81no—
131C2×Dic61—81no—
132C2×Dic61#10212yes—
133C2×Dic61#10212yes—
134C2×Dic61#10712yes—
135C4×D61—88no—
136C4×D61#9514yes—
137C4×D61#10314yes—
138C4×D61#10314yes—
139C4×D61#10614yes—
140C8×C61C2×C4211yescharacteristic
141C8×C61C2×C22⋊C411yescharacteristic
142C8×C61C2×C4⋊C411yescharacteristic
143C8×C61C2×C4⋊C411yescharacteristic
144#92418#9514yes—
145#12557—611nomaximal, Sylow
146C22×C61C2×#10211yescharacteristic
147C22×C61C2×#10211yescharacteristic
148C22×C61C2×#10711yescharacteristic
149C22×C61#50911yescharacteristic
150C22×C61#50911yescharacteristic
151C22×C61#51011yescharacteristic
152C22×C61#51011yescharacteristic
153C2×D61—22no—
154C2×D61—22no—
155C2×D61—22no—
156C2×D61—22no—
157C2×D61—22no—
158C2×D61—22no—
159C2×D61#34014yes—
160C4×C61—82no—
161C4×C61—82no—
162C4×C61C2×#9511yescharacteristic
163C4×C61C2×#10311yescharacteristic
164C4×C61C2×#10311yescharacteristic
165C4×C61C2×#10611yescharacteristic
166Dic61—82no—
167Dic61—82no—
168Dic61#34014yes—
169C22×M32—611no—
170C2×#427—611no—
171C22×#107—611no—
172#1380—611no—
173#1380—611no—
174#1380—611no—
175#1380D6111yescharacteristic, complemented
176C2×C61C2×#34011yescharacteristic
177C2×C61#213911yescharacteristic
178C2×C61#213911yescharacteristic
179C2×C61#214011yescharacteristic
180C2×C61#214111yescharacteristic
181C2×C61#215911yescharacteristic
182C2×C61#215911yescharacteristic
183D61—24no—
184C23×C8—611no—
185C2×#117—1221no—
186C22×C4⋊C4—1221no—
187C2×M32—614no—
188C2×M32—1222no—
189C2×M32—1222no—
190C22×C16—611no—
191C22×C16C2×D6111yescharacteristic
192#427—614no—
193#427—1222no—
194C2×#107—611no—
195C2×#107—611no—
196C2×#107—614no—
197C2×#107—1222no—
198C2×#107—1221no—
199C2×#107C2×D6111yescharacteristic
200C2×#107C2×D6111yescharacteristic
201C2×#106—1221no—
202C61#1255711yescharacteristic, complemented, Sylow
203C23×C4—611no—
204C23×C4—2441no—
205C22×C8—611no—
206C22×C8—612no—
207C22×C8—612no—
208C22×C8—612no—
209C22×C8—612no—
210C22×C8—612no—
211C22×C8—612no—
212C22×C8C22×D6111yescharacteristic
213#117—1224no—
214#117—2442no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—1224no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—2442no—
221C2×C4⋊C4—2441no—
222C2×C22⋊C4—2441no—
223C2×C42—2441no—
224M32—1228no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—612no—
228C2×C16—1221no—
229C2×C16—1221no—
230C2×C16C4×D6112yescomplemented
231#107—21no—
232#107—21no—
233#107—612no—
234#107—612no—
235#107—1224no—
236#107—1221no—
237#107—1221no—
238#107C4×D6112yes—
239#107C4×D6112yescomplemented
240#106—1224no—
241C24—611no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—611no—
245C22×C4—612no—
246C22×C4—612no—
247C22×C4—612no—
248C22×C4—612no—
249C22×C4—612no—
250C22×C4—612no—
251C22×C4—4882no—
252C22×C4—4882no—
253C22×C4—2444no—
254C22×C4—2441no—
255C22×C4—2441no—
256C22×C4D4×D6111yescharacteristic
257C2×C8—612no—
258C2×C8—612no—
259C2×C8—612no—
260C2×C8—614no—
261C2×C8—614no—
262C2×C8—614no—
263C2×C8—614no—
264C2×C8C2×C4×D6111yescharacteristic
265C2×C8C2×C4×D6111yescharacteristic
266C2×C8C2×C4×D6111yescharacteristic
267C2×C8D4×D6111yescharacteristic
268C2×C8D4×D6111yescharacteristic
269C2×C8Q8×D6111yescharacteristic
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—1222no—
275C4⋊C4—1222no—
276C4⋊C4—2444no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C22⋊C4—2444no—
280C42—2444no—
281C16—22no—
282C16—1222no—
283C23—612no—
284C23—612no—
285C23—612no—
286C23—1221no—
287C23—1221no—
288C23—1221no—
289C23—1221no—
290C23D8×D6111yescharacteristic
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—612no—
298C2×C4—612no—
299C2×C4—612no—
300C2×C4—614no—
301C2×C4—614no—
302C2×C4—614no—
303C2×C4—614no—
304C2×C4—4888no—
305C2×C4—4881no—
306C2×C4—4881no—
307C2×C4—4881no—
308C2×C4—4881no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4C22⋊C4×D6111yescharacteristic
312C2×C4C22⋊C4×D6111yescharacteristic
313C2×C4C4⋊C4×D6111yescharacteristic
314C2×C4SD16×D6111yescharacteristic
315C2×C4SD16×D6111yescharacteristic
316C2×C4Q16×D6111yescharacteristic
317C8—614no—
318C8C42×D6111yescharacteristic
319C8C22⋊C4×D6111yescharacteristic
320C8C4⋊C4×D6111yescharacteristic
321C8C4⋊C4×D6111yescharacteristic
322C22—614no—
323C22—1222no—
324C22—1222no—
325C22—1222no—
326C22—1222no—
327C22—1222no—
328C22—1222no—
329C22#102×D6111yescharacteristic
330C22#102×D6111yescharacteristic
331C22#107×D6111yescharacteristic
332C22#1684019611yescharacteristic
333C22#1684019911yescharacteristic
334C22#1684020111yescharacteristic
335C22#1684020411yescharacteristic
336C4—82no—
337C4—82no—
338C4—614no—
339C4—4882no—
340C4—4882no—
341C4#95×D6111yescharacteristic
342C4#103×D6111yescharacteristic
343C4#103×D6111yescharacteristic
344C4#106×D6111yescharacteristic
345C2—1224no—
346C2#340×D6111yescharacteristic
347C2#2144752811yescharacteristic
348C2#2144753011yescharacteristic
349C2#2144753311yescharacteristic
350C2#2144753711yescharacteristic
351C2#2144760411yescharacteristic
352C2#2144760611yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC8×C61C2×C42
FrattiniC22×C8C22×D61
FittingC61×#1380C2
socleC23×C61C2×D8
prime radicalC23×D61D8
cosocleC22×C8×C61C23
hypercenter#1380D61
hypocenterC61#12557
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
4. #21443911C2semidirect
5. #21443910C2semidirect
6. #21443910C2semidirect
7. C61×#1380C2semidirect
13. C2×#14519716C4semidirect
16. #16839799C4semidirect
21. #107×D61C4semidirect
47. C2×#11758185C2×C4semidirect
51. #14519711C2×C4semidirect
59. C2×C16×C61C2×C4semidirect
63. #107×C61C2×C4semidirect
175. #1380D61semidirect
202. C61#12557semidirect
230. C2×C16C4×D61semidirect
239. #107C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197162
C2×#168397992
C2×#107×D612
#214439112
#214439102
C61×#13802
#1255761