Finite Group Databasebeta

← #30600725

Subgroups — #30600725

Order 15616.

422 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 76 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#21462708C211yesmaximal, characteristic
2#21461213C211yesmaximal, characteristic, complemented
3#21461208C211yesmaximal, characteristic
4#21461183C211yesmaximal, characteristic, complemented
5#21460676C211yesmaximal, characteristic, complemented
6#21460642C211yesmaximal, characteristic, complemented
7#21456952C211yesmaximal, characteristic
8#16840797—21no—
9#16840793—22no—
10D4×#92420—21no—
11#16840784C2211yescharacteristic
12#16840784C2211yescharacteristic
13#16840780C2211yescharacteristic
14#16840735—22no—
15#16840698C2211yescharacteristic
16#16840616—22no—
17#16840613—22no—
18#16840612—21no—
19#16840612—21no—
20#16840317C2211yescharacteristic
21#16840310C2211yescharacteristic
22#16840165C411yescharacteristic, complemented
23#16840160C411yescharacteristic, complemented
24#16840108C411yescharacteristic, complemented
25#16840085—22no—
26#16840064—22no—
27#16839813C2211yescharacteristic
28C61×#486C411yescharacteristic, complemented
29C2×#11758217—22no—
30C22×#92420—22no—
31C2×C4×#6923—21no—
32#14519844—21no—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35#14519834—21no—
36#14519834—21no—
37#14519834D412yes—
38C22⋊#92420—22no—
39C22⋊#92420—22no—
40#14519832—21no—
41#14519832—21no—
42C4×#92420D411yescharacteristic
43C4×#92420D411yescharacteristic
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4—21no—
46C2×#11758224—21no—
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519768—21no—
50#14519764—21no—
51#14519757—21no—
52#14519753—21no—
53#14519748—21no—
54#14519748D412yes—
55Q8×Dic61—21no—
56#14519742—21no—
57D4×Dic61—21no—
58D4×Dic61D412yes—
59#14519735—21no—
60C42×D61C2311yescharacteristic
61#14519710—22no—
62#14519709—22no—
63#14519706—21no—
64#14519698—22no—
65#14519689—21no—
66C4×#92418C2×C411yescharacteristic
67C4⋊Q8×C61C2×C411yescharacteristic
68#127×C61C2×C411yescharacteristic
69#102×C61—22no—
70C4×C8×C61C2×C411yescharacteristic
71C22×#6923—41no—
72C2×Q8×C61—21no—
73C2×D4×C61—21no—
74C2×D4×C61—21no—
75C2×D4×C61C22⋊C412yes—
76C2×C22⋊D61—22no—
77C22×Dic61—22no—
78#11758224—82no—
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×D411yescharacteristic
81C2×C4×D61#4111yescharacteristic
82C2×Dic122—21no—
83C2×Dic122—21no—
84#11758217—82no—
85C2×#92420—21no—
86C2×#92420—21no—
87C2×#92420—22no—
88C2×#92420—41no—
89C2×#92420—82no—
90C4×#6923—22no—
91C4×#6923—41no—
92C2×C8×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C42×C61C22×C411yescharacteristic
96C22⋊Dic61—22no—
97C22⋊Dic61—22no—
98#11758198—41no—
99C4⋊Dic61—21no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102C4⋊Dic61—41no—
103C4⋊Dic61C22⋊C412yes—
104#11758196—41no—
105C4×Dic61—41no—
106C4×Dic61C2×D411yescharacteristic
107C4×Dic61C2×D411yescharacteristic
108C4×Dic61#4111yescharacteristic
109C2×#92418—21no—
110C23×C61—22no—
111C22×D61#12111yescharacteristic
112C2×#6923—42no—
113C2×#6923—81no—
114C2×#6923—81no—
115Q8×C61—81no—
116D4×C61—42no—
117D4×C61—41no—
118D4×C61—41no—
119D4×C61—81no—
120C2×C4×C61—41no—
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×C4×C61C4×D411yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C22⋊D61—82no—
125C2×Dic61—22no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—41no—
129C2×Dic61#12011yescharacteristic
130C2×Dic61#12111yescharacteristic
131C2×Dic61#12311yescharacteristic
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—41no—
135C4×D61—41no—
136Dic122—41no—
137Dic122—41no—
138Dic122—81no—
139#92420—41no—
140#92420—42no—
141#92420—41no—
142#92420—81no—
143C8×C61—41no—
144#92418—41no—
145#9279—611nomaximal, Sylow
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61#38611yescharacteristic
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—21no—
153#6923—82no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—81no—
157C4×C61#44311yescharacteristic
158C4×C61#44411yescharacteristic
159Dic61—21no—
160Dic61—21no—
161Dic61—21no—
162Dic61—42no—
163Dic61—81no—
164#3228—611no—
165#2981—611no—
166#1506—611no—
167#1505—611no—
168#1497—611no—
169#1287—611no—
170#1274—611no—
171C2×C61—82no—
172C2×C61#187611yescharacteristic
173C2×C61#190411yescharacteristic
174C2×C61#190911yescharacteristic
175D61—41no—
176#533—611no—
177#532—611no—
178#500—611no—
179#498—611no—
180#486—611no—
181#486#692311yescharacteristic, complemented
182#482—1222no—
183#478—1222no—
184#436—1221no—
185#435—1222no—
186C8×D4—1221no—
187#432—611no—
188#432—611no—
189#431—611no—
190#431—611no—
191#408—1222no—
192#376—611no—
193#328—1222no—
194#327—1222no—
195Q8⋊C8—1221no—
196Q8⋊C8—1221no—
197C61#927911yescharacteristic, complemented, Sylow
198C2×#41—1221no—
199C2×M16—1222no—
200C22×C8—1222no—
201C4⋊Q8—611no—
202C4⋊Q8—611no—
203C4⋊Q8C2×#692311yescharacteristic
204#127C2×#692311yescharacteristic
205#125—1221no—
206#124—1221no—
207C22⋊Q8—1221no—
208C22⋊Q8—1221no—
209#121—612no—
210#121—1221no—
211C4×Q8—1221no—
212C4×Q8—1221no—
213C4×D4—612no—
214C4×D4—1221no—
215#117—1221no—
216C2×C42—611no—
217C2×C42—1221no—
218#107—1222no—
219#106—1222no—
220C4⋊C8—1221no—
221C4⋊C8—1221no—
222C4⋊C8—1221no—
223C4⋊C8—1221no—
224#102—22no—
225#102—1222no—
226C22⋊C8—612no—
227C22⋊C8—1221no—
228C22⋊C8—1221no—
229C22⋊C8—1222no—
230C22⋊C8—1222no—
231C22⋊C8—1221no—
232#97—1221no—
233#97—1221no—
234C4×C8—611no—
235C4×C8—611no—
236C4×C8—611no—
237C4×C8C2×#692311yescharacteristic
238#95—1221no—
239#95—1221no—
240#41—4882no—
241C2×Q8—21no—
242C2×Q8—1221no—
243C2×Q8—1221no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—1222no—
247C2×D4C22⋊#692312yes—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—1222no—
252C22×C4—2441no—
253M16—4882no—
254C2×C8—21no—
255C2×C8—1221no—
256C2×C8—1221no—
257C2×C8—1222no—
258C2×C8—1221no—
259C2×C8—4882no—
260C2×C8—2441no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—612no—
264C4⋊C4—1221no—
265C4⋊C4—1221no—
266C4⋊C4—1221no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C22⋊C4—1222no—
270C22⋊C4—1222no—
271C22⋊C4—2441no—
272C42—611no—
273C42—611no—
274C42—611no—
275C42—1222no—
276C42—2441no—
277C42—2441no—
278C42C22×#692311yescharacteristic
279C23—22no—
280C23—611no—
281Q8—81no—
282Q8—4881no—
283Q8—2441no—
284Q8—2441no—
285D4—42no—
286D4—41no—
287D4—41no—
288D4—81no—
289D4—4882no—
290C2×C4—41no—
291C2×C4—611no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1222no—
297C2×C4—4881no—
298C2×C4—4881no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2441no—
305C2×C4C2×C22⋊#692311yescharacteristic
306C2×C4D4×#692311yescharacteristic
307C2×C4D4×#692311yescharacteristic
308C8—41no—
309C8—4881no—
310C8—2441no—
311C8—2441no—
312C8—2442no—
313C8—2441no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—1221no—
318C22—1221no—
319C22—1221no—
320C22#1684071011yescharacteristic
321C4—21no—
322C4—21no—
323C4—81no—
324C4—1221no—
325C4—1221no—
326C4—1221no—
327C4—4882no—
328C4—4881no—
329C4—2442no—
330C4#1684080511yescharacteristic
331C4#1684080711yescharacteristic
332C2—82no—
333C2—2441no—
334C2#2146180811yescharacteristic
335C2#2146185311yescharacteristic
336C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#486C4
socleC22×C61#386
prime radical#16840310C22
cosocleC42×D61C23
hypercenter#486#6923
hypocenterC61#9279
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #21461213C2semidirect
4. #21461183C2semidirect
5. #21460676C2semidirect
6. #21460642C2semidirect
22. #16840165C4semidirect
23. #16840160C4semidirect
24. #16840108C4semidirect
28. C61×#486C4semidirect
181. #486#6923semidirect
197. C61#9279semidirect

Maximal subgroups

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

subgroupindex
#214627082
#214612132
#214612082
#214611832
#214606762
#214606422
#214569522
#927961