Finite Group Databasebeta

← #29577231

Subgroups — #29577231

Order 15616.

390 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 85 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#21457061C211yesmaximal, characteristic
2#21454799C211yesmaximal, characteristic
3#21444397C211yesmaximal, characteristic
4#21444371C211yesmaximal, characteristic, complemented
5#21444239C211yesmaximal, characteristic, complemented
6#21443685C211yesmaximal, characteristic, complemented
7C61×#1287C211yesmaximal, characteristic, complemented
8#16840319C2211yescharacteristic
9#16840296C2211yescharacteristic
10#16840171C411yescharacteristic, complemented
11#16840171C411yescharacteristic, complemented
12#16840122C411yescharacteristic, complemented
13#16840122C411yescharacteristic, complemented
14#16840104—21no—
15#16840087—21no—
16#16840070—21no—
17#16840063—21no—
18#16839891—21no—
19#16839837—21no—
20#16839836—21no—
21#16839832—21no—
22#16839824—21no—
23#16839719—21no—
24C4×#11758194C2211yescharacteristic
25#16839630C2211yescharacteristic
26#16839500—21no—
27#16839500—21no—
28#16839500—21no—
29#16839497—21no—
30#16839492—21no—
31#16839492—21no—
32C61×#432C2211yescharacteristic
33C61×#432C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C2×#11758194—21no—
36C2×#11758189—21no—
37C2×C8×D61—21no—
38#14519777C2×C411yescharacteristic
39#14519777C2×C411yescharacteristic
40#14519771—21no—
41#14519769—21no—
42#14519769C2×C411yescharacteristic
43#14519765C2×C411yescharacteristic
44#14519760—21no—
45#14519756—21no—
46#14519756—21no—
47#14519756D411yescharacteristic
48#14519751D411yescharacteristic
49C4×Dic122—21no—
50C4×Dic122—21no—
51C4×Dic122D411yescharacteristic
52C4×D244D411yescharacteristic
53C4×C22⋊D61—21no—
54#14519736—21no—
55#14519734—21no—
56#14519711—21no—
57#14519711—21no—
58#14519709—21no—
59#14519709—21no—
60#14519706—21no—
61#14519706—21no—
62#14519702—21no—
63#14519702—21no—
64#14519699—21no—
65#14519699—21no—
66#14519689—21no—
67#14519689—21no—
68#14519688—21no—
69#14519687—21no—
70C4×#92418D411yescharacteristic
71C4×#92418D411yescharacteristic
72C8×Dic61—21no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61—21no—
80C22⋊C8×C61—21no—
81C22⋊C8×C61—21no—
82C22⋊C8×C61D411yescharacteristic
83C22⋊C8×C61D411yescharacteristic
84#97×C61—21no—
85#97×C61C2×C411yescharacteristic
86#97×C61C2×C411yescharacteristic
87C4×C8×C61—21no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C2×C22⋊D61—21no—
92C22×Dic61—41no—
93C2×D244C22⋊C411yescharacteristic
94C2×C4×D61—21no—
95C2×Dic122—21no—
96C2×Dic122—21no—
97C2×Dic122C22⋊C411yescharacteristic
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C2×C8×C61—21no—
101C2×C8×C61—21no—
102C2×C8×C61—41no—
103C42×C61C22×C411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61#4111yescharacteristic
107C22⋊Dic61—21no—
108C22⋊Dic61—41no—
109#11758198—21no—
110#11758198—21no—
111#11758198—21no—
112#11758198—21no—
113C4⋊Dic61—21no—
114C4⋊Dic61—21no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61C22⋊C411yescharacteristic
118C4⋊Dic61C22⋊C411yescharacteristic
119#11758196—21no—
120#11758196—21no—
121#11758196—21no—
122#11758196—21no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127#11758194—42no—
128C2×#92418—21no—
129C2×#92418—21no—
130#11758189—42no—
131C8×D61—42no—
132C23×C61#12111yescharacteristic
133C22×D61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C22⋊D61—81no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—81no—
150C2×Dic61—81no—
151C2×Dic61—81no—
152D244—41no—
153C4×D61—41no—
154C4×D61—41no—
155Dic122—41no—
156Dic122—42no—
157Dic122—81no—
158C8×C61—41no—
159C8×C61—41no—
160C8×C61—41no—
161C8×C61—41no—
162C8×C61—81no—
163#92418—22no—
164#92418—41no—
165#9369—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61#38611yescharacteristic
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179Dic61—41no—
180Dic61—41no—
181Dic61—41no—
182Dic61—81no—
183Dic61—81no—
184Dic61—161no—
185#3240—611no—
186#2967—611no—
187#1521—611no—
188#1515—611no—
189#1485—611no—
190#1288—611no—
191#1287D6111yescharacteristic, complemented
192C2×C61—41no—
193C2×C61#188111yescharacteristic
194C2×C61#190611yescharacteristic
195C2×C61#190911yescharacteristic
196D61—81no—
197#533—611no—
198#529—611no—
199#501—611no—
200#501—611no—
201#489—611no—
202#489—611no—
203#485—1221no—
204#482—1221no—
205#479—1221no—
206#478—1221no—
207#446—1221no—
208#436—1221no—
209#435—1221no—
210#435—1221no—
211#433—1221no—
212#432C2×D6111yescharacteristic
213#432C2×D6111yescharacteristic
214#431C2×D6111yescharacteristic
215#405—1221no—
216C4×M16—611no—
217#383—611no—
218#328—1221no—
219#328—1221no—
220#328—1221no—
221#327—1221no—
222Q8⋊C8—1221no—
223Q8⋊C8—1221no—
224C61#936911yescharacteristic, complemented, Sylow
225C2×M16—1221no—
226C2×M16—1221no—
227C22×C8—1221no—
228C4⋊Q8—611no—
229C4⋊Q8—611no—
230#126—1221no—
231#125—611no—
232#125—1221no—
233#124—611no—
234#123—1221no—
235C22⋊Q8—611no—
236C22⋊Q8—1221no—
237C22⋊Q8—1221no—
238#121—611no—
239C4×Q8—611no—
240C4×Q8—1221no—
241C4×Q8—1221no—
242C4×D4—611no—
243C4×D4—1221no—
244#117—1221no—
245#117—1221no—
246C2×C42C22×D6111yescharacteristic
247#107—1221no—
248#107—1221no—
249#106—1221no—
250#106—1221no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—1221no—
255C4⋊C8—1221no—
256#103—1221no—
257#103—1221no—
258#102—1221no—
259#102—1221no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—1221no—
263C22⋊C8—1221no—
264C22⋊C8D24411yescharacteristic
265C22⋊C8D24411yescharacteristic
266#97—21no—
267#97—1221no—
268#97—1221no—
269#97C4×D6111yescharacteristic
270#97C4×D6111yescharacteristic
271C4×C8—21no—
272C4×C8—611no—
273C4×C8—611no—
274C4×C8—1221no—
275#95—1221no—
276#95—1221no—
277#95—1221no—
278C2×Q8—611no—
279C2×Q8—1221no—
280C2×Q8—1221no—
281C2×D4—611no—
282C2×D4—1221no—
283C22×C4—1221no—
284C22×C4—2441no—
285C22×C4C2×D24411yescharacteristic
286C22×C4D4×D6111yescharacteristic
287C22×C4#1175822611yescharacteristic
288M16—2442no—
289M16—2442no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—1221no—
296C2×C8—1221no—
297C2×C8—2442no—
298C4⋊C4—611no—
299C4⋊C4—611no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—1221no—
308C4⋊C4—2441no—
309C4⋊C4—2441no—
310C22⋊C4—1221no—
311C22⋊C4—1221no—
312C22⋊C4—1221no—
313C22⋊C4—1221no—
314C22⋊C4—1221no—
315C22⋊C4—2441no—
316C42—1221no—
317C42—2441no—
318C42C2×C4×D6111yescharacteristic
319C42#1175822211yescharacteristic
320C42D4×D6111yescharacteristic
321C42D4×D6111yescharacteristic
322C23—1221no—
323C23#1451975211yescharacteristic
324Q8—4881no—
325Q8—2442no—
326Q8—2441no—
327D4—4881no—
328D4—2441no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—1221no—
333C2×C4—1221no—
334C2×C4—1221no—
335C2×C4—4881no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4C22⋊C4×D6111yescharacteristic
343C2×C4#1451973911yescharacteristic
344C2×C4C4×D24411yescharacteristic
345C2×C4#1451974711yescharacteristic
346C2×C4#1451975011yescharacteristic
347C2×C4#1451976211yescharacteristic
348C8—41no—
349C8—41no—
350C8—41no—
351C8—41no—
352C8—81no—
353C8—1222no—
354C8—2441no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—2441no—
359C22—2441no—
360C22—2441no—
361C22#1683964011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—9761no—
369C4—4881no—
370C4—4881no—
371C4—2441no—
372C4—2441no—
373C4—2441no—
374C2—41no—
375C2—4881no—
376C2#2144618211yescharacteristic
377C2#2144632111yescharacteristic
378C2#2144634011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1287C2
socleC22×C61#386
prime radical#14519751D4
cosocleC2×C42×C61C23
hypercenter#1287D61
hypocenterC61#9369
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

Complements

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

NH ≅ G/Ntype
4. #21444371C2semidirect
5. #21444239C2semidirect
6. #21443685C2semidirect
7. C61×#1287C2semidirect
10. #16840171C4semidirect
11. #16840171C4semidirect
12. #16840122C4semidirect
13. #16840122C4semidirect
191. #1287D61semidirect
224. C61#9369semidirect

Maximal subgroups

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

subgroupindex
#214570612
#214547992
#214443972
#214443712
#214442392
#214436852
C61×#12872
#936961