Finite Group Databasebeta

← #30600951

Subgroups — #30600951

Order 15616.

390 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 72 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#21462639C211yesmaximal, characteristic
2#21461260C211yesmaximal, characteristic
3#21461242C211yesmaximal, characteristic, complemented
4#21461142C211yesmaximal, characteristic, complemented
5#21460680C211yesmaximal, characteristic, complemented
6#21460675C211yesmaximal, characteristic, complemented
7#21457056C211yesmaximal, characteristic
8#16840813—21no—
9#16840795—21no—
10#16840793—21no—
11#16840791—21no—
12#16840786—21no—
13#16840785C2211yescharacteristic
14#16840781C2211yescharacteristic
15#16840726—21no—
16C4×#11758213C2211yescharacteristic
17#16840707C2211yescharacteristic
18#16840616—21no—
19#16840615—21no—
20#16840615—21no—
21#16840613—21no—
22Q8⋊#92420—21no—
23Q8⋊#92420—21no—
24#16840317C2211yescharacteristic
25#16840291C2211yescharacteristic
26#16840171C411yescharacteristic, complemented
27#16840168C411yescharacteristic, complemented
28#16840121C411yescharacteristic, complemented
29#16840102—21no—
30#16840085—21no—
31#16840067—21no—
32#16840060—21no—
33#16839817C2211yescharacteristic
34C61×#489C411yescharacteristic, complemented
35C2×#11758213—21no—
36C2×#11758217—21no—
37C22×#92420—21no—
38#14519844—21no—
39#14519844—21no—
40C4⋊#92420—21no—
41C4⋊#92420—21no—
42#14519834—21no—
43#14519834D411yescharacteristic
44#14519834D411yescharacteristic
45C22⋊#92420—21no—
46C22⋊#92420—21no—
47#14519832—21no—
48#14519832—21no—
49#14519832—21no—
50C4×#92420—21no—
51C4×#92420—21no—
52C4×#92420D411yescharacteristic
53C4×#92420D411yescharacteristic
54(C2×#6923)⋊C4—21no—
55(C2×#6923)⋊C4—21no—
56(C2×#6923)⋊C4—21no—
57#14519777C2×C411yescharacteristic
58#14519770—21no—
59#14519769C2×C411yescharacteristic
60#14519768—21no—
61#14519759—21no—
62#14519757—21no—
63#14519753—21no—
64#14519753D411yescharacteristic
65#14519748D411yescharacteristic
66Q8×Dic61—21no—
67Q8×Dic61D411yescharacteristic
68#14519742—21no—
69D4×Dic61D411yescharacteristic
70#14519737—21no—
71#14519735—21no—
72#14519733—21no—
73C42×D61C2311yescharacteristic
74#14519711—21no—
75#14519710—21no—
76#14519709—21no—
77#14519708—21no—
78#14519706—21no—
79#14519701—21no—
80#14519698—21no—
81#14519689—21no—
82#14519688C2×C411yescharacteristic
83C4⋊Q8×C61C2×C411yescharacteristic
84#124×C61C2×C411yescharacteristic
85#103×C61—21no—
86#102×C61—21no—
87#97×C61C2×C411yescharacteristic
88C22×#6923—41no—
89C2×Q8×C61—21no—
90C2×Q8×C61C22⋊C411yescharacteristic
91C2×D4×C61C22⋊C411yescharacteristic
92C2×C22⋊D61—21no—
93C22×Dic61—21no—
94C2×C4×D61C2×D411yescharacteristic
95C2×C4×D61C2×D411yescharacteristic
96C2×C4×D61#4111yescharacteristic
97C2×Dic122—21no—
98#11758217—42no—
99C2×#92420—21no—
100C2×#92420—21no—
101C2×#92420—21no—
102C2×#92420—21no—
103C2×#92420—42no—
104C2×#92420—41no—
105#11758213—42no—
106C2×C8×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C22⋊C4×C61—21no—
110C22⋊C4×C61—21no—
111C42×C61C22×C411yescharacteristic
112C22⋊Dic61—21no—
113C22⋊Dic61—21no—
114#11758198—21no—
115#11758198—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—41no—
120C4⋊Dic61C22⋊C411yescharacteristic
121C4⋊Dic61C22⋊C411yescharacteristic
122#11758196—21no—
123#11758196—21no—
124#11758196—21no—
125#11758196—41no—
126C4×Dic61—21no—
127C4×Dic61—41no—
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C4×Dic61#4111yescharacteristic
131C2×#92418—21no—
132C23×C61—21no—
133C22×D61#12111yescharacteristic
134C2×#6923—81no—
135C2×#6923—81no—
136C2×#6923—81no—
137Q8×C61—41no—
138Q8×C61—42no—
139D4×C61—41no—
140C2×C4×C61—21no—
141C2×C4×C61—41no—
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61C4×D411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C22⋊D61—81no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61#12011yescharacteristic
152C2×Dic61#12111yescharacteristic
153C2×Dic61#12311yescharacteristic
154C4×D61—41no—
155C4×D61—41no—
156C4×D61—41no—
157Dic122—81no—
158#92420—22no—
159#92420—41no—
160#92420—41no—
161#92420—41no—
162#92420—81no—
163C8×C61—41no—
164#92418—41no—
165#9369—611nomaximal, Sylow
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61#38611yescharacteristic
170C2×D61—21no—
171C2×D61—21no—
172C2×D61—21no—
173#6923—161no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—41no—
178C4×C61—81no—
179Dic61—21no—
180Dic61—21no—
181Dic61—21no—
182Dic61—41no—
183Dic61—41no—
184Dic61—81no—
185#3240—611no—
186#2967—611no—
187#1521—611no—
188#1515—611no—
189#1485—611no—
190#1288—611no—
191#1287—611no—
192C2×C61—81no—
193C2×C61#188111yescharacteristic
194C2×C61#190611yescharacteristic
195C2×C61#190911yescharacteristic
196D61—41no—
197#533—611no—
198#529—611no—
199#501—611no—
200#501—611no—
201#489—611no—
202#489#692311yescharacteristic, complemented
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#432—611no—
213#432—611no—
214#431—611no—
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⋊Q8C2×#692311yescharacteristic
230#126—1221no—
231#125—611no—
232#125—1221no—
233#124C2×#692311yescharacteristic
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×C42—611no—
247#107—1221no—
248#107—1221no—
249#106—1221no—
250#106—1221no—
251C4⋊C8—1221no—
252C4⋊C8—1221no—
253C4⋊C8—1221no—
254C4⋊C8—1221no—
255C4⋊C8—1221no—
256#103—21no—
257#103—1221no—
258#102—21no—
259#102—1221no—
260C22⋊C8—611no—
261C22⋊C8—611no—
262C22⋊C8—1221no—
263C22⋊C8—1221no—
264C22⋊C8—1221no—
265C22⋊C8—1221no—
266#97—611no—
267#97—1221no—
268#97—1221no—
269#97—1221no—
270#97C2×#692311yescharacteristic
271C4×C8—611no—
272C4×C8—611no—
273C4×C8—1221no—
274C4×C8—1221no—
275#95—1221no—
276#95—1221no—
277#95—1221no—
278C2×Q8—21no—
279C2×Q8—1221no—
280C2×Q8C22⋊#692311yescharacteristic
281C2×D4—1221no—
282C2×D4C22⋊#692311yescharacteristic
283C22×C4—611no—
284C22×C4—611no—
285C22×C4—611no—
286C22×C4—1221no—
287C22×C4—2441no—
288M16—2442no—
289M16—2442no—
290C2×C8—21no—
291C2×C8—1221no—
292C2×C8—1221no—
293C2×C8—1221no—
294C2×C8—1221no—
295C2×C8—1221no—
296C2×C8—2442no—
297C2×C8—2441no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—611no—
301C4⋊C4—611no—
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—21no—
311C22⋊C4—21no—
312C22⋊C4—1221no—
313C22⋊C4—1221no—
314C22⋊C4—1221no—
315C22⋊C4—2441no—
316C42—611no—
317C42—611no—
318C42—611no—
319C42—1221no—
320C42—2441no—
321C42C22×#692311yescharacteristic
322C23—21no—
323C23—611no—
324Q8—41no—
325Q8—42no—
326Q8—4881no—
327D4—41no—
328D4—4881no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—1221no—
335C2×C4—1221no—
336C2×C4—4881no—
337C2×C4—4881no—
338C2×C4—4881no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4C2×C22⋊#692311yescharacteristic
346C2×C4D4×#692311yescharacteristic
347C2×C4D4×#692311yescharacteristic
348C8—41no—
349C8—1222no—
350C8—4881no—
351C8—2441no—
352C8—2441no—
353C8—2441no—
354C8—2441no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—1221no—
359C22—1221no—
360C22—1221no—
361C22#1684071011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—81no—
367C4—1221no—
368C4—1221no—
369C4—1221no—
370C4—9761no—
371C4—4881no—
372C4—2441no—
373C4—2441no—
374C2—81no—
375C2—2441no—
376C2#2146181711yescharacteristic
377C2#2146185711yescharacteristic
378C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#489C4
socleC22×C61#386
prime radical#14519748D4
cosocleC42×D61C23
hypercenter#489#6923
hypocenterC61#9369
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
3. #21461242C2semidirect
4. #21461142C2semidirect
5. #21460680C2semidirect
6. #21460675C2semidirect
26. #16840171C4semidirect
27. #16840168C4semidirect
28. #16840121C4semidirect
34. C61×#489C4semidirect
202. #489#6923semidirect
224. C61#9369semidirect

Maximal subgroups

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

subgroupindex
#214626392
#214612602
#214612422
#214611422
#214606802
#214606752
#214570562
#936961