Finite Group Databasebeta

← #30607561

Subgroups — #30607561

Order 15616.

516 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 90 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#21462472C211yesmaximal, characteristic, complemented
2#21462141C212yesmaximal, complemented
3D61⋊#349C211yesmaximal, characteristic, complemented
4#21460886C212yesmaximal, complemented
5#21455430C211yesmaximal, characteristic
6C2×#14519856—21no—
7D61⋊M32C2212yes—
8#16840773—24no—
9#16840772—21no—
10#16840772C2212yes—
11#16840679—24no—
12#16840659—21no—
13#16840658—21no—
14C2×#14519800C2211yescharacteristic
15C2×M16×D61C2211yescharacteristic
16#16840275—21no—
17#16840017C411yescharacteristic, complemented
18#16840016C411yescharacteristic, complemented
19#16840015C411yescharacteristic, complemented
20#16840011—21no—
21#16840010—21no—
22#16839797C2211yescharacteristic
23#16839757—21no—
24C61×#470C411yescharacteristic, complemented
25#14519856—22no—
26#14519856—22no—
27#14519856—41no—
28#14519856—41no—
29C2×C4⋊#6923—22no—
30C2×C22⋊#6923—41no—
31C2×C4×#6923—41no—
32#14519852—22no—
33#14519851—22no—
34D61⋊C16—22no—
35#14519846—22no—
36#14519846—22no—
37#14519845—22no—
38#14519845D414yescomplemented
39C2×#11758226—21no—
40C22×C4×D61D411yescharacteristic
41#14519800—22no—
42#14519800—22no—
43C2×#11758201—21no—
44C2×D488C2×C411yescharacteristic, complemented
45C2×#11758194D411yescharacteristic
46C2×#11758189C2×C411yescharacteristic
47C2×#11758189D411yescharacteristic
48M16×D61—22no—
49M16×D61—22no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52#14519762—41no—
53#14519753—21no—
54#14519752—21no—
55#14519751—21no—
56#14519747—41no—
57#14519739—41no—
58#14519737—41no—
59#14519735—21no—
60#14519710—21no—
61#14519709C2×C411yescharacteristic, complemented
62#14519701—21no—
63#14519699—21no—
64#14519697—21no—
65C2×Q16×C61C2×C411yescharacteristic, complemented
66C2×M16×C61C2×C411yescharacteristic
67C22⋊Q8×C61—21no—
68#106×C61C2×C411yescharacteristic, complemented
69#103×C61—21no—
70C23×D61D811yescharacteristic
71C22×#6923—42no—
72C2×Q8×C61—21no—
73C22×C4×C61C22⋊C411yescharacteristic
74C2×C22⋊D61—41no—
75C22×Dic61SD1611yescharacteristic
76#11758226—41no—
77#11758226—41no—
78#11758226—41no—
79#11758226—41no—
80C2×D244—21no—
81C2×D244—41no—
82C2×D244—41no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—41no—
88C2×C4×D61C22⋊C411yescharacteristic
89C2×C4×D61D811yescharacteristic
90C2×C4×D61SD1611yescharacteristic
91C2×C4×D61C2×D411yescharacteristic
92C22⋊#6923—42no—
93C22⋊#6923—42no—
94C4⋊#6923—24no—
95C4⋊#6923—42no—
96C4×#6923—42no—
97C4×#6923—42no—
98Q16×C61—22no—
99M16×C61—22no—
100C2×C8×C61C22⋊C411yescharacteristic
101C2×C8×C61C22×C411yescharacteristic
102C4⋊C4×C61—21no—
103C4⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105#11758201—42no—
106#11758198—41no—
107#11758198—41no—
108#11758198—41no—
109#11758198—41no—
110C4⋊Dic61—21no—
111#11758196—41no—
112C4×Dic61—41no—
113#11758194—22no—
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C2×D411yescharacteristic
116D488—22no—
117#11758189—22no—
118#11758189—22no—
119C8×D61—21no—
120C8×D61—21no—
121C8×D61C2×D412yes—
122C8×D61#4112yes—
123#11758187—42no—
124C23×C61#10211yescharacteristic
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—41no—
128C22×D61—81no—
129C22×D61#10211yescharacteristic
130C22×D61C2×D811yescharacteristic
131C2×#6923—44no—
132C2×#6923—84no—
133Q8×C61—41no—
134Q8×C61—41no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61#10211yescharacteristic
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C22⋊D61—82no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—41no—
145C2×Dic61#10211yescharacteristic
146C2×Dic61C2×SD1611yescharacteristic
147D244—41no—
148D244—41no—
149D244—82no—
150D244—81no—
151D244—81no—
152C4×D61—21no—
153C4×D61—21no—
154C4×D61—21no—
155C4×D61—21no—
156C4×D61—21no—
157C4×D61—21no—
158C4×D61—81no—
159C4×D61—81no—
160C4×D61#12011yescharacteristic
161C4×D61#12311yescharacteristic
162C4×D61AGL(1,ℤ/8)11yescharacteristic
163C4×D61#13711yescharacteristic
164C8×C61—21no—
165C8×C61C4×D412yes—
166#92418—21no—
167#92418#12112yes—
168#13008—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—41no—
171C22×C61C2×#10211yescharacteristic
172C2×D61—21no—
173C2×D61—41no—
174C2×D61—42no—
175C2×D61—41no—
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C2×D61—81no—
180C2×D61#44711yescharacteristic
181C2×D61#48011yescharacteristic
182#6923—84no—
183C4×C61—21no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61#38611yescharacteristic
187C4×C61#41711yescharacteristic
188Dic61—21no—
189Dic61—81no—
190Dic61#44811yescharacteristic
191Dic61#48111yescharacteristic
192#3054—611no—
193#2911—611no—
194#2145—612no—
195C2×#349—611no—
196#1384—612no—
197C2×C61—21no—
198C2×C61—41no—
199C2×C61#187411yescharacteristic
200D61—21no—
201D61—21no—
202D61—41no—
203D61—161no—
204C2×#135—611no—
205C22×M16—611no—
206#525—1221no—
207C2×#117—1221no—
208C2×M32—612no—
209#470—611no—
210#470#692311yescharacteristic, complemented
211#469—611no—
212#469—611no—
213#468—1221no—
214#468—1221no—
215#428—1224no—
216#427—612no—
217#427—611no—
218#427—1221no—
219#416—1221no—
220#360—1224no—
221#349—1221no—
222#349—1221no—
223C61#1300811yescharacteristic, complemented, Sylow
224C2×#41—1221no—
225C23×C4—611no—
226#135—1222no—
227#135—1222no—
228C2×Q16C2×#692311yescharacteristic, complemented
229C2×SD16—1221no—
230C2×D8—611no—
231C2×M16—611no—
232C2×M16—611no—
233C2×M16—611no—
234C2×M16—1222no—
235C2×M16—1222no—
236C2×M16C2×#692311yescharacteristic
237C22×C8—611no—
238C22×C8—611no—
239#123—2441no—
240C22⋊Q8—21no—
241C22⋊Q8—1221no—
242#121—1221no—
243#121—1221no—
244#120—2441no—
245C4×D4—2441no—
246C4×D4—2441no—
247#117—1221no—
248#117—1222no—
249#117—1222no—
250#117—2441no—
251#117—2441no—
252C2×C4⋊C4—1222no—
253C2×C22⋊C4—2441no—
254C2×C42—2441no—
255M32—1222no—
256M32—1222no—
257C2×C16—1222no—
258#107—1221no—
259#107—1222no—
260#107—1222no—
261#106—614no—
262#106—611no—
263#106—1222no—
264#106C2×#692311yescharacteristic, complemented
265#103—21no—
266#103—1221no—
267#102—1221no—
268#102—1221no—
269C24—611no—
270#41—2441no—
271#41—2441no—
272#41—2441no—
273#41—2441no—
274C2×Q8—21no—
275C2×D4—1221no—
276C2×D4—2441no—
277C2×D4—2441no—
278C2×D4—2441no—
279C22×C4—611no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—2441no—
289C22×C4—2442no—
290C22×C4C22⋊#692311yescharacteristic
291Q16—22no—
292SD16—2442no—
293D8—1222no—
294M16—22no—
295M16—1222no—
296M16—1222no—
297M16—1222no—
298C2×C8—611no—
299C2×C8—612no—
300C2×C8—612no—
301C2×C8—611no—
302C2×C8—1221no—
303C2×C8—1221no—
304C2×C8C22⋊#692311yescharacteristic
305C2×C8C22×#692311yescharacteristic
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—1224no—
309C4⋊C4—1221no—
310C4⋊C4—2441no—
311C4⋊C4—2442no—
312C22⋊C4—41no—
313C22⋊C4—2441no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C22⋊C4—2442no—
317C22⋊C4—2442no—
318C22⋊C4—2441no—
319C42—2441no—
320C42—2442no—
321C42—2442no—
322C16—2442no—
323C23—611no—
324C23—611no—
325C23—1221no—
326C23—1221no—
327C23—4881no—
328C23—2441no—
329C23#1451983911yescharacteristic
330Q8—41no—
331Q8—41no—
332D4—4881no—
333D4—4881no—
334D4—4882no—
335D4—4882no—
336D4—2441no—
337D4—2441no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—611no—
346C2×C4—611no—
347C2×C4—611no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—4881no—
357C2×C4—4881no—
358C2×C4—4884no—
359C2×C4—2444no—
360C2×C4—2441no—
361C2×C4#1451983911yescharacteristic
362C2×C4C2×C22⋊#692311yescharacteristic
363C8—21no—
364C8—612no—
365C8—1221no—
366C8D4×#692312yes—
367C22—21no—
368C22—41no—
369C22—611no—
370C22—611no—
371C22—1221no—
372C22—4881no—
373C22—4881no—
374C22—4881no—
375C22—4881no—
376C22—2441no—
377C22—2441no—
378C22—2442no—
379C22C2×#1451983911yescharacteristic
380C4—21no—
381C4—81no—
382C4—81no—
383C4—611no—
384C4—611no—
385C4—1221no—
386C4—4881no—
387C4—4884no—
388C4#1684071011yescharacteristic
389C4#1684075311yescharacteristic
390C2—21no—
391C2—41no—
392C2—1221no—
393C2—1221no—
394C2—9761no—
395C2—2441no—
396C2#2146180611yescharacteristic
subgroupG/H
centerC2#21461806
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#470C4
socleC2×C61#1874
prime radical#21455430C2
cosocleC2×C8×D61C23
hypercenter#470#6923
hypocenterC61#13008
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #21462472C2semidirect
2. #21462141C2semidirect
3. D61⋊#349C2semidirect
4. #21460886C2semidirect
17. #16840017C4semidirect
18. #16840016C4semidirect
19. #16840015C4semidirect
24. C61×#470C4semidirect
38. #14519845D4semidirect
44. C2×D488C2×C4semidirect
61. #14519709C2×C4semidirect
65. C2×Q16×C61C2×C4semidirect
68. #106×C61C2×C4semidirect
210. #470#6923semidirect
223. C61#13008semidirect
228. C2×Q16C2×#6923semidirect
264. #106C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624722
#214621412
D61⋊#3492
#214608862
#214554302
#1300861