Finite Group Databasebeta

← #30607562

Subgroups — #30607562

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#21462140C212yesmaximal, complemented
3D61⋊#349C211yesmaximal, characteristic, complemented
4#21460886C212yesmaximal, complemented
5#21455426C211yesmaximal, characteristic
6C2×#14519856—21no—
7D61⋊M32C2212yes—
8#16840773—24no—
9#16840772—21no—
10#16840772C2212yes—
11#16840678—24no—
12#16840659—21no—
13#16840658—21no—
14C2×#14519796C2211yescharacteristic
15C2×M16×D61C2211yescharacteristic
16#16840272—21no—
17#16840021C411yescharacteristic, complemented
18#16840020C411yescharacteristic, complemented
19#16840012C411yescharacteristic, complemented
20#16840007—21no—
21#16840006—21no—
22#16839797C2211yescharacteristic
23#16839755—21no—
24C61×#469C411yescharacteristic, 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×#11758224—21no—
40C22×C4×D61D411yescharacteristic
41#14519796—22no—
42#14519796—22no—
43C2×Dic244C2×C411yescharacteristic, complemented
44C2×#11758200—21no—
45C2×#11758194D411yescharacteristic
46C2×#11758189C2×C411yescharacteristic
47C2×#11758189D411yescharacteristic
48M16×D61—22no—
49M16×D61—22no—
50C2×C8×D61D411yescharacteristic
51C2×C8×D61C2311yescharacteristic
52#14519759—41no—
53#14519757—21no—
54#14519756—21no—
55#14519748—21no—
56#14519745—41no—
57D4×Dic61—41no—
58#14519737—41no—
59#14519735—21no—
60#14519710—21no—
61#14519709C2×C411yescharacteristic, complemented
62#14519702—21no—
63#14519700—21no—
64#14519698—21no—
65C2×D8×C61C2×C411yescharacteristic, complemented
66C2×M16×C61C2×C411yescharacteristic
67#121×C61—21no—
68#106×C61C2×C411yescharacteristic, complemented
69#102×C61—21no—
70C23×D61D811yescharacteristic
71C22×#6923—42no—
72C2×D4×C61—21no—
73C2×D4×C61—41no—
74C22×C4×C61C22⋊C411yescharacteristic
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C22×Dic61—41no—
78C22×Dic61SD1611yescharacteristic
79#11758224—41no—
80#11758224—41no—
81#11758224—41no—
82#11758224—41no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61C22⋊C411yescharacteristic
88C2×C4×D61D811yescharacteristic
89C2×C4×D61SD1611yescharacteristic
90C2×C4×D61C2×D411yescharacteristic
91C2×Dic122—21no—
92C22⋊#6923—42no—
93C22⋊#6923—42no—
94C4⋊#6923—24no—
95C4⋊#6923—42no—
96C4×#6923—42no—
97C4×#6923—42no—
98D8×C61—22no—
99M16×C61—22no—
100C2×C8×C61C22⋊C411yescharacteristic
101C2×C8×C61C22×C411yescharacteristic
102C4⋊C4×C61—21no—
103C22⋊C4×C61—41no—
104C22⋊Dic61—41no—
105C22⋊Dic61—41no—
106#11758200—42no—
107#11758198—41no—
108#11758198—41no—
109C4⋊Dic61—21no—
110C4⋊Dic61—41no—
111#11758196—41no—
112C4×Dic61—41no—
113#11758194—22no—
114C2×#92418C22⋊C411yescharacteristic
115C2×#92418C2×D411yescharacteristic
116Dic244—22no—
117#11758189—22no—
118#11758189—22no—
119C8×D61—21no—
120C8×D61—21no—
121C8×D61C2×D412yes—
122C8×D61#4112yes—
123#11758187—42no—
124C23×C61—41no—
125C23×C61#10211yescharacteristic
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—81no—
129C22×D61#10211yescharacteristic
130C22×D61C2×D811yescharacteristic
131C2×#6923—44no—
132C2×#6923—84no—
133D4×C61—41no—
134D4×C61—41no—
135D4×C61—82no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—41no—
139C2×C4×C61#10211yescharacteristic
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C22⋊D61—82no—
142C22⋊D61—81no—
143C22⋊D61—81no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—81no—
149C2×Dic61—81no—
150C2×Dic61#10211yescharacteristic
151C2×Dic61C2×SD1611yescharacteristic
152C4×D61—21no—
153C4×D61—21no—
154C4×D61—21no—
155C4×D61—21no—
156C4×D61—21no—
157C4×D61—21no—
158C4×D61#12011yescharacteristic
159C4×D61#12311yescharacteristic
160C4×D61AGL(1,ℤ/8)11yescharacteristic
161C4×D61#13711yescharacteristic
162Dic122—41no—
163Dic122—41no—
164C8×C61—21no—
165C8×C61C4×D412yes—
166#92418—21no—
167#92418#12112yes—
168#13008—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—41no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61C2×#10211yescharacteristic
175C2×D61—21no—
176C2×D61—41no—
177C2×D61—42no—
178C2×D61—41no—
179C2×D61—81no—
180C2×D61#44711yescharacteristic
181C2×D61#48011yescharacteristic
182#6923—84no—
183C4×C61—21no—
184C4×C61—81no—
185C4×C61#38611yescharacteristic
186C4×C61#41711yescharacteristic
187Dic61—21no—
188Dic61—81no—
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—161no—
200C2×C61#187411yescharacteristic
201D61—21no—
202D61—21no—
203D61—41no—
204C2×#135—611no—
205C22×M16—611no—
206#525—1221no—
207C2×#117—1221no—
208C2×M32—612no—
209#470—611no—
210#470—611no—
211#469—611no—
212#469#692311yescharacteristic, complemented
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×Q16—611no—
229C2×SD16—1221no—
230C2×D8C2×#692311yescharacteristic, complemented
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—1221no—
241C22⋊Q8—1221no—
242#121—21no—
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—1221no—
266#103—1221no—
267#102—21no—
268#102—1221no—
269C24—611no—
270#41—2441no—
271#41—2441no—
272#41—2441no—
273#41—2441no—
274C2×Q8—1221no—
275C2×D4—21no—
276C2×D4—41no—
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—1222no—
292SD16—2442no—
293D8—22no—
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—1224no—
308C4⋊C4—1221no—
309C4⋊C4—2441no—
310C4⋊C4—2442no—
311C4⋊C4—2441no—
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—41no—
324C23—611no—
325C23—611no—
326C23—1221no—
327C23—1221no—
328C23—4881no—
329C23#1451983911yescharacteristic
330Q8—2441no—
331Q8—2441no—
332D4—41no—
333D4—41no—
334D4—82no—
335D4—4881no—
336D4—4881no—
337D4—4882no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—611no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—611no—
346C2×C4—611no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—4881no—
356C2×C4—4881no—
357C2×C4—4884no—
358C2×C4—2444no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4#1451983911yescharacteristic
362C2×C4C2×C22⋊#692311yescharacteristic
363C8—21no—
364C8—612no—
365C8—1221no—
366C8D4×#692312yes—
367C22—21no—
368C22—41no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—611no—
373C22—611no—
374C22—1221no—
375C22—4881no—
376C22—2441no—
377C22—2441no—
378C22—2442no—
379C22C2×#1451983911yescharacteristic
380C4—21no—
381C4—81no—
382C4—611no—
383C4—611no—
384C4—1221no—
385C4—4881no—
386C4—4884no—
387C4—4881no—
388C4#1684071011yescharacteristic
389C4#1684075311yescharacteristic
390C2—21no—
391C2—41no—
392C2—161no—
393C2—1221no—
394C2—1221no—
395C2—2441no—
396C2#2146180611yescharacteristic
subgroupG/H
centerC2#21461806
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#469C4
socleC2×C61#1874
prime radical#21455426C2
cosocleC2×C8×D61C23
hypercenter#469#6923
hypocenterC61#13008
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #21462472C2semidirect
2. #21462140C2semidirect
3. D61⋊#349C2semidirect
4. #21460886C2semidirect
17. #16840021C4semidirect
18. #16840020C4semidirect
19. #16840012C4semidirect
24. C61×#469C4semidirect
38. #14519845D4semidirect
43. C2×Dic244C2×C4semidirect
61. #14519709C2×C4semidirect
65. C2×D8×C61C2×C4semidirect
68. #106×C61C2×C4semidirect
212. #469#6923semidirect
223. C61#13008semidirect
230. C2×D8C2×#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
#214621402
D61⋊#3492
#214608862
#214554262
#1300861