Finite Group Databasebeta

← #29667890

Subgroups — #29667890

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21456947C211yesmaximal, characteristic
3#21456379C211yesmaximal, characteristic
4#21456278C211yesmaximal, characteristic
5C2×#16840112C211yesmaximal, characteristic, complemented
6#21445453C212yesmaximal, complemented
7#21445442C212yesmaximal, complemented
8#21445421C212yesmaximal, complemented
9#21445417C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3165C211yesmaximal, characteristic, complemented
12C2×#14519810—22no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840158C2212yes—
16#16840158C2212yes—
17#16840158C2211yescharacteristic
18#16840156—22no—
19#16840142C2212yes—
20#16840136C2211yescharacteristic
21#16840135C2212yes—
22#16840112—22no—
23#16840112C2212yes—
24#16840112C2212yes—
25#16840112C2211yescharacteristic
26#16840112C2212yes—
27#16840110C2211yescharacteristic
28#16840110C2212yes—
29#16840090—22no—
30#16840080—22no—
31#16840049—22no—
32#16840047—22no—
33#16839993—22no—
34C4×#11758201—22no—
35#16839815C2211yescharacteristic
36C2×#14519701—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yes—
39C8×#92418C2212yes—
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#501—22no—
43C61×#500C2212yes—
44C61×#498C2212yes—
45C61×#428—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49#14519810—48no—
50C2×Dic244—24no—
51C2×#11758190—24no—
52C2×#11758201—22no—
53C22×#92418D412yes—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777C2312yes—
58#14519777C2311yescharacteristic
59#14519774C2311yescharacteristic
60#14519765—21no—
61#14519751—21no—
62#14519751—21no—
63C4×D244—21no—
64C4×D244—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67#14519708—22no—
68#14519708—24no—
69#14519708—24no—
70C4⋊#92418D412yes—
71#14519701—22no—
72#14519701—24no—
73#14519701—24no—
74#14519701—24no—
75#14519697—22no—
76#14519697—24no—
77C22⋊#92418D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C22×Q8×C61—21no—
83C2×M16×C61D412yes—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2312yes—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95#117×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—24no—
100#106×C61—24no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—44no—
107C2×Q8×C61—41no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C2×C22⋊D61—41no—
113C22×Dic61—41no—
114#11758222—44no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×C4×D61—41no—
119C2×Dic122—22no—
120C2×Dic122—21no—
121C2×Dic122—44no—
122C2×Dic122—41no—
123M16×C61SD1618yescomplemented
124C2×C8×C61C2×D414yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2411yescharacteristic
141#11758201—44no—
142#11758198—41no—
143#11758198—41no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—41no—
149C4⋊Dic61—41no—
150C2×#92418—42no—
151C2×#92418C2×D412yes—
152C2×#92418C2×D412yes—
153C2×#92418C2×D414yes—
154Dic244—48no—
155#11758190—48no—
156C23×C61#12711yescharacteristic
157C22×D61—41no—
158Q8×C61—42no—
159Q8×C61—44no—
160Q8×C61—42no—
161Q8×C61—82no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—82no—
168C2×C4×C61#12711yescharacteristic
169C2×C4×C61#12711yescharacteristic
170C2×C4×C61#12711yescharacteristic
171C2×C4×C61C2×SD1614yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C22⋊D61—82no—
176C2×Dic61—42no—
177C2×Dic61—41no—
178C2×Dic61—82no—
179D244—42no—
180D244—42no—
181D244—42no—
182C4×D61—82no—
183Dic122—44no—
184Dic122—42no—
185Dic122—82no—
186C8×C61C2×SD1618yes—
187#92418—22no—
188#92418—22no—
189#92418—24no—
190#19624—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61#49212yes—
194C22×C61C2×#12711yescharacteristic
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—82no—
201C4×C61—81no—
202C4×C61—81no—
203C4×C61#49212yes—
204C4×C61C22×SD1612yes—
205C4×C61#57512yes—
206C4×C61#57812yes—
207Dic61—82no—
208Dic61—81no—
209#3229—611no—
210#3227—611no—
211#3167—611no—
212#3165D6111yescharacteristic, complemented
213#3152—611no—
214C2×#487—611no—
215#1716—612no—
216#1714—612no—
217#1708—612no—
218#1707—612no—
219C8×M16—611no—
220C2×C61—22no—
221C2×C61C2×#49211yescharacteristic
222C2×C61#315311yescharacteristic
223C2×C61#315611yescharacteristic
224D61—161no—
225C2×#137—1222no—
226#533C2×D6111yescharacteristic
227#532—611no—
228C2×C4⋊Q8—611no—
229C2×C4⋊Q8C2×D6111yescharacteristic
230#501—22no—
231#500C2×D6112yes—
232#498—611no—
233#498—612no—
234#498—612no—
235#498C2×D6112yes—
236#497—1222no—
237#494—612no—
238#492—611no—
239#492—612no—
240#487—612no—
241#487—612no—
242#487—611no—
243#487—612no—
244#487—1222no—
245#486—611no—
246#486—612no—
247#483—1222no—
248#481—1222no—
249#476—1222no—
250#475—1222no—
251#465—1222no—
252C4×SD16—1222no—
253#431—611no—
254#428—22no—
255C2×#103—1222no—
256C4×M16C2×D6111yescharacteristic
257C2×C4×C8—611no—
258#322—612no—
259C82—612no—
260C61#1962411yescharacteristic, complemented, Sylow
261C2×#41—1221no—
262C22×Q8—21no—
263C22×Q8—1221no—
264#137—2448no—
265C2×Q16—1224no—
266C2×SD16—1222no—
267C2×SD16—1224no—
268C2×M16C22⋊D6112yes—
269C22×C8—612no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—612no—
274C4⋊Q8—611no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278C4⋊Q8C22×D6112yes—
279C4⋊Q8C22×D6111yescharacteristic
280C4⋊Q8C22×D6111yescharacteristic
281#127—611no—
282#125—21no—
283#124—1221no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286#121—1221no—
287#121—1221no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×D4—1221no—
291C4×D4—1221no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C42C22×D6111yescharacteristic
298#107—24no—
299#106—24no—
300#106—1222no—
301#106—1224no—
302#106—1224no—
303C4⋊C8—612no—
304#103—1224no—
305#103—1224no—
306#103—1222no—
307#103—1224no—
308#102—1222no—
309#102—1224no—
310C22⋊C8—612no—
311#97C22⋊D6112yes—
312C4×C8—612no—
313C4×C8—611no—
314C4×C8—612no—
315C4×C8—611no—
316C4×C8C22×D6112yes—
317#41—2444no—
318C2×Q8—21no—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—1221no—
324C2×Q8—1222no—
325C2×Q8—2444no—
326C2×Q8—2441no—
327C2×D4—1221no—
328C2×D4—1221no—
329C2×D4—1221no—
330C2×D4—2441no—
331C22×C4—41no—
332C22×C4—2441no—
333C22×C4—2441no—
334C22×C4D4×D6111yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337Q16—2448no—
338SD16—2444no—
339SD16—2448no—
340M16#1175820118yescomplemented
341C2×C8—612no—
342C2×C8—612no—
343C2×C8—614no—
344C2×C8—2442no—
345C2×C8C2×C22⋊D6114yes—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—1222no—
357C4⋊C4—1222no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—2441no—
361C4⋊C4—2441no—
362C22⋊C4—41no—
363C22⋊C4—2441no—
364C22⋊C4—2441no—
365C42—41no—
366C42D4×D6111yescharacteristic
367C42D4×D6111yescharacteristic
368C42C2×C22⋊D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—2441no—
371C23#1451977311yescharacteristic
372Q8—42no—
373Q8—44no—
374Q8—42no—
375Q8—82no—
376Q8—4882no—
377Q8—2444no—
378Q8—2442no—
379D4—4882no—
380D4—2442no—
381D4—2442no—
382D4—2442no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—4882no—
390C2×C4—4882no—
391C2×C4—2441no—
392C2×C4—2442no—
393C2×C4#1451977311yescharacteristic
394C2×C4#1451977311yescharacteristic
395C2×C4#1451977311yescharacteristic
396C2×C4C2×#1175820114yes—
397C2×C4C2×D4×D6111yescharacteristic
398C2×C4C2×D4×D6111yescharacteristic
399C2×C4C22×C22⋊D6111yescharacteristic
400C8—1222no—
401C8—1222no—
402C8—1224no—
403C8C2×#1175820118yes—
404C22—21no—
405C22—21no—
406C22—4881no—
407C22—4881no—
408C22—4881no—
409C22#1684013612yes—
410C22C2×#1451977311yescharacteristic
411C4—21no—
412C4—21no—
413C4—82no—
414C4—81no—
415C4—81no—
416C4—4881no—
417C4—4882no—
418C4#1684013612yes—
419C4C22×#1175820112yes—
420C4#1684055812yes—
421C4#1684057812yes—
422C2—22no—
423C2—9761no—
424C2C2×#1684013611yescharacteristic
425C2#2145628111yescharacteristic
426C2#2145631211yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenter#3165D61
hypocenterC61#19624
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
5. C2×#16840112C2semidirect
6. #21445453C2semidirect
7. #21445442C2semidirect
8. #21445421C2semidirect
9. #21445417C2semidirect
10. M16×#92418C2semidirect
11. C61×#3165C2semidirect
123. M16×C61SD16semidirect
212. #3165D61semidirect
260. C61#19624semidirect
340. M16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214569472
#214563792
#214562782
C2×#168401122
#214454532
#214454422
#214454212
#214454172
M16×#924182
C61×#31652
#1962461