Finite Group Databasebeta

← #29667876

Subgroups — #29667876

Order 15616.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic, complemented
2#21456379C211yesmaximal, characteristic, complemented
3#21456366C211yesmaximal, characteristic, complemented
4#21456269C211yesmaximal, characteristic
5C2×#16840108C211yesmaximal, characteristic, complemented
6#21445449C212yesmaximal, complemented
7#21445439C212yesmaximal, complemented
8#21445419C212yesmaximal, complemented
9#21445409C212yesmaximal, complemented
10#21444100C211yesmaximal, characteristic, complemented
11C61×#3208C211yesmaximal, characteristic, complemented
12C2×#14519803—22no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840172—22no—
16#16840165C2212yescomplemented
17#16840158C2212yescomplemented
18#16840158C2211yescharacteristic, complemented
19#16840149—22no—
20#16840138C2212yes—
21#16840136C2211yescharacteristic, complemented
22#16840135C2212yescomplemented
23#16840134C2212yes—
24#16840113C2212yescomplemented
25#16840112C2211yescharacteristic, complemented
26#16840110C2211yescharacteristic, complemented
27#16840108—22no—
28#16840108C2212yes—
29#16840108C2212yes—
30#16839993—22no—
31#16839968—22no—
32#16839903—22no—
33C4×#11758201—22no—
34#16839805—22no—
35C2×#14519698—22no—
36C4×#11758194C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839484C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#492C2212yes—
43C61×#486C2212yes—
44C61×#460—22no—
45C61×#450—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49#14519803—48no—
50C2×#11758190—24no—
51C2×#11758200—24no—
52C2×#11758201—22no—
53C2×#11758199—24no—
54C2×#11758194D412yes—
55C22×#92418D412yes—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777C2312yes—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69#14519710—24no—
70#14519708—22no—
71#14519708—24no—
72#14519702—24no—
73#14519701—22no—
74#14519698—24no—
75#14519698—24no—
76#14519697—22no—
77#14519688D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C2×SD16×C61—24no—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61—21no—
89#127×C61—21no—
90#127×C61—21no—
91#127×C61C2312yes—
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97#117×C61—21no—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61D412yes—
100#102×C61—24no—
101C22⋊C8×C61D412yes—
102C4×C8×C61C2312yes—
103C24×C61—41no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—44no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—44no—
114C2×D4×C61—82no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C2×C22⋊D61—41no—
119C22×Dic61—41no—
120#11758222—44no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×C4×D61—41no—
125C2×Dic122—22no—
126C2×Dic122—21no—
127C2×Dic122—41no—
128C2×Dic122—82no—
129SD16×C61—84no—
130C2×C8×C61C2×D414yes—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61—41no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61C2411yescharacteristic
143#11758201—44no—
144#11758200—48no—
145#11758199—48no—
146#11758198—41no—
147#11758198—41no—
148C4⋊Dic61—22no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—22no—
152C4⋊Dic61—41no—
153C4⋊Dic61—41no—
154#11758194SD1618yescomplemented
155C2×#92418—42no—
156C2×#92418C2×D412yes—
157C2×#92418C2×D412yes—
158C2×#92418C2×D414yes—
159#11758190—84no—
160C23×C61—42no—
161C23×C61—82no—
162C23×C61—82no—
163C23×C61—82no—
164C23×C61#12711yescharacteristic
165C22×D61—41no—
166Q8×C61—42no—
167Q8×C61—42no—
168D4×C61—44no—
169D4×C61—44no—
170D4×C61—44no—
171D4×C61—82no—
172D4×C61—82no—
173D4×C61—82no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61#12711yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C2×C4×C61#12711yescharacteristic
181C2×C4×C61C2×SD1614yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C22⋊D61—82no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—82no—
189D244—42no—
190D244—42no—
191D244—42no—
192C4×D61—82no—
193Dic122—42no—
194Dic122—44no—
195Dic122—82no—
196C8×C61—24no—
197#92418—22no—
198#92418—22no—
199#92418C2×SD1618yes—
200#19623—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—82no—
204C22×C61—82no—
205C22×C61—82no—
206C22×C61—164no—
207C22×C61#49212yes—
208C22×C61C2×#12711yescharacteristic
209C2×D61—81no—
210C2×D61—81no—
211C2×D61—81no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—81no—
215C4×C61—81no—
216C4×C61#49212yes—
217C4×C61C22×SD1612yes—
218C4×C61#57512yes—
219C4×C61#57612yes—
220Dic61—81no—
221Dic61—82no—
222#3208D6111yescharacteristic, complemented
223#3206—611no—
224#3167—611no—
225#3165—611no—
226#3151—611no—
227C2×#486—611no—
228#1715—612no—
229#1714—612no—
230#1708—612no—
231#1706—612no—
232C8×M16—611no—
233C2×C61—22no—
234C2×C61—162no—
235C2×C61C2×#49211yescharacteristic
236C2×C61#315311yescharacteristic
237C2×C61#315511yescharacteristic
238D61—161no—
239C2×AGL(1,ℤ/8)—1222no—
240#533C2×D6111yescharacteristic
241#532—611no—
242C2×C4⋊Q8—611no—
243C2×#127C2×D6111yescharacteristic
244#501—1222no—
245#500—612no—
246#498—612no—
247#498—611no—
248#496—1222no—
249#493—612no—
250#492—611no—
251#492—612no—
252#492—612no—
253#492C2×D6112yes—
254#487—611no—
255#487—612no—
256#486—611no—
257#486—612no—
258#486—612no—
259#486—1222no—
260#486C2×D6112yes—
261#465—1222no—
262#461—1222no—
263#460—22no—
264#450—22no—
265#448—1222no—
266C4×SD16—1222no—
267#431C2×D6111yescharacteristic
268#428—1222no—
269C2×#102—1222no—
270C4×M16—611no—
271C2×C4×C8—611no—
272#322—612no—
273C82—612no—
274C61#1962311yescharacteristic, complemented, Sylow
275C2×#41—1221no—
276C22×Q8—1221no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280AGL(1,ℤ/8)—2448no—
281C2×SD16—24no—
282C2×SD16—1224no—
283C2×SD16—1222no—
284C2×SD16—1224no—
285C2×D8—1224no—
286C2×M16—612no—
287C22×C8—612no—
288C4⋊Q8—612no—
289C4⋊Q8—611no—
290C4⋊Q8—1221no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293C4⋊Q8C22×D6111yescharacteristic
294C4⋊Q8C22×D6111yescharacteristic
295#127—21no—
296#127—21no—
297#127—21no—
298#127—611no—
299#127C22×D6112yes—
300#125—21no—
301#124—1221no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1221no—
305#121—1221no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×D4—1221no—
309C4×D4—1221no—
310#117—21no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1224no—
315#106—1224no—
316#106—1222no—
317C4⋊C8C22⋊D6112yes—
318#103—1222no—
319#103—1224no—
320#102—24no—
321#102—1222no—
322#102—1224no—
323#102—1224no—
324C22⋊C8C22⋊D6112yes—
325#97—612no—
326C4×C8—612no—
327C4×C8—611no—
328C4×C8—612no—
329C4×C8—611no—
330C4×C8C22×D6112yes—
331C24—41no—
332#41—2444no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—1221no—
336C2×Q8—1222no—
337C2×Q8—4882no—
338C2×Q8—2441no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—82no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4—1221no—
351C2×D4—2441no—
352C22×C4—2441no—
353C22×C4—2441no—
354C22×C4D4×D6111yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357SD16—84no—
358SD16—4884no—
359SD16—2448no—
360SD16—2444no—
361D8—2448no—
362M16—618no—
363C2×C8—614no—
364C2×C8—612no—
365C2×C8—612no—
366C2×C8—2442no—
367C2×C8C2×C22⋊D6114yes—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1222no—
375C4⋊C4—1221no—
376C4⋊C4—1222no—
377C4⋊C4—1221no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C22⋊C4—41no—
381C22⋊C4—2441no—
382C22⋊C4—2441no—
383C42—41no—
384C42D4×D6111yescharacteristic
385C42D4×D6111yescharacteristic
386C42C2×C22⋊D6111yescharacteristic
387C42C23×D6111yescharacteristic
388C23—42no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—2441no—
393C23#1451977311yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—4882no—
397Q8—2442no—
398Q8—2444no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—4882no—
406D4—2442no—
407D4—2442no—
408D4—2442no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—4882no—
414C2×C4—4882no—
415C2×C4—2441no—
416C2×C4—2442no—
417C2×C4#1451977311yescharacteristic
418C2×C4#1451977311yescharacteristic
419C2×C4#1451977311yescharacteristic
420C2×C4SD16×D6114yes—
421C2×C4C2×D4×D6111yescharacteristic
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C22×C22⋊D6111yescharacteristic
424C8—24no—
425C8—618no—
426C8—1222no—
427C8—1222no—
428C22—21no—
429C22—21no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—164no—
434C22—4881no—
435C22—4881no—
436C22—4881no—
437C22#1684013312yes—
438C22C2×#1451977311yescharacteristic
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—4881no—
444C4—4882no—
445C4#1684013312yes—
446C4C2×SD16×D6112yes—
447C4#1684055812yes—
448C4#1684056512yes—
449C2—22no—
450C2—162no—
451C2—9761no—
452C2C2×#1684013311yescharacteristic
453C2#2145628111yescharacteristic
454C2#2145630211yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3208C2
socleC22×C61C2×#127
prime radical#21456269C2
hypercenter#3208D61
hypocenterC61#19623
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21456746C2semidirect
2. #21456379C2semidirect
3. #21456366C2semidirect
5. C2×#16840108C2semidirect
6. #21445449C2semidirect
7. #21445439C2semidirect
8. #21445419C2semidirect
9. #21445409C2semidirect
10. #21444100C2semidirect
11. C61×#3208C2semidirect
16. #16840165C22semidirect
17. #16840158C22semidirect
18. #16840158C22semidirect
21. #16840136C22semidirect
22. #16840135C22semidirect
24. #16840113C22semidirect
25. #16840112C22semidirect
26. #16840110C22semidirect
38. #16839484C22semidirect
39. C8×#92418C22semidirect
154. #11758194SD16semidirect
222. #3208D61semidirect
274. C61#19623semidirect

Maximal subgroups

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

subgroupindex
#214567462
#214563792
#214563662
#214562692
C2×#168401082
#214454492
#214454392
#214454192
#214454092
#214441002
C61×#32082
#1962361