Finite Group Databasebeta

← #29669352

Subgroups — #29669352

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456553C211yesmaximal, characteristic
2#21456538C211yesmaximal, characteristic, complemented
3#21456213C211yesmaximal, characteristic, complemented
4#21456212C211yesmaximal, characteristic, complemented
5#21456201C211yesmaximal, characteristic, complemented
6#21445475C211yesmaximal, characteristic, complemented
7#21445470C211yesmaximal, characteristic, complemented
8#21445466C211yesmaximal, characteristic, complemented
9#21445464C211yesmaximal, characteristic, complemented
10#21445459C211yesmaximal, characteristic, complemented
11#21445456C211yesmaximal, characteristic, complemented
12#21445432C211yesmaximal, characteristic, complemented
13#21445430C211yesmaximal, characteristic, complemented
14#21444122C211yesmaximal, characteristic, complemented
15C61×#3182C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18#16840165C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840138C2211yescharacteristic
21#16840136C2211yescharacteristic
22#16840132—22no—
23#16840122—22no—
24#16840121—22no—
25#16840114C2211yescharacteristic
26#16840114C2211yescharacteristic
27#16840114C2211yescharacteristic
28#16840114C2211yescharacteristic
29#16840113C2211yescharacteristic
30#16840113C2211yescharacteristic
31#16840112C2211yescharacteristic
32#16840112C2211yescharacteristic
33#16840112C2211yescharacteristic
34#16840110C2211yescharacteristic
35#16840110C2211yescharacteristic
36#16840109C2211yescharacteristic
37#16840109C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840108C2211yescharacteristic
41#16840013—22no—
42#16840010—22no—
43#16839872—22no—
44#16839860—22no—
45#16839770—22no—
46#16839764—22no—
47#16839763—22no—
48C4×#11758194C2211yescharacteristic
49C4×#11758194C2211yescharacteristic
50#16839484C2211yescharacteristic
51#16839484C2211yescharacteristic
52#16839483C2211yescharacteristic
53C8×#92418C2211yescharacteristic
54C61×#533C2211yescharacteristic
55C61×#532C2211yescharacteristic
56C61×#500C2211yescharacteristic
57C61×#492C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#486C2211yescharacteristic
60C61×#468—22no—
61C61×#440—22no—
62C4×M16×C61C2211yescharacteristic
63C2×#11758222—21no—
64C22×Dic122—21no—
65C2×#11758201—22no—
66C2×#11758199—22no—
67C2×#11758194D412yes—
68C2×#11758194D412yes—
69#14519777—21no—
70#14519777—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519774C2311yescharacteristic
76#14519765—21no—
77#14519751—21no—
78#14519751—21no—
79C4×D244—21no—
80C4×D244—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83#14519710—22no—
84#14519708—22no—
85#14519702—22no—
86#14519702—22no—
87#14519701—22no—
88#14519701—22no—
89#14519701—22no—
90#14519700—22no—
91#14519700—22no—
92#14519700—22no—
93#14519700—22no—
94#14519699—22no—
95#14519699—22no—
96#14519698—22no—
97#14519698—22no—
98#14519698—22no—
99#14519697—22no—
100#14519697—22no—
101#14519688D412yes—
102#14519688D412yes—
103C4×#92418C2311yescharacteristic
104C4×#92418C2311yescharacteristic
105C4×#92418C2311yescharacteristic
106C4×#92418C2311yescharacteristic
107C2×#41×C61—21no—
108C2×SD16×C61—22no—
109C2×M16×C61D412yes—
110C4⋊Q8×C61C2311yescharacteristic
111C4⋊Q8×C61C2311yescharacteristic
112C4⋊Q8×C61C2311yescharacteristic
113#127×C61C2311yescharacteristic
114#125×C61—21no—
115#124×C61—21no—
116C22⋊Q8×C61—21no—
117C22⋊Q8×C61—21no—
118#121×C61—21no—
119#121×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124#117×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#107×C61—22no—
127#103×C61—22no—
128#102×C61—22no—
129#97×C61D412yes—
130C4×C8×C61C2311yescharacteristic
131C4×C8×C61C2311yescharacteristic
132#41×C61—44no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C2×C22⋊D61—41no—
145C22×Dic61—41no—
146#11758222—44no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×C4×D61—41no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—44no—
155C2×Dic122—41no—
156SD16×C61—44no—
157M16×C61—42no—
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×D412yes—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C4⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61—41no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2411yescharacteristic
176#11758201—44no—
177#11758199—44no—
178#11758198—41no—
179#11758198—41no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—41no—
187C4⋊Dic61—41no—
188#11758194—42no—
189#11758194—42no—
190C2×#92418C2×D412yes—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193C2×#92418C2×D412yes—
194C23×C61—41no—
195C23×C61#12711yescharacteristic
196C22×D61—41no—
197Q8×C61—42no—
198Q8×C61—42no—
199Q8×C61—42no—
200D4×C61—42no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—82no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—82no—
211C2×C4×C61#12711yescharacteristic
212C2×C4×C61#12711yescharacteristic
213C2×C4×C61#12711yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C22⋊D61—82no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—82no—
222D244—42no—
223D244—42no—
224D244—42no—
225C4×D61—82no—
226Dic122—42no—
227Dic122—42no—
228Dic122—42no—
229Dic122—82no—
230C8×C61—22no—
231C8×C61—22no—
232#92418—22no—
233#92418—22no—
234#92418—22no—
235#92418—22no—
236#19754—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61C2×#12711yescharacteristic
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—81no—
251C4×C61—81no—
252C4×C61—81no—
253C4×C61#57512yes—
254C4×C61#57512yes—
255C4×C61#57512yes—
256Dic61—81no—
257Dic61—81no—
258Dic61—81no—
259#3184—611no—
260#3182—611no—
261#3182D6111yescharacteristic, complemented
262#3141—611no—
263#3141—611no—
264#3139—611no—
265#1719—611no—
266#1719—611no—
267#1718—611no—
268#1718—611no—
269#1717—611no—
270#1717—611no—
271#1711—611no—
272#1711—611no—
273#1455—611no—
274C2×C61—41no—
275C2×C61—161no—
276C2×C61#315311yescharacteristic
277C2×C61#315311yescharacteristic
278C2×C61#315311yescharacteristic
279D61—161no—
280#533C2×D6111yescharacteristic
281#532—611no—
282#532C2×D6111yescharacteristic
283C2×C4⋊Q8—611no—
284#500—611no—
285#500C2×D6111yescharacteristic
286#498—611no—
287#493—611no—
288#492—611no—
289#492C2×D6111yescharacteristic
290#491—1222no—
291#489—1222no—
292#489—1222no—
293#487—611no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487—611no—
299#487—611no—
300#487—611no—
301#487—611no—
302#487C2×D6111yescharacteristic
303#486—611no—
304#486—611no—
305#486—611no—
306#486—611no—
307#486—611no—
308#486—611no—
309#486—611no—
310#486C2×D6111yescharacteristic
311#469—1222no—
312#468—22no—
313#468—1222no—
314#442—1222no—
315#440—22no—
316#440—1222no—
317#419—1222no—
318#417—1222no—
319#417—1222no—
320C4×M16—611no—
321C4×M16—611no—
322C4×M16C2×D6111yescharacteristic
323#322—611no—
324#322—611no—
325#322—611no—
326C82—611no—
327C61#1975411yescharacteristic, complemented, Sylow
328C2×#41—21no—
329C2×#41—1221no—
330C22×Q8—1221no—
331C2×SD16—22no—
332C2×SD16—1222no—
333C2×D8—1222no—
334C2×M16—612no—
335C2×M16—612no—
336C2×M16C22⋊D6112yes—
337C4⋊Q8—611no—
338C4⋊Q8—611no—
339C4⋊Q8—611no—
340C4⋊Q8—1221no—
341C4⋊Q8—1221no—
342C4⋊Q8—1221no—
343C4⋊Q8C22×D6111yescharacteristic
344C4⋊Q8C22×D6111yescharacteristic
345C4⋊Q8C22×D6111yescharacteristic
346#127—611no—
347#127C22×D6111yescharacteristic
348#125—21no—
349#124—21no—
350#124—1221no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353#121—21no—
354#121—21no—
355#121—1221no—
356#121—1221no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—1221no—
362C4×D4—1221no—
363#117—21no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C42C22×D6111yescharacteristic
367#107—22no—
368#107—1222no—
369#106—1222no—
370#103—22no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#102—22no—
381#102—1222no—
382#102—1222no—
383#102—1222no—
384#102—1222no—
385#102—1222no—
386#102—1222no—
387#102—1222no—
388#97—612no—
389#97—612no—
390#97C22⋊D6112yes—
391C4×C8—611no—
392C4×C8—611no—
393C4×C8—611no—
394C4×C8—611no—
395C4×C8C22×D6111yescharacteristic
396C4×C8C22×D6111yescharacteristic
397#41—44no—
398#41—2444no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—2444no—
406C2×Q8—2441no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—2441no—
415C22×C4—41no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421SD16—44no—
422SD16—2444no—
423D8—2444no—
424M16—42no—
425M16—2442no—
426M16—2442no—
427C2×C8—612no—
428C2×C8—612no—
429C2×C8—612no—
430C2×C8—612no—
431C2×C8C2×C22⋊D6112yes—
432C2×C8C2×C22⋊D6112yes—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—1221no—
446C4⋊C4—1221no—
447C4⋊C4—2441no—
448C4⋊C4—2441no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—2441no—
453C22⋊C4—2441no—
454C42—41no—
455C42D4×D6111yescharacteristic
456C42D4×D6111yescharacteristic
457C42C2×C22⋊D6111yescharacteristic
458C42C23×D6111yescharacteristic
459C23—41no—
460C23—2441no—
461C23#1451977311yescharacteristic
462Q8—42no—
463Q8—42no—
464Q8—42no—
465Q8—4882no—
466Q8—2442no—
467Q8—2442no—
468Q8—2442no—
469D4—42no—
470D4—42no—
471D4—42no—
472D4—82no—
473D4—4882no—
474D4—2442no—
475D4—2442no—
476D4—2442no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—82no—
484C2×C4—4882no—
485C2×C4—4882no—
486C2×C4—2441no—
487C2×C4—2441no—
488C2×C4—2441no—
489C2×C4#1451977311yescharacteristic
490C2×C4#1451977311yescharacteristic
491C2×C4#1451977311yescharacteristic
492C2×C4C2×D4×D6111yescharacteristic
493C2×C4C2×D4×D6111yescharacteristic
494C2×C4C22×C22⋊D6111yescharacteristic
495C8—22no—
496C8—22no—
497C8—1222no—
498C8—1222no—
499C8—1222no—
500C8—1222no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22C2×#1451977311yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—81no—
515C4—81no—
516C4—81no—
517C4—4881no—
518C4—4881no—
519C4—4881no—
520C4#1684055812yes—
521C4#1684056012yes—
522C4#1684056112yes—
523C2—41no—
524C2—161no—
525C2—9761no—
526C2#2145628111yescharacteristic
527C2#2145628611yescharacteristic
528C2#2145628711yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3182C2
socleC22×C61C2×#127
prime radical#21456553C2
hypercenter#3182D61
hypocenterC61#19754
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
2. #21456538C2semidirect
3. #21456213C2semidirect
4. #21456212C2semidirect
5. #21456201C2semidirect
6. #21445475C2semidirect
7. #21445470C2semidirect
8. #21445466C2semidirect
9. #21445464C2semidirect
10. #21445459C2semidirect
11. #21445456C2semidirect
12. #21445432C2semidirect
13. #21445430C2semidirect
14. #21444122C2semidirect
15. C61×#3182C2semidirect
261. #3182D61semidirect
327. C61#19754semidirect

Maximal subgroups

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

subgroupindex
#214565532
#214565382
#214562132
#214562122
#214562012
#214454752
#214454702
#214454662
#214454642
#214454592
#214454562
#214454322
#214454302
#214441222
C61×#31822
#1975461