Finite Group Databasebeta

← #29669335

Subgroups — #29669335

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456560C211yesmaximal, characteristic, complemented
2#21456533C211yesmaximal, characteristic
3#21456366C211yesmaximal, characteristic, complemented
4#21456212C211yesmaximal, characteristic
5#21456200C211yesmaximal, characteristic, complemented
6#21445495C211yesmaximal, characteristic, complemented
7#21445480C211yesmaximal, characteristic, complemented
8#21445470C211yesmaximal, characteristic, complemented
9#21445464C211yesmaximal, characteristic, complemented
10#21445462C211yesmaximal, characteristic, complemented
11#21445439C211yesmaximal, characteristic, complemented
12#21445427C211yesmaximal, characteristic, complemented
13#21445419C211yesmaximal, characteristic, complemented
14#21444122C211yesmaximal, characteristic, complemented
15C61×#3182C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18#16840172—22no—
19#16840165C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840158C2211yescharacteristic
23#16840158C2211yescharacteristic
24#16840158C2211yescharacteristic
25#16840141C2211yescharacteristic
26#16840134C2211yescharacteristic
27#16840131—22no—
28#16840121—22no—
29#16840114C2211yescharacteristic
30#16840114C2211yescharacteristic
31#16840113C2211yescharacteristic
32#16840113C2211yescharacteristic
33#16840113C2211yescharacteristic
34#16840113C2211yescharacteristic
35#16840112C2211yescharacteristic
36#16840112C2211yescharacteristic
37#16840112C2211yescharacteristic
38#16840108C2211yescharacteristic
39#16840108C2211yescharacteristic
40#16840108C2211yescharacteristic
41#16840018—22no—
42#16840006—22no—
43#16839867—22no—
44#16839862—22no—
45#16839805—22no—
46#16839772—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
63C22×Dic122—21no—
64C2×#11758202—22no—
65C2×#11758200—22no—
66C2×#11758194D412yes—
67C2×#11758194D412yes—
68#14519777—21no—
69#14519777—21no—
70#14519777—21no—
71#14519777C2311yescharacteristic
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519769—21no—
76#14519756—21no—
77#14519756—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519736—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83#14519710—22no—
84#14519710—22no—
85#14519710—22no—
86#14519708—22no—
87#14519708—22no—
88#14519708—22no—
89#14519702—22no—
90#14519702—22no—
91#14519702—22no—
92#14519702—22no—
93#14519701—22no—
94#14519701—22no—
95#14519701—22no—
96#14519700—22no—
97#14519700—22no—
98#14519698—22no—
99#14519698—22no—
100#14519698—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
144C22×Dic61—41no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122—44no—
150C2×Dic122—41no—
151SD16×C61—44no—
152M16×C61—42no—
153C2×C8×C61C2×D412yes—
154C2×C8×C61C2×D412yes—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—41no—
162C4⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C22⋊C4×C61—41no—
165C22⋊C4×C61—41no—
166C42×C61—41no—
167C42×C61C2×D411yescharacteristic
168C42×C61C2×D411yescharacteristic
169C42×C61C2×D411yescharacteristic
170C42×C61C2411yescharacteristic
171C22⋊Dic61—41no—
172#11758202—44no—
173#11758200—44no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—41no—
183C4⋊Dic61—41no—
184#11758196—41no—
185#11758196—41no—
186C4×Dic61—41no—
187#11758194—24no—
188#11758194—42no—
189C2×#92418C2×D412yes—
190C2×#92418C2×D412yes—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193C23×C61—41no—
194C23×C61#12711yescharacteristic
195Q8×C61—42no—
196Q8×C61—42no—
197Q8×C61—42no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—82no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—82no—
209C2×C4×C61#12711yescharacteristic
210C2×C4×C61#12711yescharacteristic
211C2×C4×C61#12711yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—82no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—82no—
225C8×C61—22no—
226C8×C61—22no—
227#92418—22no—
228#92418—22no—
229#92418—22no—
230#92418#13714yes—
231#19752—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61C2×#12711yescharacteristic
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61C2×#13712yes—
246C4×C61#57512yes—
247C4×C61#57512yes—
248Dic61—81no—
249Dic61—81no—
250Dic61—81no—
251Dic61—81no—
252#3185—611no—
253#3182—611no—
254#3182D6111yescharacteristic
255#3165—611no—
256#3141—611no—
257#3139—611no—
258#1723—611no—
259#1720—611no—
260#1719—611no—
261#1718—611no—
262#1718—611no—
263#1714—611no—
264#1710—611no—
265#1708—611no—
266#1455—611no—
267C2×C61—41no—
268C2×C61—161no—
269C2×C61#315211yescharacteristic
270C2×C61#315311yescharacteristic
271C2×C61#315311yescharacteristic
272#533—611no—
273#533C2×D6111yescharacteristic
274#532C2×D6111yescharacteristic
275C2×C4⋊Q8—611no—
276#501—1222no—
277#500—611no—
278#500—611no—
279#500—611no—
280#500C2×D6111yescharacteristic
281#498—611no—
282#498—611no—
283#498—611no—
284#494—611no—
285#492—611no—
286#492C2×D6111yescharacteristic
287#491—1222no—
288#489—1222no—
289#487—611no—
290#487—611no—
291#487—611no—
292#487—611no—
293#487—611no—
294#487—611no—
295#487—611no—
296#487—611no—
297#487—611no—
298#487C2×D6111yescharacteristic
299#486—611no—
300#486—611no—
301#486—611no—
302#486C2×D6111yescharacteristic
303#470—1222no—
304#468—22no—
305#468—1222no—
306#441—1222no—
307#440—22no—
308#440—1222no—
309#428—1222no—
310#419—1222no—
311#417—1222no—
312C4×M16—611no—
313C4×M16—611no—
314C4×M16C2×D6111yescharacteristic
315#322—611no—
316#322—611no—
317#322—611no—
318C82—611no—
319C61#1975211yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×Q8—1221no—
322C2×Q16—1222no—
323C2×SD16—22no—
324C2×SD16—1222no—
325C2×M16—612no—
326C2×M16—612no—
327C2×M16C22⋊D6112yes—
328C4⋊Q8—611no—
329C4⋊Q8—611no—
330C4⋊Q8—611no—
331C4⋊Q8—611no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8C22×D6111yescharacteristic
336C4⋊Q8C22×D6111yescharacteristic
337C4⋊Q8C22×D6111yescharacteristic
338#127C22×D6111yescharacteristic
339#125—21no—
340#125—1221no—
341#124—21no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—1221no—
345C22⋊Q8—1221no—
346#121—21no—
347#121—21no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—1221no—
351C4×Q8—1221no—
352C4×D4—21no—
353C4×D4—21no—
354#117—21no—
355#117—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4—1221no—
358C2×C42C22×D6111yescharacteristic
359#107—22no—
360#107—1222no—
361#107—1222no—
362#107—1222no—
363#106—1222no—
364#106—1222no—
365#106—1222no—
366#103—22no—
367#103—1222no—
368#103—1222no—
369#103—1222no—
370#103—1222no—
371#103—1222no—
372#103—1222no—
373#103—1222no—
374#103—1222no—
375#103—1222no—
376#102—22no—
377#102—1222no—
378#102—1222no—
379#102—1222no—
380#97—612no—
381#97—612no—
382#97C22⋊D6112yes—
383C4×C8—611no—
384C4×C8—611no—
385C4×C8—611no—
386C4×C8—611no—
387C4×C8C22×D6111yescharacteristic
388C4×C8C22×D6111yescharacteristic
389#41—44no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—1221no—
394C2×Q8—1221no—
395C2×Q8—1221no—
396C2×Q8—1221no—
397C2×Q8—2444no—
398C2×Q8—2441no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—41no—
403C22×C4—41no—
404C22×C4—2441no—
405C22×C4D4×D6111yescharacteristic
406C22×C4D4×D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—2444no—
409SD16—44no—
410SD16—2444no—
411M16—42no—
412M16—1224no—
413M16—2442no—
414C2×C8—612no—
415C2×C8—612no—
416C2×C8—612no—
417C2×C8—612no—
418C2×C8C2×C22⋊D6112yes—
419C2×C8C2×C22⋊D6112yes—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—1221no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—2441no—
444C42—41no—
445C42—2441no—
446C42D4×D6111yescharacteristic
447C42D4×D6111yescharacteristic
448C42C2×C22⋊D6111yescharacteristic
449C42C23×D6111yescharacteristic
450C23—41no—
451C23#1451977311yescharacteristic
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—4882no—
456Q8—2442no—
457Q8—2442no—
458Q8—2442no—
459Q8—2442no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—82no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—4882no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451977311yescharacteristic
477C2×C4#1451977311yescharacteristic
478C2×C4#1451977311yescharacteristic
479C2×C4C2×D4×D6111yescharacteristic
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4C22×C22⋊D6111yescharacteristic
482C8—22no—
483C8—22no—
484C8—614no—
485C8—1222no—
486C8—1222no—
487C8—1222no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22C2×#1451977311yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—4881no—
502C4—4881no—
503C4—4881no—
504C4—4881no—
505C4#137×D6112yes—
506C4#1684055812yes—
507C4#1684056012yes—
508C2—41no—
509C2—161no—
510C2#2145627411yescharacteristic
511C2#2145628111yescharacteristic
512C2#2145628711yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3182C2
socleC22×C61C2×#127
prime radicalC61×#532C22
hypercenter#3182D61
hypocenterC61#19752
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
1. #21456560C2semidirect
3. #21456366C2semidirect
5. #21456200C2semidirect
6. #21445495C2semidirect
7. #21445480C2semidirect
8. #21445470C2semidirect
9. #21445464C2semidirect
10. #21445462C2semidirect
11. #21445439C2semidirect
12. #21445427C2semidirect
13. #21445419C2semidirect
14. #21444122C2semidirect
319. C61#19752semidirect

Maximal subgroups

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

subgroupindex
#214565602
#214565332
#214563662
#214562122
#214562002
#214454952
#214454802
#214454702
#214454642
#214454622
#214454392
#214454272
#214454192
#214441222
C61×#31822
#1975261