Finite Group Databasebeta

← #29699457

Subgroups — #29699457

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14519797C211yesmaximal, characteristic
2#21454485C211yesmaximal, characteristic, complemented
3#21454388C211yesmaximal, characteristic, complemented
4#21446527C212yesmaximal, complemented
5#21446416C212yesmaximal, complemented
6#21446143C212yesmaximal
7#21445956C211yesmaximal, characteristic, complemented
8#21445840C211yesmaximal, characteristic, complemented
9#21445688C211yesmaximal, characteristic, complemented
10#21445660C212yesmaximal, complemented
11C61×#1806C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519785C2212yes—
14C4×#11758222C2211yescharacteristic
15C4×#11758202C411yescharacteristic, complemented
16C4×#11758202C411yescharacteristic, complemented
17C4×#11758201C411yescharacteristic, complemented
18C4×#11758201C411yescharacteristic, complemented
19C4×#11758200C411yescharacteristic, complemented
20C4×#11758200C411yescharacteristic, complemented
21C4×#11758199C411yescharacteristic, complemented
22C4×#11758199C411yescharacteristic, complemented
23#16839823—22no—
24#16839819—21no—
25#16839815—21no—
26#16839798C2211yescharacteristic
27#16839790C2212yes—
28#16839787—22no—
29#16839786C2211yescharacteristic
30#16839784—21no—
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33C2×#14519705C2212yes—
34#16839758C2211yescharacteristic
35#16839756C2211yescharacteristic
36#16839729C2212yes—
37#16839718C2212yes—
38C2×#14519688C2212yes—
39C2×C4×#92418C2211yescharacteristic
40#16839526C2212yes—
41#16839526C2212yes—
42#16839522C2212yes—
43#16839517C2212yes—
44C4×#41×C61C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×#104×C61C2212yes—
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#339C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519797—22no—
51#14519797—22no—
52C2×#11758202C2×C411yescharacteristic
53C2×#11758200C2×C411yescharacteristic
54C2×#11758201C2×C411yescharacteristic
55C2×#11758199C2×C411yescharacteristic
56#14519785—24no—
57#14519785—42no—
58C2×#11758194C2312yes—
59C2×#11758189—22no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2312yes—
63C2×C8×D61—22no—
64C4×Dic122C2×C411yescharacteristic
65C4×Dic122C2×C411yescharacteristic
66C4×D244C2×C411yescharacteristic
67C4×D244C2×C411yescharacteristic
68C4×C22⋊D61—21no—
69C4×C22⋊D61—21no—
70#14519736C2311yescharacteristic
71#14519734—21no—
72C42×D61—21no—
73#14519710C2×C411yescharacteristic
74#14519710C2×C411yescharacteristic
75#14519708C2×C411yescharacteristic
76#14519708C2×C411yescharacteristic
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79C4⋊#92418D411yescharacteristic
80C4⋊#92418D411yescharacteristic
81C4⋊#92418D412yes—
82#14519706—22no—
83#14519706—22no—
84#14519705—24no—
85#14519701C2×C411yescharacteristic
86#14519701C2×C411yescharacteristic
87#14519700C2×C411yescharacteristic
88#14519700C2×C411yescharacteristic
89#14519698C2×C411yescharacteristic
90#14519698C2×C411yescharacteristic
91#14519697C2×C411yescharacteristic
92#14519697C2×C411yescharacteristic
93C22⋊#92418—22no—
94C22⋊#92418—22no—
95#14519688—22no—
96#14519688—42no—
97C4×#92418—22no—
98C4×#92418C2×C411yescharacteristic
99C4×#92418C2×C411yescharacteristic
100C4×#92418C2×C411yescharacteristic
101C4×#92418C2×C411yescharacteristic
102C2×#41×C61C2311yescharacteristic
103C2×M16×C61C2312yes—
104C4×Q8×C61C2×C411yescharacteristic
105C4×Q8×C61C2×C411yescharacteristic
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61C2×C411yescharacteristic
109C4×D4×C61C2×C411yescharacteristic
110#117×C61—21no—
111#117×C61C2311yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2312yes—
115C2×C42×C61C2311yescharacteristic
116C4⋊C8×C61—21no—
117C4⋊C8×C61—21no—
118#104×C61—42no—
119#95×C61—21no—
120#41×C61—21no—
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124C2×Q8×C61C22×C411yescharacteristic
125C2×D4×C61—21no—
126C2×D4×C61C22×C411yescharacteristic
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61#4111yescharacteristic
132C22×C4×C61#4111yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—21no—
135#11758222—21no—
136#11758222—21no—
137#11758222#4112yes—
138#11758222#4112yes—
139C2×D244C22×C411yescharacteristic
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×Dic122C22×C411yescharacteristic
144M16×C61—22no—
145M16×C61—22no—
146C2×C8×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—42no—
150C4⋊C4×C61C22×C411yescharacteristic
151C4⋊C4×C61C22×C411yescharacteristic
152C22⋊C4×C61—21no—
153C22⋊C4×C61—21no—
154C22⋊C4×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61—22no—
159C42×C61—22no—
160C42×C61C22×C411yescharacteristic
161C42×C61C22×C411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C22⋊Dic61—21no—
165#11758202—22no—
166#11758201—22no—
167#11758200—22no—
168#11758199—22no—
169#11758198—21no—
170#11758198—21no—
171C4⋊Dic61C22×C411yescharacteristic
172C4⋊Dic61C22×C411yescharacteristic
173#11758196—21no—
174#11758196—21no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178#11758194—22no—
179#11758194—22no—
180C2×#92418—22no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—22no—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418C22×C411yescharacteristic
188C2×#92418C22×C411yescharacteristic
189C2×#92418C22×C411yescharacteristic
190C2×#92418C22×C411yescharacteristic
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193#11758189—44no—
194C8×D61—44no—
195C23×C61—21no—
196C23×C61C2×#4111yescharacteristic
197C22×D61—21no—
198Q8×C61—21no—
199Q8×C61—21no—
200D4×C61—21no—
201D4×C61—21no—
202D4×C61—41no—
203D4×C61—41no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—81no—
216C2×C4×C61C4×D411yescharacteristic
217C2×C4×C61C4×D411yescharacteristic
218C2×C4×C61C4×D411yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C23×C411yescharacteristic
221C2×C4×C61C22×D411yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C22⋊D61—22no—
227C22⋊D61—22no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231D244—21no—
232D244#11712yes—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—42no—
236C4×D61—81no—
237Dic122—21no—
238Dic122#11712yes—
239C8×C61—42no—
240#92418—21no—
241#92418—21no—
242#92418—21no—
243#92418—21no—
244#92418—22no—
245#92418—22no—
246#92418—42no—
247#22782—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61C2×C4×D411yescharacteristic
254C22×C61#55211yescharacteristic
255C22×C61#55311yescharacteristic
256C2×D61—22no—
257C2×D61—41no—
258C2×D61—41no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—41no—
265C4×C61—82no—
266C4×C61#44412yes—
267C4×C61C2×#11711yescharacteristic
268C4×C61#52011yescharacteristic
269C4×C61#54611yescharacteristic
270C4×C61#54811yescharacteristic
271Dic61—22no—
272Dic61—41no—
273Dic61—41no—
274C4×#135—611no—
275#2925—611no—
276#2906—611no—
277#1938—612no—
278#1922—612no—
279#1876—612no—
280#1837—611no—
281#1806—611no—
282#1806D6111yescharacteristic, complemented
283#1770—611no—
284#1765—612no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—81no—
288C2×C61#235211yescharacteristic
289C2×C61C2×#44411yescharacteristic
290C2×C61#295711yescharacteristic
291D61—42no—
292C2×#135—611no—
293C2×#131—612no—
294C4×#41—611no—
295C4×#41C2×D6111yescharacteristic
296C4×Q16—611no—
297C4×Q16—611no—
298C4×SD16—611no—
299C4×SD16—611no—
300C4×SD16—611no—
301C4×SD16—611no—
302C4×D8—611no—
303C4×D8—611no—
304#433—1222no—
305#432—1221no—
306#431—1221no—
307#427—611no—
308#424—612no—
309#424—1222no—
310#423—611no—
311#423—1221no—
312#423C2×D6111yescharacteristic
313C2×C4⋊C8—611no—
314C2×C4⋊C8—1221no—
315C2×#104—612no—
316C2×#104C2×D6112yes—
317#416—611no—
318#416—611no—
319#408—612no—
320#405—612no—
321C2×#97—612no—
322C2×C4×C8—611no—
323C4×C4⋊C4C2×D6111yescharacteristic
324#341—612no—
325#341—612no—
326#339—612no—
327#339C2×D6112yes—
328#336—612no—
329C61#2278211yescharacteristic, complemented, Sylow
330C2×#41—611no—
331C2×#41C22×D6111yescharacteristic
332#135—1222no—
333#135—1222no—
334C2×Q16—611no—
335C2×SD16—611no—
336C2×SD16—611no—
337C2×D8—611no—
338#131—1224no—
339#131—2442no—
340C2×M16—612no—
341C2×M16—1222no—
342C2×M16C22×D6112yes—
343C22×C8—612no—
344C22×C8—611no—
345C22×C8—611no—
346C22×C8—1222no—
347C4×Q8—611no—
348C4×Q8—611no—
349C4×Q8C4×D6111yescharacteristic
350C4×Q8C4×D6111yescharacteristic
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—611no—
354C4×D4—611no—
355C4×D4—1221no—
356C4×D4—1221no—
357C4×D4C4×D6111yescharacteristic
358C4×D4C4×D6111yescharacteristic
359#117—21no—
360#117—611no—
361#117—1221no—
362#117C22×D6111yescharacteristic
363C2×C4⋊C4—21no—
364C2×C42—21no—
365C2×C42—1221no—
366C2×C42C22×D6112yes—
367C2×C42C22×D6111yescharacteristic
368#107—611no—
369#107—611no—
370#106—611no—
371#106—611no—
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—611no—
375C4⋊C8—611no—
376C4⋊C8—612no—
377C4⋊C8—1222no—
378C4⋊C8—1222no—
379C4⋊C8—1221no—
380C4⋊C8—1221no—
381#104—42no—
382#104—1224no—
383#103—611no—
384#103—611no—
385#103—611no—
386#103—611no—
387#102—611no—
388#102—611no—
389#102—611no—
390#102—611no—
391C22⋊C8—1222no—
392C22⋊C8—1222no—
393#97—1222no—
394#97—2442no—
395C4×C8—611no—
396C4×C8—611no—
397C4×C8—611no—
398C4×C8—611no—
399C4×C8—1222no—
400#95—21no—
401#41—21no—
402#41—21no—
403#41—21no—
404#41—21no—
405#41—612no—
406#41—612no—
407#41—1221no—
408#41—1221no—
409C2×Q8—611no—
410C2×Q8C2×C4×D6111yescharacteristic
411C2×D4—21no—
412C2×D4—611no—
413C2×D4—1221no—
414C2×D4C2×C4×D6111yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4#1175822411yescharacteristic
423C22×C4#1175822411yescharacteristic
424C22×C4C23×D6111yescharacteristic
425Q16—1222no—
426SD16—1222no—
427SD16—1222no—
428D8—1222no—
429M16—22no—
430M16—22no—
431M16—1222no—
432M16—1222no—
433M16—2444no—
434C2×C8—22no—
435C2×C8—611no—
436C2×C8—611no—
437C2×C8—611no—
438C2×C8—611no—
439C2×C8—612no—
440C2×C8—612no—
441C2×C8—1222no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1222no—
446C2×C8—1221no—
447C2×C8—1222no—
448C2×C8—2444no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—611no—
453C4⋊C4—611no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4C2×C4×D6111yescharacteristic
457C4⋊C4C2×C4×D6111yescharacteristic
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—1221no—
462C22⋊C4—1221no—
463C22⋊C4—1221no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42C2×C4×D6111yescharacteristic
473C42C2×C4×D6111yescharacteristic
474C42D4×D6111yescharacteristic
475C42D4×D6111yescharacteristic
476C23—21no—
477C23—1221no—
478C23C2×#1175822411yescharacteristic
479Q8—21no—
480Q8—21no—
481Q8—612no—
482Q8—1221no—
483D4—21no—
484D4—21no—
485D4—41no—
486D4—41no—
487D4—612no—
488D4—1222no—
489D4—1222no—
490D4—1221no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—81no—
503C2×C4—1221no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—4881no—
509C2×C4—2442no—
510C2×C4#1451973711yescharacteristic
511C2×C4#1451973711yescharacteristic
512C2×C4#1451973711yescharacteristic
513C2×C4#1451973711yescharacteristic
514C2×C4C22×C4×D6111yescharacteristic
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4#41×D6111yescharacteristic
517C2×C4#41×D6111yescharacteristic
518C2×C4#1451982211yescharacteristic
519C2×C4#1451982511yescharacteristic
520C8—42no—
521C8—1221no—
522C8—1221no—
523C8—1221no—
524C8—1221no—
525C8—1222no—
526C8—1222no—
527C8—2442no—
528C22—21no—
529C22—21no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—1222no—
534C22—2441no—
535C22—2441no—
536C22C2×#1451973711yescharacteristic
537C22#1684044611yescharacteristic
538C22#1684045911yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—82no—
546C4—1222no—
547C4—2441no—
548C4—2441no—
549C4#1683988112yes—
550C4#117×D6111yescharacteristic
551C4#1684024311yescharacteristic
552C4#1684040911yescharacteristic
553C4#1684042211yescharacteristic
554C2—21no—
555C2—21no—
556C2—81no—
557C2—2442no—
558C2#2144874111yescharacteristic
559C2C2×#1683988111yescharacteristic
560C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1806C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1806D61
hypocenterC61#22782
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #21454485C2semidirect
3. #21454388C2semidirect
4. #21446527C2semidirect
5. #21446416C2semidirect
7. #21445956C2semidirect
8. #21445840C2semidirect
9. #21445688C2semidirect
10. #21445660C2semidirect
11. C61×#1806C2semidirect
15. C4×#11758202C4semidirect
16. C4×#11758202C4semidirect
17. C4×#11758201C4semidirect
18. C4×#11758201C4semidirect
19. C4×#11758200C4semidirect
20. C4×#11758200C4semidirect
21. C4×#11758199C4semidirect
22. C4×#11758199C4semidirect
282. #1806D61semidirect
329. C61#22782semidirect

Maximal subgroups

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

subgroupindex
C4×#145197972
#214544852
#214543882
#214465272
#214464162
#214461432
#214459562
#214458402
#214456882
#214456602
C61×#18062
#2278261