Finite Group Databasebeta

← #30619325

Subgroups — #30619325

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462310C211yesmaximal, characteristic, complemented
2#21462309C211yesmaximal, characteristic
3#21461948C211yesmaximal, characteristic
4#21461946C211yesmaximal, characteristic, complemented
5#21461815C212yesmaximal
6#21461813C212yesmaximal, complemented
7#21461813C212yesmaximal, complemented
8#21461812C212yesmaximal
9#21461742C211yesmaximal, characteristic, complemented
10#21461539C211yesmaximal, characteristic, complemented
11#497×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2211yescharacteristic
13C2×Q8×#6923C2212yes—
14C2×D4×#6923C2211yescharacteristic
15C2×#14519840C2212yes—
16C2×#14519840C2212yes—
17C2×#14519840C2212yes—
18C2×#14519839C2212yes—
19C4×D61⋊C8C2211yescharacteristic
20(C2×#6923)⋊Q8C2211yescharacteristic
21(C2×#6923)⋊Q8C2212yes—
22#16840714C2211yescharacteristic
23#16840712—22no—
24#16840711—22no—
25#16840711—22no—
26#16840709—22no—
27#16840708C2211yescharacteristic
28#16840706C2211yescharacteristic
29#16840698C2211yescharacteristic
30#16840636C2212yes—
31#16840636C2212yes—
32(C2×#6923)⋊C8C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C2×Q16×D61C2212yes—
35C2×SD16×D61C2212yes—
36C4⋊Q8×D61C2211yescharacteristic
37#124×D61C2211yescharacteristic
38#16840157C411yescharacteristic, complemented
39#16840156C411yescharacteristic, complemented
40#16840155C412yescomplemented
41#16840154C411yescharacteristic, complemented
42#16840153C412yescomplemented
43#97×D61C2211yescharacteristic
44C61×#497C411yescharacteristic, complemented
45C23×#6923—21no—
46C2×D61⋊C8C2312yes—
47Q8×#6923—42no—
48Q8×#6923—42no—
49Q8×#6923—42no—
50D4×#6923—42no—
51C2×C4⋊#6923—21no—
52C2×C4⋊#6923—22no—
53C2×C4⋊#6923—22no—
54C2×C4⋊#6923C2312yes—
55C2×C4⋊#6923C2311yescharacteristic
56C2×C4⋊#6923C2311yescharacteristic
57C2×C22⋊#6923—21no—
58C2×C22⋊#6923—22no—
59C2×C4×#6923—22no—
60C2×C4×#6923—21no—
61C2×C4×#6923C2311yescharacteristic
62C2×C4×#6923C2311yescharacteristic
63#14519840—42no—
64#14519840—42no—
65#14519840—42no—
66#14519839—42no—
67C4×#92420—22no—
68(C2×#6923)⋊C4—22no—
69(C2×#6923)⋊C4—22no—
70(C2×#6923)⋊C4—22no—
71(C2×#6923)⋊C4—22no—
72(C2×#6923)⋊C4D412yes—
73(C2×#6923)⋊C4D412yes—
74C2×Q8×D61C2311yescharacteristic
75C2×Q8×D61C2312yes—
76C2×D4×D61C2311yescharacteristic
77C2×Dic244C2×C412yescomplemented
78C2×#11758202C2×C412yescomplemented
79C2×#11758202C2×C412yescomplemented
80Q16×D61—42no—
81Q16×D61—42no—
82C2×#11758190C2×C412yes—
83C2×#11758200C2×C412yes—
84C2×#11758201C2×C412yes—
85SD16×D61—42no—
86SD16×D61—42no—
87C2×C8×D61C2312yes—
88#14519777C2×C411yescharacteristic, complemented
89#14519776—22no—
90#14519775C2×C412yescomplemented
91#14519766C2×C411yescharacteristic
92#14519765C2×C411yescharacteristic
93#14519764C2×C411yescharacteristic
94#14519763—22no—
95C4⋊C4×D61—22no—
96C22⋊C4×D61—22no—
97C42×D61C2311yescharacteristic
98#14519688C2×C411yescharacteristic, complemented
99#14519687C2×C412yescomplemented
100C2×Q16×C61C2×C412yescomplemented
101C2×SD16×C61C2×C412yes—
102C4⋊Q8×C61C2×C411yescharacteristic, complemented
103#124×C61C2×C411yescharacteristic
104#97×C61C2×C411yescharacteristic, complemented
105C23×D61—21no—
106C22×#6923—22no—
107C22×#6923—21no—
108C22×#6923—21no—
109C22×#6923—42no—
110C22×#6923—82no—
111C22×#6923C2×D411yescharacteristic
112C22×#6923C2×D411yescharacteristic
113C22×#6923C2×D412yes—
114C2×Q8×C61C22×C411yescharacteristic
115C2×Q8×C61C22×C412yes—
116C2×D4×C61C22×C411yescharacteristic
117C2×C22⋊D61—21no—
118Q8×D61—22no—
119Q8×D61—22no—
120Q8×D61—22no—
121Q8×D61—22no—
122Q8×D61—22no—
123Q8×D61—22no—
124D4×D61—22no—
125D4×D61—22no—
126C2×D244C22×C411yescharacteristic
127C2×C4×D61—21no—
128C2×C4×D61—22no—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2411yescharacteristic
131C2×Dic122—21no—
132C2×Dic122—22no—
133C2×Dic122C22×C411yescharacteristic
134C2×Dic122C22×C412yes—
135C22⋊#6923—42no—
136C22⋊#6923—82no—
137C2×#92420—22no—
138C4⋊#6923—22no—
139C4⋊#6923—22no—
140C4⋊#6923—22no—
141C4⋊#6923—22no—
142C4⋊#6923—44no—
143C4⋊#6923—42no—
144C4⋊#6923—82no—
145C4×#6923—22no—
146C4×#6923—22no—
147C4×#6923—44no—
148C4×#6923—42no—
149D61⋊C8—22no—
150D61⋊C8—22no—
151Q16×C61—42no—
152SD16×C61—42no—
153C2×C8×C61C22×C412yes—
154C4⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61C22×C411yescharacteristic
157C22⋊Dic61—22no—
158#11758202—42no—
159#11758202—42no—
160#11758201—42no—
161#11758200—42no—
162#11758198—22no—
163#11758198—22no—
164C4⋊Dic61—22no—
165#11758196—24no—
166C4×Dic61C22×C411yescharacteristic
167C4×Dic61#4112yes—
168C2×#92418C22×C412yes—
169Dic244—42no—
170#11758190—42no—
171C8×D61—22no—
172C8×D61—22no—
173C23×C61—21no—
174C22×D61—21no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61C22×D411yescharacteristic
180C2×#6923—22no—
181C2×#6923—21no—
182C2×#6923—21no—
183C2×#6923—22no—
184C2×#6923—22no—
185C2×#6923—22no—
186C2×#6923—22no—
187C2×#6923—41no—
188C2×#6923—42no—
189C2×#6923—41no—
190C2×#6923—42no—
191C2×#6923—42no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—84no—
195Q8×C61—22no—
196Q8×C61—22no—
197Q8×C61—22no—
198D4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—21no—
201C2×C4×C61C4×D412yes—
202C2×C4×C61C23×C411yescharacteristic
203C22⋊D61—42no—
204C2×Dic61—22no—
205C2×Dic61—21no—
206C2×Dic61C4×D412yes—
207C2×Dic61C2×#4111yescharacteristic
208D244—22no—
209C4×D61—22no—
210C4×D61—22no—
211C4×D61—41no—
212C4×D61—42no—
213C4×D61—42no—
214C4×D61—41no—
215C4×D61H(5,𝔽2)12yes—
216C4×D61H(5,𝔽2)12yes—
217Dic122—22no—
218Dic122—22no—
219Dic122—22no—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223#92420—42no—
224C8×C61—22no—
225#92418—22no—
226#22278—611nomaximal, Sylow
227C22×C61—41no—
228C22×C61—42no—
229C22×C61C2×C4×D411yescharacteristic
230C2×D61—41no—
231C2×D61—42no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—82no—
235C2×D61#53511yescharacteristic
236C2×D61#53511yescharacteristic
237C2×D61#57712yes—
238C2×D61#57812yes—
239#6923—41no—
240#6923—41no—
241#6923—42no—
242#6923—81no—
243#6923—81no—
244#6923—82no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—41no—
248C4×C61#51812yes—
249Dic61—22no—
250Dic61—42no—
251Dic61—41no—
252Dic61#54012yes—
253C2×#497—611no—
254#2327—611no—
255#2326—611no—
256#1958—611no—
257#1956—611no—
258#1880—612no—
259#1878—612no—
260#1878—612no—
261#1877—612no—
262#1836—611no—
263#1746—611no—
264C2×C61—81no—
265C2×C61#232811yescharacteristic
266C2×C61#294712yes—
267D61—81no—
268D61#322211yescharacteristic
269D61#322311yescharacteristic
270D61#322512yes—
271C22×Q16—612no—
272C22×SD16—612no—
273C2×C4⋊Q8—611no—
274C2×#124—611no—
275C2×C4×Q8—611no—
276C2×C4×Q8—612no—
277C2×C4×D4—611no—
278#497—612no—
279#497—612no—
280#497—611no—
281#497—611no—
282#497—611no—
283#497#692311yescharacteristic, complemented
284C2×#103—612no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#102—612no—
288C2×#97—611no—
289C2×C4×C8—611no—
290#391—611no—
291#391—612no—
292#390—611no—
293#388—1222no—
294#387—1222no—
295#387—1222no—
296#385—1222no—
297#384—611no—
298#382—611no—
299#376—611no—
300#340—612no—
301#340—612no—
302#336—612no—
303#336—612no—
304C61#2227811yescharacteristic, complemented, Sylow
305C22×Q8—611no—
306C22×Q8—612no—
307C22×D4—611no—
308C23×C4—1221no—
309C2×Q16—612no—
310C2×Q16—612no—
311C2×Q16—612no—
312C2×Q16—2442no—
313C2×Q16—2442no—
314C2×Q16C2×#692312yescomplemented
315C2×SD16—612no—
316C2×SD16—612no—
317C2×SD16—612no—
318C2×SD16—2442no—
319C2×SD16—2442no—
320C2×SD16C2×#692312yes—
321C22×C8—612no—
322C22×C8—612no—
323C4⋊Q8—612no—
324C4⋊Q8—611no—
325C4⋊Q8—1222no—
326C4⋊Q8C2×#692311yescharacteristic, complemented
327#124—611no—
328#124—611no—
329#124—611no—
330#124—1222no—
331#124C2×#692311yescharacteristic
332C4×Q8—2442no—
333C4×Q8—2442no—
334C4×Q8—2442no—
335C4×D4—2442no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1222no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1221no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—611no—
349C2×C42—1222no—
350C2×C42—1221no—
351#103—2442no—
352#103—2442no—
353#103—2442no—
354#102—2442no—
355#97—612no—
356#97—611no—
357#97C2×#692311yescharacteristic, complemented
358C4×C8—1222no—
359#95—612no—
360#95—612no—
361#95—1222no—
362#95—1222no—
363#95—1222no—
364#95—1222no—
365C24—1221no—
366C2×Q8—611no—
367C2×Q8—612no—
368C2×Q8—1221no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—1222no—
374C2×Q8—1222no—
375C2×Q8—1222no—
376C2×Q8C22×#692311yescharacteristic
377C2×Q8C22×#692312yes—
378C2×D4—611no—
379C2×D4—1222no—
380C2×D4—1222no—
381C2×D4—1221no—
382C2×D4C22×#692311yescharacteristic
383C22×C4—611no—
384C22×C4—612no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—612no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—4882no—
394C22×C4—2442no—
395Q16—42no—
396Q16—2442no—
397Q16—2442no—
398Q16—2442no—
399SD16—42no—
400SD16—2442no—
401SD16—2442no—
402SD16—2442no—
403C2×C8—612no—
404C2×C8—1222no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8C22×#692312yes—
410C4⋊C4—22no—
411C4⋊C4—1224no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—4882no—
418C4⋊C4—2444no—
419C4⋊C4—2442no—
420C22⋊C4—22no—
421C22⋊C4—1222no—
422C22⋊C4—1222no—
423C22⋊C4—1222no—
424C22⋊C4—4882no—
425C22⋊C4—2442no—
426C42—612no—
427C42—611no—
428C42—1222no—
429C42—1222no—
430C42—2442no—
431C42—2444no—
432C42C22×#692311yescharacteristic
433C23—21no—
434C23—611no—
435C23—1221no—
436C23—2442no—
437C23—2441no—
438C23—2441no—
439C23—2442no—
440Q8—22no—
441Q8—22no—
442Q8—22no—
443Q8—1222no—
444Q8—1222no—
445Q8—1222no—
446Q8—2442no—
447Q8—2442no—
448Q8—2442no—
449D4—22no—
450D4—1222no—
451D4—2442no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—611no—
455C2×C4—612no—
456C2×C4—612no—
457C2×C4—612no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1221no—
461C2×C4—1222no—
462C2×C4—1221no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—1221no—
468C2×C4—1222no—
469C2×C4—4884no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2441no—
473C2×C4—2442no—
474C2×C4—2442no—
475C2×C4—2441no—
476C2×C4—2442no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4D4×#692312yes—
482C2×C4C23×#692311yescharacteristic
483C8—22no—
484C8—1222no—
485C8—2442no—
486C22—41no—
487C22—42no—
488C22—611no—
489C22—611no—
490C22—612no—
491C22—612no—
492C22—4881no—
493C22—4882no—
494C22—4881no—
495C22—2441no—
496C22—2442no—
497C22C2×D4×#692311yescharacteristic
498C4—22no—
499C4—41no—
500C4—42no—
501C4—612no—
502C4—1222no—
503C4—4881no—
504C4—4881no—
505C4—4882no—
506C4—2441no—
507C4—2442no—
508C4—2441no—
509C4—2441no—
510C4—2442no—
511C4#1684082912yes—
512C2—81no—
513C2—611no—
514C2—612no—
515C2—611no—
516C2—4881no—
517C2#2146231111yescharacteristic
518C2#2146257712yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#497C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#497#6923
hypocenterC61#22278
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #21462310C2semidirect
4. #21461946C2semidirect
6. #21461813C2semidirect
7. #21461813C2semidirect
9. #21461742C2semidirect
10. #21461539C2semidirect
38. #16840157C4semidirect
39. #16840156C4semidirect
40. #16840155C4semidirect
41. #16840154C4semidirect
42. #16840153C4semidirect
44. C61×#497C4semidirect
77. C2×Dic244C2×C4semidirect
78. C2×#11758202C2×C4semidirect
79. C2×#11758202C2×C4semidirect
88. #14519777C2×C4semidirect
90. #14519775C2×C4semidirect
98. #14519688C2×C4semidirect
99. #14519687C2×C4semidirect
100. C2×Q16×C61C2×C4semidirect
102. C4⋊Q8×C61C2×C4semidirect
104. #97×C61C2×C4semidirect
283. #497#6923semidirect
304. C61#22278semidirect
314. C2×Q16C2×#6923semidirect
326. C4⋊Q8C2×#6923semidirect
357. #97C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623102
#214623092
#214619482
#214619462
#214618152
#214618132
#214618122
#214617422
#214615392
#497×D612
#2227861