Finite Group Databasebeta

← #29835351

Subgroups — #29835351

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456352C211yesmaximal, characteristic, complemented
2#21456307C211yesmaximal, characteristic
3#21448064C211yesmaximal, characteristic, complemented
4#21448063C211yesmaximal, characteristic, complemented
5#21448058C211yesmaximal, characteristic, complemented
6#21448057C211yesmaximal, characteristic, complemented
7#21448052C211yesmaximal, characteristic, complemented
8#21448051C211yesmaximal, characteristic, complemented
9#21448045C211yesmaximal, characteristic, complemented
10#21447537C211yesmaximal, characteristic, complemented
11#21447530C211yesmaximal, characteristic
12#21447525C211yesmaximal, characteristic
13#358×D61C211yesmaximal, characteristic, complemented
14#21447414C211yesmaximal, characteristic, complemented
15C61×#2241C211yesmaximal, characteristic, complemented
16C2×#14519809—21no—
17C2×#14519800C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic
19#16840389—21no—
20#16840337—21no—
21C2×#14519716C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic
23#16840170—21no—
24#16840168—21no—
25#16840166C2211yescharacteristic
26#16840164C2211yescharacteristic
27#16840163C2211yescharacteristic
28#16840155—21no—
29#16840154—21no—
30#16840147C2211yescharacteristic
31#16840146C2211yescharacteristic
32#16840140C2211yescharacteristic
33#16839803—21no—
34#16839799C2211yescharacteristic
35#107×D61C2211yescharacteristic
36#16839716C2211yescharacteristic
37#16839567C2211yescharacteristic
38#16839567C2211yescharacteristic
39#16839566C2211yescharacteristic
40#16839566C2211yescharacteristic
41#16839566C2211yescharacteristic
42#16839566C2211yescharacteristic
43#16839565C2211yescharacteristic
44#16839565C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839538C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C61×#500C2211yescharacteristic
53C61×#494C2211yescharacteristic
54C61×#358C2211yescharacteristic
55C61×#358C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C4×C16×C61C2211yescharacteristic
59C2×#11758226—21no—
60C2×Q8×D61—21no—
61#14519809—44no—
62#14519800—22no—
63#14519800—22no—
64C2×Dic244C2311yescharacteristic
65C2×#11758202—21no—
66C2×#11758202—21no—
67Q16×D61—22no—
68Q16×D61—22no—
69C2×#11758190—21no—
70C2×#11758201—21no—
71C2×#11758201—21no—
72C2×D488C2311yescharacteristic
73C2×#11758189D411yescharacteristic
74C2×C8×D61C2311yescharacteristic
75#14519776—21no—
76#14519775—21no—
77#14519775—41no—
78#14519769—21no—
79#14519768—21no—
80#14519767—41no—
81#14519766—21no—
82#14519765—21no—
83#14519762—41no—
84#14519762—41no—
85#14519755—41no—
86#14519755—41no—
87#14519755—41no—
88#14519753—41no—
89#14519735—21no—
90#14519734D411yescharacteristic
91C4⋊C4×D61—21no—
92#14519716—22no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61—22no—
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519710—21no—
100#14519709—21no—
101#14519708—21no—
102#14519708—21no—
103#14519688D411yescharacteristic
104#14519687D411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×Q16×C61—21no—
107C2×Q16×C61C2311yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109C4⋊Q8×C61—21no—
110C4⋊Q8×C61—21no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113#107×C61—21no—
114#107×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116C4×C8×C61C2311yescharacteristic
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—41no—
120#11758226—42no—
121#11758226—42no—
122Q8×D61—42no—
123Q8×D61—42no—
124C2×D244—21no—
125C2×D244—41no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61C2×D411yescharacteristic
130C2×Dic122—21no—
131C2×Dic122—41no—
132C2×Dic122—41no—
133Q16×C61—21no—
134Q16×C61—21no—
135Q16×C61—21no—
136Q16×C61—21no—
137Q16×C61—42no—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2411yescharacteristic
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C42×C61C2×D411yescharacteristic
146#11758202—42no—
147#11758202—42no—
148#11758201—42no—
149#11758201—42no—
150#11758198—41no—
151#11758198—41no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198D811yescharacteristic
156#11758198D811yescharacteristic
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—41no—
160#11758196—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196D811yescharacteristic
168#11758196D811yescharacteristic
169C4×Dic61—41no—
170C4×Dic61C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173Dic244—21no—
174Dic244—21no—
175D488—21no—
176D488—21no—
177#11758190—42no—
178#11758189SD1614yes—
179C8×D61#4112yes—
180C8×D61#4112yes—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C22×D61—41no—
186C22×D61C2×D811yescharacteristic
187Q8×C61—41no—
188Q8×C61—41no—
189Q8×C61—41no—
190Q8×C61—41no—
191Q8×C61—161no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C2×D811yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61C2×D811yescharacteristic
203C2×Dic61C2×D811yescharacteristic
204C2×Dic61C2×D811yescharacteristic
205D244—41no—
206D244—41no—
207D244—82no—
208C4×D61—82no—
209C4×D61—82no—
210C4×D61—161no—
211C4×D61#12412yes—
212C4×D61C2×SD1612yes—
213Dic122—41no—
214Dic122—41no—
215Dic122—82no—
216Dic122—161no—
217C8×C61C2×SD1612yes—
218C8×C61C2×#4111yescharacteristic
219C8×C61C2×#4111yescharacteristic
220#92418#12412yes—
221#92418C2×SD1612yes—
222#33703—611nomaximal, Sylow
223C22×C61C22×D811yescharacteristic
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—81no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61#48612yes—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61C2×#12411yescharacteristic
237C4×C61C22×SD1611yescharacteristic
238Dic61—21no—
239Dic61—21no—
240Dic61—81no—
241Dic61—81no—
242Dic61—81no—
243Dic61#48612yes—
244#3164—611no—
245#3156—611no—
246#2245—611no—
247#2245—611no—
248#2243—611no—
249#2243—611no—
250#2242—611no—
251#2242—611no—
252#2241—611no—
253#2241D6111yescharacteristic, complemented
254#2141—611no—
255#2139—611no—
256#2138—611no—
257C2×#358—611no—
258#2108—611no—
259C2×C61C2×#48611yescharacteristic
260C2×C61#341611yescharacteristic
261C2×C61#341711yescharacteristic
262D61—22no—
263D61—161no—
264C2×#137—1221no—
265C2×#135—611no—
266C22×Q16—611no—
267#544—1221no—
268#536—1221no—
269C2×M32—611no—
270C22×C16—611no—
271#501—1221no—
272#501—1221no—
273#500—611no—
274#500C2×D6111yescharacteristic
275#499—611no—
276#499—611no—
277#497—1221no—
278#497—1221no—
279#495—611no—
280#495—611no—
281#494—611no—
282#494C2×D6111yescharacteristic
283#428—1221no—
284#427—611no—
285C2×#107—611no—
286#405—611no—
287#358—611no—
288#358—611no—
289#358—611no—
290#358—611no—
291#358—611no—
292#358—611no—
293#358—611no—
294#358—611no—
295#358C2×D6111yescharacteristic
296#358C2×D6111yescharacteristic
297#358C2×D6111yescharacteristic
298#358C2×D6111yescharacteristic
299#357—611no—
300#357—611no—
301#357—611no—
302#357—611no—
303#346—611no—
304#346—611no—
305C4×C16—611no—
306C4×C16C2×D6111yescharacteristic
307C61#3370311yescharacteristic, complemented, Sylow
308C2×#41—1221no—
309C22×Q8—1221no—
310#137—2444no—
311#135—1222no—
312#135—1222no—
313C2×Q16—21no—
314C2×Q16—611no—
315C2×Q16—1222no—
316C2×Q16—1222no—
317C2×Q16—1221no—
318C2×Q16—1221no—
319C2×Q16C22×D6111yescharacteristic
320C2×Q16C22×D6111yescharacteristic
321C2×SD16—1221no—
322C2×SD16—1221no—
323C2×SD16—1221no—
324C2×D8—611no—
325C2×M16—611no—
326C22×C8—611no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—1221no—
330C4⋊Q8—1221no—
331C4⋊Q8—2441no—
332#125—1221no—
333#125—1221no—
334#125—2441no—
335#124—1221no—
336#124—1221no—
337#123—2441no—
338#123—2441no—
339C22⋊Q8—2441no—
340C22⋊Q8—2441no—
341C22⋊Q8—2441no—
342C22⋊Q8—2441no—
343#117—611no—
344#117—1221no—
345C2×C4⋊C4—1221no—
346M32—1222no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16—1222no—
350C2×C16C22×D6111yescharacteristic
351C2×C16C22×D6111yescharacteristic
352#107—21no—
353#107—611no—
354#107—611no—
355#107—1221no—
356#107—1221no—
357#107C22×D6111yescharacteristic
358#107C22×D6111yescharacteristic
359#106—1221no—
360#106—1221no—
361#106—1221no—
362#97—611no—
363#97—611no—
364C4×C8—611no—
365C4×C8C22×D6111yescharacteristic
366#41—2442no—
367#41—2442no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—41no—
371C2×Q8—1221no—
372C2×Q8—2441no—
373C2×Q8—2441no—
374C2×Q8—2442no—
375C2×Q8—2442no—
376C2×D4—1221no—
377C2×D4—2441no—
378C22×C4—611no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4—2441no—
382Q16—21no—
383Q16—21no—
384Q16—21no—
385Q16—21no—
386Q16—42no—
387Q16—1221no—
388Q16—1221no—
389Q16—2442no—
390Q16—2442no—
391SD16—2442no—
392SD16—2442no—
393SD16—2442no—
394D8—1221no—
395D8—1221no—
396M16—614no—
397C2×C8—611no—
398C2×C8—611no—
399C2×C8—612no—
400C2×C8—612no—
401C2×C8D4×D6111yescharacteristic
402C2×C8C23×D6111yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—611no—
409C4⋊C4—611no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C4⋊C4—2441no—
417C4⋊C4—2441no—
418C4⋊C4—2441no—
419C4⋊C4—2441no—
420C22⋊C4—611no—
421C22⋊C4—611no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2441no—
427C42—611no—
428C42—2441no—
429C42D4×D6111yescharacteristic
430C16—21no—
431C16—21no—
432C16—1221no—
433C16—1221no—
434C23—611no—
435C23—2441no—
436Q8—41no—
437Q8—41no—
438Q8—41no—
439Q8—41no—
440Q8—161no—
441Q8—9761no—
442Q8—4882no—
443Q8—2441no—
444Q8—2441no—
445D4—4882no—
446D4—2441no—
447D4—2441no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—611no—
453C2×C4—612no—
454C2×C4—612no—
455C2×C4—611no—
456C2×C4—611no—
457C2×C4—9761no—
458C2×C4—4882no—
459C2×C4—4882no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4D8×D6111yescharacteristic
464C2×C4D8×D6111yescharacteristic
465C2×C4C2×D4×D6111yescharacteristic
466C8—612no—
467C8—612no—
468C8SD16×D6112yes—
469C8#41×D6111yescharacteristic
470C8#41×D6111yescharacteristic
471C22—612no—
472C22—1221no—
473C22—1221no—
474C22—4881no—
475C22—4881no—
476C22—4881no—
477C22C2×D8×D6111yescharacteristic
478C4—21no—
479C4—21no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—612no—
485C4—1221no—
486C4—1221no—
487C4—4881no—
488C4—4881no—
489C4—4881no—
490C4#124×D6111yescharacteristic
491C4C2×SD16×D6111yescharacteristic
492C2—1222no—
493C2—9761no—
494C2#486×D6111yescharacteristic
495C2#2145896111yescharacteristic
496C2#2145896811yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2241C2
socleC22×C61C22×D8
prime radicalC2×#14519800C22
hypercenter#2241D61
hypocenterC61#33703
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
1. #21456352C2semidirect
3. #21448064C2semidirect
4. #21448063C2semidirect
5. #21448058C2semidirect
6. #21448057C2semidirect
7. #21448052C2semidirect
8. #21448051C2semidirect
9. #21448045C2semidirect
10. #21447537C2semidirect
13. #358×D61C2semidirect
14. #21447414C2semidirect
15. C61×#2241C2semidirect
253. #2241D61semidirect
307. C61#33703semidirect

Maximal subgroups

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

subgroupindex
#214563522
#214563072
#214480642
#214480632
#214480582
#214480572
#214480522
#214480512
#214480452
#214475372
#214475302
#214475252
#358×D612
#214474142
C61×#22412
#3370361