Finite Group Databasebeta

← #29730392

Subgroups — #29730392

Order 15616.

764 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#21456213C211yesmaximal, characteristic, complemented
2C2×#16840042C211yesmaximal, characteristic, complemented
3#21455283C211yesmaximal, characteristic
4#21454565C211yesmaximal, characteristic, complemented
5#21447263C212yesmaximal, complemented
6#21447111C212yesmaximal, complemented
7#21446808C211yesmaximal, characteristic, complemented
8#21446594C211yesmaximal, characteristic, complemented
9#21446324C212yesmaximal, complemented
10#21445882C212yesmaximal, complemented
11C4×#100×C61C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519785C2212yes—
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840122—22no—
17#16840113—21no—
18#16840109—21no—
19#16840042—21no—
20#16840042—22no—
21#16840042—21no—
22#16839977—22no—
23#16839967—21no—
24#16839957—21no—
25#16839841—21no—
26#16839835—22no—
27C8×D244—21no—
28C2×#14519712C2212yes—
29C2×#14519710C2212yes—
30#16839786C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839764C2212yes—
33#16839764C2212yes—
34#16839761C2212yes—
35C2×#14519698C2212yes—
36#16839730C2212yes—
37#16839530C2212yes—
38#16839530C2212yes—
39#16839526C2212yes—
40C2×C4×D4×C61C2211yescharacteristic
41C2×#100×C61C2211yescharacteristic
42C2×#100×C61C2211yescharacteristic
43C4×M16×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#343C2211yescharacteristic
46C61×#343C2211yescharacteristic
47C2×#11758222C2312yes—
48C22×Dic122—21no—
49#14519803—24no—
50#14519803—42no—
51C2×#11758200—22no—
52C2×#11758199—22no—
53#14519785—24no—
54#14519785—42no—
55C2×#11758194C2312yes—
56C2×#11758189—22no—
57C22×#92418C2312yes—
58C2×C8×D61—22no—
59#14519777—21no—
60#14519777—21no—
61#14519777—21no—
62#14519777—21no—
63#14519777D411yescharacteristic
64#14519774D411yescharacteristic
65#14519765D412yes—
66#14519751—21no—
67#14519751—21no—
68C4×D244—21no—
69C4×D244—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72#14519712—42no—
73#14519710—22no—
74#14519710—22no—
75C4⋊#92418—21no—
76C4⋊#92418—21no—
77C4⋊#92418D411yescharacteristic
78C4⋊#92418D412yes—
79C4⋊#92418D411yescharacteristic
80#14519702—22no—
81#14519702—22no—
82#14519700—22no—
83#14519699—22no—
84#14519699—22no—
85#14519698—22no—
86#14519698—22no—
87#14519697—22no—
88#14519689—22no—
89#14519689—22no—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61—21no—
92C2×M16×C61C2311yescharacteristic
93C2×M16×C61C2311yescharacteristic
94C2×M16×C61C2311yescharacteristic
95C2×M16×C61C2311yescharacteristic
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#100×C61—22no—
104#100×C61—22no—
105#97×C61—21no—
106#97×C61—21no—
107C4×C8×C61—21no—
108C4×C8×C61—21no—
109C24×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61#4112yes—
116C22×C4×C61—21no—
117C22×C4×C61—41no—
118C22×C4×C61—41no—
119C22×C4×C61—82no—
120C22×C4×C61#4111yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—22no—
124C22×Dic61—22no—
125#11758222—22no—
126#11758222—22no—
127#11758222—22no—
128#11758222—22no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244C2×D412yes—
132C2×C4×D61—22no—
133C2×Dic122—41no—
134C2×Dic122—41no—
135C2×Dic122—82no—
136C2×Dic122C2×D412yes—
137M16×C61—21no—
138M16×C61—21no—
139M16×C61—21no—
140M16×C61—21no—
141M16×C61—21no—
142M16×C61—21no—
143M16×C61—21no—
144M16×C61#4112yes—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157#11758200—44no—
158#11758199—44no—
159#11758198—22no—
160#11758198—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—41no—
166C4⋊Dic61—41no—
167#11758194—22no—
168#11758194—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418C2×D412yes—
173C2×#92418C2×D412yes—
174#11758189—44no—
175C8×D61—44no—
176C23×C61—22no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—81no—
180C23×C61—81no—
181C23×C61C2×#4111yescharacteristic
182C22×D61—22no—
183D4×C61—22no—
184D4×C61—22no—
185D4×C61—41no—
186D4×C61—41no—
187C2×C4×C61—22no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—82no—
191C2×C4×C61—82no—
192C2×C4×C61—81no—
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)11yescharacteristic
202C2×C4×C612−1+411yescharacteristic
203C22⋊D61—42no—
204C22⋊D61—42no—
205C2×Dic61—22no—
206C2×Dic61—42no—
207C2×Dic61—82no—
208D244—22no—
209D244—22no—
210D244—42no—
211D244—81no—
212C4×D61—42no—
213C4×D61—42no—
214Dic122—22no—
215Dic122—22no—
216Dic122—82no—
217Dic122—82no—
218C8×C61—22no—
219C8×C61—41no—
220C8×C61—41no—
221C8×C61—41no—
222#92418—22no—
223#92418—22no—
224#92418—42no—
225#25358—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61C2×#12111yescharacteristic
234C22×C61#52911yescharacteristic
235C22×C61#54211yescharacteristic
236C2×D61—42no—
237C2×D61—42no—
238C2×D61—42no—
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—21no—
242C4×C61—81no—
243C4×C61#47212yes—
244C4×C61#54611yescharacteristic
245C4×C61#54611yescharacteristic
246C4×C61#54711yescharacteristic
247C4×C61#54811yescharacteristic
248Dic61—42no—
249Dic61—82no—
250#3141—611no—
251C2×#474—611no—
252#3036—611no—
253#2935—611no—
254#2078—612no—
255#2051—612no—
256#1997—611no—
257#1950—611no—
258#1907—612no—
259#1816—612no—
260C4×#100D6111yescharacteristic, complemented
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—82no—
264C2×C61#242711yescharacteristic
265C2×C61C2×#47211yescharacteristic
266C2×C61#306911yescharacteristic
267D61—82no—
268C2×AGL(1,ℤ/8)—612no—
269C2×#131—612no—
270#532—611no—
271C2×C4⋊Q8—611no—
272C2×C4×D4C2×D6111yescharacteristic
273#489—1222no—
274#487—1221no—
275#486—1221no—
276#474—1221no—
277#474—1222no—
278#474—1221no—
279#463—1222no—
280#461—1221no—
281#459—1221no—
282#436—1221no—
283#435—1222no—
284C8×D4—1221no—
285C2×#108—612no—
286C2×#107—612no—
287#423—611no—
288C2×C4⋊C8—611no—
289#417—612no—
290#417—612no—
291#417—612no—
292C2×#102—612no—
293C2×#100C2×D6111yescharacteristic
294C2×#100C2×D6111yescharacteristic
295#408—612no—
296C4×M16C2×D6111yescharacteristic
297C4×M16C2×D6111yescharacteristic
298#343—612no—
299#343—612no—
300#343C2×D6111yescharacteristic
301#343C2×D6111yescharacteristic
302#341—612no—
303C61#2535811yescharacteristic, complemented, Sylow
304C2×#41—612no—
305C22×Q8—1221no—
306C22×D4C22×D6111yescharacteristic
307C23×C4—21no—
308AGL(1,ℤ/8)—1224no—
309AGL(1,ℤ/8)—2442no—
310C2×SD16—1222no—
311C2×D8—1222no—
312#131—1224no—
313#131—2442no—
314C2×M16—612no—
315C2×M16—1222no—
316C2×M16C22×D6111yescharacteristic
317C2×M16C22×D6111yescharacteristic
318C2×M16C22×D6111yescharacteristic
319C2×M16C22×D6111yescharacteristic
320C22×C8—612no—
321C22×C8—1222no—
322C4⋊Q8—611no—
323C4⋊Q8—1221no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8—1221no—
327#127—611no—
328#124—612no—
329#121—1221no—
330#121—1221no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—1221no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C42C22×D6111yescharacteristic
342#108—2442no—
343#107—1222no—
344#107—1222no—
345C4⋊C8—611no—
346C4⋊C8—612no—
347C4⋊C8—611no—
348C4⋊C8—1221no—
349C4⋊C8—1221no—
350#103—1222no—
351#103—1222no—
352#103—1222no—
353#102—1222no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—22no—
359#100—22no—
360C22⋊C8—1222no—
361C22⋊C8—1222no—
362#97—21no—
363#97—21no—
364C4×C8—21no—
365C4×C8—21no—
366C24—21no—
367#41—1222no—
368#41—1222no—
369#41—1222no—
370#41—1222no—
371C2×Q8—612no—
372C2×Q8—4882no—
373C2×Q8—2441no—
374C2×Q8—2441no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—612no—
381C2×D4—1222no—
382C2×D4—1221no—
383C2×D4—1221no—
384C2×D4#1175822212yes—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—82no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4#1175822211yescharacteristic
392C22×C4#1175822211yescharacteristic
393C22×C4C23×D6111yescharacteristic
394SD16—2444no—
395D8—2444no—
396M16—21no—
397M16—21no—
398M16—21no—
399M16—21no—
400M16—21no—
401M16—21no—
402M16—21no—
403M16—1222no—
404M16—1222no—
405M16—2444no—
406M16#1175822212yes—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—612no—
412C2×C8—612no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—2444no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—1221no—
420C4⋊C4—1221no—
421C4⋊C4—1222no—
422C4⋊C4—1222no—
423C4⋊C4—2441no—
424C4⋊C4—2441no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—1222no—
428C22⋊C4—1222no—
429C42D4×D6111yescharacteristic
430C42D4×D6111yescharacteristic
431C42D4×D6111yescharacteristic
432C42D4×D6111yescharacteristic
433C23—22no—
434C23—41no—
435C23—41no—
436C23—81no—
437C23—81no—
438C23—1222no—
439C23C2×#1175822211yescharacteristic
440Q8—1222no—
441Q8—1222no—
442Q8—4882no—
443Q8—4882no—
444D4—22no—
445D4—22no—
446D4—41no—
447D4—41no—
448D4—1222no—
449D4—1222no—
450D4—4881no—
451D4—2442no—
452D4—2442no—
453D4—2442no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—1222no—
461C2×C4—4882no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4#1451975011yescharacteristic
466C2×C4#1451975011yescharacteristic
467C2×C4#1451975011yescharacteristic
468C2×C4#1451975011yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C2×D4×D6111yescharacteristic
471C2×C4C2×#1175822211yescharacteristic
472C2×C4C2×#1175822211yescharacteristic
473C2×C4#1451982311yescharacteristic
474C2×C4#1451982411yescharacteristic
475C8—22no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—1222no—
480C8—1222no—
481C8—2442no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—82no—
486C22—82no—
487C22—81no—
488C22—81no—
489C22—2442no—
490C22—2442no—
491C22—2442no—
492C22C2×#1451975011yescharacteristic
493C22#1684029611yescharacteristic
494C22#1684038311yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—81no—
499C4—4882no—
500C4—2442no—
501C4#1684003112yes—
502C4#1684040811yescharacteristic
503C4#1684040811yescharacteristic
504C4#1684041711yescharacteristic
505C4#1684042111yescharacteristic
506C2—21no—
507C2—21no—
508C2—82no—
509C2—4882no—
510C2#2144939611yescharacteristic
511C2C2×#1684003111yescharacteristic
512C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#100×C61C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC4×#100D61
hypocenterC61#25358
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
1. #21456213C2semidirect
2. C2×#16840042C2semidirect
4. #21454565C2semidirect
5. #21447263C2semidirect
6. #21447111C2semidirect
7. #21446808C2semidirect
8. #21446594C2semidirect
9. #21446324C2semidirect
10. #21445882C2semidirect
11. C4×#100×C61C2semidirect
260. C4×#100D61semidirect
303. C61#25358semidirect

Maximal subgroups

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

subgroupindex
#214562132
C2×#168400422
#214552832
#214545652
#214472632
#214471112
#214468082
#214465942
#214463242
#214458822
C4×#100×C612
#2535861