Finite Group Databasebeta

← #29730388

Subgroups — #29730388

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#21456215C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic
3#21454562C211yesmaximal, characteristic, complemented
4#21447260C212yesmaximal, complemented
5#21447112C212yesmaximal, complemented
6#21446809C211yesmaximal, characteristic, complemented
7#21446595C211yesmaximal, characteristic, complemented
8#21446330C212yesmaximal, complemented
9#21445884C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C2×C61×#474C211yesmaximal, characteristic, complemented
12C2×#14519803C2212yes—
13C2×#14519783C2212yes—
14#16840312C2211yescharacteristic
15#16840124—22no—
16#16840114—21no—
17#16840110—21no—
18#16839977—22no—
19#16839967—21no—
20#16839957—21no—
21#16839839—21no—
22#16839832—22no—
23#16839827—21no—
24C2×#14519713C2212yes—
25#16839786C2211yescharacteristic
26#16839761C2212yes—
27#16839761C2212yes—
28#16839761C2212yes—
29C2×#14519694C2211yescharacteristic
30C2×#14519694C2211yescharacteristic
31#16839728C2212yes—
32C4×#11758194C2211yescharacteristic
33C4×#11758194C2211yescharacteristic
34#16839531C2211yescharacteristic
35#16839531C2212yes—
36#16839531C2212yes—
37#16839531C2211yescharacteristic
38#16839527C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#474—21no—
42C61×#474—22no—
43C61×#474—21no—
44C2×#107×C61C2212yes—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C2×#11758222C2312yes—
48#14519803—24no—
49#14519803—42no—
50C2×#11758200—22no—
51C2×#11758199—22no—
52#14519783—24no—
53#14519783—42no—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2311yescharacteristic
56C2×#11758194C2311yescharacteristic
57C2×#11758194C2312yes—
58C2×#11758194C2311yescharacteristic
59C2×#11758189—22no—
60C2×C8×D61—22no—
61#14519777D411yescharacteristic
62#14519774D411yescharacteristic
63#14519765D412yes—
64#14519751—21no—
65#14519751—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519713—42no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71#14519700—22no—
72#14519700—22no—
73#14519700—22no—
74#14519697—22no—
75#14519697—22no—
76#14519697—22no—
77#14519694—22no—
78#14519694—22no—
79#14519689—22no—
80#14519689—22no—
81#14519688—21no—
82#14519688—21no—
83C4×#92418—21no—
84C4×#92418—21no—
85C22×Q8×C61—21no—
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C22×C8×C61C2312yes—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#107×C61—22no—
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D412yes—
106C4⋊C8×C61D411yescharacteristic
107#102×C61—22no—
108#102×C61—22no—
109C24×C61—21no—
110C2×Q8×C61—41no—
111C2×Q8×C61—41no—
112C2×Q8×C61—82no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61#4112yes—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61—82no—
124C22×C4×C61#4111yescharacteristic
125C22×C4×C61#4111yescharacteristic
126C22×C4×C61C2411yescharacteristic
127C2×C22⋊D61—22no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244C2×D412yes—
135C2×C4×D61—22no—
136C2×Dic122C2×D412yes—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61C2×D412yes—
140C2×C8×C61C2×D412yes—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153#11758200—44no—
154#11758199—44no—
155#11758198—22no—
156#11758198—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159#11758194—21no—
160#11758194—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—22no—
164#11758194—21no—
165#11758194—22no—
166#11758194—21no—
167#11758194—21no—
168#11758194#4112yes—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
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—
183Q8×C61—82no—
184Q8×C61—82no—
185D4×C61—22no—
186D4×C61—22no—
187D4×C61—41no—
188D4×C61—41no—
189C2×C4×C61—22no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—42no—
193C2×C4×C61—82no—
194C2×C4×C61—82no—
195C2×C4×C61—81no—
196C2×C4×C61—82no—
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×C4×C61H(5,𝔽2)11yescharacteristic
206C2×C4×C612−1+411yescharacteristic
207C22⋊D61—42no—
208C22⋊D61—42no—
209C2×Dic61—22no—
210D244—22no—
211D244—22no—
212D244—42no—
213D244—81no—
214C4×D61—42no—
215C4×D61—42no—
216Dic122—22no—
217Dic122—22no—
218C8×C61—22no—
219C8×C61—22no—
220#92418—22no—
221#92418—41no—
222#92418—42no—
223#92418—41no—
224#92418—41no—
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—82no—
244C4×C61#47212yes—
245C4×C61#54611yescharacteristic
246C4×C61#54611yescharacteristic
247C4×C61#54711yescharacteristic
248C4×C61#54811yescharacteristic
249Dic61—42no—
250#3141—611no—
251C2×#474D6111yescharacteristic, complemented
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×#100—611no—
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⋊Q8C2×D6111yescharacteristic
272C2×C4×D4C2×D6111yescharacteristic
273#489—1222no—
274#487—1221no—
275#486—1221no—
276#474—21no—
277#474—22no—
278#474—21no—
279#463—1222no—
280#461—1221no—
281#459—1221no—
282#436—1221no—
283#435—1222no—
284C8×D4—1221no—
285C2×#108—612no—
286C2×#107C2×D6112yes—
287#423—611no—
288C2×C4⋊C8C2×D6111yescharacteristic
289#417—612no—
290#417—612no—
291#417—612no—
292C2×#102C2×D6112yes—
293C2×#100—611no—
294C2×#100—611no—
295#408—612no—
296C4×M16—611no—
297C4×M16—611no—
298#343—612no—
299#343—611no—
300#343—612no—
301#343—611no—
302#341—612no—
303C61#2535811yescharacteristic, complemented, Sylow
304C2×#41—612no—
305C22×Q8—21no—
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—611no—
315C2×M16—612no—
316C2×M16—611no—
317C2×M16—611no—
318C2×M16—611no—
319C2×M16—1222no—
320C22×C8—1222no—
321C22×C8C22×D6112yes—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—611no—
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—21no—
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C4⋊C4C22×D6112yes—
340C2×C22⋊C4—21no—
341C2×C42C22×D6111yescharacteristic
342#108—2442no—
343#107—22no—
344#107—22no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8C22⋊D6111yescharacteristic
348C4⋊C8C22⋊D6112yes—
349C4⋊C8C22⋊D6111yescharacteristic
350#103—1222no—
351#103—1222no—
352#103—1222no—
353#102—22no—
354#102—22no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—1222no—
359#100—1222no—
360C22⋊C8—1222no—
361C22⋊C8—1222no—
362#97—1221no—
363#97—1221no—
364C4×C8—1221no—
365C4×C8—1221no—
366C24—21no—
367#41—1222no—
368#41—1222no—
369#41—1222no—
370#41—1222no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—82no—
374C2×Q8—612no—
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#1175822612yes—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—82no—
390C22×C4—1222no—
391C22×C4#1175822411yescharacteristic
392C22×C4#1175822411yescharacteristic
393C22×C4C23×D6111yescharacteristic
394SD16—2444no—
395D8—2444no—
396M16—612no—
397M16—1221no—
398M16—1221no—
399M16—1221no—
400M16—1221no—
401M16—1222no—
402M16—1221no—
403M16—1222no—
404M16—1221no—
405M16—1221no—
406M16—2444no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—1221no—
410C2×C8—1221no—
411C2×C8—1221no—
412C2×C8—1222no—
413C2×C8—1221no—
414C2×C8—2444no—
415C2×C8C2×C22⋊D6112yes—
416C2×C8C2×C22⋊D6112yes—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—1222no—
428C22⋊C4—1222no—
429C42D4×D6111yescharacteristic
430C42D4×D6111yescharacteristic
431C42C2×C22⋊D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C23—22no—
434C23—41no—
435C23—41no—
436C23—81no—
437C23—81no—
438C23—1222no—
439C23C2×#1175822411yescharacteristic
440Q8—82no—
441Q8—82no—
442Q8—1222no—
443Q8—1222no—
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—42no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—1222no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4#1451974811yescharacteristic
466C2×C4#1451974811yescharacteristic
467C2×C4#1451974811yescharacteristic
468C2×C4#1451974811yescharacteristic
469C2×C4C2×D4×D6111yescharacteristic
470C2×C4C22×C22⋊D6111yescharacteristic
471C2×C4#41×D6111yescharacteristic
472C2×C4C2×#1175822611yescharacteristic
473C2×C4#1451982311yescharacteristic
474C2×C4#1451982511yescharacteristic
475C8—22no—
476C8—22no—
477C8—1222no—
478C8—2441no—
479C8—2442no—
480C8—2441no—
481C8—2441no—
482C22—21no—
483C22—21no—
484C22—42no—
485C22—82no—
486C22—82no—
487C22—81no—
488C22—81no—
489C22—2442no—
490C22—2442no—
491C22—2442no—
492C22C2×#1451974811yescharacteristic
493C22#1684029811yescharacteristic
494C22#1684038511yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—81no—
499C4—82no—
500C4—2442no—
501C4#1684002912yes—
502C4#1684040011yescharacteristic
503C4#1684040311yescharacteristic
504C4#1684041411yescharacteristic
505C4#1684042311yescharacteristic
506C2—21no—
507C2—21no—
508C2—82no—
509C2—4882no—
510C2#2144939211yescharacteristic
511C2C2×#1684002911yescharacteristic
512C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC2×#474D61
hypocenterC61#25358
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #21456215C2semidirect
3. #21454562C2semidirect
4. #21447260C2semidirect
5. #21447112C2semidirect
6. #21446809C2semidirect
7. #21446595C2semidirect
8. #21446330C2semidirect
9. #21445884C2semidirect
10. C4×#14519694C2semidirect
11. C2×C61×#474C2semidirect
251. C2×#474D61semidirect
303. C61#25358semidirect

Maximal subgroups

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

subgroupindex
#214562152
#214552832
#214545622
#214472602
#214471122
#214468092
#214465952
#214463302
#214458842
C4×#145196942
C2×C61×#4742
#2535861