Finite Group Databasebeta

← #29733681

Subgroups — #29733681

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840383C211yesmaximal, characteristic
2C2×#16840154C211yesmaximal, characteristic, complemented
3C2×#16840112C211yesmaximal, characteristic, complemented
4#21446827C212yesmaximal, complemented
5#21446820C212yesmaximal
6#21446791C211yesmaximal, characteristic, complemented
7#21446628C211yesmaximal, characteristic, complemented
8#21446163C212yesmaximal, complemented
9#21446156C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic, complemented
11C61×#2621C211yesmaximal, characteristic, complemented
12C22×#11758202C2212yes—
13C22×#11758201C2212yes—
14#16840383—24no—
15#16840383—24no—
16C2×#14519777C2211yescharacteristic
17C2×#14519772—21no—
18C2×#14519765C2211yescharacteristic
19C2×#14519763—21no—
20C2×#14519759—21no—
21C2×#14519756C2211yescharacteristic
22C2×#14519755—21no—
23C2×#14519751C2211yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26#16840154—24no—
27#16840112—24no—
28C2×#14519701C2212yes—
29C2×#14519701C2212yes—
30C2×#14519700C2212yes—
31C2×#14519697C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×#14519688C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37#16839517C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C22⋊Q8×C61C2212yes—
40C2×#117×C61C2211yescharacteristic
41C61×#398—21no—
42C61×#398—21no—
43C61×#397—21no—
44C61×#397—21no—
45C61×#391C2212yes—
46C61×#382C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61—21no—
52C2×#11758202—24no—
53C2×#11758202—24no—
54C2×#11758202—42no—
55C2×#11758201—24no—
56C2×#11758201—24no—
57C2×#11758201—42no—
58C22×#92418C2312yes—
59C22×#92418C2312yes—
60#14519777—22no—
61#14519777D414yes—
62#14519772—28no—
63#14519765—22no—
64#14519765D414yes—
65#14519763—28no—
66#14519759—28no—
67#14519756—22no—
68#14519756—22no—
69#14519755—28no—
70#14519751—22no—
71#14519751—22no—
72C4×Dic122—22no—
73C4×Dic122—22no—
74C4×D244—22no—
75C4×D244—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87#14519701—24no—
88#14519701—24no—
89#14519700—42no—
90#14519697—42no—
91C22⋊#92418—24no—
92#14519688—22no—
93C4×#92418—22no—
94C22×Q8×C61C2312yes—
95C23×C4×C61C2311yescharacteristic
96C4⋊Q8×C61—22no—
97C4⋊Q8×C61—22no—
98C22⋊Q8×C61—42no—
99C22⋊Q8×C61—42no—
100#117×C61—22no—
101#117×C61—22no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2312yes—
108C2×C22⋊C4×C61—22no—
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—22no—
116#95×C61—22no—
117C24×C61C2×D411yescharacteristic
118C23×D61—21no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—24no—
138C2×C22⋊D61—82no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C2×D244—22no—
143C2×D244—22no—
144C2×D244—21no—
145C2×D244C2×D412yes—
146C2×C4×D61—24no—
147C2×C4×D61—82no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—21no—
151C2×Dic122—24no—
152C2×Dic122—82no—
153C2×Dic122C2×D412yes—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—44no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—44no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—24no—
169#11758202—48no—
170#11758201—48no—
171#11758198—24no—
172#11758198—24no—
173#11758198—24no—
174#11758198—24no—
175C4⋊Dic61—24no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C2×D412yes—
179C4⋊Dic61C2×D412yes—
180#11758196—24no—
181#11758196—24no—
182C4×Dic61—24no—
183C2×#92418—22no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418#4114yes—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61C22×D411yescharacteristic
195C22×D61—22no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—42no—
199C22×D61—42no—
200Q8×C61—42no—
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—42no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61#12012yes—
222C2×C4×C61#12012yes—
223C2×C4×C61C22×D411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C2×#4112yes—
226C2×C4×C61H(5,𝔽2)12yes—
227C22⋊D61—84no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—24no—
236D244—42no—
237C4×D61—84no—
238Dic122—24no—
239Dic122—42no—
240Dic122—84no—
241#92418—24no—
242#92418—42no—
243#25613—611nomaximal, Sylow
244C22×C61—42no—
245C22×C61—41no—
246C22×C61—42no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#53211yescharacteristic
251C22×C61#53511yescharacteristic
252C22×C61#57712yes—
253C22×C61#57812yes—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61—82no—
258C2×D61—82no—
259C4×C61—22no—
260C4×C61—22no—
261C4×C61—41no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—82no—
265C4×C61—82no—
266C4×C61#54612yes—
267C4×C61#55012yes—
268Dic61—42no—
269Dic61—42no—
270Dic61—42no—
271C2×#542—611no—
272C2×#497—611no—
273C2×#487—611no—
274#2621D6111yescharacteristic, complemented
275#2001—612no—
276#2000—612no—
277#1993—611no—
278#1958—611no—
279#1878—612no—
280#1877—612no—
281#1750—611no—
282C2×C61—81no—
283C2×C61#243111yescharacteristic
284C2×C61#300911yescharacteristic
285C2×C61#301111yescharacteristic
286C2×C61#317812yes—
287C2×C61#322512yes—
288D61—82no—
289C22×Q16—612no—
290C22×SD16—612no—
291#542—1224no—
292#542—1224no—
293C2×C4⋊Q8—611no—
294C2×C4⋊Q8C2×D6111yescharacteristic
295C2×#126—1221no—
296C2×#124—611no—
297C2×#124—1221no—
298C2×#123—1221no—
299C2×C22⋊Q8—611no—
300C2×C22⋊Q8—1221no—
301C2×C22⋊Q8C2×D6112yes—
302C2×#121—611no—
303C2×C4×Q8—611no—
304C2×C4×D4—611no—
305C2×#117C2×D6111yescharacteristic
306#497—1224no—
307#487—1224no—
308C2×#103—612no—
309C2×#103—612no—
310C2×#103—612no—
311C2×#102—612no—
312C2×C22⋊C8—612no—
313C2×#97—611no—
314C2×C4×C8—611no—
315#398—21no—
316#398—21no—
317#397—21no—
318#397—21no—
319#391C2×D6112yes—
320#382C2×D6111yescharacteristic
321#340—612no—
322#340—612no—
323#336—612no—
324C61#2561311yescharacteristic, complemented, Sylow
325C22×Q8—611no—
326C22×Q8—1221no—
327C22×Q8C22×D6112yes—
328C22×D4—611no—
329C22×D4—1221no—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332C2×Q16—1224no—
333C2×Q16—1224no—
334C2×Q16—2442no—
335C2×SD16—1224no—
336C2×SD16—1224no—
337C2×SD16—2442no—
338C22×C8—612no—
339C22×C8—612no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—614no—
343C4⋊Q8—1222no—
344#126—1228no—
345#124—614no—
346#124—1228no—
347#124—1222no—
348#123—1228no—
349C22⋊Q8—42no—
350C22⋊Q8—42no—
351C22⋊Q8—1228no—
352C22⋊Q8—1222no—
353C22⋊Q8—1222no—
354#121—1222no—
355#121—1222no—
356C4×Q8—1222no—
357C4×Q8—1222no—
358C4×D4—1222no—
359C4×D4—1222no—
360#117—22no—
361#117—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4C22⋊D6111yescharacteristic
371C2×C4⋊C4C22⋊D6111yescharacteristic
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—1221no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4C22⋊D6111yescharacteristic
380C2×C22⋊C4C22⋊D6111yescharacteristic
381C2×C42—1221no—
382C2×C42C22×D6111yescharacteristic
383C2×C42C22×D6111yescharacteristic
384#103—1224no—
385#103—1224no—
386#103—2442no—
387#102—2442no—
388C22⋊C8—1224no—
389#97—1222no—
390C4×C8—1222no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395C24—1221no—
396C24C2×C22⋊D6111yescharacteristic
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—612no—
404C2×Q8—1221no—
405C2×Q8—1224no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—4882no—
409C2×D4—612no—
410C2×D4—1224no—
411C2×D4—1221no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—4882no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—1224no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—4882no—
426C22×C4C2×C22⋊D6111yescharacteristic
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6111yescharacteristic
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432Q16—2448no—
433SD16—2448no—
434C2×C8—614no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C2×C8—1222no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—612no—
448C4⋊C4—612no—
449C4⋊C4—1224no—
450C4⋊C4—1224no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1224no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—44no—
457C22⋊C4—1224no—
458C22⋊C4—1224no—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C42—21no—
463C42—21no—
464C42—1224no—
465C42D4×D6112yes—
466C42D4×D6112yes—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—1222no—
473C23—2441no—
474C23—2442no—
475C23—2442no—
476C23—2441no—
477C23C22×C22⋊D6111yescharacteristic
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—1224no—
483Q8—4884no—
484Q8—2442no—
485D4—1224no—
486D4—4884no—
487D4—2442no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—4884no—
509C2×C4—2441no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4#1451974512yes—
514C2×C4#1451974512yes—
515C2×C4C2×D4×D6111yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C2×C4#41×D6112yes—
518C2×C4#1451982212yes—
519C8—1224no—
520C8—2442no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—41no—
525C22—41no—
526C22—4882no—
527C22—4882no—
528C22—2442no—
529C22—2442no—
530C22—2442no—
531C22C2×#1451974511yescharacteristic
532C22#1684030811yescharacteristic
533C22#1684032911yescharacteristic
534C22#1684057012yes—
535C22#1684057712yes—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—82no—
542C4—82no—
543C4—2442no—
544C4—2442no—
545C4—2442no—
546C4#1684040912yes—
547C4#1684043012yes—
548C2—81no—
549C2—4882no—
550C2#2144943911yescharacteristic
551C2#2145509411yescharacteristic
552C2#2145511211yescharacteristic
553C2#2145648912yes—
554C2#2145692812yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2621C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#2621D61
hypocenterC61#25613
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. C2×#16840154C2semidirect
3. C2×#16840112C2semidirect
4. #21446827C2semidirect
6. #21446791C2semidirect
7. #21446628C2semidirect
8. #21446163C2semidirect
9. #21446156C2semidirect
10. #21445585C2semidirect
11. C61×#2621C2semidirect
274. #2621D61semidirect
324. C61#25613semidirect

Maximal subgroups

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

subgroupindex
C2×#168403832
C2×#168401542
C2×#168401122
#214468272
#214468202
#214467912
#214466282
#214461632
#214461562
#214455852
C61×#26212
#2561361