Finite Group Databasebeta

← #30619304

Subgroups — #30619304

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840797C211yesmaximal, characteristic, complemented
2C2×#16840795C211yesmaximal, characteristic, complemented
3#21461944C211yesmaximal, characteristic, complemented
4#21461943C211yesmaximal, characteristic, complemented
5#21461798C212yesmaximal, complemented
6#21461797C212yesmaximal, complemented
7#21461796C212yesmaximal, complemented
8#21461796C212yesmaximal, complemented
9#21461737C211yesmaximal, characteristic, complemented
10#21461538C211yesmaximal, characteristic, complemented
11#21449097C211yesmaximal, characteristic
12C22×#11758213C2211yescharacteristic
13C22×#11758217C2211yescharacteristic
14C22×#11758217C2212yes—
15#16840797—24no—
16#16840795—24no—
17C2×#14519844C2211yescharacteristic
18C2×C4⋊#92420C2211yescharacteristic
19C2×#14519834C2211yescharacteristic
20C2×#14519834C2212yes—
21C2×C22⋊#92420C2211yescharacteristic
22C2×C22⋊#92420C2212yes—
23C2×C22⋊#92420C2212yes—
24C2×C22⋊#92420C2212yes—
25C2×C22⋊#92420C2212yes—
26C2×#14519832C2211yescharacteristic
27C2×C4×#92420C2211yescharacteristic
28#16840628C2212yes—
29#16840628C2212yes—
30#16840628C2212yes—
31#16840628C2212yes—
32C2×D4×Dic61C2211yescharacteristic
33C2×D4×Dic61C2212yes—
34C2×#14519737C2211yescharacteristic
35C2×#14519737C2212yes—
36#16839699C411yescharacteristic
37#16839698C411yescharacteristic
38#16839698C412yes—
39#16839670C412yes—
40#16839670C411yescharacteristic
41#16839607C2211yescharacteristic
42C61×#392C411yescharacteristic
43C2×#11758213—22no—
44C2×#11758213—42no—
45C2×#11758217—22no—
46C2×#11758217—24no—
47C2×#11758217—42no—
48C2×#11758217—42no—
49C2×#11758217—42no—
50C22×#92420C2312yes—
51C22×#92420C2311yescharacteristic
52C22×#92420C2312yes—
53C22×#92420C2311yescharacteristic
54C22×#92420C2311yescharacteristic
55C22×#92420C2311yescharacteristic
56#14519844—22no—
57C4⋊#92420—22no—
58#14519834—22no—
59#14519834—42no—
60C22⋊#92420—22no—
61C22⋊#92420—24no—
62C22⋊#92420—42no—
63C22⋊#92420—42no—
64C22⋊#92420—42no—
65#14519832—22no—
66C4×#92420D414yescomplemented
67C22×C22⋊D61C2×C411yescharacteristic
68C22×C22⋊D61C2×C412yes—
69C23×Dic61C2311yescharacteristic
70C23×Dic61C2312yes—
71C22×C4×D61C2311yescharacteristic
72D4×Dic61—22no—
73D4×Dic61—22no—
74D4×Dic61—42no—
75D4×Dic61—42no—
76#14519737—22no—
77#14519737—22no—
78#14519737—42no—
79#14519737—42no—
80C2×C4⋊Dic61C2×C411yescharacteristic
81C2×C4⋊Dic61C2×C412yes—
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C412yes—
84C2×C22⋊Dic61C2×C412yes—
85C2×C22⋊Dic61C2×C412yes—
86C2×C22⋊Dic61C2×C412yes—
87C2×#11758198C2×C411yescharacteristic
88C2×#11758198C2×C412yes—
89C2×C4×Dic61C2312yes—
90C2×C4×Dic61C2311yescharacteristic
91#14519684C2×C411yescharacteristic
92#14519684C2×C412yes—
93C22×D4×C61C2×C411yescharacteristic
94C22×D4×C61C2×C412yes—
95C2×C22⋊C4×C61C2×C411yescharacteristic
96C2×C22⋊C4×C61C2×C412yes—
97#95×C61C2×C411yescharacteristic
98C24×C61C22×C412yes—
99C24×C61C22×C411yescharacteristic
100C23×D61C22×C411yescharacteristic
101C2×D4×C61—22no—
102C2×D4×C61—42no—
103C2×D4×C61—42no—
104C2×D4×C61—42no—
105C2×D4×C61—42no—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C22×C412yes—
108C22×C4×C61C22×C411yescharacteristic
109C2×C22⋊D61—22no—
110C2×C22⋊D61—41no—
111C2×C22⋊D61—41no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C2×C22⋊D61—42no—
115C22×Dic61—21no—
116C22×Dic61—22no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C412yes—
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C2411yescharacteristic
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134#11758217—44no—
135#11758217—48no—
136C2×#92420—22no—
137C2×#92420—22no—
138C2×#92420—22no—
139C2×#92420—21no—
140C2×#92420—21no—
141C2×#92420—22no—
142C2×#92420—22no—
143C2×#92420—22no—
144C2×#92420—22no—
145C2×#92420—22no—
146C2×#92420—22no—
147C2×#92420—22no—
148C2×#92420C2×D412yes—
149C2×#92420C2×D412yes—
150#11758213—44no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—42no—
153C22⋊Dic61—24no—
154C22⋊Dic61—42no—
155C22⋊Dic61—42no—
156#11758198—22no—
157#11758198—42no—
158C4⋊Dic61—22no—
159C4⋊Dic61—42no—
160#11758196—22no—
161#11758196—42no—
162C4×Dic61—22no—
163C4×Dic61—22no—
164C4×Dic61C2×D412yes—
165C4×Dic61#4112yes—
166C23×C61—22no—
167C23×C61—21no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—21no—
172C23×C61—22no—
173C23×C61—21no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61C23×C411yescharacteristic
179C22×D61—21no—
180C22×D61—22no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—22no—
184D4×C61—44no—
185D4×C61—48no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61C4×D412yes—
193C22⋊D61—44no—
194C22⋊D61—48no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61C4×D412yes—
213C2×Dic61C22×D411yescharacteristic
214C2×Dic61C2×#4111yescharacteristic
215C2×Dic61H(5,𝔽2)12yes—
216C2×Dic61H(5,𝔽2)12yes—
217C4×D61—41no—
218C4×D61—41no—
219C4×D61—42no—
220#92420—22no—
221#92420—22no—
222#92420—41no—
223#92420—42no—
224#92420—41no—
225#92420—42no—
226#22265—611nomaximal, Sylow
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—41no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61C2×C4×D411yescharacteristic
240C22×C61#51812yes—
241C22×C61#56811yescharacteristic
242C22×C61#56812yes—
243C22×C61#56811yescharacteristic
244C2×D61—41no—
245C2×D61—41no—
246C2×D61—41no—
247C2×D61—42no—
248C2×D61—42no—
249C4×C61—22no—
250C4×C61—41no—
251C4×C61—42no—
252Dic61—22no—
253Dic61—41no—
254Dic61—42no—
255Dic61—42no—
256Dic61#53511yescharacteristic
257Dic61#53511yescharacteristic
258Dic61#54012yes—
259C2×#436—611no—
260C2×#436—611no—
261#2397—611no—
262#1955—611no—
263#1955—611no—
264#1870—612no—
265#1870—612no—
266#1870—612no—
267#1870—612no—
268#1832—611no—
269#1746—611no—
270C2×C61—81no—
271C2×C61—82no—
272C2×C61#232811yescharacteristic
273C2×C61#293011yescharacteristic
274C2×C61#293011yescharacteristic
275C2×C61#297812yes—
276C2×C61#297812yes—
277D61—81no—
278C22×M16—611no—
279C22×M16—612no—
280C22×M16—611no—
281C2×C4×D4—611no—
282C2×C4×D4—612no—
283C2×C4×D4—611no—
284C2×C4×D4—612no—
285#436—1224no—
286#436—1224no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—611no—
289C2×C22⋊C8—611no—
290C2×C22⋊C8—612no—
291C2×C22⋊C8—611no—
292C2×C22⋊C8—612no—
293C2×C22⋊C8—612no—
294C2×C22⋊C8—612no—
295C2×C22⋊C8—612no—
296C2×#97—611no—
297C2×C4×C8—611no—
298#399—612no—
299#399—611no—
300#399—611no—
301#392—612no—
302#392—611no—
303#392#692311yescharacteristic
304#376—611no—
305#336—612no—
306#336—612no—
307#336—612no—
308#336—612no—
309C61#2226511yescharacteristic, complemented, Sylow
310C22×D4—611no—
311C22×D4—612no—
312C22×D4C2×#692311yescharacteristic
313C22×D4C2×#692312yes—
314C23×C4—612no—
315C23×C4—611no—
316C23×C4—611no—
317C2×M16—1222no—
318C2×M16—1224no—
319C2×M16—1222no—
320C2×M16—2442no—
321C2×M16—2442no—
322C2×M16—2442no—
323C2×M16—2442no—
324C22×C8—611no—
325C22×C8—611no—
326C22×C8—611no—
327C22×C8—612no—
328C22×C8—611no—
329C22×C8—612no—
330C4×D4—1222no—
331C4×D4—1222no—
332C4×D4—1222no—
333C4×D4—1222no—
334C4×D4—2442no—
335C4×D4—2442no—
336C4×D4—2442no—
337C4×D4—2442no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—612no—
342C2×C22⋊C4—612no—
343C2×C22⋊C4—611no—
344C2×C22⋊C4—612no—
345C2×C22⋊C4—612no—
346C2×C22⋊C4—612no—
347C2×C22⋊C4C2×#692311yescharacteristic
348C2×C22⋊C4C2×#692312yes—
349C2×C42—611no—
350C2×C42—612no—
351C4⋊C8—1222no—
352C4⋊C8—1222no—
353C22⋊C8—1222no—
354C22⋊C8—1224no—
355C22⋊C8—1222no—
356C22⋊C8—2442no—
357C22⋊C8—2442no—
358C22⋊C8—2442no—
359C22⋊C8—2442no—
360#97—1222no—
361C4×C8—614no—
362#95—612no—
363#95—611no—
364#95C2×#692311yescharacteristic
365C24—611no—
366C24C22×#692311yescharacteristic
367C24C22×#692312yes—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—1222no—
374C2×D4—2442no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2442no—
378C2×D4—2442no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—611no—
384C22×C4—1221no—
385C22×C4—1222no—
386C22×C4—1221no—
387C22×C4—1222no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4C22×#692312yes—
399C22×C4C22×#692311yescharacteristic
400C22×C4C22×#692311yescharacteristic
401M16—2444no—
402M16—2444no—
403M16—2448no—
404C2×C8—612no—
405C2×C8—612no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1221no—
411C2×C8—1221no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—2442no—
421C4⋊C4—2442no—
422C22⋊C4—22no—
423C22⋊C4—42no—
424C22⋊C4—1224no—
425C22⋊C4—1222no—
426C22⋊C4—2442no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C42—612no—
430C42—612no—
431C42—1222no—
432C42—1222no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—22no—
445C23—1221no—
446C23—1222no—
447C23—1221no—
448C23—1222no—
449C23—1221no—
450C23C23×#692311yescharacteristic
451D4—44no—
452D4—48no—
453D4—2444no—
454D4—2448no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—611no—
462C2×C4—611no—
463C2×C4—612no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—1221no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—1222no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2442no—
479C2×C4—2441no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4D4×#692312yes—
487C8—1222no—
488C8—1222no—
489C8—2442no—
490C8—2441no—
491C8—2441no—
492C8—2442no—
493C22—41no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—2441no—
506C22—2441no—
507C22—2441no—
508C22—2442no—
509C22—2442no—
510C22C2×D4×#692311yescharacteristic
511C22#1684082912yes—
512C22#1684084011yescharacteristic
513C22#1684084012yes—
514C22#1684084111yescharacteristic
515C4—22no—
516C4—41no—
517C4—42no—
518C4—611no—
519C4—611no—
520C4—612no—
521C4—1222no—
522C4—2441no—
523C4—2442no—
524C4—2442no—
525C2—81no—
526C2—82no—
527C2—4881no—
528C2#2146231111yescharacteristic
529C2#2146253711yescharacteristic
530C2#2146253811yescharacteristic
531C2#2146269512yes—
532C2#2146269612yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#392C4
prime radical#21449097C2
cosocleC22×Dic61C24
hypercenter#392#6923
hypocenterC61#22265
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#16840797C2semidirect
2. C2×#16840795C2semidirect
3. #21461944C2semidirect
4. #21461943C2semidirect
5. #21461798C2semidirect
6. #21461797C2semidirect
7. #21461796C2semidirect
8. #21461796C2semidirect
9. #21461737C2semidirect
10. #21461538C2semidirect
66. C4×#92420D4semidirect
309. C61#22265semidirect

Maximal subgroups

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

subgroupindex
C2×#168407972
C2×#168407952
#214619442
#214619432
#214617982
#214617972
#214617962
#214617372
#214615382
#214490972
#2226561