Finite Group Databasebeta

← #29608736

Subgroups — #29608736

Order 15616.

1398 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839669C211yesmaximal, characteristic
2C2×#16839669C211yesmaximal, characteristic
3#21444083C212yesmaximal
4#21444082C212yesmaximal
5#21444071C212yesmaximal
6#21444052C212yesmaximal
7#21444050C212yesmaximal
8#21444047C212yesmaximal
9C2×C61×#391C211yesmaximal, characteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C22×#11758196C2212yes—
13C22×#11758196C2212yes—
14C22×#11758196C2212yes—
15C22×C4×Dic61C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839669—24no—
18#16839669C418yescomplemented
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C23×Q8×C61C2211yescharacteristic
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C42×C61C2211yescharacteristic
31C61×#391—24no—
32C2×#95×C61C2212yes—
33C2×#95×C61C2212yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61—41no—
40C2×C4⋊Dic61—41no—
41C2×C4⋊Dic61—41no—
42C2×C4⋊Dic61—41no—
43C2×C4⋊Dic61C2×C414yes—
44C2×#11758196—24no—
45C2×#11758196—42no—
46C2×#11758196—42no—
47C2×#11758196—42no—
48C2×#11758196—42no—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×#11758196—42no—
52C2×#11758196Q818yescomplemented
53C2×C4×Dic61—22no—
54C2×C4×Dic61—41no—
55C2×C4×Dic61—41no—
56C2×C4×Dic61—41no—
57C2×C4×Dic61—41no—
58C2×C4×Dic61C2×C414yes—
59#14519684—24no—
60#14519684—24no—
61#14519684—24no—
62#14519684—24no—
63#14519684—42no—
64#14519684—42no—
65#14519684—42no—
66#14519684C2×C418yes—
67#14519684C2×C418yes—
68C22×Q8×C61—41no—
69C22×Q8×C61—41no—
70C22×Q8×C61—41no—
71C22×Q8×C61—41no—
72C22×Q8×C61C2×C414yes—
73C22×Q8×C61D414yes—
74C22×Q8×C61Q814yes—
75C23×C4×C61C2312yes—
76C23×C4×C61C2312yes—
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—41no—
82C2×C4⋊C4×C61—41no—
83C2×C42×C61—22no—
84C2×C42×C61—41no—
85C2×C42×C61—41no—
86#95×C61—24no—
87#95×C61—24no—
88C24×C61C2411yescharacteristic
89C2×Q8×C61—44no—
90C2×Q8×C61—44no—
91C2×Q8×C61—44no—
92C2×Q8×C61—44no—
93C2×Q8×C61—44no—
94C2×Q8×C61—44no—
95C2×Q8×C61C4⋊C4116yescomplemented
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C412yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×Q814yes—
128C22×C4×C61C2×Q812yes—
129C22×C4×C61C2×Q812yes—
130C22×C4×C61C2×Q812yes—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61C22×C414yes—
155C22×Dic61C22×C414yes—
156C22×Dic61C2×Q814yes—
157C22×Dic61C2×Q814yes—
158C22×Dic61#4114yes—
159C4⋊C4×C61—44no—
160C42×C61—44no—
161C4⋊Dic61—44no—
162C4⋊Dic61—44no—
163#11758196—48no—
164#11758196—48no—
165#11758196—48no—
166C4×Dic61—44no—
167C4×Dic61—44no—
168C23×C61C23×C411yescharacteristic
169C23×C61C22×D411yescharacteristic
170C23×C61C22×Q811yescharacteristic
171C23×C61C22×Q812yes—
172C23×C61C2×#4112yes—
173C23×C61H(5,𝔽2)11yescharacteristic
174C23×C61H(5,𝔽2)11yescharacteristic
175C23×C61H(5,𝔽2)11yescharacteristic
176C23×C61H(5,𝔽2)11yescharacteristic
177C23×C61H(5,𝔽2)12yes—
178C23×C612−1+412yes—
179Q8×C61—416no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61C2×C4⋊C414yes—
213C2×C4×C61C2×C4⋊C414yes—
214C2×C4×C61C2×C4⋊C414yes—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—24no—
221C2×Dic61—24no—
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—24no—
225C2×Dic61—24no—
226C2×Dic61—24no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61C4×Q818yes—
243#14467—611nomaximal, Sylow
244C22×C61C22×C4⋊C411yescharacteristic
245C22×C61C2×C4×Q812yes—
246C22×C61#51811yescharacteristic
247C22×C61#51811yescharacteristic
248C22×C61#52012yes—
249C22×C61#53411yescharacteristic
250C22×C61#53411yescharacteristic
251C22×C61#53812yes—
252C22×C61#53912yes—
253C22×C61#54312yes—
254C22×C61#54311yescharacteristic
255C22×C61#54311yescharacteristic
256C22×C61#54412yes—
257C22×C61D4×Q812yes—
258C22×C61#55012yes—
259C22×C61#55412yes—
260C22×C61#55511yescharacteristic
261C22×C61#55712yes—
262C22×C61Q8211yescharacteristic
263C22×C61Q8212yes—
264C22×C61H(3,𝔽4)11yescharacteristic
265C22×C61#56212yes—
266C22×C61#56311yescharacteristic
267C4×C61—24no—
268C4×C61—24no—
269C4×C61—24no—
270C4×C61—44no—
271C4×C61—82no—
272Dic61—28no—
273Dic61—44no—
274Dic61—44no—
275Dic61—82no—
276C2×#391—611no—
277C2×#391—611no—
278C2×#391D6111yescharacteristic
279#1444—612no—
280#1444—612no—
281#1441—612no—
282#1436—612no—
283#1436—612no—
284C4⋊#95—612no—
285C2×C61#231511yescharacteristic
286C2×C61#232912yes—
287C2×C61Q8×C4⋊C412yes—
288C2×C61#236911yescharacteristic
289C2×C61#237911yescharacteristic
290C2×C61#273212yes—
291C2×C61#273311yescharacteristic
292C2×C61#273411yescharacteristic
293C2×C61#280612yes—
294C2×C61#280911yescharacteristic
295C2×C61#281111yescharacteristic
296C23×Q8C2×D6111yescharacteristic
297C22×C4⋊C4—612no—
298C22×C4⋊C4—611no—
299C22×C4⋊C4—612no—
300C22×C4⋊C4—611no—
301C22×C4⋊C4—612no—
302C22×C4⋊C4C2×D6111yescharacteristic
303C22×C42—611no—
304C22×C42—611no—
305C22×C42C2×D6111yescharacteristic
306#391—24no—
307#391—618no—
308#391—1224no—
309C2×#95—612no—
310C2×#95—612no—
311C2×#95—612no—
312C2×#95—612no—
313C2×#95—612no—
314C2×#95—612no—
315C2×#95—612no—
316C2×#95—612no—
317C2×#95—612no—
318C2×#95C2×D6112yes—
319C2×#95C2×D6112yes—
320C61#1446711yescharacteristic, complemented, Sylow
321C22×Q8—41no—
322C22×Q8—41no—
323C22×Q8—41no—
324C22×Q8—41no—
325C22×Q8Dic12214yes—
326C22×Q8C4×D6114yes—
327C22×Q8C22⋊D6114yes—
328C23×C4—612no—
329C23×C4—612no—
330C23×C4—612no—
331C23×C4—612no—
332C23×C4C22×D6112yes—
333C23×C4C22×D6112yes—
334C23×C4C22×D6111yescharacteristic
335C23×C4C22×D6111yescharacteristic
336C23×C4C22×D6111yescharacteristic
337C2×C4⋊C4—22no—
338C2×C4⋊C4—41no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—614no—
341C2×C4⋊C4—618no—
342C2×C4⋊C4—1224no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—2442no—
345C2×C4⋊C4—2442no—
346C2×C4⋊C4—2442no—
347C2×C4⋊C4—2442no—
348C2×C4⋊C4—2442no—
349C2×C4⋊C4—2442no—
350C2×C4⋊C4—2442no—
351C2×C4⋊C4—2441no—
352C2×C4⋊C4—2441no—
353C2×C4⋊C4—2441no—
354C2×C4⋊C4—2441no—
355C2×C42—22no—
356C2×C42—41no—
357C2×C42—41no—
358C2×C42—614no—
359C2×C42—1222no—
360C2×C42—2441no—
361C2×C42—2441no—
362C2×C42—2441no—
363C2×C42—2441no—
364#95—24no—
365#95—24no—
366#95—618no—
367#95—618no—
368#95—1224no—
369#95—1224no—
370#95—1224no—
371#95—1224no—
372#95—2442no—
373#95—2442no—
374#95—2442no—
375C24C23×D6111yescharacteristic
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8#11758196116yescomplemented
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—614no—
409C22×C4—614no—
410C22×C4—614no—
411C22×C4—614no—
412C22×C4—614no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4C2×Dic12212yes—
437C22×C4C2×Dic12212yes—
438C22×C4C2×Dic12212yes—
439C22×C4C2×C4×D6112yes—
440C22×C4C2×C4×D6112yes—
441C22×C4C2×C4×D6112yes—
442C22×C4Q8×D6114yes—
443C22×C4C2×C22⋊D6112yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6112yes—
446C4⋊C4—44no—
447C4⋊C4—2448no—
448C4⋊C4—2444no—
449C4⋊C4—2444no—
450C4⋊C4—2448no—
451C4⋊C4—2448no—
452C42—44no—
453C42—2444no—
454C42—2444no—
455C23C22×C4×D6111yescharacteristic
456C23C22×C22⋊D6111yescharacteristic
457C23C2×Q8×D6112yes—
458C23C22×Dic12211yescharacteristic
459C23#41×D6112yes—
460C23#1451982211yescharacteristic
461C23#1451982211yescharacteristic
462C23#1451982211yescharacteristic
463C23#1451982211yescharacteristic
464C23#1451982312yes—
465C23#1451982512yes—
466Q8—416no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—618no—
500C2×C4—1224no—
501C2×C4—1224no—
502C2×C4—1224no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—1224no—
506C2×C4—1224no—
507C2×C4—1224no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4C2×#1175819614yes—
528C2×C4C2×#1175819614yes—
529C2×C4C2×#1175819614yes—
530C22C22×#1175819611yescharacteristic
531C22C4×Q8×D6112yes—
532C22#1684023511yescharacteristic
533C22#1684023511yescharacteristic
534C22#1684024412yes—
535C22#1684032111yescharacteristic
536C22#1684032111yescharacteristic
537C22#1684034612yes—
538C22#1684035812yes—
539C22#1684038712yes—
540C22#1684038811yescharacteristic
541C22#1684038811yescharacteristic
542C22#1684038912yes—
543C22Q8×C22⋊D6112yes—
544C22#1684043112yes—
545C22#1684046612yes—
546C22#1684047411yescharacteristic
547C22#1684048812yes—
548C22#1684049211yescharacteristic
549C22Q8×Dic12212yes—
550C22#1684050411yescharacteristic
551C22#1684050812yes—
552C22#1684051111yescharacteristic
553C4—24no—
554C4—24no—
555C4—24no—
556C4—44no—
557C4—82no—
558C4—1228no—
559C4—4882no—
560C4—2444no—
561C4—2444no—
562C2#2144843111yescharacteristic
563C2#2144855112yes—
564C2Q8×#1175819612yes—
565C2#2144890711yescharacteristic
566C2#2144899111yescharacteristic
567C2#2145283412yes—
568C2#2145284311yescharacteristic
569C2#2145284911yescharacteristic
570C2#2145377912yes—
571C2#2145380511yescharacteristic
572C2#2145381211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#391C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#391D61
hypocenterC61#14467
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
18. #16839669C4semidirect
52. C2×#11758196Q8semidirect
95. C2×Q8×C61C4⋊C4semidirect
320. C61#14467semidirect
382. C2×Q8#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#168396692
#214440832
#214440822
#214440712
#214440522
#214440502
#214440472
C2×C61×#3912
#1446761