Finite Group Databasebeta

← #29608197

Subgroups — #29608197

Order 15616.

1506 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839656C212yesmaximal
2C2×#16839622C211yesmaximal, characteristic
3#21444074C212yesmaximal, complemented
4#21444071C212yesmaximal, complemented
5#21444066C212yesmaximal
6#21444053C212yesmaximal, complemented
7#21444043C212yesmaximal, complemented
8#21444042C211yesmaximal, characteristic
9C61×C22⋊#95C211yesmaximal, characteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C22×#11758196C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2212yes—
15C22×C4×Dic61C2212yes—
16#16839656—28no—
17#16839622C418yescomplemented
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C24×C4×C61C2211yescharacteristic
29C22×C22⋊C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2211yescharacteristic
31C2×#95×C61C2212yes—
32C2×#95×C61C2212yes—
33C23×Dic61C2312yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C2×C4⋊Dic61—41no—
38C2×C4⋊Dic61—42no—
39C2×C4⋊Dic61—41no—
40C2×C4⋊Dic61C2×C414yes—
41C2×C4⋊Dic61C2×C414yes—
42C2×#11758196—24no—
43C2×#11758196—42no—
44C2×#11758196—42no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—24no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—24no—
49C2×C22⋊Dic61—42no—
50C2×C22⋊Dic61C2×C418yes—
51C2×C4×Dic61—24no—
52C2×C4×Dic61—24no—
53C2×C4×Dic61—42no—
54#14519684—24no—
55#14519684—24no—
56#14519684—24no—
57#14519684—24no—
58#14519684—41no—
59#14519684—42no—
60#14519684—42no—
61#14519684—42no—
62#14519684—41no—
63#14519684C2×C418yes—
64C25×C61C2311yescharacteristic
65C23×C4×C61—24no—
66C23×C4×C61—42no—
67C23×C4×C61—41no—
68C23×C4×C61—41no—
69C23×C4×C61C2×C414yes—
70C23×C4×C61C2312yes—
71C23×C4×C61C2311yescharacteristic
72C23×C4×C61C2312yes—
73C23×C4×C61C2311yescharacteristic
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61—41no—
76C2×C22⋊C4×C61—41no—
77C2×C22⋊C4×C61—41no—
78C2×C22⋊C4×C61—41no—
79C2×C22⋊C4×C61D414yes—
80#95×C61—24no—
81#95×C61—42no—
82C24×C61—22no—
83C24×C61—22no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—21no—
90C24×C61C22×C412yes—
91C24×C61C2×D412yes—
92C24×C61#4112yes—
93C24×C61C2411yescharacteristic
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—28no—
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—44no—
116C22×C4×C61—44no—
117C22×C4×C61—48no—
118C22×C4×C61—48no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C412yes—
121C22×C4×C61C2×D414yes—
122C22×C4×C61#4114yes—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
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×Dic61C22×C414yes—
149C22×Dic61C22×C414yes—
150C22×Dic61#4114yes—
151C22⋊C4×C61—44no—
152C22⋊C4×C61—44no—
153C22⋊Dic61—216no—
154C22⋊Dic61—48no—
155C4⋊Dic61—44no—
156C4⋊Dic61—44no—
157#11758196—48no—
158C4×Dic61—48no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—22no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—24no—
172C23×C61—24no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61C4×D414yes—
182C23×C61C23×C411yescharacteristic
183C23×C61C22×D411yescharacteristic
184C23×C61C2×#4112yes—
185C23×C61C2×#4111yescharacteristic
186C23×C61C2×#4112yes—
187C23×C61H(5,𝔽2)11yescharacteristic
188C23×C61H(5,𝔽2)11yescharacteristic
189C23×C61H(5,𝔽2)12yes—
190C23×C61H(5,𝔽2)11yescharacteristic
191C23×C612−1+412yes—
192C23×C612−1+411yescharacteristic
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—416no—
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×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61#11714yes—
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×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246#14422—611nomaximal, Sylow
247C22×C61—24no—
248C22×C61—24no—
249C22×C61—24no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—44no—
255C22×C61—44no—
256C22×C61—44no—
257C22×C61—44no—
258C22×C61C2×#11711yescharacteristic
259C22×C61C2×C4×D411yescharacteristic
260C22×C61C4×#4111yescharacteristic
261C22×C61#51811yescharacteristic
262C22×C61#51811yescharacteristic
263C22×C61#51911yescharacteristic
264C22×C61#52011yescharacteristic
265C22×C61#53411yescharacteristic
266C22×C61#53711yescharacteristic
267C22×C61#53812yes—
268C22×C61#53812yes—
269C22×C61#53811yescharacteristic
270C22×C61#53912yes—
271C22×C61#53911yescharacteristic
272C22×C61#54212yes—
273C22×C61#54612yes—
274C22×C61#54712yes—
275C22×C61#55212yes—
276C22×C61#55212yes—
277C22×C61#55312yes—
278C22×C61#55512yes—
279C22×C61#56212yes—
280C22×C61#56312yes—
281C4×C61—24no—
282C4×C61—42no—
283C4×C61—44no—
284C4×C61—82no—
285Dic61—44no—
286Dic61—44no—
287Dic61—44no—
288Dic61—82no—
289C2×#388—612no—
290C2×#380—611no—
291#1442—612no—
292#1441—612no—
293#1440—612no—
294#1436—612no—
295#1434—612no—
296C22⋊#95—611no—
297C22⋊#95D6111yescharacteristic
298C2×C61—44no—
299C2×C61#231211yescharacteristic
300C2×C61#232511yescharacteristic
301C2×C61#233211yescharacteristic
302C2×C61#234411yescharacteristic
303C2×C61#234611yescharacteristic
304C2×C61#236411yescharacteristic
305C2×C61#237811yescharacteristic
306C2×C61#268612yes—
307C2×C61#269512yes—
308C2×C61#277012yes—
309C2×C61#278112yes—
310C24×C4C2×D6111yescharacteristic
311C22×C4⋊C4—611no—
312C22×C4⋊C4—611no—
313C22×C4⋊C4—612no—
314C22×C22⋊C4—612no—
315C22×C22⋊C4—612no—
316C22×C22⋊C4C2×D6111yescharacteristic
317C22×C22⋊C4C2×D6111yescharacteristic
318C22×C42—612no—
319#388—1228no—
320#380—618no—
321C2×#95—612no—
322C2×#95—612no—
323C2×#95—612no—
324C2×#95—612no—
325C2×#95—612no—
326C2×#95—611no—
327C2×#95—611no—
328C2×#95—612no—
329C2×#95—612no—
330C2×#95—612no—
331C2×#95C2×D6112yes—
332C2×#95C2×D6112yes—
333C61#1442211yescharacteristic, complemented, Sylow
334C25C22×D6111yescharacteristic
335C23×C4—24no—
336C23×C4—42no—
337C23×C4—41no—
338C23×C4—41no—
339C23×C4—612no—
340C23×C4—612no—
341C23×C4—612no—
342C23×C4—612no—
343C23×C4C4×D6114yes—
344C23×C4C22×D6112yes—
345C23×C4C22×D6111yescharacteristic
346C23×C4C22×D6112yes—
347C23×C4C22×D6111yescharacteristic
348C2×C4⋊C4—614no—
349C2×C4⋊C4—614no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—2442no—
352C2×C4⋊C4—2442no—
353C2×C4⋊C4—2441no—
354C2×C4⋊C4—2442no—
355C2×C4⋊C4—2441no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—41no—
358C2×C22⋊C4—41no—
359C2×C22⋊C4—41no—
360C2×C22⋊C4—41no—
361C2×C22⋊C4—618no—
362C2×C22⋊C4—1224no—
363C2×C22⋊C4—1224no—
364C2×C22⋊C4—1224no—
365C2×C22⋊C4—1224no—
366C2×C22⋊C4—2442no—
367C2×C22⋊C4D24414yes—
368C2×C42—1224no—
369C2×C42—1224no—
370C2×C42—2442no—
371#95—24no—
372#95—42no—
373#95—618no—
374#95—1224no—
375#95—1224no—
376#95—1224no—
377#95—1224no—
378#95—2441no—
379#95—2441no—
380#95—2442no—
381#95—2442no—
382#95—2442no—
383C24—22no—
384C24—22no—
385C24—21no—
386C24—21no—
387C24—21no—
388C24—22no—
389C24—22no—
390C24—21no—
391C24C2×C4×D6112yes—
392C24C2×D24412yes—
393C24#1175822212yes—
394C24C23×D6111yescharacteristic
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—614no—
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×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4C2×C4×D6112yes—
449C22×C4C2×C4×D6112yes—
450C22×C4C2×D24414yes—
451C22×C4#1175822414yes—
452C4⋊C4—2444no—
453C4⋊C4—2444no—
454C4⋊C4—2448no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—12216no—
458C22⋊C4—2448no—
459C42—2448no—
460C23—24no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23C4×D24414yes—
483C23C22×C4×D6111yescharacteristic
484C23C22×D24411yescharacteristic
485C23#41×D6112yes—
486C23C2×#1175822412yes—
487C23C2×#1175822211yescharacteristic
488C23#1451982212yes—
489C23#1451982211yescharacteristic
490C23#1451982211yescharacteristic
491C23#1451982211yescharacteristic
492C23#1451982412yes—
493C23#1451982511yescharacteristic
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—416no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1224no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4#1451973314yes—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—42no—
551C22—44no—
552C22—42no—
553C22—44no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22C2×#1451973311yescharacteristic
559C22C2×C4×D24411yescharacteristic
560C22#1684023011yescharacteristic
561C22#1684023511yescharacteristic
562C22#1684023511yescharacteristic
563C22#1684023911yescharacteristic
564C22#1684024311yescharacteristic
565C22#1684032411yescharacteristic
566C22#1684034311yescharacteristic
567C22#1684034712yes—
568C22#1684034712yes—
569C22#1684035011yescharacteristic
570C22#1684035512yes—
571C22#1684035911yescharacteristic
572C22#1684038012yes—
573C22#1684040712yes—
574C22#1684041612yes—
575C22#1684044612yes—
576C22#1684045212yes—
577C22#1684046112yes—
578C22#1684047612yes—
579C22#1684050712yes—
580C22#1684051312yes—
581C4—24no—
582C4—44no—
583C4—42no—
584C4—82no—
585C4—4882no—
586C4—2444no—
587C4—2444no—
588C4—2444no—
589C2—44no—
590C2#2144841511yescharacteristic
591C2#2144852211yescharacteristic
592C2#2144857211yescharacteristic
593C2#2144868611yescharacteristic
594C2#2144870611yescharacteristic
595C2#2144886911yescharacteristic
596C2#2144898211yescharacteristic
597C2#2145222312yes—
598C2#2145235212yes—
599C2#2145332012yes—
600C2#2145348512yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C22⋊#95C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC22⋊#95D61
hypocenterC61#14422
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. #21444074C2semidirect
4. #21444071C2semidirect
6. #21444053C2semidirect
7. #21444043C2semidirect
17. #16839622C4semidirect
333. C61#14422semidirect

Maximal subgroups

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

subgroupindex
C2×#168396562
C2×#168396222
#214440742
#214440712
#214440662
#214440532
#214440432
#214440422
C61×C22⋊#952
#1442261