Finite Group Databasebeta

← #30617801

Subgroups — #30617801

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840787C212yesmaximal, complemented
2#21461945C212yesmaximal, complemented
3#21461944C212yesmaximal, complemented
4#21461942C212yesmaximal, complemented
5#21461941C212yesmaximal, complemented
6#21461644C212yesmaximal, complemented
7#21461642C211yesmaximal, characteristic, complemented
8#21461642C211yesmaximal, characteristic, complemented
9C2×#16840275C211yesmaximal, characteristic
10#16840787—28no—
11C2×#14519844C2212yescomplemented
12C2×C4⋊#92420C2212yescomplemented
13C2×#14519834C2212yes—
14C2×#14519834C2212yes—
15C2×#14519834C2211yescharacteristic
16C2×#14519834C2212yes—
17C2×#14519834C2211yescharacteristic
18C2×#14519834C2212yescomplemented
19C2×#14519834C2212yescomplemented
20C2×#14519834C2212yescomplemented
21C2×#14519834C2212yescomplemented
22C2×C22⋊#92420C2211yescharacteristic
23C2×C22⋊#92420C2212yes—
24C2×C22⋊#92420C2211yescharacteristic
25C2×#14519832C2212yescomplemented
26C2×C4×#92420C2212yescomplemented
27C22×#11758226C2211yescharacteristic
28C23×C4×D61C2211yescharacteristic
29#16840275—28no—
30C2×#14519762C412yes—
31C2×#14519753C411yescharacteristic
32C2×#14519752C411yescharacteristic
33C2×#14519751C411yescharacteristic
34C2×#14519747C412yes—
35C2×#14519739C2212yes—
36C2×#14519737C2212yes—
37C2×#14519735C2211yescharacteristic
38C2×C22⋊Q8×C61C411yescharacteristic
39C22×#92420C2312yes—
40C22×#92420C2312yes—
41C22×#92420C2312yes—
42C22×#92420C2312yes—
43#14519844—24no—
44C4⋊#92420—24no—
45#14519834—24no—
46#14519834—24no—
47#14519834—41no—
48#14519834—42no—
49#14519834—41no—
50#14519834—42no—
51#14519834—42no—
52#14519834—42no—
53#14519834—42no—
54C22⋊#92420—41no—
55C22⋊#92420—42no—
56C22⋊#92420—41no—
57#14519832—24no—
58C4×#92420—24no—
59C24×D61C2×C411yescharacteristic
60C2×#11758226—24no—
61C2×#11758226—41no—
62C2×#11758226—41no—
63C2×#11758226—42no—
64C22×D244C2×C412yes—
65C22×D244C2×C412yes—
66C22×D244C2×C411yescharacteristic
67C22×C22⋊D61C2×C412yes—
68C23×Dic61C2311yescharacteristic
69C22×C4×D61—24no—
70C22×C4×D61—42no—
71C22×C4×D61—41no—
72C22×C4×D61—41no—
73C22×C4×D61C2×C411yescharacteristic
74C22×C4×D61C2×C411yescharacteristic
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2312yes—
77C22×C4×D61C2311yescharacteristic
78#14519762—28no—
79#14519753—24no—
80#14519752—24no—
81#14519751—24no—
82#14519747—28no—
83#14519739—24no—
84#14519739—24no—
85#14519737—24no—
86#14519737—24no—
87#14519735D418yes—
88C2×C4⋊Dic61C2×C411yescharacteristic
89C2×#11758196C2×C412yes—
90C2×#11758198C2×C412yes—
91C2×#11758198C2×C412yes—
92C2×#11758198C2×C412yes—
93C2×#11758198C2×C412yes—
94C2×C4×Dic61C2312yes—
95C22×Q8×C61C2×C411yescharacteristic
96C23×C4×C61C2×C411yescharacteristic
97C22⋊Q8×C61—24no—
98C2×C4⋊C4×C61C2×C412yes—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C22⋊C4×C61C2×C412yes—
101C24×C61C22×C411yescharacteristic
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—22no—
105C23×D61—41no—
106C23×D61—41no—
107C23×D61—41no—
108C23×D61—41no—
109C23×D61C22×C412yes—
110C23×D61C22×C411yescharacteristic
111C23×D61C22×C411yescharacteristic
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61—41no—
115C22×C4×C61—22no—
116C22×C4×C61—41no—
117C22×C4×C61—41no—
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C412yes—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—42no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C412yes—
131C22×Dic61C2411yescharacteristic
132#11758226—48no—
133#11758226—48no—
134C2×D244—24no—
135C2×D244—24no—
136C2×D244—22no—
137C2×D244—24no—
138C2×D244—42no—
139C2×D244—41no—
140C2×D244—41no—
141C2×D244—42no—
142C2×D244—42no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—48no—
159C2×C4×D61—48no—
160C2×C4×D61—41no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×C4×D61C2×D414yes—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420—22no—
175C2×#92420—22no—
176C2×#92420—22no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61C22⋊C414yes—
179C22⋊C4×C61—24no—
180#11758198—24no—
181#11758198—24no—
182#11758198—24no—
183#11758198C22⋊C418yes—
184C4⋊Dic61C22⋊C414yes—
185#11758196—24no—
186C4×Dic61C2×D414yes—
187C4×Dic61C2×D414yes—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61#13112yes—
195C23×C61C23×C411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61—44no—
221C22×D61—44no—
222C22×D61C2×C22⋊C412yes—
223C22×D61#13112yes—
224Q8×C61—44no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—44no—
236C2×C4×C61C2×C22⋊C414yes—
237C2×C4×C61C2×C22⋊C412yes—
238C22⋊D61—48no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C2×C22⋊C414yes—
247C2×Dic61C22×D412yes—
248C2×Dic61H(5,𝔽2)11yescharacteristic
249C2×Dic61H(5,𝔽2)11yescharacteristic
250C2×Dic61H(5,𝔽2)12yes—
251D244—48no—
252D244—44no—
253D244—48no—
254C4×D61—24no—
255C4×D61—24no—
256C4×D61—44no—
257C4×D61—44no—
258C4×D61—42no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—42no—
262C4×D61—42no—
263C4×D61—42no—
264#92420—42no—
265#92420—42no—
266#92420—42no—
267#92420—42no—
268#21792—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61C22×C22⋊C411yescharacteristic
277C22×C61#51811yescharacteristic
278C22×C61#51811yescharacteristic
279C22×C61C2×#13111yescharacteristic
280C22×C61C2×#13111yescharacteristic
281C22×C61#56811yescharacteristic
282C22×C61#56811yescharacteristic
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—416no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C4×C61—22no—
306C4×C61—24no—
307C4×C61—41no—
308C4×C61—42no—
309Dic61—24no—
310Dic61—41no—
311Dic61#53512yes—
312Dic61#53512yes—
313C2×#525—611no—
314C2×#433—612no—
315#1955—612no—
316#1955—612no—
317#1954—612no—
318#1954—612no—
319#1780—612no—
320#1779—611no—
321#1779—611no—
322C2×C61—42no—
323C2×C61#230911yescharacteristic
324C2×C61#288011yescharacteristic
325C2×C61#288011yescharacteristic
326C2×C61#297611yescharacteristic
327C2×C61#297611yescharacteristic
328C2×C61#297711yescharacteristic
329C2×C61#297811yescharacteristic
330D61—42no—
331D61—42no—
332D61—82no—
333C22×#41—611no—
334C24×C4—611no—
335#525—1228no—
336C2×#123—612no—
337C2×C22⋊Q8—611no—
338C2×C22⋊Q8#692311yescharacteristic
339C2×#121—611no—
340C2×#121—611no—
341C2×#120—612no—
342C2×C4×D4—612no—
343C2×C4×D4—612no—
344C2×#117—611no—
345#433—1228no—
346C2×C4⋊C8—612no—
347C2×C4⋊C8—612no—
348C2×C22⋊C8—612no—
349C2×C22⋊C8—612no—
350C2×C22⋊C8—611no—
351C2×C22⋊C8—612no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—612no—
355C2×C22⋊C8—611no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8—612no—
358C2×C22⋊C8—612no—
359C2×C22⋊C8—612no—
360C2×#97—612no—
361C2×C4×C8—612no—
362C61#2179211yescharacteristic, complemented, Sylow
363C25—611no—
364C2×#41—1224no—
365C2×#41—2441no—
366C2×#41—2441no—
367C2×#41—2442no—
368C22×Q8C2×#692311yescharacteristic
369C22×D4—612no—
370C22×D4—612no—
371C22×D4—612no—
372C22×D4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—612no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—1224no—
380C23×C4—2442no—
381C23×C4—2441no—
382C23×C4—2441no—
383C23×C4C2×#692311yescharacteristic
384C22×C8—612no—
385C22×C8—612no—
386C22×C8—612no—
387C22×C8—612no—
388#123—1228no—
389C22⋊Q8—24no—
390C22⋊Q8—1224no—
391#121—1224no—
392#121—1224no—
393#120—1228no—
394C4×D4—1224no—
395C4×D4—1224no—
396C4×D4—1224no—
397C4×D4—1224no—
398#117—618no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—612no—
401C2×C4⋊C4C2×#692312yes—
402C2×C4⋊C4C2×#692311yescharacteristic
403C2×C22⋊C4—612no—
404C2×C22⋊C4—612no—
405C2×C22⋊C4—612no—
406C2×C22⋊C4—612no—
407C2×C22⋊C4C2×#692312yes—
408C2×C42—612no—
409C4⋊C8—1224no—
410C4⋊C8—1224no—
411C22⋊C8—1224no—
412C22⋊C8—1224no—
413C22⋊C8—2441no—
414C22⋊C8—2442no—
415C22⋊C8—2441no—
416C22⋊C8—2441no—
417C22⋊C8—2442no—
418C22⋊C8—2441no—
419C22⋊C8—2442no—
420C22⋊C8—2442no—
421C22⋊C8—2442no—
422C22⋊C8—2442no—
423#97—1224no—
424C4×C8—1224no—
425C24—611no—
426C24—611no—
427C24—612no—
428C24—1222no—
429C24—1222no—
430C24—1222no—
431C24—2441no—
432C24—2441no—
433C24—2441no—
434C24—2441no—
435C24C22×#692311yescharacteristic
436#41—2448no—
437#41—2448no—
438C2×Q8—22no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×D4—1224no—
442C2×D4—1224no—
443C2×D4—1224no—
444C2×D4—1224no—
445C2×D4—1224no—
446C2×D4—1222no—
447C2×D4—2441no—
448C2×D4—2441no—
449C2×D4—2442no—
450C2×D4—2442no—
451C2×D4—2442no—
452C2×D4—2442no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—611no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—611no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—2441no—
480C22×C4—2441no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4—2448no—
484C22×C4—2448no—
485C22×C4C22×#692311yescharacteristic
486C22×C4C22×#692312yes—
487C22×C4C22×#692312yes—
488C22×C4C22×#692311yescharacteristic
489C2×C8—1222no—
490C2×C8—1222no—
491C2×C8—1222no—
492C2×C8—1222no—
493C2×C8—1222no—
494C2×C8—1222no—
495C2×C8—1222no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C4⋊C4—24no—
502C4⋊C4—614no—
503C4⋊C4—1224no—
504C4⋊C4C22⋊#692314yes—
505C22⋊C4—24no—
506C22⋊C4—618no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C42—614no—
511C42—614no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—612no—
519C23—612no—
520C23—1221no—
521C23—1221no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23—1221no—
532C23—1222no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—1222no—
538C23—1222no—
539C23—2444no—
540C23—2444no—
541C23—2444no—
542C23—2444no—
543C23—2444no—
544C23—2444no—
545C23—2444no—
546C23#1451986512yes—
547C23C23×#692311yescharacteristic
548Q8—44no—
549D4—2448no—
550D4—2444no—
551D4—2448no—
552D4—2448no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—612no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—612no—
568C2×C4—614no—
569C2×C4—1224no—
570C2×C4—1224no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—2444no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2444no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4C2×C22⋊#692314yes—
587C2×C4C2×C22⋊#692312yes—
588C8—2442no—
589C8—2442no—
590C8—2442no—
591C8—2442no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—1222no—
600C22—1222no—
601C22—1222no—
602C22—1222no—
603C22—1222no—
604C22—1222no—
605C22—24416no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22—2442no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22—2441no—
617C22—2441no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22C22×C22⋊#692311yescharacteristic
622C22#1684082911yescharacteristic
623C22#1684082911yescharacteristic
624C22D61⋊#13111yescharacteristic
625C22C2×#1451986511yescharacteristic
626C22#1684084111yescharacteristic
627C22#1684084211yescharacteristic
628C4—22no—
629C4—24no—
630C4—41no—
631C4—42no—
632C4—612no—
633C4—612no—
634C4—1224no—
635C4—2441no—
636C2—42no—
637C2—4882no—
638C2—2442no—
639C2—2442no—
640C2#2146229211yescharacteristic
641C2#2146240811yescharacteristic
642C2#2146240911yescharacteristic
643C2#2146268911yescharacteristic
644C2#2146269011yescharacteristic
645C2#2146269411yescharacteristic
646C2#2146269811yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C22⋊Q8×C61C4
prime radicalC2×#16840275C2
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#21792
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840787C2semidirect
2. #21461945C2semidirect
3. #21461944C2semidirect
4. #21461942C2semidirect
5. #21461941C2semidirect
6. #21461644C2semidirect
7. #21461642C2semidirect
8. #21461642C2semidirect
11. C2×#14519844C22semidirect
12. C2×C4⋊#92420C22semidirect
18. C2×#14519834C22semidirect
19. C2×#14519834C22semidirect
20. C2×#14519834C22semidirect
21. C2×#14519834C22semidirect
25. C2×#14519832C22semidirect
26. C2×C4×#92420C22semidirect
362. C61#21792semidirect

Maximal subgroups

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

subgroupindex
C2×#168407872
#214619452
#214619442
#214619422
#214619412
#214616442
#214616422
C2×#168402752
#2179261