Finite Group Databasebeta

← #30609365

Subgroups — #30609365

Order 15616.

1246 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840711C212yesmaximal
2#21461008C212yesmaximal
3#21461004C212yesmaximal
4#21461004C212yesmaximal
5#21460998C212yesmaximal
6#21460998C212yesmaximal
7#21460995C211yesmaximal, characteristic
8#21460995C211yesmaximal, characteristic
9#384×D61C211yesmaximal, characteristic
10C22×C4⋊#6923C2212yes—
11C22×C4×#6923C2212yes—
12#16840711—28no—
13C2×(C2×#6923)⋊C4C2212yes—
14C2×(C2×#6923)⋊C4C2212yes—
15C2×(C2×#6923)⋊C4C2212yes—
16C2×(C2×#6923)⋊C4C2211yescharacteristic
17C2×(C2×#6923)⋊C4C2211yescharacteristic
18C2×(C2×#6923)⋊C4C2212yes—
19C2×(C2×#6923)⋊C4C2212yes—
20C2×(C2×#6923)⋊C4C2212yes—
21C2×(C2×#6923)⋊C4C2212yes—
22C2×(C2×#6923)⋊C4C2212yes—
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2211yescharacteristic
26C2×(C2×#6923)⋊C4C2211yescharacteristic
27C2×C4⋊C4×D61C2212yes—
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C4⋊C4×D61C2212yes—
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C42×D61C2211yescharacteristic
32#16839632C411yescharacteristic, complemented
33#16839631C412yescomplemented
34#16839631C411yescharacteristic, complemented
35#16839631C411yescharacteristic, complemented
36#16839631C412yescomplemented
37C61×#384C411yescharacteristic, complemented
38C23×#6923C2312yes—
39C23×#6923C2312yes—
40C23×#6923C2312yes—
41C23×#6923C2312yes—
42C2×C4⋊#6923—24no—
43C2×C4⋊#6923—42no—
44C2×C4⋊#6923—42no—
45C2×C4×#6923—24no—
46C2×C4×#6923—42no—
47C2×C4×#6923—42no—
48(C2×#6923)⋊C4—24no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—24no—
52(C2×#6923)⋊C4—24no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—24no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—42no—
57(C2×#6923)⋊C4—42no—
58(C2×#6923)⋊C4—42no—
59(C2×#6923)⋊C4—42no—
60(C2×#6923)⋊C4D414yes—
61(C2×#6923)⋊C4Q814yes—
62C22×C4×D61C2312yes—
63C22×C4×D61C2312yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C2×C4⋊Dic61C2×C412yes—
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×C4⋊Dic61C2×C411yescharacteristic
70C2×C4⋊Dic61C2×C412yes—
71C2×#11758196C2×C412yes—
72C2×#11758196C2×C412yes—
73C2×#11758196C2×C412yes—
74C2×#11758196C2×C412yes—
75C2×#11758196C2×C412yes—
76C2×#11758196C2×C412yes—
77C4⋊C4×D61—24no—
78C4⋊C4×D61—24no—
79C4⋊C4×D61—42no—
80C4⋊C4×D61—42no—
81C4⋊C4×D61—42no—
82C4⋊C4×D61—42no—
83C2×C4×Dic61C2×C411yescharacteristic
84C2×C4×Dic61C2×C411yescharacteristic
85C2×C4×Dic61C2×C411yescharacteristic
86C42×D61—24no—
87C2×C4⋊C4×C61C2×C412yes—
88C2×C4⋊C4×C61C2×C411yescharacteristic
89C2×C4⋊C4×C61C2×C411yescharacteristic
90C2×C4⋊C4×C61C2×C412yes—
91C2×C42×C61C2×C411yescharacteristic
92C23×D61C2411yescharacteristic
93C22×#6923—22no—
94C22×#6923—22no—
95C22×#6923—22no—
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923—22no—
115C22×#6923—22no—
116C22×#6923—22no—
117C22×#6923—22no—
118C22×#6923—22no—
119C22×#6923C2×D414yes—
120C22×#6923C2×Q814yes—
121C22×C4×C61C22×C412yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×Dic61C22×C412yes—
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C411yescharacteristic
130C22×Dic61C22×C411yescharacteristic
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61C2×D412yes—
151C2×C4×D61C2×Q812yes—
152C2×C4×D61#4114yes—
153C2×C4×D61#4112yes—
154C2×C4×D61#4112yes—
155C4⋊#6923—48no—
156C4×#6923—48no—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—41no—
159C4⋊C4×C61—41no—
160C4⋊C4×C61—42no—
161C42×C61—22no—
162C4⋊Dic61—24no—
163C4⋊Dic61—41no—
164C4⋊Dic61—41no—
165C4⋊Dic61—42no—
166#11758196—24no—
167#11758196—24no—
168#11758196—42no—
169#11758196—42no—
170#11758196—42no—
171#11758196—42no—
172C4×Dic61—22no—
173C4×Dic61—22no—
174C4×Dic61—22no—
175C23×C61C23×C411yescharacteristic
176C22×D61C22×D411yescharacteristic
177C22×D61C22×Q811yescharacteristic
178C22×D61C2×#4112yes—
179C22×D61C2×#4111yescharacteristic
180C22×D61C2×#4111yescharacteristic
181C22×D61H(5,𝔽2)12yes—
182C22×D61H(5,𝔽2)12yes—
183C22×D612−1+411yescharacteristic
184C22×D612−1+411yescharacteristic
185C22×D612−1+412yes—
186C2×#6923—24no—
187C2×#6923—24no—
188C2×#6923—24no—
189C2×#6923—24no—
190C2×#6923—24no—
191C2×#6923—24no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—42no—
195C2×#6923—42no—
196C2×#6923—42no—
197C2×#6923—42no—
198C2×#6923—42no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210C2×#6923—42no—
211C2×#6923—42no—
212C2×#6923—42no—
213C2×#6923—42no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61#11712yes—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×Q812yes—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61#11712yes—
242C2×Dic61C4×D412yes—
243C2×Dic61C4×Q812yes—
244C4×D61—24no—
245C4×D61—24no—
246C4×D61—24no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—42no—
255C4×D61—42no—
256C4×D61—42no—
257C4×D61—42no—
258#14425—611nomaximal, Sylow
259C22×C61C2×#11711yescharacteristic
260C22×C61C2×C4×D411yescharacteristic
261C22×C61C2×C4×Q811yescharacteristic
262C22×C61#51811yescharacteristic
263C22×C61#51811yescharacteristic
264C22×C61#51911yescharacteristic
265C22×C61#51911yescharacteristic
266C2×D61#53512yes—
267C2×D61#53912yes—
268C2×D61#53912yes—
269C2×D61#54012yes—
270C2×D61#54212yes—
271C2×D61#54412yes—
272C2×D61#54711yescharacteristic
273C2×D61#54712yes—
274C2×D61D4×Q811yescharacteristic
275C2×D61#55212yes—
276C2×D61#55312yes—
277C2×D61#55412yes—
278C2×D61#55611yescharacteristic
279C2×D61#55711yescharacteristic
280C2×D61#56312yes—
281C2×D61#56312yes—
282#6923—44no—
283#6923—44no—
284#6923—82no—
285#6923—82no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—42no—
290C4×C61—42no—
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—42no—
295Dic61—42no—
296C2×#387—612no—
297C2×#384—611no—
298#1445—612no—
299#1441—612no—
300#1441—612no—
301#1436—612no—
302#1436—612no—
303#1434—611no—
304#1434—611no—
305C2×C61#231311yescharacteristic
306C2×C61#232811yescharacteristic
307C2×C61#233111yescharacteristic
308C2×C61#234411yescharacteristic
309C2×C61#234511yescharacteristic
310C2×C61#236511yescharacteristic
311C2×C61#238111yescharacteristic
312D61#271812yes—
313D61#272012yes—
314D61#279112yes—
315D61#279312yes—
316C22×C4⋊C4—612no—
317C22×C4⋊C4—611no—
318C22×C4⋊C4—612no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—612no—
321C22×C42—611no—
322C22×C42—612no—
323#387—1228no—
324#384—612no—
325#384—611no—
326#384—611no—
327#384—612no—
328#384—611no—
329#384#692311yescharacteristic, complemented
330C2×#95—611no—
331C2×#95—611no—
332C2×#95—612no—
333C2×#95—612no—
334C2×#95—612no—
335C2×#95—611no—
336C2×#95—611no—
337C2×#95—612no—
338C2×#95—612no—
339C2×#95—612no—
340C2×#95—612no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95—612no—
344C61#1442511yescharacteristic, complemented, Sylow
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—612no—
351C23×C4—612no—
352C23×C4—612no—
353C23×C4—612no—
354C2×C4⋊C4—612no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—612no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—2442no—
368C2×C4⋊C4—2442no—
369C2×C4⋊C4—2442no—
370C2×C4⋊C4—2442no—
371C2×C4⋊C4—2442no—
372C2×C4⋊C4—2442no—
373C2×C4⋊C4C2×#692312yes—
374C2×C4⋊C4C2×#692311yescharacteristic
375C2×C4⋊C4C2×#692311yescharacteristic
376C2×C4⋊C4C2×#692312yes—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—1224no—
381C2×C42—1224no—
382C2×C42—2442no—
383C2×C42—2442no—
384C2×C42C2×#692311yescharacteristic
385#95—614no—
386#95—614no—
387#95—1224no—
388#95—1224no—
389#95—1222no—
390#95—1224no—
391#95—1222no—
392#95—1224no—
393#95—1224no—
394#95—1224no—
395#95—2442no—
396#95—2442no—
397#95—2442no—
398#95—2442no—
399C24—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—614no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—614no—
409C22×C4—614no—
410C22×C4—612no—
411C22×C4—612no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1221no—
417C22×C4—1221no—
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×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×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4C22×#692312yes—
458C22×C4C22×#692312yes—
459C22×C4C22×#692311yescharacteristic
460C22×C4C22×#692311yescharacteristic
461C22×C4C22×#692311yescharacteristic
462C4⋊C4—24no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
468C4⋊C4—1224no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4—2442no—
472C4⋊C4—2442no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C4⋊C4—2442no—
476C4⋊C4—2448no—
477C42—22no—
478C42—1222no—
479C42—1222no—
480C42—1222no—
481C42—2448no—
482C23—612no—
483C23—611no—
484C23—611no—
485C23—611no—
486C23—611no—
487C23—612no—
488C23—612no—
489C23—611no—
490C23—611no—
491C23—612no—
492C23C23×#692311yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—1224no—
509C2×C4—1224no—
510C2×C4—1224no—
511C2×C4—1224no—
512C2×C4—1224no—
513C2×C4—1224no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1224no—
521C2×C4—1224no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
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—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4#1451985612yes—
563C2×C4D4×#692312yes—
564C2×C4Q8×#692312yes—
565C22—612no—
566C22—612no—
567C22—612no—
568C22—612no—
569C22—612no—
570C22—612no—
571C22—612no—
572C22—612no—
573C22—612no—
574C22—612no—
575C22—611no—
576C22—611no—
577C22—611no—
578C22—611no—
579C22—612no—
580C22—612no—
581C22C2×#1451985611yescharacteristic
582C22C2×D4×#692311yescharacteristic
583C22C2×Q8×#692311yescharacteristic
584C22#1684082911yescharacteristic
585C22#1684082911yescharacteristic
586C22#1684083011yescharacteristic
587C22#1684083011yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—1222no—
594C4—1222no—
595C4—1222no—
596C4—4882no—
597C4—4882no—
598C4—2444no—
599C4—2444no—
600C4—2442no—
601C4—2442no—
602C2—612no—
603C2—612no—
604C2—612no—
605C2—612no—
606C2#2146229611yescharacteristic
607C2#2146231111yescharacteristic
608C2#2146231411yescharacteristic
609C2#2146232711yescharacteristic
610C2#2146232811yescharacteristic
611C2#2146234811yescharacteristic
612C2#2146236411yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#384C4
prime radical, cosocleC23×D61C24
hypercenter#384#6923
hypocenterC61#14425
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
32. #16839632C4semidirect
33. #16839631C4semidirect
34. #16839631C4semidirect
35. #16839631C4semidirect
36. #16839631C4semidirect
37. C61×#384C4semidirect
329. #384#6923semidirect
344. C61#14425semidirect

Maximal subgroups

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

subgroupindex
C2×#168407112
#214610082
#214610042
#214609982
#214609952
#384×D612
#1442561