Finite Group Databasebeta

← #29679483

Subgroups — #29679483

Order 15616.

1514 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic61C211yesmaximal, characteristic
2C22×#14519706C211yesmaximal, characteristic, complemented
3C2×#16839758C211yesmaximal, characteristic
4#21448222C211yesmaximal, characteristic, complemented
5#21445832C212yesmaximal, complemented
6#21445827C212yesmaximal, complemented
7#21445822C212yesmaximal, complemented
8#21445817C212yesmaximal, complemented
9#21445536C211yesmaximal, characteristic, complemented
10#21445535C211yesmaximal, characteristic, complemented
11C2×C61×#416C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13#41×Dic61—24no—
14#41×Dic61C418yescomplemented
15C2×Q8×Dic61C2211yescharacteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61—21no—
18C2×D4×Dic61—21no—
19C2×D4×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×#14519736—21no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24C22×C4×Dic61—21no—
25C22×C4×Dic61C2211yescharacteristic
26C2×#14519706—24no—
27C2×#14519706C2212yes—
28C2×#14519706C2212yes—
29#16839758—24no—
30C2×#14519701C2212yes—
31C2×#14519698C2212yes—
32#16839631C2212yes—
33#16839627C2212yes—
34#16839620—21no—
35#16839620—21no—
36C4×#11758196—21no—
37C4×#11758196—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40#16839524C2212yes—
41#16839524C2212yes—
42C22×#41×C61C2211yescharacteristic
43C23×C8×C61C2211yescharacteristic
44C2×#117×C61C2211yescharacteristic
45C61×#416—24no—
46C2×#103×C61C2212yes—
47C2×#102×C61C2212yes—
48C23×Dic61—21no—
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C23×Dic61D411yescharacteristic
52C22×#92418—22no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418C2312yes—
56Q8×Dic61—22no—
57Q8×Dic61—22no—
58Q8×Dic61C2×C414yes—
59Q8×Dic61C2×C414yes—
60D4×Dic61—28no—
61D4×Dic61—28no—
62D4×Dic61—22no—
63D4×Dic61—22no—
64D4×Dic61C2×C414yes—
65D4×Dic61C2×C414yes—
66#14519736—22no—
67#14519736—28no—
68#14519736—22no—
69#14519736—22no—
70#14519736C2×C414yes—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—28no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2×C414yes—
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D414yes—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519706—24no—
94#14519706—24no—
95#14519701—24no—
96#14519698—24no—
97#14519684—22no—
98#14519684—22no—
99C2×#41×C61—22no—
100C2×#41×C61—22no—
101C2×#41×C61—22no—
102C2×#41×C61C2×C414yes—
103C2×#41×C61D414yes—
104C2×#41×C61Q814yes—
105C22×Q8×C61C2311yescharacteristic
106C22×D4×C61—21no—
107C22×D4×C61C2311yescharacteristic
108C23×C4×C61—21no—
109C23×C4×C61C2311yescharacteristic
110C22×C8×C61—22no—
111C22×C8×C61—22no—
112C22×C8×C61—22no—
113C22×C8×C61C2312yes—
114#117×C61—22no—
115#117×C61—22no—
116C2×C4⋊C4×C61C2312yes—
117C2×C22⋊C4×C61—21no—
118C2×C42×C61—21no—
119#103×C61—24no—
120#102×C61—24no—
121C24×C61—21no—
122C24×C61C2×D411yescharacteristic
123#41×C61—28no—
124#41×C61—28no—
125#41×C61—28no—
126#41×C61C4⋊C4116yescomplemented
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61C22×C412yes—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×Q812yes—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—24no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C22×C412yes—
140C2×D4×C61C2×D412yes—
141C2×D4×C61C2×Q812yes—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61C22×C412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×Q812yes—
152C22×C4×C61#4112yes—
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—24no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—24no—
159C22×Dic61—24no—
160C22×Dic61—22no—
161C22×Dic61—42no—
162C22×Dic61—44no—
163C22×Dic61—44no—
164C22×Dic61C22⋊C414yes—
165C22×Dic61C22⋊C414yes—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—44no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C22⋊C4×C61—42no—
177C42×C61—42no—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C22⋊Dic61—42no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—24no—
185C4⋊Dic61—24no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61C22×C412yes—
189C4⋊Dic61C22×C412yes—
190#11758196—44no—
191#11758196—44no—
192C4×Dic61—24no—
193C4×Dic61—24no—
194C4×Dic61—24no—
195C4×Dic61—42no—
196C4×Dic61C22⋊C414yes—
197C4×Dic61C22⋊C414yes—
198C4×Dic61C22×C412yes—
199C4×Dic61C22×C412yes—
200C4×Dic61C2×D412yes—
201C4×Dic61C2×D412yes—
202C2×#92418—22no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—44no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61C2×C22⋊C412yes—
214C23×C61#12112yes—
215C23×C61C22⋊Q812yes—
216C23×C61C22×D411yescharacteristic
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61C2×C4⋊C414yes—
221D4×C61—216no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—22no—
225D4×C61C2×C4⋊C414yes—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—216no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C4⋊C414yes—
235C2×C4×C61#11714yes—
236C2×C4×C61#12114yes—
237C2×C4×C61#12112yes—
238C2×C4×C61C22⋊Q814yes—
239C2×C4×C61C22⋊Q812yes—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—28no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—44no—
253C2×Dic61—48no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—48no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61#13514yes—
265C8×C61—42no—
266#92418—42no—
267#20645—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—44no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—44no—
277C22×C61#38014yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C22×C61C2×#12111yescharacteristic
280C22×C61C2×C22⋊Q811yescharacteristic
281C22×C61C2×#13512yes—
282C22×C61#57512yes—
283C4×C61—24no—
284C4×C61—82no—
285C4×C61#38014yes—
286C4×C61C22×C4⋊C411yescharacteristic
287C4×C61C2×#11711yescharacteristic
288C4×C61C2×#12111yescharacteristic
289C4×C61C2×C22⋊Q811yescharacteristic
290Dic61—22no—
291Dic61—22no—
292Dic61—22no—
293Dic61—42no—
294Dic61—42no—
295Dic61—82no—
296Dic61#41614yescomplemented
297C2×C4×#41—611no—
298C22×C4⋊C8—611no—
299C2×#416—611no—
300C2×#416D6111yescharacteristic
301#2285—611no—
302#1805—612no—
303#1804—612no—
304#1803—612no—
305#1802—612no—
306#1735—611no—
307#1735—611no—
308C2×C61—24no—
309C2×C61—44no—
310C2×C61C2×#38011yescharacteristic
311C2×C61C2×#41611yescharacteristic
312C2×C61#289611yescharacteristic
313C2×C61#303512yes—
314C2×C61#307612yes—
315C22×#41C2×D6111yescharacteristic
316C23×C8—611no—
317C23×C8C2×D6111yescharacteristic
318C4×#41—618no—
319C4×#41—1224no—
320C2×C4×Q8—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4—1221no—
325C2×C4×D4—1221no—
326C2×#117—611no—
327C2×#117—611no—
328C2×#117—1221no—
329C2×#117C2×D6111yescharacteristic
330C22×C42—611no—
331C22×C42—1221no—
332C2×C4⋊C8—612no—
333C2×C4⋊C8—612no—
334C2×C4⋊C8—1224no—
335#416—24no—
336#416—1224no—
337C2×#103—612no—
338C2×#103C2×D6112yes—
339C2×#102—612no—
340C2×#102C2×D6112yes—
341#384—612no—
342#382—612no—
343#380—1221no—
344#380—1221no—
345C4×C4⋊C4—1221no—
346C4×C4⋊C4—1221no—
347#340—612no—
348#340—612no—
349#340—612no—
350#340—612no—
351C61#2064511yescharacteristic, complemented, Sylow
352C2×#41—22no—
353C2×#41—22no—
354C2×#41—22no—
355C2×#41Dic12214yes—
356C2×#41C4×D6114yes—
357C2×#41C22⋊D6114yes—
358C22×Q8C22×D6111yescharacteristic
359C22×D4—21no—
360C22×D4C22×D6111yescharacteristic
361C23×C4—21no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—1221no—
365C23×C4—1221no—
366C23×C4C22×D6111yescharacteristic
367C22×C8—22no—
368C22×C8—22no—
369C22×C8—22no—
370C22×C8—612no—
371C22×C8—1222no—
372C22×C8—1222no—
373C22×C8—1222no—
374C22×C8C22×D6112yes—
375C4×Q8—614no—
376C4×Q8—614no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×D4—614no—
380C4×D4—614no—
381C4×D4—1222no—
382C4×D4—1222no—
383C4×D4—1228no—
384C4×D4—1228no—
385#117—22no—
386#117—22no—
387#117—614no—
388#117—1228no—
389#117—1222no—
390#117—1222no—
391#117—1222no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4C22×D6112yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C42—21no—
406C2×C42—614no—
407C2×C42—614no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—1228no—
413C2×C42—1221no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42—1221no—
417C4⋊C8—1224no—
418C4⋊C8—1224no—
419#103—24no—
420#103—1224no—
421#102—24no—
422#102—1224no—
423#95—1222no—
424#95—1222no—
425C24—21no—
426C24D4×D6111yescharacteristic
427#41—28no—
428#41—28no—
429#41—28no—
430#41#11758196116yescomplemented
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C2×Dic12212yes—
435C2×Q8C2×C4×D6112yes—
436C2×Q8C2×C22⋊D6112yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4C2×Dic12212yes—
444C2×D4C2×C4×D6112yes—
445C2×D4C2×C22⋊D6112yes—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—614no—
456C22×C4—614no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—2444no—
464C22×C4—2442no—
465C22×C4—2444no—
466C22×C4C2×Dic12212yes—
467C22×C4C2×C4×D6112yes—
468C22×C4D4×D6112yes—
469C22×C4D4×D6111yescharacteristic
470C22×C4#1175822412yes—
471C22×C4#1175822412yes—
472C22×C4C2×C22⋊D6112yes—
473C22×C4C23×D6111yescharacteristic
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—2444no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—2444no—
493C4⋊C4—2444no—
494C22⋊C4—42no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4—1224no—
498C22⋊C4—2442no—
499C42—42no—
500C42—612no—
501C42—612no—
502C42—612no—
503C42—612no—
504C42—614no—
505C42—614no—
506C42—1224no—
507C42—1224no—
508C42—1224no—
509C42—2442no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23C22⋊C4×D6112yes—
518C23#1451974912yes—
519C23#1451975412yes—
520C23C2×D4×D6111yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—22no—
524Q8C2×#1175819614yes—
525D4—216no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4C2×#1175819614yes—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—216no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—612no—
538C2×C4—614no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—1228no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—2448no—
549C2×C4—2444no—
550C2×C4—2448no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4C22⋊C4×D6112yes—
557C2×C4C2×#1175819614yes—
558C2×C4#1451973314yes—
559C2×C4#1451974914yes—
560C2×C4#1451974912yes—
561C2×C4#1451975414yes—
562C2×C4#1451975412yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C22×C22⋊D6111yescharacteristic
566C2×C4C22×Dic12211yescharacteristic
567C2×C4C2×#1175822411yescharacteristic
568C2×C4C2×#1175822411yescharacteristic
569C8—42no—
570C8—2442no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22#1683962014yes—
581C22C2×C22⋊C4×D6111yescharacteristic
582C22C2×#1451974911yescharacteristic
583C22C2×#1451975411yescharacteristic
584C22#135×D6112yes—
585C22#1684055912yes—
586C4—24no—
587C4—82no—
588C4—614no—
589C4—1222no—
590C4—1222no—
591C4—1222no—
592C4—4882no—
593C4—2442no—
594C4—2442no—
595C4#1683962014yes—
596C4C22×#1175819611yescharacteristic
597C4C2×#1451973311yescharacteristic
598C4C2×#1451974911yescharacteristic
599C4C2×#1451975411yescharacteristic
600C2—24no—
601C2—44no—
602C2C2×#1683962011yescharacteristic
603C2#416×D6111yescharacteristic
604C2#2145433411yescharacteristic
605C2#2145527012yes—
606C2#2145560312yes—
subgroupG/H
centerC22×C4D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#416C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC2×#416D61
hypocenterC61#20645
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#416

Complements

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

NH ≅ G/Ntype
2. C22×#14519706C2semidirect
4. #21448222C2semidirect
5. #21445832C2semidirect
6. #21445827C2semidirect
7. #21445822C2semidirect
8. #21445817C2semidirect
9. #21445536C2semidirect
10. #21445535C2semidirect
14. #41×Dic61C4semidirect
126. #41×C61C4⋊C4semidirect
296. Dic61#416semidirect
351. C61#20645semidirect
430. #41#11758196semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic612
C22×#145197062
C2×#168397582
#214482222
#214458322
#214458272
#214458222
#214458172
#214455362
#214455352
C2×C61×#4162
#2064561