Finite Group Databasebeta

← #29711105

Subgroups — #29711105

Order 15616.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C22⋊D61C211yesmaximal, characteristic
2C2×#16839918C211yesmaximal, characteristic, complemented
3C2×#16839916C212yesmaximal, complemented
4#21449178C211yesmaximal, characteristic
5#21446917C212yesmaximal, complemented
6#21446826C212yesmaximal, complemented
7#21446242C211yesmaximal, characteristic, complemented
8#21446162C212yesmaximal, complemented
9#21446111C211yesmaximal, characteristic
10#21445754C211yesmaximal, characteristic, complemented
11C23×Q16×C61C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×#11758202C2212yes—
14Q8×C22⋊D61—24no—
15Q8×C22⋊D61—24no—
16C2×#14519775—21no—
17C2×#14519775C2211yescharacteristic
18C2×#14519756—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519753—21no—
21C2×#14519753C2211yescharacteristic
22C2×#14519753C2212yes—
23C2×Q8×Dic61C2211yescharacteristic
24C2×C4×C22⋊D61—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26#16839918—24no—
27#16839916—28no—
28C2×#14519702C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519701C2212yes—
32C2×C22⋊#92418C2211yescharacteristic
33C2×#14519689C2212yes—
34#16839681—21no—
35#16839673—21no—
36#16839669C2212yes—
37#16839641—21no—
38#16839635—21no—
39#16839622C2211yescharacteristic
40#16839525C2212yes—
41#16839516C2212yes—
42C23×Q8×C61C2211yescharacteristic
43C23×Q8×C61C2211yescharacteristic
44C22×Q16×C61—24no—
45C22×Q16×C61—28no—
46C22×Q16×C61C2212yes—
47C22×Q16×C61C2212yes—
48C23×C8×C61C2211yescharacteristic
49C22×Dic122—21no—
50C22×Dic122C2311yescharacteristic
51C2×Q8×D61—22no—
52C2×Q8×D61—22no—
53C2×Q8×D61—24no—
54C2×Q8×D61—42no—
55C22×C22⋊D61D411yescharacteristic
56C22×C4×D61—21no—
57C22×C4×D61C2311yescharacteristic
58C2×#11758202—24no—
59C2×#11758202—24no—
60C2×#11758202—42no—
61C22×#92418C2312yes—
62#14519775—28no—
63#14519775—22no—
64#14519775—22no—
65#14519756—28no—
66#14519756—22no—
67#14519756D414yes—
68#14519753—28no—
69#14519753—22no—
70#14519753—22no—
71#14519753—24no—
72#14519753—42no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75C4×C22⋊D61—28no—
76C4×C22⋊D61—22no—
77C4×C22⋊D61—22no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2312yes—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—22no—
86C2×#11758196D411yescharacteristic
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61D411yescharacteristic
90C2×#11758198—21no—
91C2×#11758198—22no—
92C2×#11758198D411yescharacteristic
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95#14519702—24no—
96#14519701—24no—
97#14519701—24no—
98#14519701—42no—
99C22⋊#92418—22no—
100#14519689—24no—
101#14519684—21no—
102#14519684—22no—
103#14519684—22no—
104C22×Q8×C61—21no—
105C22×Q8×C61—22no—
106C22×Q8×C61—21no—
107C22×Q8×C61—24no—
108C22×Q8×C61—24no—
109C22×Q8×C61—22no—
110C22×Q8×C61—22no—
111C22×Q8×C61—42no—
112C22×Q8×C61—42no—
113C22×Q8×C61D414yes—
114C22×Q8×C61C2311yescharacteristic
115C22×Q8×C61C2312yes—
116C22×Q8×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C23×C4×C61—21no—
119C23×C4×C61C2311yescharacteristic
120C2×Q16×C61—232no—
121C2×Q16×C61—24no—
122C2×Q16×C61—24no—
123C2×Q16×C61—24no—
124C2×Q16×C61—24no—
125C2×Q16×C61—24no—
126C2×Q16×C61D418yescomplemented
127C22×C8×C61—22no—
128C22×C8×C61—24no—
129C22×C8×C61C2312yes—
130C24×C61C2×D411yescharacteristic
131C23×D61C2×D411yescharacteristic
132C2×Q8×C61—24no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—28no—
138C2×Q8×C61—28no—
139C2×Q8×C61—22no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—22no—
143C2×Q8×C61—22no—
144C2×Q8×C61—22no—
145C2×Q8×C61—44no—
146C2×Q8×C61—44no—
147C2×Q8×C61—44no—
148C2×Q8×C61—48no—
149C2×Q8×C61—48no—
150C2×Q8×C61—44no—
151C2×Q8×C61—42no—
152C2×Q8×C61C2×D412yes—
153C2×Q8×C61C2×D414yes—
154C2×Q8×C61C2×D412yes—
155C2×Q8×C61C2×D414yes—
156C22×C4×C61—21no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—24no—
160C22×C4×C61—22no—
161C22×C4×C61—21no—
162C22×C4×C61—44no—
163C22×C4×C61—44no—
164C22×C4×C61—42no—
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D412yes—
167C22×C4×C61C2×D412yes—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—22no—
170C2×C22⋊D61—22no—
171C2×C22⋊D61Q1614yescomplemented
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—22no—
175C22×Dic61C2×D411yescharacteristic
176C22×Dic61C2×D411yescharacteristic
177C22×Dic61C2×D411yescharacteristic
178Q8×D61—48no—
179Q8×D61—44no—
180Q8×D61—44no—
181C2×C4×D61—24no—
182C2×C4×D61—22no—
183C2×C4×D61—21no—
184C2×C4×D61—21no—
185C2×C4×D61—22no—
186C2×C4×D61—44no—
187C2×Dic122—22no—
188C2×Dic122—22no—
189C2×Dic122—24no—
190C2×Dic122—21no—
191C2×Dic122—44no—
192C2×Dic122C2×D412yes—
193Q16×C61—216no—
194Q16×C61—216no—
195C2×C8×C61—28no—
196C2×C8×C61—22no—
197C2×C8×C61—22no—
198C2×C8×C61C2×D414yes—
199C22⋊Dic61—24no—
200C22⋊Dic61—22no—
201C22⋊Dic61—42no—
202#11758202—48no—
203#11758198—24no—
204#11758198—22no—
205#11758198—44no—
206C4⋊Dic61—24no—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61C2×D412yes—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214#11758196—24no—
215#11758196—22no—
216#11758196—44no—
217C4×Dic61—24no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C2×#92418—22no—
221C2×#92418—22no—
222C2×#92418—22no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61#12012yes—
226C23×C61C2×Q1612yes—
227C23×C61#13712yes—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—22no—
231C22×D61—21no—
232C22×D61—22no—
233C22×D61C2×Q1612yescomplemented
234Q8×C61—22no—
235Q8×C61—24no—
236Q8×C61—22no—
237Q8×C61—24no—
238Q8×C61—22no—
239Q8×C61—24no—
240Q8×C61—48no—
241Q8×C61—48no—
242Q8×C61—42no—
243Q8×C61—42no—
244Q8×C61—42no—
245Q8×C61#12014yes—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—24no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—48no—
255C2×C4×C61—48no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61#12012yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D412yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61H(5,𝔽2)12yes—
264C22⋊D61—28no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61C2×Q1612yescomplemented
279C4×D61—48no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—42no—
283Dic122—24no—
284Dic122—48no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—24no—
288#92418—42no—
289#23856—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61#45114yescomplemented
295C22×C61C2×#12011yescharacteristic
296C22×C61D4212yes—
297C22×C61C22×Q1611yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61#57812yes—
300C2×D61—22no—
301C2×D61—24no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—42no—
305C4×C61—24no—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—44no—
309C4×C61C2×#12011yescharacteristic
310C4×C61#53411yescharacteristic
311C4×C61D4212yes—
312Dic61—22no—
313Dic61—42no—
314Dic61—41no—
315Dic61—41no—
316Dic61—42no—
317Dic61—82no—
318C23×Q16D6111yescharacteristic, complemented
319C2×D4×Q8—611no—
320C2×#451—611no—
321C2×#451—612no—
322#2409—611no—
323#2016—612no—
324#2001—612no—
325#1892—611no—
326#1878—612no—
327#1869—611no—
328#1787—611no—
329C2×C61—24no—
330C2×C61#240311yescharacteristic
331C2×C61C2×#45111yescharacteristic
332C2×C61#301111yescharacteristic
333C2×C61#327012yes—
334C2×C61D4×Q1612yes—
335D61—42no—
336C23×Q8—611no—
337C23×Q8C2×D6111yescharacteristic
338C23×Q8C2×D6111yescharacteristic
339C22×Q16—24no—
340C22×Q16—28no—
341C22×Q16—612no—
342C22×Q16C2×D6112yes—
343C22×Q16C2×D6112yes—
344C23×C8C2×D6111yescharacteristic
345D4×Q8—1224no—
346D4×Q8—1224no—
347C2×C4⋊Q8—611no—
348C2×C4⋊Q8—1221no—
349C2×C22⋊Q8—611no—
350C2×C22⋊Q8—611no—
351C2×C22⋊Q8—612no—
352C2×C22⋊Q8—1221no—
353C2×C22⋊Q8—1221no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356C2×C4×D4—1221no—
357#451—1224no—
358#451—1228no—
359C2×#103—612no—
360C2×#103—612no—
361C2×#103—612no—
362C2×#103—612no—
363C2×C22⋊C8—611no—
364C2×C22⋊C8—612no—
365#395—1221no—
366#393—1221no—
367#391—612no—
368#386—1221no—
369#385—1221no—
370#380—611no—
371#340—612no—
372#336—612no—
373C61#2385611yescharacteristic, complemented, Sylow
374C22×Q8—21no—
375C22×Q8—22no—
376C22×Q8—21no—
377C22×Q8—24no—
378C22×Q8—24no—
379C22×Q8—22no—
380C22×Q8—22no—
381C22×Q8—42no—
382C22×Q8—42no—
383C22×Q8—611no—
384C22×Q8—1221no—
385C22×Q8—1222no—
386C22×Q8—1222no—
387C22×Q8—1224no—
388C22×Q8—2442no—
389C22×Q8C22⋊D6114yes—
390C22×Q8C22×D6111yescharacteristic
391C22×Q8C22×D6112yes—
392C22×Q8C22×D6111yescharacteristic
393C22×D4—611no—
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—611no—
397C23×C4—1221no—
398C23×C4C22×D6111yescharacteristic
399C2×Q16—232no—
400C2×Q16—24no—
401C2×Q16—24no—
402C2×Q16—24no—
403C2×Q16—24no—
404C2×Q16—24no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×Q16—2442no—
408C2×Q16C22⋊D6118yescomplemented
409C22×C8—22no—
410C22×C8—24no—
411C22×C8—612no—
412C22×C8C22×D6112yes—
413C4⋊Q8—1222no—
414C4⋊Q8—1228no—
415C4⋊Q8—1222no—
416C22⋊Q8—614no—
417C22⋊Q8—1222no—
418C22⋊Q8—1228no—
419C22⋊Q8—1228no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—1224no—
423C22⋊Q8—2442no—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×D4—1228no—
427C4×D4—1222no—
428C4×D4—1222no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1222no—
438C2×C22⋊C4—611no—
439C2×C22⋊C4—611no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1221no—
444C2×C42—611no—
445C2×C42—1221no—
446#103—1224no—
447#103—1224no—
448#103—1224no—
449#103—2442no—
450C22⋊C8—1222no—
451C22⋊C8—1224no—
452#95—1222no—
453#95—1222no—
454#95—1221no—
455C24—611no—
456C24D4×D6111yescharacteristic
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—28no—
463C2×Q8—28no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—44no—
471C2×Q8—44no—
472C2×Q8—44no—
473C2×Q8—48no—
474C2×Q8—48no—
475C2×Q8—44no—
476C2×Q8—42no—
477C2×Q8—612no—
478C2×Q8—1224no—
479C2×Q8—1221no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—2444no—
483C2×Q8—2448no—
484C2×Q8—2444no—
485C2×Q8—2444no—
486C2×Q8C2×C22⋊D6112yes—
487C2×Q8C2×C22⋊D6114yes—
488C2×Q8C2×C22⋊D6114yes—
489C2×Q8C2×C22⋊D6112yes—
490C2×D4—614no—
491C2×D4—1222no—
492C2×D4—1222no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—24no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1224no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—1221no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—2444no—
514C22×C4D4×D6112yes—
515C22×C4D4×D6111yescharacteristic
516C22×C4C2×C22⋊D6112yes—
517C22×C4C23×D6111yescharacteristic
518Q16—216no—
519Q16—216no—
520Q16—2448no—
521C2×C8—28no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—1222no—
525C2×C8—1222no—
526C2×C8—1222no—
527C2×C8C2×C22⋊D6114yes—
528C4⋊C4—612no—
529C4⋊C4—1224no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1221no—
534C4⋊C4—1222no—
535C4⋊C4—1224no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—2444no—
539C22⋊C4—1224no—
540C22⋊C4—1222no—
541C22⋊C4—1224no—
542C22⋊C4—1222no—
543C22⋊C4—2444no—
544C22⋊C4—2442no—
545C42—1224no—
546C42—1221no—
547C42—1221no—
548C23—22no—
549C23—22no—
550C23—612no—
551C23—1221no—
552C23—1221no—
553C23—1222no—
554C23—1222no—
555C23#1451974512yes—
556C23Q16×D6112yes—
557C23#1451980912yes—
558C23C2×D4×D6111yescharacteristic
559Q8—22no—
560Q8—24no—
561Q8—22no—
562Q8—24no—
563Q8—22no—
564Q8—24no—
565Q8—48no—
566Q8—48no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—1224no—
571Q8—2448no—
572Q8—2441no—
573Q8—2441no—
574Q8#1451974614yes—
575D4—1228no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—612no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—2448no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2442no—
607C2×C4#1451974512yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4C22×C22⋊D6112yes—
611C2×C4#1451982212yes—
612C8—24no—
613C8—2442no—
614C22—22no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—1224no—
619C22—1222no—
620C22—2441no—
621C22—2441no—
622C22—2442no—
623C22#1683991614yescomplemented
624C22C2×#1451974511yescharacteristic
625C22D4×C22⋊D6112yes—
626C22C2×Q16×D6111yescharacteristic
627C22C2×#1451980911yescharacteristic
628C22#1684057512yes—
629C4—22no—
630C4—24no—
631C4—42no—
632C4—44no—
633C4—1222no—
634C4—4882no—
635C4—2441no—
636C4—2442no—
637C4—2441no—
638C4—2442no—
639C4C2×#1451974611yescharacteristic
640C4#1684032111yescharacteristic
641C4D4×C22⋊D6112yes—
642C2—24no—
643C2—2442no—
644C2#2144912911yescharacteristic
645C2C2×#1683991611yescharacteristic
646C2#2145511011yescharacteristic
647C2#2145738912yes—
648C2Q16×C22⋊D6112yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC23×Q16×C61C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenterC23×Q16D61
hypocenterC61#23856
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#16839918C2semidirect
3. C2×#16839916C2semidirect
5. #21446917C2semidirect
6. #21446826C2semidirect
7. #21446242C2semidirect
8. #21446162C2semidirect
10. #21445754C2semidirect
11. C23×Q16×C61C2semidirect
126. C2×Q16×C61D4semidirect
171. C2×C22⋊D61Q16semidirect
233. C22×D61C2×Q16semidirect
278. C2×Dic61C2×Q16semidirect
294. C22×C61#451semidirect
318. C23×Q16D61semidirect
373. C61#23856semidirect
408. C2×Q16C22⋊D61semidirect
623. C22#16839916semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊D612
C2×#168399182
C2×#168399162
#214491782
#214469172
#214468262
#214462422
#214461622
#214461112
#214457542
C23×Q16×C612
#2385661