Finite Group Databasebeta

← #29711108

Subgroups — #29711108

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
1C23×#11758202C211yesmaximal, characteristic, complemented
2C2×D4×Dic122C211yesmaximal, characteristic
3C2×#16839918C211yesmaximal, characteristic, complemented
4C2×#16839917C212yesmaximal, complemented
5#21446918C212yesmaximal, complemented
6#21446827C212yesmaximal, complemented
7#21446240C211yesmaximal, characteristic, complemented
8#21446163C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic
10#21445753C211yesmaximal, characteristic, complemented
11C61×#2409C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14C22×#11758202—28no—
15C22×#11758202—24no—
16C22×#11758202C2212yes—
17C22×#11758202C2212yes—
18C22×#11758202C2212yes—
19C23×#92418C2211yescharacteristic
20D4×Dic122—24no—
21D4×Dic122—24no—
22C2×#14519777C2211yescharacteristic
23C2×#14519776—21no—
24C2×#14519756C2211yescharacteristic
25C2×#14519756C2211yescharacteristic
26C2×#14519754—21no—
27C2×#14519754—21no—
28C2×C4×Dic122C2211yescharacteristic
29C2×D4×Dic61—21no—
30#16839918—24no—
31#16839917—28no—
32C2×#14519701C2212yes—
33C2×#14519700C2212yes—
34C2×#14519700C2212yes—
35C2×#14519700C2212yes—
36C2×C22⋊#92418C2211yescharacteristic
37C2×C22⋊#92418C2212yes—
38#16839524C2212yes—
39#16839517C2212yes—
40C23×Q8×C61C2211yescharacteristic
41C2×C22⋊Q8×C61C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#395—21no—
44C61×#393—21no—
45C61×#391C2212yes—
46C61×#386—21no—
47C61×#385—21no—
48C61×#380C2211yescharacteristic
49C22×Dic122—24no—
50C22×Dic122—21no—
51C22×Dic122—21no—
52C22×Dic122—22no—
53C22×Dic122—22no—
54C22×Dic122—22no—
55C22×Dic122—24no—
56C22×Dic122—42no—
57C22×Dic122—42no—
58C22×Dic122D414yes—
59C22×Dic122C2311yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C2×#11758202—232no—
65C2×#11758202—24no—
66C2×#11758202—24no—
67C2×#11758202—24no—
68C2×#11758202—24no—
69C2×#11758202—24no—
70C2×#11758202—24no—
71C2×#11758202—24no—
72C2×#11758202—42no—
73C2×#11758202D418yescomplemented
74C22×#92418—22no—
75C22×#92418—24no—
76C22×#92418C2312yes—
77C22×#92418C2312yes—
78#14519777—22no—
79#14519777—22no—
80#14519776—28no—
81#14519756—22no—
82#14519756—22no—
83#14519756—22no—
84#14519756D414yes—
85#14519754—28no—
86#14519754—28no—
87C4×Dic122—22no—
88C4×Dic122—22no—
89D4×Dic61—28no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×C4×Dic61—21no—
99#14519701—24no—
100#14519700—24no—
101#14519700—24no—
102#14519700—42no—
103C22⋊#92418—22no—
104C22⋊#92418—24no—
105C22×Q8×C61—21no—
106C22×Q8×C61—24no—
107C22×Q8×C61—22no—
108C22×Q8×C61—22no—
109C22×Q8×C61—42no—
110C22×Q8×C61C2312yes—
111C22×D4×C61D411yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114C23×C4×C61C2311yescharacteristic
115C22⋊Q8×C61—24no—
116C22⋊Q8×C61—42no—
117C4×D4×C61—22no—
118C4×D4×C61—22no—
119C2×C4⋊C4×C61—22no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61C2312yes—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—22no—
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—22no—
129#95×C61—22no—
130C24×C61C2×D411yescharacteristic
131C24×C61C2×D411yescharacteristic
132C2×Q8×C61—22no—
133C2×Q8×C61—28no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—44no—
139C2×Q8×C61—48no—
140C2×Q8×C61—44no—
141C2×Q8×C61—42no—
142C2×Q8×C61C2×D414yes—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61Q1614yescomplemented
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61—44no—
155C22×C4×C61—42no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2411yescharacteristic
163C22×Dic61—21no—
164C22×Dic61—24no—
165C22×Dic61—24no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—44no—
170C22×Dic61—44no—
171C2×Dic122—21no—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122—22no—
177C2×Dic122—28no—
178C2×Dic122—24no—
179C2×Dic122—24no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—44no—
183C2×Dic122—48no—
184C2×Dic122—48no—
185C2×Dic122—44no—
186C2×Dic122—44no—
187C2×Dic122—44no—
188C2×Dic122—44no—
189C2×Dic122C2×D412yes—
190C2×Dic122C2×D412yes—
191C2×Dic122C2×D414yes—
192C2×Dic122C2×D412yes—
193C4⋊C4×C61—22no—
194C4⋊C4×C61—22no—
195C4⋊C4×C61—22no—
196C4⋊C4×C61—44no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—44no—
200C22⋊C4×C61—42no—
201C42×C61—21no—
202C42×C61—21no—
203C22⋊Dic61—24no—
204C22⋊Dic61—24no—
205#11758202—216no—
206#11758202—216no—
207#11758202—48no—
208C4⋊Dic61—24no—
209C4⋊Dic61—24no—
210C4⋊Dic61—21no—
211C4⋊Dic61C2×D412yes—
212#11758196—24no—
213#11758196—24no—
214#11758196—24no—
215C4×Dic61—24no—
216C2×#92418—28no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—22no—
222C2×#92418C2×D414yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—21no—
226C23×C61—22no—
227C23×C61—21no—
228C23×C61—22no—
229C23×C61#12012yes—
230C23×C61C2×Q1612yes—
231C23×C61C2×Q1612yescomplemented
232C23×C61#13712yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—24no—
235Q8×C61—24no—
236Q8×C61—48no—
237Q8×C61—42no—
238Q8×C61—42no—
239D4×C61—28no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—24no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—48no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61#12012yes—
261C2×C4×C61C2×Q1612yescomplemented
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D412yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61H(5,𝔽2)12yes—
266C2×Dic61—22no—
267C2×Dic61—24no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—48no—
272C2×Dic61—48no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276Dic122—22no—
277Dic122—24no—
278Dic122—22no—
279Dic122—22no—
280Dic122—24no—
281Dic122—48no—
282Dic122—48no—
283Dic122—41no—
284Dic122—41no—
285Dic122—42no—
286Dic122#12014yes—
287#92418—24no—
288#92418—42no—
289#23856—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61—22no—
295C22×C61—24no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61#45114yescomplemented
300C22×C61C2×#12011yescharacteristic
301C22×C61D4212yes—
302C22×C61C22×Q1611yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C22×C61#57812yes—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—41no—
308C4×C61—44no—
309C4×C61—41no—
310C4×C61—82no—
311C4×C61C2×#12011yescharacteristic
312C4×C61#53411yescharacteristic
313C4×C61D4212yes—
314Dic61—24no—
315Dic61—42no—
316Dic61—42no—
317Dic61—42no—
318C23×Q16—611no—
319C2×D4×Q8—611no—
320C2×#451—611no—
321C2×#451—612no—
322#2409D6111yescharacteristic
323#2016—612no—
324#2001—612no—
325#1892—611no—
326#1878—612no—
327#1869—611no—
328#1787—611no—
329C2×C61—24no—
330C2×C61—42no—
331C2×C61#240311yescharacteristic
332C2×C61C2×#45111yescharacteristic
333C2×C61#301111yescharacteristic
334C2×C61#327012yes—
335C2×C61D4×Q1612yes—
336C23×Q8—611no—
337C23×Q8—611no—
338C23×Q8C2×D6111yescharacteristic
339C22×Q16—612no—
340C22×Q16—612no—
341C22×Q16—612no—
342C22×Q16—1228no—
343C22×Q16—1224no—
344C23×C8—611no—
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—1221no—
352C2×C22⋊Q8—1221no—
353C2×C22⋊Q8C2×D6112yes—
354C2×C4×Q8—611no—
355C2×C4×D4—1221no—
356C2×C4×D4C2×D6111yescharacteristic
357#451—1228no—
358#451—1224no—
359C2×#103—612no—
360C2×#103—612no—
361C2×#103—612no—
362C2×#103—612no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8—611no—
365#395—21no—
366#393—21no—
367#391C2×D6112yes—
368#386—21no—
369#385—21no—
370#380C2×D6111yescharacteristic
371#340—612no—
372#336—612no—
373C61#2385611yescharacteristic, complemented, Sylow
374C22×Q8—21no—
375C22×Q8—24no—
376C22×Q8—22no—
377C22×Q8—22no—
378C22×Q8—42no—
379C22×Q8—614no—
380C22×Q8—611no—
381C22×Q8—611no—
382C22×Q8—611no—
383C22×Q8—1222no—
384C22×Q8—1222no—
385C22×Q8—1224no—
386C22×Q8—1224no—
387C22×Q8—1222no—
388C22×Q8—1221no—
389C22×Q8—1221no—
390C22×Q8—2442no—
391C22×Q8—2442no—
392C22×Q8C22×D6112yes—
393C22×D4C22⋊D6111yescharacteristic
394C23×C4—21no—
395C23×C4—1221no—
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C2×Q16—618no—
400C2×Q16—1224no—
401C2×Q16—1224no—
402C2×Q16—1224no—
403C2×Q16—1224no—
404C2×Q16—12232no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×Q16—1224no—
408C2×Q16—2442no—
409C22×C8—612no—
410C22×C8—612no—
411C22×C8—1224no—
412C22×C8—1222no—
413C4⋊Q8—1222no—
414C4⋊Q8—1222no—
415C4⋊Q8—1228no—
416C22⋊Q8—24no—
417C22⋊Q8—42no—
418C22⋊Q8—614no—
419C22⋊Q8—1228no—
420C22⋊Q8—1228no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C4×Q8—1222no—
425C4×Q8—1222no—
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—1228no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C4⋊C4C22×D6112yes—
438C2×C22⋊C4—22no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4C22⋊D6111yescharacteristic
443C2×C22⋊C4C22⋊D6111yescharacteristic
444C2×C42—1221no—
445C2×C42C22×D6111yescharacteristic
446#103—1224no—
447#103—1224no—
448#103—1224no—
449#103—2442no—
450C22⋊C8—1224no—
451C22⋊C8—1222no—
452#95—21no—
453#95—22no—
454#95—22no—
455C24C2×C22⋊D6111yescharacteristic
456C24C2×C22⋊D6111yescharacteristic
457C2×Q8—22no—
458C2×Q8—28no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—44no—
464C2×Q8—48no—
465C2×Q8—44no—
466C2×Q8—42no—
467C2×Q8—614no—
468C2×Q8—612no—
469C2×Q8—612no—
470C2×Q8—612no—
471C2×Q8—1224no—
472C2×Q8—1228no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1222no—
477C2×Q8—1224no—
478C2×Q8—1222no—
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—2444no—
483C2×Q8—2444no—
484C2×Q8—2444no—
485C2×Q8—2448no—
486C2×Q8—2444no—
487C2×Q8—2448no—
488C2×Q8—2444no—
489C2×Q8D4×D6114yes—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4#1175820214yescomplemented
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—44no—
502C22×C4—42no—
503C22×C4—1221no—
504C22×C4—1224no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1224no—
509C22×C4—2444no—
510C22×C4—2444no—
511C22×C4D4×D6112yes—
512C22×C4D4×D6112yes—
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518Q16—12216no—
519Q16—12216no—
520Q16—2448no—
521C2×C8—614no—
522C2×C8—1228no—
523C2×C8—1222no—
524C2×C8—1222no—
525C2×C8—1222no—
526C2×C8—1222no—
527C2×C8—1222no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—44no—
532C4⋊C4—612no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1221no—
538C4⋊C4—1224no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C42—21no—
546C42—21no—
547C42—1224no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23#1451974512yes—
555C23C2×#1175820212yes—
556C23C2×#1175820212yescomplemented
557C23#1451980712yes—
558C23C22×C22⋊D6111yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—614no—
565Q8—1222no—
566Q8—1224no—
567Q8—1222no—
568Q8—1222no—
569Q8—1224no—
570Q8—2448no—
571Q8—2448no—
572Q8—2441no—
573Q8—2441no—
574Q8—2442no—
575D4—28no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—1222no—
597C2×C4—1224no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—2448no—
602C2×C4—2448no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4#1451974512yes—
607C2×C4C2×#1175820212yescomplemented
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4C2×D4×D6112yes—
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4#1451982212yes—
612C8—1224no—
613C8—2442no—
614C22—22no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22#1683991714yescomplemented
624C22C2×#1451974511yescharacteristic
625C22D4×C22⋊D6112yes—
626C22C22×#1175820211yescharacteristic
627C22C2×#1451980711yescharacteristic
628C22#1684057712yes—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—44no—
634C4—82no—
635C4—1224no—
636C4—2442no—
637C4—2442no—
638C4—2442no—
639C4#120×D6111yescharacteristic
640C4#1684032211yescharacteristic
641C4#1684039412yes—
642C2—24no—
643C2—42no—
644C2#2144913011yescharacteristic
645C2C2×#1683991711yescharacteristic
646C2#2145511211yescharacteristic
647C2#2145738512yes—
648C2D4×#1175820212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2409C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2409D61
hypocenterC61#23856
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#11758202C2semidirect
3. C2×#16839918C2semidirect
4. C2×#16839917C2semidirect
5. #21446918C2semidirect
6. #21446827C2semidirect
7. #21446240C2semidirect
8. #21446163C2semidirect
10. #21445753C2semidirect
73. C2×#11758202D4semidirect
145. C2×D4×C61Q16semidirect
231. C23×C61C2×Q16semidirect
261. C2×C4×C61C2×Q16semidirect
299. C22×C61#451semidirect
373. C61#23856semidirect
492. C2×D4#11758202semidirect
556. C23C2×#11758202semidirect
607. C2×C4C2×#11758202semidirect
623. C22#16839917semidirect

Maximal subgroups

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

subgroupindex
C23×#117582022
C2×D4×Dic1222
C2×#168399182
C2×#168399172
#214469182
#214468272
#214462402
#214461632
#214461122
#214457532
C61×#24092
#2385661