Finite Group Databasebeta

← #29686386

Subgroups — #29686386

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839982C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3#21448531C211yesmaximal, characteristic
4#21448184C211yesmaximal, characteristic, complemented
5#21446750C211yesmaximal, characteristic
6#21446574C211yesmaximal, characteristic, complemented
7#21446459C211yesmaximal, characteristic, complemented
8#21446442C211yesmaximal, characteristic
9#21446176C211yesmaximal, characteristic
10#21446158C211yesmaximal, characteristic, complemented
11#21445822C211yesmaximal, characteristic, complemented
12#21445679C211yesmaximal, characteristic, complemented
13#21445650C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic
15C2×C61×#440C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21#16839982C418yescomplemented
22C4×#11758202—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×C4×#92418C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839665C2211yescharacteristic
35#16839650—21no—
36#16839650—21no—
37#16839635—21no—
38#16839635—21no—
39#16839631C2211yescharacteristic
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×C4⋊Dic61—21no—
45C4×C4⋊Dic61C2211yescharacteristic
46#16839608—21no—
47#16839608C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4×D4×C61C2211yescharacteristic
57C61×#440—24no—
58C2×#107×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C2×#97×C61C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202C2×C414yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519764—22no—
69#14519764C2×C414yes—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61C2×C414yescomplemented
74C4×Dic122—22no—
75C4×Dic122—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—22no—
97#14519706—22no—
98#14519706C2×C414yescomplemented
99#14519702—22no—
100#14519702C2×C414yescomplemented
101#14519701—22no—
102#14519700—22no—
103#14519698—22no—
104#14519698C2×C414yes—
105C4×#92418—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×D4×C61C2311yescharacteristic
117C23×C4×C61—21no—
118C2×SD16×C61—22no—
119C2×SD16×C61—22no—
120C2×SD16×C61C2×C414yes—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×Q8×C61—22no—
124C4×Q8×C61D414yes—
125C4×D4×C61—22no—
126C4×D4×C61—22no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—22no—
134#103×C61—22no—
135#102×C61—22no—
136#97×C61—22no—
137C24×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61#4112yes—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—42no—
154C22×C4×C61—81no—
155C22×C4×C61—81no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2411yescharacteristic
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173SD16×C61—28no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C412yes—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C2×D412yes—
185C22⋊C4×C61—42no—
186C42×C61—24no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189C22⋊Dic61—24no—
190C22⋊Dic61—24no—
191C22⋊Dic61—81no—
192C22⋊Dic61—81no—
193#11758202—28no—
194C4⋊Dic61—24no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C22×C412yes—
201C4⋊Dic61#4112yes—
202#11758196—42no—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—24no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—24no—
209C4×Dic61—42no—
210C4×Dic61—81no—
211C4×Dic61—81no—
212C4×Dic61—81no—
213C4×Dic61C22×C412yes—
214C4×Dic61#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61C22×D411yescharacteristic
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61C4×D414yes—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—41no—
236D4×C61—41no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—84no—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61C4×D412yes—
282C2×Dic61C4×D412yes—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—22no—
288C8×C61—41no—
289#92418—22no—
290#92418—41no—
291#21512—611nomaximal, Sylow
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61C2×C4×D411yescharacteristic
300C22×C61C2×#12111yescharacteristic
301C22×C61#52511yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57711yescharacteristic
304C22×C61#57811yescharacteristic
305C22×C61#57811yescharacteristic
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—24no—
309C4×C61—42no—
310C4×C61—81no—
311C4×C61C2×C4×D411yescharacteristic
312C4×C61C4×#4111yescharacteristic
313C4×C61#52911yescharacteristic
314C4×C61#53211yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—41no—
318Dic61—41no—
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323C2×#464—611no—
324C2×#440D6111yescharacteristic
325C2×C4×Q16—611no—
326#2326—611no—
327#2276—611no—
328#1983—611no—
329#1947—611no—
330#1928—611no—
331#1925—611no—
332#1880—611no—
333#1878—611no—
334#1803—611no—
335#1768—611no—
336#1763—611no—
337C4×#102—611no—
338C2×C61—82no—
339C2×C61C4×#12111yescharacteristic
340C2×C61#294711yescharacteristic
341C2×C61#295111yescharacteristic
342C2×C61#304411yescharacteristic
343C2×C61#304511yescharacteristic
344C2×C61#312111yescharacteristic
345C2×C61#312411yescharacteristic
346C22×Q16—611no—
347C22×SD16C2×D6111yescharacteristic
348C2×#124—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8C2×D6111yescharacteristic
353C2×C4×D4C2×D6111yescharacteristic
354#464—618no—
355#440—24no—
356C4×Q16—1224no—
357C2×#107—611no—
358C2×#107C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×C4⋊C8—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103C2×D6111yescharacteristic
366C2×#102—611no—
367C2×#102—611no—
368C2×#102C2×D6111yescharacteristic
369C2×#97C2×D6111yescharacteristic
370C2×C4×C8—611no—
371#391—611no—
372#390—611no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#385—1221no—
377#384—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C4⋊C4—1221no—
383C4×C4⋊C4—1221no—
384#376—611no—
385#376—1221no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336—611no—
391#336—611no—
392C61#2151211yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×Q8C22×D6111yescharacteristic
395C22×D4C22×D6111yescharacteristic
396C23×C4—21no—
397C2×Q16—614no—
398C2×Q16—1222no—
399C2×Q16—1222no—
400C2×SD16—22no—
401C2×SD16—22no—
402C2×SD16C4×D6114yes—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#124—614no—
408#124—1222no—
409C4×Q8—22no—
410C4×Q8—614no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8C22⋊D6114yes—
417C4×D4—22no—
418C4×D4—22no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—21no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—22no—
446#107—1222no—
447C4⋊C8—614no—
448C4⋊C8—1222no—
449#103—22no—
450#103—614no—
451#103—1222no—
452#103—1222no—
453#103—1222no—
454#102—22no—
455#102—614no—
456#102—1222no—
457#97—22no—
458C4×C8—1222no—
459#95—611no—
460#95—611no—
461#95—611no—
462#95—611no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—612no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8C2×C4×D6112yes—
479C2×Q8#1175822212yes—
480C2×Q8C2×C22⋊D6112yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D6112yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—1228no—
505SD16—28no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—612no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8C2×C4×D6112yes—
518C4⋊C4—21no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—1224no—
525C4⋊C4—1224no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—4881no—
531C4⋊C4—4881no—
532C4⋊C4—2442no—
533C4⋊C4C2×C22⋊D6112yes—
534C22⋊C4—42no—
535C22⋊C4—1224no—
536C22⋊C4—1224no—
537C22⋊C4—4881no—
538C22⋊C4—4881no—
539C42—24no—
540C42—612no—
541C42—612no—
542C42—1224no—
543C42—1224no—
544C42—1221no—
545C42—1221no—
546C42—4881no—
547C42—4881no—
548C42—4881no—
549C42—2442no—
550C42#1175822412yes—
551C42C2×C22⋊D6112yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—1222no—
564Q8—1222no—
565Q8—2441no—
566Q8—2441no—
567Q8C4×C22⋊D6114yes—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4#1451974912yes—
611C2×C4#1451974912yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4C2×#1175822411yescharacteristic
617C2×C4C2×#1175822211yescharacteristic
618C8—22no—
619C8—41no—
620C8—1222no—
621C8—2441no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C4×D4×D6111yescharacteristic
630C22C2×#1451974911yescharacteristic
631C22#1684027111yescharacteristic
632C22#1684056611yescharacteristic
633C22#1684057211yescharacteristic
634C22#1684057811yescharacteristic
635C22#1684057811yescharacteristic
636C4—22no—
637C4—24no—
638C4—22no—
639C4—42no—
640C4—81no—
641C4—1222no—
642C4—1222no—
643C4—4881no—
644C4—2442no—
645C4—2442no—
646C4—2441no—
647C4—2441no—
648C4—2442no—
649C4C2×C4×C22⋊D6111yescharacteristic
650C4C4×#1175822411yescharacteristic
651C4#1684029211yescharacteristic
652C4#1684030811yescharacteristic
653C2—82no—
654C2C4×#1451974911yescharacteristic
655C2#2145465911yescharacteristic
656C2#2145469111yescharacteristic
657C2#2145535111yescharacteristic
658C2#2145536211yescharacteristic
659C2#2145604611yescharacteristic
660C2#2145607911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#440C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#440D61
hypocenterC61#21512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
4. #21448184C2semidirect
6. #21446574C2semidirect
7. #21446459C2semidirect
10. #21446158C2semidirect
11. #21445822C2semidirect
12. #21445679C2semidirect
13. #21445650C2semidirect
21. #16839982C4semidirect
65. C2×#11758202C2×C4semidirect
73. Q8×Dic61C2×C4semidirect
98. #14519706C2×C4semidirect
100. #14519702C2×C4semidirect
392. C61#21512semidirect

Maximal subgroups

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

subgroupindex
C2×#168399822
C2×C4×#117582022
#214485312
#214481842
#214467502
#214465742
#214464592
#214464422
#214461762
#214461582
#214458222
#214456792
#214456502
C4×#145196982
C2×C61×#4402
#2151261