Finite Group Databasebeta

← #29686387

Subgroups — #29686387

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×#16839986C211yesmaximal, characteristic
2C2×#16839865C211yesmaximal, characteristic
3C2×Q16×Dic61C211yesmaximal, characteristic, complemented
4#21448533C211yesmaximal, characteristic
5#21448184C211yesmaximal, characteristic, complemented
6#21446748C211yesmaximal, characteristic
7#21446576C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446440C211yesmaximal, characteristic
10#21446174C211yesmaximal, characteristic
11#21445822C211yesmaximal, characteristic, complemented
12#21445675C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic
15C61×#1878C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22#16839986C418yescomplemented
23#16839865—24no—
24Q16×Dic61—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839651—21no—
39#16839651—21no—
40#16839636—21no—
41#16839636—21no—
42#16839631C2211yescharacteristic
43C4⋊C4×Dic61—21no—
44C4⋊C4×Dic61—21no—
45C4⋊C4×Dic61C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47C4×C4⋊Dic61—21no—
48C4×C4⋊Dic61C2211yescharacteristic
49#16839608—21no—
50#16839608C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839517C2211yescharacteristic
55C22×Q16×C61C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C2×#11758201—22no—
65C2×#11758201—22no—
66C2×#11758201C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519766—22no—
70#14519766C2×C414yes—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61—22no—
74Q8×Dic61—22no—
75Q8×Dic61C2×C414yescomplemented
76Q8×Dic61D414yes—
77#14519739—22no—
78#14519739—22no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519711—22no—
101#14519710—22no—
102#14519706—22no—
103#14519706C2×C414yescomplemented
104#14519701—22no—
105#14519701—22no—
106#14519701C2×C414yescomplemented
107#14519699—22no—
108#14519699C2×C414yes—
109#14519697—22no—
110#14519687—22no—
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×Q8×C61C2311yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C2×Q16×C61—22no—
122C2×Q16×C61—22no—
123C2×Q16×C61C2×C414yescomplemented
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C4×Q8×C61—22no—
127C4×Q8×C61—22no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#103×C61—22no—
134#103×C61—22no—
135#95×C61—21no—
136#95×C61—21no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C22×C412yes—
147C2×Q8×C61C22×C412yes—
148C2×Q8×C61C2×D412yes—
149C2×Q8×C61#4112yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244C22×C412yes—
170C2×C4×D61—42no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173Q16×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×C61#4112yes—
185C42×C61—24no—
186C42×C61—21no—
187C42×C61—21no—
188#11758201—28no—
189#11758198—24no—
190#11758198—24no—
191#11758198—42no—
192#11758198—81no—
193#11758198—81no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—24no—
197C4⋊Dic61—24no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—42no—
201C4⋊Dic61C22×C412yes—
202C4⋊Dic61C2×D412yes—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—24no—
206C4×Dic61—24no—
207C4×Dic61—42no—
208C4×Dic61—81no—
209C4×Dic61—81no—
210C4×Dic61—81no—
211C4×Dic61C22×C412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C4×Dic61#4112yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—22no—
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61C4×D414yes—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61C4×D412yes—
279C2×Dic61C4×D412yes—
280C2×Dic61#12112yes—
281C2×Dic61#12112yes—
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286C4×D61—84no—
287C8×C61—22no—
288C8×C61—41no—
289#92418—22no—
290#92418—41no—
291#21512—611nomaximal, Sylow
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C2×#12111yescharacteristic
294C22×C61#52511yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57711yescharacteristic
297C22×C61#57811yescharacteristic
298C22×C61#57811yescharacteristic
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C4×C61—24no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—81no—
312C4×C61C2×C4×D411yescharacteristic
313C4×C61C4×#4111yescharacteristic
314C4×C61#52911yescharacteristic
315C4×C61#53211yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323C2×#464—611no—
324C2×#440—611no—
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#1878D6111yescharacteristic, complemented
334#1803—611no—
335#1768—611no—
336#1763—611no—
337C4×#102—611no—
338C2×C61C4×#12111yescharacteristic
339C2×C61#294711yescharacteristic
340C2×C61#295111yescharacteristic
341C2×C61#304411yescharacteristic
342C2×C61#304511yescharacteristic
343C2×C61#312111yescharacteristic
344C2×C61#312411yescharacteristic
345D61—82no—
346C22×Q16C2×D6111yescharacteristic
347C22×SD16—611no—
348C2×#124—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8C2×D6111yescharacteristic
353C2×C4×D4—611no—
354#464—618no—
355#440—1224no—
356C4×Q16—1224no—
357C2×#107—611no—
358C2×#107—611no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103C2×D6111yescharacteristic
365C2×#103C2×D6111yescharacteristic
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#97—611no—
370C2×C4×C8—611no—
371#391C2×D6111yescharacteristic
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#340C2×D6111yescharacteristic
390#336—611no—
391#336C2×D6111yescharacteristic
392C61#2151211yescharacteristic, complemented, Sylow
393C22×Q8C22×D6111yescharacteristic
394C22×Q8C22×D6111yescharacteristic
395C22×D4—611no—
396C23×C4—1221no—
397C2×Q16—22no—
398C2×Q16—22no—
399C2×Q16C4×D6114yescomplemented
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#124—614no—
408#124—1222no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—614no—
412C4×Q8—614no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
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—1221no—
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—1222no—
446#107—1222no—
447C4⋊C8—614no—
448C4⋊C8—1222no—
449#103—22no—
450#103—22no—
451#103—614no—
452#103—1222no—
453#103—1222no—
454#102—614no—
455#102—1222no—
456#102—1222no—
457#97—1222no—
458C4×C8—1222no—
459#95—21no—
460#95—21no—
461#95—611no—
462#95—611no—
463#95—611no—
464#95—611no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468C24—1221no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8C2×C4×D6112yes—
478C2×Q8C2×C4×D6112yes—
479C2×Q8D4×D6112yes—
480C2×Q8#1175822412yes—
481C2×D4—612no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—4881no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—28no—
505SD16—1228no—
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—24no—
519C4⋊C4—21no—
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⋊C4#1175822412yes—
534C22⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—4881no—
537C22⋊C4—4881no—
538C22⋊C4—2442no—
539C42—24no—
540C42—21no—
541C42—21no—
542C42—612no—
543C42—612no—
544C42—612no—
545C42—612no—
546C42—1224no—
547C42—1224no—
548C42—4881no—
549C42—4881no—
550C42—4881no—
551C42—2442no—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C2×D4×D6111yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8#1451973714yes—
568D4—1222no—
569D4—1222no—
570D4—2441no—
571D4—2441no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—4884no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4C2×#1175822411yescharacteristic
617C2×C4C2×#1175822411yescharacteristic
618C8—22no—
619C8—41no—
620C8—1222no—
621C8—2441no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22C4×D4×D6111yescharacteristic
630C22#121×D6111yescharacteristic
631C22#1684026811yescharacteristic
632C22#1684056411yescharacteristic
633C22#1684057111yescharacteristic
634C22#1684057511yescharacteristic
635C22#1684057811yescharacteristic
636C4—24no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—81no—
642C4—1222no—
643C4—1222no—
644C4—1222no—
645C4—1222no—
646C4—4881no—
647C4—2442no—
648C4—2442no—
649C4C2×#1451973711yescharacteristic
650C4C4×#1175822411yescharacteristic
651C4#1684028911yescharacteristic
652C4#1684030711yescharacteristic
653C2—4882no—
654C2#2144832911yescharacteristic
655C2#2145466111yescharacteristic
656C2#2145469411yescharacteristic
657C2#2145534411yescharacteristic
658C2#2145536111yescharacteristic
659C2#2145604911yescharacteristic
660C2#2145608211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1878C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1878D61
hypocenterC61#21512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
3. C2×Q16×Dic61C2semidirect
5. #21448184C2semidirect
7. #21446576C2semidirect
8. #21446457C2semidirect
11. #21445822C2semidirect
12. #21445675C2semidirect
13. #21445651C2semidirect
15. C61×#1878C2semidirect
22. #16839986C4semidirect
75. Q8×Dic61C2×C4semidirect
103. #14519706C2×C4semidirect
106. #14519701C2×C4semidirect
123. C2×Q16×C61C2×C4semidirect
333. #1878D61semidirect
392. C61#21512semidirect
399. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399862
C2×#168398652
C2×Q16×Dic612
#214485332
#214481842
#214467482
#214465762
#214464572
#214464402
#214461742
#214458222
#214456752
#214456512
#214456332
C61×#18782
#2151261