Finite Group Databasebeta

← #29729758

Subgroups — #29729758

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839959C211yesmaximal, characteristic
2C2×#16839893C211yesmaximal, characteristic, complemented
3#21451185C211yesmaximal, characteristic, complemented
4#21447207C211yesmaximal, characteristic, complemented
5#21447193C211yesmaximal, characteristic, complemented
6#21447136C211yesmaximal, characteristic, complemented
7#21447136C211yesmaximal, characteristic, complemented
8#21446699C211yesmaximal, characteristic, complemented
9#21446672C211yesmaximal, characteristic, complemented
10#21446644C211yesmaximal, characteristic
11#21446587C211yesmaximal, characteristic, complemented
12#21446519C211yesmaximal, characteristic, complemented
13#21446505C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2548C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19#16839959—24no—
20#16839893—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839700—21no—
38#16839681—21no—
39#16839668C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839643—21no—
44#16839643—21no—
45#16839628C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50C2×#127×C61C2211yescharacteristic
51C2×#124×C61—21no—
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#394—21no—
54C61×#394—21no—
55C61×#390—21no—
56C61×#390C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758200—22no—
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—41no—
67C2×#11758200—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×Dic122—22no—
73C4×Dic122—22no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C4×Dic61—21no—
81#14519710—41no—
82#14519708—41no—
83C4⋊#92418—22no—
84C4⋊#92418—22no—
85C4⋊#92418D414yes—
86#14519700—22no—
87#14519700—22no—
88#14519698—22no—
89#14519698—22no—
90#14519698—22no—
91#14519698—22no—
92#14519698—41no—
93#14519698—41no—
94#14519688—22no—
95#14519688—22no—
96C4×#92418—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61D411yescharacteristic
106C22×D4×C61—21no—
107C22×D4×C61—21no—
108C22×D4×C61C2311yescharacteristic
109C22×D4×C61C2311yescharacteristic
110#127×C61—22no—
111#127×C61—22no—
112#124×C61—44no—
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#95×C61—21no—
133C24×C61—21no—
134C24×C61—21no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—41no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—42no—
151C2×D4×C61—42no—
152C2×D4×C61—42no—
153C2×D4×C61—42no—
154C2×D4×C61—81no—
155C2×D4×C61—81no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2411yescharacteristic
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122#4112yes—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C22⋊C4×C61—42no—
183C22⋊C4×C61—42no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—81no—
187C22⋊C4×C61—81no—
188C22⋊C4×C61—81no—
189C22⋊C4×C61—81no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—42no—
195C42×C61C2×D412yes—
196C42×C61#4112yes—
197#11758200—44no—
198#11758200—44no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208C4×Dic61—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D412yes—
219C2×#92418C2×D412yes—
220C2×#92418#4112yes—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61C22×D411yescharacteristic
236Q8×C61—44no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61—41no—
245D4×C61—84no—
246D4×C61—84no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61#12112yes—
269C2×C4×C61#12112yes—
270C2×C4×C61AGL(1,ℤ/8)12yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61H(5,𝔽2)11yescharacteristic
276C2×C4×C612−1+411yescharacteristic
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289Dic122—22no—
290Dic122—22no—
291Dic122—41no—
292Dic122—41no—
293#92418—22no—
294#92418—22no—
295#92418—22no—
296#92418—41no—
297#25310—611nomaximal, Sylow
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61—81no—
302C22×C61—81no—
303C22×C61—81no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61—81no—
312C22×C61C2×#12111yescharacteristic
313C22×C61#54811yescharacteristic
314C22×C61#55011yescharacteristic
315C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57711yescharacteristic
318C22×C61#57711yescharacteristic
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—22no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61#52911yescharacteristic
328C4×C61#54211yescharacteristic
329C4×C61#54611yescharacteristic
330C4×C61#54811yescharacteristic
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335C2×#460—611no—
336C2×#446—611no—
337#2581—611no—
338#2548D6111yescharacteristic
339#2067—611no—
340#2065—611no—
341#2055—611no—
342#2055—611no—
343#1973—611no—
344#1968—611no—
345#1961—611no—
346#1949—611no—
347#1936—611no—
348#1934—611no—
349#1763—611no—
350C2×C61—161no—
351C2×C61—161no—
352C2×C61#242611yescharacteristic
353C2×C61#305511yescharacteristic
354C2×C61#305711yescharacteristic
355C2×C61#336811yescharacteristic
356C2×C61#337111yescharacteristic
357C2×C61#339711yescharacteristic
358C2×C61#340111yescharacteristic
359C22×SD16—611no—
360C22×SD16—611no—
361C2×#127C2×D6111yescharacteristic
362C2×#124—21no—
363C2×C4×Q8—611no—
364C2×C4×Q8C2×D6111yescharacteristic
365#460—1224no—
366#446—1224no—
367C2×#107—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×C4⋊C8—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#97—611no—
381C2×#97—611no—
382C2×C4×C8—611no—
383#400—1221no—
384#395—1221no—
385#394—21no—
386#394—21no—
387#391—611no—
388#391—611no—
389#390—21no—
390#390C2×D6111yescharacteristic
391#390C2×D6111yescharacteristic
392#390C2×D6111yescharacteristic
393#390C2×D6111yescharacteristic
394#389—611no—
395#389—611no—
396#387—1221no—
397#387—1221no—
398#382—611no—
399C4×C4⋊C4C2×D6111yescharacteristic
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404C61#2531011yescharacteristic, complemented, Sylow
405C22×Q8—611no—
406C22×Q8C22⋊D6111yescharacteristic
407C22×D4—21no—
408C22×D4—21no—
409C22×D4C22×D6111yescharacteristic
410C22×D4C22×D6111yescharacteristic
411C2×SD16—1222no—
412C2×SD16—1222no—
413C2×SD16—1222no—
414C2×SD16—1222no—
415C2×SD16—2441no—
416C2×SD16—2441no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8—611no—
421#127—22no—
422#127—22no—
423#124—44no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22⋊D6111yescharacteristic
435C2×C4⋊C4C22⋊D6111yescharacteristic
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C42—21no—
448C2×C42—1221no—
449C2×C42C22×D6111yescharacteristic
450C2×C42C22×D6111yescharacteristic
451C2×C42C22×D6111yescharacteristic
452#107—2441no—
453#106—2441no—
454C4⋊C8—614no—
455C4⋊C8—1222no—
456C4⋊C8—1222no—
457#103—1222no—
458#103—1222no—
459#102—1222no—
460#102—1222no—
461#102—1222no—
462#102—1222no—
463#102—2441no—
464#102—2441no—
465#97—1222no—
466#97—1222no—
467C4×C8—1222no—
468#95—21no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477C24—21no—
478C24—21no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—41no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—42no—
503C2×D4—42no—
504C2×D4—81no—
505C2×D4—81no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C23×D6111yescharacteristic
519SD16—2444no—
520SD16—2444no—
521C2×C8—612no—
522C2×C8—612no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—81no—
554C22⋊C4—81no—
555C22⋊C4—81no—
556C22⋊C4—81no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—42no—
562C42—2442no—
563C42D4×D6112yes—
564C42#1175822412yes—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C22×C22⋊D6111yescharacteristic
580Q8—44no—
581Q8—1222no—
582Q8—1222no—
583Q8—2441no—
584Q8—2441no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—84no—
594D4—84no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—1222no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4#1451974912yes—
629C2×C4#1451974912yes—
630C2×C4#1451980312yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C2×C4#1451982311yescharacteristic
636C2×C4#1451982411yescharacteristic
637C8—1222no—
638C8—1222no—
639C8—1222no—
640C8—2441no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C2×#1451974911yescharacteristic
656C22#1684041911yescharacteristic
657C22#1684043111yescharacteristic
658C22C2×#1451980311yescharacteristic
659C22#1684056011yescharacteristic
660C22#1684057311yescharacteristic
661C22#1684057311yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—81no—
669C4—81no—
670C4—4881no—
671C4—4881no—
672C4—4881no—
673C4—2442no—
674C4#1684029111yescharacteristic
675C4#1684038111yescharacteristic
676C4#1684040511yescharacteristic
677C4#1684042011yescharacteristic
678C2—161no—
679C2—161no—
680C2#2144938811yescharacteristic
681C2#2145543411yescharacteristic
682C2#2145545511yescharacteristic
683C2#2145854411yescharacteristic
684C2#2145858111yescharacteristic
685C2#2145883511yescharacteristic
686C2#2145887911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2548C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2548D61
hypocenterC61#25310
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. C2×#16839893C2semidirect
3. #21451185C2semidirect
4. #21447207C2semidirect
5. #21447193C2semidirect
6. #21447136C2semidirect
7. #21447136C2semidirect
8. #21446699C2semidirect
9. #21446672C2semidirect
11. #21446587C2semidirect
12. #21446519C2semidirect
13. #21446505C2semidirect
14. #21445652C2semidirect
404. C61#25310semidirect

Maximal subgroups

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

subgroupindex
C2×#168399592
C2×#168398932
#214511852
#214472072
#214471932
#214471362
#214466992
#214466722
#214466442
#214465872
#214465192
#214465052
#214456522
C61×#25482
#2531061