Finite Group Databasebeta

← #29729410

Subgroups — #29729410

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×#16840120C211yesmaximal, characteristic, complemented
2C2×#16840040C211yesmaximal, characteristic, complemented
3#21450787C211yesmaximal, characteristic
4#21448905C211yesmaximal, characteristic, complemented
5#21447211C211yesmaximal, characteristic, complemented
6#21447205C211yesmaximal, characteristic, complemented
7#21447058C211yesmaximal, characteristic, complemented
8#21446750C211yesmaximal, characteristic, complemented
9#21446750C211yesmaximal, characteristic, complemented
10#21446747C211yesmaximal, characteristic
11#21446541C211yesmaximal, characteristic, complemented
12#21446461C211yesmaximal, characteristic, complemented
13#21446442C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic, complemented
15C61×#2041C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20#16840120—24no—
21#16840040—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36#16839676—21no—
37#16839676—21no—
38#16839668C2211yescharacteristic
39#16839667—21no—
40#16839666—21no—
41#16839665C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839657—21no—
44#16839642—21no—
45#16839642—21no—
46C4⋊C4×Dic61C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4×#11758196C2211yescharacteristic
49C4×C4⋊Dic61C2211yescharacteristic
50#16839613—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#127×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519776—22no—
67#14519776D414yescomplemented
68#14519764—22no—
69#14519764—22no—
70#14519764—22no—
71#14519764—41no—
72D4×Dic61—22no—
73D4×Dic61—22no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—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×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519710—22no—
95C4⋊#92418—41no—
96#14519706—22no—
97#14519706—41no—
98#14519702—22no—
99#14519702—22no—
100#14519700—22no—
101#14519700—41no—
102#14519698—22no—
103#14519698—22no—
104#14519698—22no—
105#14519698—22no—
106#14519687—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×D4×C61—21no—
116C22×D4×C61—21no—
117C22×D4×C61C2311yescharacteristic
118C22×D4×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121#127×C61—22no—
122#127×C61—41no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#102×C61—22no—
128#102×C61—22no—
129#102×C61—22no—
130#102×C61—41no—
131#95×C61—21no—
132C24×C61—21no—
133C24×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
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—42no—
146C2×D4×C61—42no—
147C2×D4×C61—42no—
148C2×D4×C61—42no—
149C2×D4×C61—81no—
150C2×D4×C61—81no—
151C2×D4×C61#4112yes—
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×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—81no—
163C22×Dic61—81no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—24no—
175C2×Dic122—81no—
176C2×Dic122—81no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—81no—
187C4⋊C4×C61C2×D412yes—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195C22⋊Dic61—81no—
196C22⋊Dic61—81no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C2×D412yes—
202C4⋊Dic61#4112yes—
203#11758196—22no—
204#11758196—24no—
205#11758196—41no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—42no—
210C4×Dic61—81no—
211C4×Dic61—81no—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C23×C61—22no—
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×C61C22×D411yescharacteristic
235D4×C61—22no—
236D4×C61—22no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—41no—
240D4×C61—41no—
241D4×C61—41no—
242D4×C61—41no—
243D4×C61—84no—
244D4×C61—84no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#13712yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—84no—
286C2×Dic61#12112yes—
287C2×Dic61#12112yes—
288Dic122—41no—
289Dic122—41no—
290Dic122—41no—
291Dic122—41no—
292Dic122—84no—
293C8×C61—41no—
294C8×C61—41no—
295#92418—41no—
296#92418—41no—
297#25285—611nomaximal, Sylow
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
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#54611yescharacteristic
314C22×C61#54811yescharacteristic
315C22×C61C2×#13711yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57611yescharacteristic
318C22×C61#57611yescharacteristic
319C4×C61—22no—
320C4×C61—42no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61#52911yescharacteristic
324C4×C61#54211yescharacteristic
325C4×C61#54811yescharacteristic
326C4×C61#55011yescharacteristic
327Dic61—22no—
328Dic61—22no—
329Dic61—42no—
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
333Dic61—81no—
334Dic61—81no—
335C2×#489—611no—
336C2×#474—611no—
337#2544—611no—
338#2369—611no—
339#2068—611no—
340#2067—611no—
341#2042—611no—
342#2041D6111yescharacteristic
343#1983—611no—
344#1983—611no—
345#1982—611no—
346#1940—611no—
347#1928—611no—
348#1925—611no—
349#1762—611no—
350C2×C61—82no—
351C2×C61—161no—
352C2×C61#242611yescharacteristic
353C2×C61#303711yescharacteristic
354C2×C61#304111yescharacteristic
355C2×C61#331411yescharacteristic
356C2×C61#331911yescharacteristic
357C2×C61#337011yescharacteristic
358C2×C61#337511yescharacteristic
359C2×C4⋊Q8—611no—
360C2×#127C2×D6111yescharacteristic
361C2×#124—611no—
362C2×#124—611no—
363C2×C4×D4—611no—
364#489—1224no—
365#474—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×C4⋊C8—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102C2×D6111yescharacteristic
380C2×#102C2×D6111yescharacteristic
381C2×#102C2×D6111yescharacteristic
382C2×#102C2×D6111yescharacteristic
383C2×#97—611no—
384#394—1221no—
385#394—1221no—
386#391—611no—
387#391—1221no—
388#391—1221no—
389#390—611no—
390#389—611no—
391#389—1221no—
392#389C2×D6111yescharacteristic
393#386—1221no—
394#386—1221no—
395C4×C4⋊C4—611no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—611no—
398C4×C4⋊C4—611no—
399C4×C4⋊C4—1221no—
400#340—611no—
401#340—611no—
402#336—611no—
403#336C2×D6111yescharacteristic
404C61#2528511yescharacteristic, complemented, Sylow
405C22×Q8—611no—
406C22×Q8—1221no—
407C22×D4—21no—
408C22×D4—21no—
409C22×D4C22×D6111yescharacteristic
410C22×D4C22×D6111yescharacteristic
411C23×C4—1221no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—614no—
417C4⋊Q8—1222no—
418#127—22no—
419#127—41no—
420#124—1222no—
421#124—1222no—
422#124—1222no—
423#124—2441no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—1222no—
450#107—1222no—
451C4⋊C8—1222no—
452C4⋊C8—2441no—
453C4⋊C8—2441no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—2441no—
458#102—22no—
459#102—22no—
460#102—22no—
461#102—41no—
462#102—1222no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#97—1222no—
467#95—21no—
468#95—611no—
469#95—611no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—21no—
477C24—21no—
478C2×Q8—1224no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—4881no—
486C2×Q8—4881no—
487C2×D4—21no—
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—42no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4—42no—
502C2×D4—81no—
503C2×D4—81no—
504C2×D4#1175822412yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—4881no—
516C22×C4—4881no—
517C22×C4—2442no—
518C22×C4D4×D6111yescharacteristic
519C22×C4D4×D6111yescharacteristic
520C22×C4C23×D6111yescharacteristic
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—81no—
537C4⋊C4—612no—
538C4⋊C4—612no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1222no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—4881no—
546C4⋊C4—2441no—
547C4⋊C4D4×D6112yes—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C42—21no—
556C42—21no—
557C42—612no—
558C42—612no—
559C42—1221no—
560C42—1221no—
561C42—4881no—
562C42—4881no—
563C42—2442no—
564C23—22no—
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—
578C23C2×D4×D6111yescharacteristic
579Q8—4884no—
580Q8—2441no—
581Q8—2441no—
582Q8—2441no—
583Q8—2441no—
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—84no—
593D4—84no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—4884no—
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—2441no—
629C2×C4—2441no—
630C2×C4#1451980612yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C2×C4#1451982211yescharacteristic
636C2×C4#1451982411yescharacteristic
637C8—41no—
638C8—41no—
639C8—2441no—
640C8—2441no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22#121×D6111yescharacteristic
656C22#1684039911yescharacteristic
657C22#1684042011yescharacteristic
658C22C2×#1451980611yescharacteristic
659C22#1684055811yescharacteristic
660C22#1684056311yescharacteristic
661C22#1684056611yescharacteristic
662C4—22no—
663C4—41no—
664C4—42no—
665C4—81no—
666C4—1222no—
667C4—1222no—
668C4—4881no—
669C4—4881no—
670C4—2441no—
671C4—2442no—
672C4—2442no—
673C4—2441no—
674C4#1684028911yescharacteristic
675C4#1684038011yescharacteristic
676C4#1684041811yescharacteristic
677C4#1684043011yescharacteristic
678C2—82no—
679C2—161no—
680C2#2144937811yescharacteristic
681C2#2145528611yescharacteristic
682C2#2145531711yescharacteristic
683C2#2145790411yescharacteristic
684C2#2145796811yescharacteristic
685C2#2145856711yescharacteristic
686C2#2145863811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2041C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenter#2041D61
hypocenterC61#25285
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
1. C2×#16840120C2semidirect
2. C2×#16840040C2semidirect
4. #21448905C2semidirect
5. #21447211C2semidirect
6. #21447205C2semidirect
7. #21447058C2semidirect
8. #21446750C2semidirect
9. #21446750C2semidirect
11. #21446541C2semidirect
12. #21446461C2semidirect
13. #21446442C2semidirect
14. #21445646C2semidirect
67. #14519776D4semidirect
404. C61#25285semidirect

Maximal subgroups

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

subgroupindex
C2×#168401202
C2×#168400402
#214507872
#214489052
#214472112
#214472052
#214470582
#214467502
#214467472
#214465412
#214464612
#214464422
#214456462
C61×#20412
#2528561