Finite Group Databasebeta

← #29733362

Subgroups — #29733362

Order 15616.

1262 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840452C211yesmaximal, characteristic, complemented
2C2×#16840126C211yesmaximal, characteristic, complemented
3C2×#16839959C211yesmaximal, characteristic, complemented
4#21446822C212yesmaximal, complemented
5#21446822C212yesmaximal, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446618C211yesmaximal, characteristic, complemented
8#21446142C212yesmaximal, complemented
9#21446137C212yesmaximal, complemented
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2541C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13#16840452—24no—
14#16840452—24no—
15C2×#14519770—21no—
16C2×#14519769C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519758—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25#16840126—24no—
26#16839959—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519698C2212yes—
30C2×#14519698C2212yes—
31C2×#14519698C2212yes—
32C2×C22⋊#92418C2212yes—
33C2×C22⋊#92418C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C2×#127×C61—21no—
39C2×#127×C61C2211yescharacteristic
40C2×#121×C61C2212yes—
41C2×#121×C61C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#392—21no—
44C61×#392—21no—
45C61×#390—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61—21no—
49C2×#11758200—24no—
50C2×#11758200—42no—
51C2×#11758200—42no—
52C22×#92418C2312yes—
53C22×#92418C2312yes—
54#14519770—28no—
55#14519769—22no—
56#14519769D414yescomplemented
57#14519767—28no—
58#14519758—28no—
59#14519756—22no—
60#14519756—22no—
61#14519756—22no—
62#14519756—22no—
63C4×Dic122—22no—
64C4×Dic122D414yescomplemented
65#14519736—22no—
66#14519736—22no—
67#14519733—28no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C4⋊#92418—22no—
85#14519700—24no—
86#14519698—24no—
87#14519698—24no—
88#14519698—24no—
89C22⋊#92418—42no—
90C22⋊#92418—42no—
91#14519688—22no—
92C22×D4×C61—22no—
93C22×D4×C61—22no—
94C22×D4×C61—21no—
95C22×D4×C61—21no—
96C22×D4×C61D411yescharacteristic
97C22×D4×C61C2312yes—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100#127×C61—22no—
101#127×C61—22no—
102#127×C61—44no—
103#121×C61—24no—
104#121×C61—24no—
105#121×C61—42no—
106#121×C61—42no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61C2312yes—
111C2×C22⋊C4×C61—22no—
112C2×C22⋊C4×C61—22no—
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C22⋊C4×C61D411yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117#95×C61—21no—
118C24×C61—22no—
119C24×C61C2×D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—41no—
130C2×D4×C61—44no—
131C2×D4×C61—44no—
132C2×D4×C61—42no—
133C2×D4×C61—42no—
134C2×D4×C61—44no—
135C2×D4×C61—44no—
136C2×D4×C61—42no—
137C2×D4×C61—42no—
138C2×D4×C61—82no—
139C2×D4×C61—82no—
140C2×D4×C61—82no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—24no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122C2×D412yes—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61#4114yes—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—44no—
172C22⋊C4×C61—44no—
173C42×C61—42no—
174C42×C61C2×D412yes—
175C42×C61C2×D412yes—
176C22⋊Dic61—24no—
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179#11758200—48no—
180C4⋊Dic61—24no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61C2×D412yes—
185C4⋊Dic61C2×D412yes—
186C4⋊Dic61C2×D412yes—
187#11758196—24no—
188#11758196—24no—
189#11758196—24no—
190#11758196—24no—
191#11758196—24no—
192C4×Dic61—24no—
193C4×Dic61—24no—
194C2×#92418—22no—
195C2×#92418—22no—
196C2×#92418—22no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—21no—
201C23×C61—22no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—22no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61AGL(1,ℤ/8)12yes—
217C23×C61C22×D411yescharacteristic
218D4×C61—44no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—42no—
223D4×C61—84no—
224D4×C61—84no—
225D4×C61—88no—
226D4×C61—88no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61#12012yes—
244C2×C4×C61#12012yes—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4112yes—
248C2×C4×C61H(5,𝔽2)12yes—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—48no—
254C2×Dic61—42no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—42no—
260Dic122—22no—
261Dic122—22no—
262Dic122—42no—
263#92418—42no—
264#92418—42no—
265#25583—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—42no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61C2×#12011yescharacteristic
284C22×C61#54612yes—
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61#57511yescharacteristic
287C22×C61#57712yes—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61—44no—
293C4×C61#53211yescharacteristic
294C4×C61#53511yescharacteristic
295C4×C61#55012yes—
296Dic61—42no—
297Dic61—42no—
298Dic61—42no—
299Dic61—42no—
300C2×#552—611no—
301C2×#490—611no—
302C2×#460—611no—
303#2541D6111yescharacteristic
304#2000—612no—
305#2000—612no—
306#1989—611no—
307#1955—611no—
308#1875—612no—
309#1874—612no—
310#1763—611no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61—162no—
314C2×C61#243111yescharacteristic
315C2×C61#300611yescharacteristic
316C2×C61#301011yescharacteristic
317C2×C61#331012yes—
318C2×C61#331712yes—
319C22×SD16—612no—
320#552—1224no—
321#552—1224no—
322C2×#127—21no—
323C2×#127C2×D6111yescharacteristic
324C2×#126—1221no—
325C2×#125—611no—
326C2×#125—1221no—
327C2×#123—1221no—
328C2×C22⋊Q8—611no—
329C2×C22⋊Q8—611no—
330C2×#121C2×D6112yes—
331C2×#121C2×D6112yes—
332C2×C4×Q8—611no—
333C2×C4×D4C2×D6111yescharacteristic
334C2×#117—611no—
335C2×#117—1221no—
336C22×C4⋊C4—611no—
337#490—1224no—
338#460—1224no—
339C2×C4⋊C8—611no—
340C2×#103—612no—
341C2×#102—612no—
342C2×#102—612no—
343C2×#102—612no—
344C2×C22⋊C8—612no—
345C2×C22⋊C8—612no—
346C2×#97—611no—
347#392—21no—
348#392—21no—
349#390—21no—
350C4×C4⋊C4C2×D6111yescharacteristic
351#340—612no—
352#340—612no—
353#340—612no—
354C61#2558311yescharacteristic, complemented, Sylow
355C22×Q8—611no—
356C22×D4—22no—
357C22×D4—22no—
358C22×D4—21no—
359C22×D4—21no—
360C22×D4C22⋊D6111yescharacteristic
361C22×D4C22×D6112yes—
362C23×C4—1221no—
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6111yescharacteristic
365C2×SD16—1224no—
366C2×SD16—2442no—
367C2×SD16—2442no—
368C22×C8—612no—
369C22×C8—612no—
370#127—22no—
371#127—22no—
372#127—44no—
373#126—1228no—
374#125—614no—
375#125—1228no—
376#125—1222no—
377#123—1228no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382#121—24no—
383#121—24no—
384#121—42no—
385#121—42no—
386C4×Q8—614no—
387C4×Q8—1222no—
388C4×D4—22no—
389C4×D4—22no—
390#117—1228no—
391#117—1222no—
392#117—1222no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4C22⋊D6111yescharacteristic
405C2×C4⋊C4C22×D6112yes—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—1221no—
409C2×C22⋊C4—1221no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4C22⋊D6111yescharacteristic
412C2×C22⋊C4C22⋊D6111yescharacteristic
413C2×C42—21no—
414C2×C42—1221no—
415C2×C42—1221no—
416C2×C42C22×D6111yescharacteristic
417C4⋊C8—1222no—
418#103—1224no—
419#102—1224no—
420#102—1224no—
421#102—1224no—
422C22⋊C8—2442no—
423C22⋊C8—2442no—
424#97—1222no—
425#95—21no—
426C24—22no—
427C24C2×C22⋊D6111yescharacteristic
428C24C2×C22⋊D6111yescharacteristic
429C2×Q8—612no—
430C2×Q8—1221no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—41no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4—82no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—1224no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C23×D6111yescharacteristic
474SD16—2448no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—1222no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—612no—
484C4⋊C4—612no—
485C4⋊C4—612no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1224no—
493C4⋊C4—1224no—
494C4⋊C4—1224no—
495C4⋊C4#1175822414yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—1224no—
501C22⋊C4—1224no—
502C22⋊C4—1224no—
503C42—42no—
504C42—1224no—
505C42—1224no—
506C42D4×D6112yes—
507C42D4×D6112yes—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#1451980312yes—
525C23C22×C22⋊D6111yescharacteristic
526Q8—1222no—
527Q8—1222no—
528Q8—2442no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—84no—
535D4—84no—
536D4—88no—
537D4—88no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—2448no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4#1451974512yes—
566C2×C4#1451974512yes—
567C2×C4C2×D4×D6111yescharacteristic
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C2×#1175822412yes—
570C2×C4#1451982212yes—
571C8—2442no—
572C8—2442no—
573C22—24no—
574C22—22no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22C2×#1451974511yescharacteristic
591C22#1684040612yes—
592C22C2×#1451980311yescharacteristic
593C22#1684056011yescharacteristic
594C22#1684057312yes—
595C4—22no—
596C4—22no—
597C4—44no—
598C4—41no—
599C4—41no—
600C4—2442no—
601C4—2442no—
602C4—2442no—
603C4—2442no—
604C4#1684031011yescharacteristic
605C4#1684032511yescharacteristic
606C4#1684043212yes—
607C2—42no—
608C2—81no—
609C2—162no—
610C2#2144943511yescharacteristic
611C2#2145506911yescharacteristic
612C2#2145510311yescharacteristic
613C2#2145784012yes—
614C2#2145794512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2541C2
socleC23×C61C22×D4
hypercenter#2541D61
hypocenterC61#25583
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16840452C2semidirect
2. C2×#16840126C2semidirect
3. C2×#16839959C2semidirect
4. #21446822C2semidirect
5. #21446822C2semidirect
6. #21446776C2semidirect
7. #21446618C2semidirect
8. #21446142C2semidirect
9. #21446137C2semidirect
10. #21445652C2semidirect
56. #14519769D4semidirect
64. C4×Dic122D4semidirect
354. C61#25583semidirect

Maximal subgroups

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

subgroupindex
C2×#168404522
C2×#168401262
C2×#168399592
#214468222
#214467762
#214466182
#214461422
#214461372
#214456522
C61×#25412
#2558361