Finite Group Databasebeta

← #29729422

Subgroups — #29729422

Order 15616.

822 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840122C211yesmaximal, characteristic
2C2×#16840058C211yesmaximal, characteristic, complemented
3#21450799C211yesmaximal, characteristic
4#21448877C211yesmaximal, characteristic, complemented
5#21447213C211yesmaximal, characteristic, complemented
6#21447206C211yesmaximal, characteristic
7#21447066C211yesmaximal, characteristic, complemented
8#21447061C211yesmaximal, characteristic
9#21446749C211yesmaximal, characteristic
10#21446749C211yesmaximal, characteristic
11#21446741C211yesmaximal, characteristic, complemented
12#21446461C211yesmaximal, characteristic, complemented
13#21446440C211yesmaximal, characteristic
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1940C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519742—21no—
22#16840122—24no—
23#16840058—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38#16839684—21no—
39#16839684—21no—
40#16839668C2211yescharacteristic
41#16839667—21no—
42#16839664C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839654—21no—
46#16839654—21no—
47#16839647—21no—
48#16839647—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50C4×C4⋊Dic61C2211yescharacteristic
51C4×C4⋊Dic61C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65#14519777—22no—
66#14519777—22no—
67#14519777—41no—
68#14519777D414yes—
69#14519765—22no—
70#14519765—22no—
71#14519765—22no—
72#14519765—41no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75#14519742—44no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97#14519711—22no—
98#14519711—22no—
99#14519702—22no—
100#14519702—22no—
101#14519702—22no—
102#14519702—22no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—41no—
107#14519699—22no—
108#14519699—22no—
109#14519699—22no—
110#14519699—41no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C4⋊C8×C61—22no—
128C4⋊C8×C61—41no—
129C4⋊C8×C61—41no—
130#97×C61—22no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C23×D61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—24no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122—42no—
168C2×Dic122#4112yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C2×D412yes—
188C42×C61#4112yes—
189#11758198—42no—
190#11758198—42no—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195C4⋊Dic61—24no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61—42no—
205C4⋊Dic61C2×D412yes—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61#4112yes—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C23×C61C22×D411yescharacteristic
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61#12112yes—
239C2×C4×C61#12112yes—
240C2×C4×C61AGL(1,ℤ/8)12yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268D244—41no—
269D244—41no—
270D244—41no—
271D244—41no—
272Dic122—22no—
273Dic122—22no—
274Dic122—48no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281C8×C61—41no—
282C8×C61—41no—
283C8×C61—41no—
284C8×C61—41no—
285#25286—611nomaximal, Sylow
286C22×C61C2×#12111yescharacteristic
287C22×C61#54611yescharacteristic
288C22×C61#54811yescharacteristic
289C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C61#57511yescharacteristic
291C22×C61#57811yescharacteristic
292C22×C61#57811yescharacteristic
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61#52911yescharacteristic
307C4×C61#54211yescharacteristic
308C4×C61#54811yescharacteristic
309C4×C61#55011yescharacteristic
310Dic61—42no—
311Dic61—42no—
312Dic61—42no—
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#489—611no—
318C2×#477—611no—
319#2545—611no—
320#2365—611no—
321#2068—611no—
322#2067—611no—
323#2043—611no—
324#2042—611no—
325#1983—611no—
326#1983—611no—
327#1981—611no—
328#1940D6111yescharacteristic, complemented
329#1928—611no—
330#1925—611no—
331#1763—611no—
332C2×C61#242611yescharacteristic
333C2×C61#303611yescharacteristic
334C2×C61#304411yescharacteristic
335C2×C61#330811yescharacteristic
336C2×C61#331311yescharacteristic
337C2×C61#336711yescharacteristic
338C2×C61#337711yescharacteristic
339D61—161no—
340C2×C4⋊Q8—611no—
341C2×C4⋊Q8—611no—
342C2×#124—611no—
343C2×#124—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—1221no—
346#489—1224no—
347#477—1224no—
348C2×#107—611no—
349C2×#107—611no—
350C2×C4⋊C8C2×D6111yescharacteristic
351C2×C4⋊C8C2×D6111yescharacteristic
352C2×C4⋊C8C2×D6111yescharacteristic
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×#97C2×D6111yescharacteristic
366#396—1221no—
367#396—1221no—
368#391—611no—
369#391—1221no—
370#390—611no—
371#389—611no—
372#389—611no—
373#388—1221no—
374#388—1221no—
375#387—1221no—
376#387—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4C2×D6111yescharacteristic
381#340—611no—
382#340—611no—
383#336C2×D6111yescharacteristic
384#336C2×D6111yescharacteristic
385C61#2528611yescharacteristic, complemented, Sylow
386C22×Q8—611no—
387C22×Q8—611no—
388C22×Q8—1221no—
389C22×D4—611no—
390C22×C8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—614no—
395C4⋊Q8—1222no—
396C4⋊Q8—1222no—
397C4⋊Q8—2441no—
398#124—1222no—
399#124—1222no—
400#124—1222no—
401#124—2441no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—2444no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4D24411yescharacteristic
417C2×C4⋊C4D24411yescharacteristic
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42—1221no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#107—1222no—
433C4⋊C8—22no—
434C4⋊C8—41no—
435C4⋊C8—41no—
436#103—1222no—
437#103—1222no—
438#103—1222no—
439#103—1222no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#102—1222no—
445#102—1222no—
446#102—1222no—
447#102—2441no—
448#97—22no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95D24411yescharacteristic
457#95D24411yescharacteristic
458C24—1221no—
459C2×Q8—612no—
460C2×Q8—1221no—
461C2×Q8—1224no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—2442no—
473C2×Q8—2442no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×D24411yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×D24411yescharacteristic
492C22×C4C2×D24411yescharacteristic
493C22×C4C23×D6111yescharacteristic
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—612no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1224no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—2442no—
535C42—2442no—
536C42—2442no—
537C42—2442no—
538C42D4×D6112yes—
539C42#1175822612yes—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23C22×D24411yescharacteristic
548Q8—1222no—
549Q8—1222no—
550Q8—2448no—
551Q8—2441no—
552Q8—2441no—
553Q8—2441no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560D4—2441no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4#1451975212yes—
598C2×C4#1451975212yes—
599C2×C4#1451980512yes—
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4#41×D6111yescharacteristic
603C2×C4C2×#1175822611yescharacteristic
604C2×C4#1451982211yescharacteristic
605C2×C4#1451982511yescharacteristic
606C8—41no—
607C8—41no—
608C8—41no—
609C8—41no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22C2×#1451975211yescharacteristic
618C22#1684040711yescharacteristic
619C22#1684042411yescharacteristic
620C22C2×#1451980511yescharacteristic
621C22#1684056211yescharacteristic
622C22#1684057911yescharacteristic
623C22#1684057911yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—4881no—
631C4—4881no—
632C4—4881no—
633C4—2442no—
634C4—2442no—
635C4—2442no—
636C4—2442no—
637C4#1684029711yescharacteristic
638C4#1684038611yescharacteristic
639C4#1684042211yescharacteristic
640C4#1684043011yescharacteristic
641C2—9761no—
642C2#2144938311yescharacteristic
643C2#2145528511yescharacteristic
644C2#2145535011yescharacteristic
645C2#2145781211yescharacteristic
646C2#2145788711yescharacteristic
647C2#2145853911yescharacteristic
648C2#2145866311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1940C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1940D61
hypocenterC61#25286
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
2. C2×#16840058C2semidirect
4. #21448877C2semidirect
5. #21447213C2semidirect
7. #21447066C2semidirect
11. #21446741C2semidirect
12. #21446461C2semidirect
14. #21445650C2semidirect
15. C61×#1940C2semidirect
328. #1940D61semidirect
385. C61#25286semidirect

Maximal subgroups

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

subgroupindex
C2×#168401222
C2×#168400582
#214507992
#214488772
#214472132
#214472062
#214470662
#214470612
#214467492
#214467412
#214464612
#214464402
#214456502
C61×#19402
#2528661