Finite Group Databasebeta

← #29729418

Subgroups — #29729418

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×#16840124C211yesmaximal, characteristic
2C2×#16840054C211yesmaximal, characteristic, complemented
3#21450799C211yesmaximal, characteristic
4#21447211C211yesmaximal, characteristic, complemented
5#21447204C211yesmaximal, characteristic
6#21447065C211yesmaximal, characteristic, complemented
7#21447060C211yesmaximal, characteristic
8#21446751C211yesmaximal, characteristic
9#21446751C211yesmaximal, characteristic
10#21446742C211yesmaximal, characteristic, complemented
11#21446543C211yesmaximal, characteristic, complemented
12#21446460C211yesmaximal, characteristic, complemented
13#21446439C211yesmaximal, characteristic
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2365C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840124—24no—
21#16840054—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×#14519697C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519688C2211yescharacteristic
40#16839684—21no—
41#16839684—21no—
42#16839668C2211yescharacteristic
43#16839664C2211yescharacteristic
44#16839654—21no—
45#16839654—21no—
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61—21no—
52C61×#391—21no—
53C61×#389C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#387—21no—
56C61×#387—21no—
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777—41no—
70#14519765—22no—
71#14519765—22no—
72#14519765—22no—
73#14519765—41no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×C4×Dic61—21no—
87#14519710—22no—
88#14519710—22no—
89C4⋊#92418—22no—
90C4⋊#92418—41no—
91C4⋊#92418—41no—
92#14519700—22no—
93#14519700—22no—
94#14519700—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—22no—
98#14519700—22no—
99#14519700—41no—
100#14519697—22no—
101#14519697—22no—
102#14519697—22no—
103#14519697—41no—
104#14519688—22no—
105#14519684—21no—
106#14519684—21no—
107C22×Q8×C61—21no—
108C4⋊Q8×C61—22no—
109C4⋊Q8×C61D414yes—
110C4×Q8×C61—44no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C23×D61—21no—
134C2×Q8×C61—24no—
135C2×Q8×C61—42no—
136C2×Q8×C61—42no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122#4112yes—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61C2×D412yes—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61#4112yes—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—42no—
187C42×C61—42no—
188C42×C61—42no—
189C42×C61C2×D412yes—
190C42×C61#4112yes—
191#11758198—42no—
192#11758198—42no—
193#11758198—42no—
194#11758198—42no—
195#11758198—42no—
196#11758198—42no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201#11758196—42no—
202C4×Dic61—42no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C23×C61C22×D411yescharacteristic
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223Q8×C61—48no—
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—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61AGL(1,ℤ/8)12yes—
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C612−1+411yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269D244—41no—
270D244—41no—
271D244—41no—
272D244—41no—
273Dic122—22no—
274Dic122—22no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#92418—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—42no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—81no—
310C4×C61#52911yescharacteristic
311C4×C61#54211yescharacteristic
312C4×C61#54811yescharacteristic
313C4×C61#55011yescharacteristic
314Dic61—42no—
315Dic61—81no—
316Dic61—81no—
317C2×#489—611no—
318C2×#477—611no—
319#2545—611no—
320#2365D6111yescharacteristic, complemented
321#2068—611no—
322#2067—611no—
323#2043—611no—
324#2042—611no—
325#1983—611no—
326#1983—611no—
327#1981—611no—
328#1940—611no—
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⋊Q8C2×D6111yescharacteristic
342C2×#124—611no—
343C2×#124—611no—
344C2×C4×Q8—21no—
345C2×C4×Q8—611no—
346#489—1224no—
347#477—1224no—
348C2×#107—611no—
349C2×#107—611no—
350C2×C4⋊C8—611no—
351C2×C4⋊C8—611no—
352C2×C4⋊C8—611no—
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×#97—611no—
366#396—1221no—
367#396—1221no—
368#391—21no—
369#391—611no—
370#390—611no—
371#389C2×D6111yescharacteristic
372#389C2×D6111yescharacteristic
373#388—1221no—
374#388—1221no—
375#387—21no—
376#387—21no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C4⋊C4C2×D6111yescharacteristic
380C4×C4⋊C4C2×D6111yescharacteristic
381#340—611no—
382#340—611no—
383#336—611no—
384#336—611no—
385C61#2528611yescharacteristic, complemented, Sylow
386C22×Q8—21no—
387C22×Q8—611no—
388C22×Q8—611no—
389C22×D4—611no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394C4⋊Q8—22no—
395C4⋊Q8—1222no—
396C4⋊Q8—2441no—
397C4⋊Q8D24414yes—
398#124—1222no—
399#124—1222no—
400#124—1222no—
401#124—2441no—
402C4×Q8—44no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C4⋊C4C22⋊D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
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—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429C2×C42C22×D6111yescharacteristic
430C2×C42C22×D6111yescharacteristic
431#107—1222no—
432#107—1222no—
433C4⋊C8—1222no—
434C4⋊C8—2441no—
435C4⋊C8—2441no—
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—1222no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1221no—
455#95—1221no—
456#95C22⋊D6111yescharacteristic
457#95C22⋊D6111yescharacteristic
458C24—1221no—
459C2×Q8—24no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—612no—
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—1221no—
473C2×Q8—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—1221no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4C2×D24412yes—
522C4⋊C4C2×D24412yes—
523C4⋊C4#1175822612yes—
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—42no—
535C42—42no—
536C42—42no—
537C42—2442no—
538C42C2×D24412yes—
539C42#1175822212yes—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23C22×C22⋊D6111yescharacteristic
548Q8—48no—
549Q8—1222no—
550Q8—1222no—
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—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—1222no—
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#1451975112yes—
598C2×C4#1451975112yes—
599C2×C4#1451980312yes—
600C2×C4C22×D24411yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822211yescharacteristic
603C2×C4C2×#1175822611yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982511yescharacteristic
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C8—2441no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22C2×#1451975111yescharacteristic
618C22#1684040011yescharacteristic
619C22#1684042311yescharacteristic
620C22C2×#1451980311yescharacteristic
621C22#1684056011yescharacteristic
622C22#1684057711yescharacteristic
623C22#1684057711yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—4881no—
635C4—4881no—
636C4—2442no—
637C4#1684029911yescharacteristic
638C4#1684038411yescharacteristic
639C4#1684042411yescharacteristic
640C4#1684043411yescharacteristic
641C2—9761no—
642C2#2144937911yescharacteristic
643C2#2145528311yescharacteristic
644C2#2145534911yescharacteristic
645C2#2145780411yescharacteristic
646C2#2145787911yescharacteristic
647C2#2145853711yescharacteristic
648C2#2145866111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2365C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2365D61
hypocenterC61#25286
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. C2×#16840054C2semidirect
4. #21447211C2semidirect
6. #21447065C2semidirect
10. #21446742C2semidirect
11. #21446543C2semidirect
12. #21446460C2semidirect
14. #21445652C2semidirect
15. C61×#2365C2semidirect
320. #2365D61semidirect
385. C61#25286semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168400542
#214507992
#214472112
#214472042
#214470652
#214470602
#214467512
#214467422
#214465432
#214464602
#214464392
#214456522
C61×#23652
#2528661