Finite Group Databasebeta

← #29740170

Subgroups — #29740170

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839967C211yesmaximal, characteristic
2C2×C4×#11758200C211yesmaximal, characteristic
3#21449595C211yesmaximal, characteristic, complemented
4#21447370C211yesmaximal, characteristic, complemented
5#21447321C211yesmaximal, characteristic, complemented
6#21447238C211yesmaximal, characteristic, complemented
7#21447065C211yesmaximal, characteristic, complemented
8#21446494C211yesmaximal, characteristic, complemented
9#21446378C211yesmaximal, characteristic, complemented
10#21446153C211yesmaximal, characteristic
11#21446153C211yesmaximal, characteristic
12#21446153C211yesmaximal, characteristic
13#21446072C211yesmaximal, characteristic, complemented
14#21446066C211yesmaximal, characteristic
15C61×#2349C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839967—24no—
21C4×#11758200—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839704—21no—
35#16839695—21no—
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839649—21no—
41#16839648—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×D4×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C61×#385—21no—
56C61×#385—21no—
57C61×#382C2211yescharacteristic
58C61×#380—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C2×#95×C61—21no—
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×#11758200—22no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—22no—
67C2×#11758200—22no—
68C2×#11758200D414yes—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73#14519777—22no—
74#14519777—41no—
75C4×Dic122—22no—
76C4×Dic122—41no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C4×Dic61—21no—
84#14519710—41no—
85#14519708—22no—
86C4⋊#92418—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700—22no—
90#14519700—22no—
91#14519700—22no—
92#14519700—41no—
93#14519698—22no—
94#14519698—22no—
95C4×#92418—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61—21no—
105C23×C4×C61—21no—
106C23×C4×C61—21no—
107C4×D4×C61—22no—
108C4×D4×C61—22no—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—41no—
112C4×D4×C61—41no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C22⋊C4×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×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61C2×D412yes—
137C2×D4×C61#4112yes—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61—81no—
148C22×C4×C61—81no—
149C22×C4×C61—81no—
150C22×C4×C61—81no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C2×D412yes—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—81no—
182C4⋊C4×C61—81no—
183C4⋊C4×C61#4112yes—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—42no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193#11758200—28no—
194#11758200—44no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—42no—
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—
215C2×#92418C2×D412yes—
216C23×C61—22no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—41no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61C22×D411yescharacteristic
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—22no—
227D4×C61#12314yes—
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—22no—
235C2×C4×C61—21no—
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—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
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—41no—
252C2×C4×C61—41no—
253C2×C4×C61—84no—
254C2×C4×C61—84no—
255C2×C4×C61—84no—
256C2×C4×C61#13512yes—
257C2×C4×C61#13712yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280Dic122—41no—
281Dic122—41no—
282Dic122—41no—
283Dic122—41no—
284#92418—22no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#26170—611nomaximal, Sylow
289C22×C61—24no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61#52511yescharacteristic
297C22×C61D4211yescharacteristic
298C22×C61#54811yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57711yescharacteristic
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C2×#12311yescharacteristic
313C4×C61#53911yescharacteristic
314C4×C61#54711yescharacteristic
315C4×C61#55311yescharacteristic
316Dic61—42no—
317Dic61—81no—
318Dic61—81no—
319Dic61—81no—
320C2×#461—611no—
321C2×C4×SD16—611no—
322#2446—611no—
323#2349D6111yescharacteristic
324#2097—611no—
325#2088—611no—
326#2074—611no—
327#2043—611no—
328#1933—611no—
329#1916—611no—
330#1877—611no—
331#1877—611no—
332#1877—611no—
333#1863—611no—
334#1862—611no—
335C2×C61—44no—
336C2×C61#244311yescharacteristic
337C2×C61#310711yescharacteristic
338C2×C61#312311yescharacteristic
339C2×C61#326611yescharacteristic
340C2×C61#328211yescharacteristic
341C2×C61#336011yescharacteristic
342C2×C61#337211yescharacteristic
343C22×SD16—611no—
344C22×SD16—611no—
345C2×C4⋊Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×D4C2×D6111yescharacteristic
348C2×C4×D4C2×D6111yescharacteristic
349C2×C4×D4C2×D6111yescharacteristic
350#461—1224no—
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#106—611no—
354C2×C4⋊C8—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×C4×C8—611no—
364#400—1221no—
365#398—1221no—
366#391—611no—
367#391—611no—
368#391—611no—
369#389—611no—
370#389C2×D6111yescharacteristic
371#389C2×D6111yescharacteristic
372#387—1221no—
373#387—1221no—
374#385—21no—
375#385—21no—
376#382C2×D6111yescharacteristic
377#380—21no—
378C4×C4⋊C4C2×D6111yescharacteristic
379C2×#95—21no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#336—611no—
385#336—611no—
386#336—611no—
387#336—611no—
388C61#2617011yescharacteristic, complemented, Sylow
389C22×Q8—611no—
390C22×Q8—611no—
391C22×D4C22×D6111yescharacteristic
392C23×C4—21no—
393C23×C4—21no—
394C23×C4—21no—
395C2×SD16—614no—
396C2×SD16—1222no—
397C2×SD16—1222no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C2×SD16—1222no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8—611no—
405C4⋊Q8—1222no—
406C4⋊Q8—2441no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—22no—
413C4×D4—41no—
414C4×D4—41no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22⋊D6111yescharacteristic
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C42—1221no—
432C2×C42C22×D6111yescharacteristic
433C2×C42C22×D6111yescharacteristic
434C2×C42C22×D6111yescharacteristic
435#107—2441no—
436#106—1222no—
437C4⋊C8—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—
446C4×C8—1222no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95C22⋊D6111yescharacteristic
459#95C22⋊D6111yescharacteristic
460C24—21no—
461C2×Q8—612no—
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—1221no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4#1175822212yes—
477C2×D4D4×D6112yes—
478C2×D4#1175822612yes—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502SD16—1228no—
503SD16—2444no—
504C2×C8—612no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4#1175822212yes—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—2442no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C22×C22⋊D6111yescharacteristic
552Q8—1222no—
553Q8—1222no—
554Q8—2441no—
555Q8—2441no—
556Q8—2441no—
557Q8—2441no—
558Q8—2441no—
559Q8—2441no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4#1451976214yes—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—1222no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4#1451979712yes—
605C2×C4#1451980712yes—
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×#1175822211yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822611yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982511yescharacteristic
612C8—1222no—
613C8—2441no—
614C8—2441no—
615C8—2441no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22#1684027311yescharacteristic
624C22D4×C22⋊D6111yescharacteristic
625C22#1684042311yescharacteristic
626C22C2×#1451979711yescharacteristic
627C22C2×#1451980711yescharacteristic
628C22#1684057311yescharacteristic
629C22#1684057311yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—4881no—
640C4—4881no—
641C4—4881no—
642C4—2442no—
643C4C2×#1451976211yescharacteristic
644C4#1684035911yescharacteristic
645C4#1684041311yescharacteristic
646C4#1684046011yescharacteristic
647C2—44no—
648C2#2144955411yescharacteristic
649C2#2145589011yescharacteristic
650C2#2145607111yescharacteristic
651C2#2145732511yescharacteristic
652C2#2145752211yescharacteristic
653C2#2145846811yescharacteristic
654C2#2145860011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2349C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2349D61
hypocenterC61#26170
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #21449595C2semidirect
4. #21447370C2semidirect
5. #21447321C2semidirect
6. #21447238C2semidirect
7. #21447065C2semidirect
8. #21446494C2semidirect
9. #21446378C2semidirect
13. #21446072C2semidirect
388. C61#26170semidirect

Maximal subgroups

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

subgroupindex
C2×#168399672
C2×C4×#117582002
#214495952
#214473702
#214473212
#214472382
#214470652
#214464942
#214463782
#214461532
#214460722
#214460662
C61×#23492
#2617061