Finite Group Databasebeta

← #30614726

Subgroups — #30614726

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840787C211yesmaximal, characteristic
2C2×#16840769C211yesmaximal, characteristic
3#21461619C211yesmaximal, characteristic, complemented
4#21461618C211yesmaximal, characteristic, complemented
5#21461615C211yesmaximal, characteristic, complemented
6#21461614C211yesmaximal, characteristic, complemented
7#21461607C211yesmaximal, characteristic, complemented
8#21461607C211yesmaximal, characteristic, complemented
9#21461605C211yesmaximal, characteristic, complemented
10#21461604C211yesmaximal, characteristic, complemented
11#21461560C211yesmaximal, characteristic
12#21461560C211yesmaximal, characteristic
13#21461557C211yesmaximal, characteristic
14#21461557C211yesmaximal, characteristic
15#21448311C211yesmaximal, characteristic
16C22×#11758213C2211yescharacteristic
17C22×D61⋊C8C2211yescharacteristic
18#16840787C418yescomplemented
19#16840769C418yescomplemented
20C2×#14519844C2211yescharacteristic
21C2×#14519844C2211yescharacteristic
22C2×#14519844C2211yescharacteristic
23C2×C4⋊#92420C2211yescharacteristic
24C2×C4⋊#92420C2211yescharacteristic
25C2×C4⋊#92420C2211yescharacteristic
26C2×#14519834C2211yescharacteristic
27C2×#14519834C2211yescharacteristic
28C2×#14519832C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×#14519832C2211yescharacteristic
31C2×#14519832C2211yescharacteristic
32C2×#14519832C2211yescharacteristic
33C2×#14519832C2211yescharacteristic
34C2×#14519832C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36C2×C4×#92420C2211yescharacteristic
37C2×C4×#92420C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×#14519735C2211yescharacteristic
47#16839654C411yescharacteristic
48#16839653C411yescharacteristic
49#16839652C411yescharacteristic
50#16839651C411yescharacteristic
51#16839648C411yescharacteristic
52#16839647C411yescharacteristic
53#16839645C411yescharacteristic
54C4⋊C4×Dic61C2211yescharacteristic
55#16839613C2211yescharacteristic
56#16839609C2211yescharacteristic
57#16839609C2211yescharacteristic
58#16839607C2211yescharacteristic
59C42×Dic61C2211yescharacteristic
60C61×#387C411yescharacteristic
61C2×#11758213—41no—
62C2×#11758213—41no—
63C2×#11758213C2×C414yes—
64C2×D61⋊C8—41no—
65C2×D61⋊C8—41no—
66C2×D61⋊C8C2×C414yes—
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70C22×#92420C2311yescharacteristic
71C22×#92420C2311yescharacteristic
72C22×#92420C2311yescharacteristic
73C22×#92420C2311yescharacteristic
74C22×#92420C2311yescharacteristic
75#14519844C2×C414yes—
76#14519844C2×C414yes—
77#14519844C2×C414yes—
78C4⋊#92420C2×C414yes—
79C4⋊#92420C2×C414yes—
80C4⋊#92420C2×C414yes—
81#14519834C2×C414yes—
82#14519834C2×C414yes—
83#14519832—41no—
84#14519832—41no—
85#14519832—41no—
86#14519832—41no—
87#14519832—41no—
88#14519832—41no—
89#14519832C2×C414yes—
90C4×#92420—41no—
91C4×#92420—41no—
92C4×#92420C2×C414yes—
93C22×C4×D61C2311yescharacteristic
94#14519735C2×C414yes—
95#14519735C2×C414yes—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758198C2×C411yescharacteristic
101C2×#11758198C2×C411yescharacteristic
102C2×#11758198C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115#14519684C2×C411yescharacteristic
116#14519684C2×C411yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119#14519684C2×C411yescharacteristic
120#14519684C2×C411yescharacteristic
121#14519684C2×C411yescharacteristic
122#14519684C2×C411yescharacteristic
123C2×C4⋊C4×C61C2×C411yescharacteristic
124C2×C4⋊C4×C61C2×C411yescharacteristic
125C2×C42×C61C2×C411yescharacteristic
126#95×C61C2×C411yescharacteristic
127#95×C61C2×C411yescharacteristic
128#95×C61C2×C411yescharacteristic
129#95×C61C2×C411yescharacteristic
130C23×D61C22×C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C22×C411yescharacteristic
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C2411yescharacteristic
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61C22×C412yes—
150C2×C4×D61C22×C412yes—
151C2×#92420—21no—
152C2×#92420—21no—
153C2×#92420—21no—
154C2×#92420—21no—
155C2×#92420—21no—
156C2×#92420—21no—
157C2×#92420—21no—
158C2×#92420—21no—
159C2×#92420—21no—
160C2×#92420—21no—
161C2×#92420—21no—
162C2×#92420—21no—
163C2×#92420—21no—
164C2×#92420—21no—
165C2×#92420—21no—
166C2×#92420—21no—
167C2×#92420C22×C412yes—
168C2×#92420C22×C412yes—
169C2×#92420C22×C412yes—
170C2×#92420C22×C412yes—
171C2×#92420C22×C412yes—
172C2×#92420C22×C412yes—
173C2×#92420C22×C412yes—
174C2×#92420C22×C412yes—
175#11758213—44no—
176D61⋊C8—44no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61C4214yes—
179C42×C61—41no—
180#11758198—41no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198C4214yes—
185#11758198C4214yes—
186C4⋊Dic61—41no—
187C4⋊Dic61C4214yes—
188#11758196—41no—
189#11758196—41no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—41no—
199C4×Dic61—41no—
200C4×Dic61—41no—
201C4×Dic61C22×C412yes—
202C4×Dic61C22×C412yes—
203C4×Dic61C22×C412yes—
204C4×Dic61C22×C412yes—
205C23×C61C23×C411yescharacteristic
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61C2×C4212yes—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
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×C61C2×C4212yes—
232C2×C4×C61C2×C4212yes—
233C2×C4×C61C2×C4212yes—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61C2×C4212yes—
251C2×Dic61C2×C4212yes—
252C2×Dic61C23×C411yescharacteristic
253C2×Dic61C23×C411yescharacteristic
254C2×Dic61H(5,𝔽2)11yescharacteristic
255C2×Dic61H(5,𝔽2)11yescharacteristic
256C2×Dic612−1+411yescharacteristic
257C2×Dic612−1+411yescharacteristic
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262#92420—22no—
263#92420—22no—
264#92420—22no—
265#92420—22no—
266#92420—22no—
267#92420—22no—
268#92420—22no—
269#92420—22no—
270#20554—611nomaximal, Sylow
271C22×C61C22×C4211yescharacteristic
272C22×C61#52011yescharacteristic
273C22×C61#52011yescharacteristic
274C22×C61#56811yescharacteristic
275C22×C61#56811yescharacteristic
276C22×C61#56811yescharacteristic
277C22×C61#56811yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292Dic61—22no—
293Dic61—22no—
294Dic61—41no—
295Dic61—41no—
296Dic61—41no—
297Dic61—41no—
298Dic61#51811yescharacteristic
299Dic61#51911yescharacteristic
300Dic61#52011yescharacteristic
301Dic61#52011yescharacteristic
302C2×#433—611no—
303C2×#424—611no—
304#2298—611no—
305#1771—611no—
306#1771—611no—
307#1770—611no—
308#1770—611no—
309#1768—611no—
310#1768—611no—
311#1767—611no—
312#1767—611no—
313#1753—611no—
314#1753—611no—
315#1752—611no—
316#1752—611no—
317C2×C61#227411yescharacteristic
318C2×C61#287511yescharacteristic
319C2×C61#287511yescharacteristic
320C2×C61#299511yescharacteristic
321C2×C61#299511yescharacteristic
322C2×C61#299511yescharacteristic
323C2×C61#299511yescharacteristic
324D61—42no—
325C22×M16—611no—
326C23×C8—611no—
327C2×#117—611no—
328#433—618no—
329#424—618no—
330C2×C4⋊C8—611no—
331C2×C4⋊C8—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8—611no—
334C2×C4⋊C8—611no—
335C2×C4⋊C8—611no—
336C2×C22⋊C8—611no—
337C2×C22⋊C8—611no—
338C2×#97—611no—
339C2×#97—611no—
340C2×#97—611no—
341C2×#97—611no—
342C2×#97—611no—
343C2×#97—611no—
344C2×#97—611no—
345C2×C4×C8—611no—
346C2×C4×C8—611no—
347C2×C4×C8—611no—
348#388—611no—
349#388—611no—
350#388—611no—
351#388—611no—
352#387—611no—
353#387—611no—
354#387—611no—
355#387#692311yescharacteristic
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C22⋊C4—611no—
359C4×C22⋊C4—611no—
360#376—611no—
361C43—611no—
362#336—611no—
363#336—611no—
364#336—611no—
365#336—611no—
366#336—611no—
367#336—611no—
368#336—611no—
369#336—611no—
370C61#2055411yescharacteristic, complemented, Sylow
371C23×C4—611no—
372C2×M16—614no—
373C2×M16—2441no—
374C2×M16—2441no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—614no—
384C22×C8—2441no—
385C22×C8—2441no—
386#117—614no—
387#117—614no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4C2×#692311yescharacteristic
393C2×C4⋊C4C2×#692311yescharacteristic
394C2×C22⋊C4—611no—
395C2×C22⋊C4—611no—
396C2×C22⋊C4—611no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42C2×#692311yescharacteristic
410C4⋊C8—614no—
411C4⋊C8—614no—
412C4⋊C8—614no—
413C4⋊C8—614no—
414C4⋊C8—614no—
415C4⋊C8—614no—
416C22⋊C8—614no—
417C22⋊C8—614no—
418#97—614no—
419#97—2441no—
420#97—2441no—
421#97—2441no—
422#97—2441no—
423#97—2441no—
424#97—2441no—
425C4×C8—614no—
426C4×C8—2441no—
427C4×C8—2441no—
428#95—611no—
429#95—611no—
430#95—611no—
431#95—611no—
432#95—611no—
433#95—611no—
434#95—611no—
435#95—611no—
436#95C2×#692311yescharacteristic
437#95C2×#692311yescharacteristic
438#95C2×#692311yescharacteristic
439#95C2×#692311yescharacteristic
440C24—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—612no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4C22×#692311yescharacteristic
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460C22×C4C22×#692311yescharacteristic
461M16—2444no—
462C2×C8—612no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—612no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—612no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—2444no—
487C4⋊C4—41no—
488C4⋊C4—614no—
489C4⋊C4—2441no—
490C4⋊C4—2441no—
491C4⋊C4—2441no—
492C4⋊C4C4×#692314yes—
493C22⋊C4—614no—
494C22⋊C4—614no—
495C22⋊C4—2441no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C42—41no—
500C42—612no—
501C42—612no—
502C42—612no—
503C42—612no—
504C42—1221no—
505C42—1221no—
506C42—1221no—
507C42—1221no—
508C42—1221no—
509C42—1221no—
510C42—1221no—
511C42—1221no—
512C42—2441no—
513C42—2441no—
514C42—2441no—
515C23—612no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23—1221no—
520C23—1221no—
521C23—1221no—
522C23C23×#692311yescharacteristic
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4C2×C4×#692312yes—
570C2×C4C2×C4×#692312yes—
571C2×C4C2×C4×#692312yes—
572C8—1222no—
573C8—1222no—
574C8—1222no—
575C8—1222no—
576C8—1222no—
577C8—1222no—
578C8—1222no—
579C8—1222no—
580C22—1222no—
581C22—1222no—
582C22—1222no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22C22×C4×#692311yescharacteristic
588C22#1684083111yescharacteristic
589C22#1684083111yescharacteristic
590C22#1684084011yescharacteristic
591C22#1684084111yescharacteristic
592C22#1684084111yescharacteristic
593C22#1684084211yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—611no—
602C4—611no—
603C4—611no—
604C4—611no—
605C4—1222no—
606C4—1222no—
607C4—2441no—
608C4—2441no—
609C4—2441no—
610C4—2441no—
611C2—2442no—
612C2#2146225611yescharacteristic
613C2#2146239011yescharacteristic
614C2#2146239111yescharacteristic
615C2#2146277311yescharacteristic
616C2#2146277411yescharacteristic
617C2#2146277611yescharacteristic
618C2#2146277711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C22×C42
FittingC61×#387C4
socleC23×C61C23×C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenter#387#6923
hypocenterC61#20554
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #21461619C2semidirect
4. #21461618C2semidirect
5. #21461615C2semidirect
6. #21461614C2semidirect
7. #21461607C2semidirect
8. #21461607C2semidirect
9. #21461605C2semidirect
10. #21461604C2semidirect
18. #16840787C4semidirect
19. #16840769C4semidirect
370. C61#20554semidirect

Maximal subgroups

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

subgroupindex
C2×#168407872
C2×#168407692
#214616192
#214616182
#214616152
#214616142
#214616072
#214616052
#214616042
#214615602
#214615572
#214483112
#2055461