Finite Group Databasebeta

← #30620732

Subgroups — #30620732

Order 15616.

800 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840787C211yesmaximal, characteristic
2C2×#16840787C211yesmaximal, characteristic
3#21461918C211yesmaximal, characteristic, complemented
4#21461918C211yesmaximal, characteristic, complemented
5#21461878C211yesmaximal, characteristic, complemented
6#21461877C211yesmaximal, characteristic, complemented
7#21461874C211yesmaximal, characteristic, complemented
8#21461873C211yesmaximal, characteristic, complemented
9#21461713C211yesmaximal, characteristic
10#21461713C211yesmaximal, characteristic
11#21461676C211yesmaximal, characteristic
12#21461614C211yesmaximal, characteristic, complemented
13#21461611C211yesmaximal, characteristic, complemented
14C4×#14519834C211yesmaximal, characteristic
15#21448225C211yesmaximal, characteristic
16#16840787—24no—
17#16840787—24no—
18C2×#14519844C2211yescharacteristic
19C2×#14519844C2211yescharacteristic
20C2×#14519844C2211yescharacteristic
21C2×#14519844C2211yescharacteristic
22C2×C4⋊#92420C2211yescharacteristic
23C2×C4⋊#92420C2211yescharacteristic
24C2×#14519834C2211yescharacteristic
25C2×#14519834C2211yescharacteristic
26C2×#14519834C2211yescharacteristic
27C2×#14519834C2211yescharacteristic
28C2×#14519832C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×C4×#92420C2211yescharacteristic
31C2×C4×#92420C2211yescharacteristic
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34#16840628C2211yescharacteristic
35#16840628C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×#14519735C2211yescharacteristic
47C2×#14519735C2211yescharacteristic
48C2×C42×D61C2211yescharacteristic
49#16839631C411yescharacteristic
50#16839631C411yescharacteristic
51#16839628C411yescharacteristic
52#16839621C411yescharacteristic
53#16839621C411yescharacteristic
54#16839621C411yescharacteristic
55#16839621C411yescharacteristic
56#16839613C2211yescharacteristic
57#16839613C2211yescharacteristic
58#16839613C2211yescharacteristic
59#16839613C2211yescharacteristic
60C61×#382C411yescharacteristic
61C22×#92420C2311yescharacteristic
62C22×#92420C2311yescharacteristic
63C22×#92420C2311yescharacteristic
64C22×#92420C2311yescharacteristic
65C22×#92420C2311yescharacteristic
66C22×#92420C2311yescharacteristic
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69#14519844—22no—
70#14519844—22no—
71#14519844—22no—
72#14519844—22no—
73C4⋊#92420—22no—
74C4⋊#92420—22no—
75#14519834—22no—
76#14519834—22no—
77#14519834—22no—
78#14519834D414yes—
79#14519832—22no—
80#14519832—22no—
81C4×#92420—22no—
82C4×#92420—22no—
83C4×#92420—22no—
84C4×#92420—22no—
85C22×C4×D61C2×C411yescharacteristic
86C22×C4×D61C2×C411yescharacteristic
87C22×C4×D61C2311yescharacteristic
88#14519735—22no—
89#14519735—22no—
90#14519735—22no—
91#14519735—22no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×#11758196C2×C411yescharacteristic
95C2×#11758196C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2×C411yescharacteristic
101C2×#11758196C2×C411yescharacteristic
102C2×#11758198C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61—22no—
114C42×D61—22no—
115#14519684C2×C411yescharacteristic
116#14519684C2×C411yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119C2×C4⋊C4×C61C2×C411yescharacteristic
120C2×C4⋊C4×C61C2×C411yescharacteristic
121C2×C42×C61C2×C411yescharacteristic
122#95×C61C2×C411yescharacteristic
123#95×C61C2×C411yescharacteristic
124#95×C61C2×C411yescharacteristic
125#95×C61C2×C411yescharacteristic
126C23×D61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C2411yescharacteristic
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—41no—
151C2×C4×D61C2×D412yes—
152C2×C4×D61#4112yes—
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×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420C2×D412yes—
176C2×#92420C2×D412yes—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C42×C61—22no—
180#11758198—22no—
181#11758198—22no—
182#11758198—22no—
183#11758198—22no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—22no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61#4112yes—
206C4×Dic61#4112yes—
207C23×C61C23×C411yescharacteristic
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61C4×D412yes—
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×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61#11712yes—
235C2×C4×C61C4×D412yes—
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×Dic61#11712yes—
251C2×Dic61#13112yes—
252C2×Dic61#13112yes—
253C2×Dic61#13112yes—
254C2×Dic61C22×D411yescharacteristic
255C2×Dic61C2×#4111yescharacteristic
256C2×Dic61C2×#4111yescharacteristic
257C2×Dic61C2×#4111yescharacteristic
258C2×Dic612−1+411yescharacteristic
259C2×Dic612−1+411yescharacteristic
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—44no—
265C4×D61—44no—
266#92420—22no—
267#92420—22no—
268#92420—41no—
269#92420—41no—
270#92420—41no—
271#92420—41no—
272#92420—41no—
273#92420—41no—
274#22600—611nomaximal, Sylow
275C22×C61C2×#11711yescharacteristic
276C22×C61C2×C4×D411yescharacteristic
277C22×C61#51911yescharacteristic
278C22×C61C2×#13111yescharacteristic
279C22×C61C2×#13111yescharacteristic
280C22×C61C2×#13111yescharacteristic
281C22×C61#56811yescharacteristic
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61—41no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—41no—
295C4×C61—41no—
296Dic61—22no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300Dic61—41no—
301Dic61#43314yescomplemented
302Dic61#54711yescharacteristic
303Dic61#54711yescharacteristic
304Dic61#55211yescharacteristic
305Dic61#55211yescharacteristic
306C2×#433—611no—
307C2×#433—611no—
308#2285—611no—
309#1939—611no—
310#1939—611no—
311#1917—611no—
312#1917—611no—
313#1916—611no—
314#1916—611no—
315#1818—611no—
316#1818—611no—
317#1797—611no—
318#1770—611no—
319C8×C4⋊C4—611no—
320C4×C22⋊C8—611no—
321C2×C61#234411yescharacteristic
322C2×C61C2×#43311yescharacteristic
323C2×C61#292411yescharacteristic
324C2×C61#293111yescharacteristic
325C2×C61#293511yescharacteristic
326C2×C61#298511yescharacteristic
327C2×C61#298611yescharacteristic
328D61—42no—
329C2×#117—611no—
330C2×#117—611no—
331C22×C42—611no—
332#433—1224no—
333#433—1224no—
334C2×C4⋊C8—611no—
335C2×C4⋊C8—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8—611no—
339C2×C4⋊C8—611no—
340C2×C22⋊C8—611no—
341C2×C22⋊C8—611no—
342C2×C22⋊C8—611no—
343C2×C22⋊C8—611no—
344C2×#97—611no—
345C2×#97—611no—
346C2×C4×C8—611no—
347C2×C4×C8—611no—
348C2×C4×C8—611no—
349C2×C4×C8—611no—
350#384—611no—
351#384—611no—
352#382—611no—
353#382#692311yescharacteristic
354#380—611no—
355#380—611no—
356#380—611no—
357#380—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—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—
370#336—611no—
371#336—611no—
372#336—611no—
373#336—611no—
374C61#2260011yescharacteristic, complemented, Sylow
375C23×C4—611no—
376C23×C4—611no—
377C23×C4—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C22×C8—611no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—611no—
386#117—1222no—
387#117—1222no—
388#117—1222no—
389#117—1222no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4C2×#692311yescharacteristic
401C2×C4⋊C4C2×#692311yescharacteristic
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—611no—
405C2×C22⋊C4—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—1222no—
414C2×C42—1222no—
415C2×C42C2×#692311yescharacteristic
416C4⋊C8—1222no—
417C4⋊C8—1222no—
418C4⋊C8—1222no—
419C4⋊C8—1222no—
420C4⋊C8—1222no—
421C4⋊C8—1222no—
422C22⋊C8—614no—
423C22⋊C8—1222no—
424C22⋊C8—1222no—
425C22⋊C8—1222no—
426#97—1222no—
427#97—1222no—
428C4×C8—1222no—
429C4×C8—1222no—
430C4×C8—1222no—
431C4×C8—1222no—
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×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—2441no—
460C22×C4C22×#692311yescharacteristic
461C22×C4C22×#692311yescharacteristic
462C22×C4C22×#692311yescharacteristic
463C22×C4C22×#692311yescharacteristic
464C22×C4C22×#692311yescharacteristic
465C22×C4C22×#692311yescharacteristic
466C22×C4C22×#692311yescharacteristic
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
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—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—2441no—
502C4⋊C4—2441no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C42—22no—
508C42—612no—
509C42—612no—
510C42—1221no—
511C42—1221no—
512C42—1221no—
513C42—1221no—
514C42—1222no—
515C42—1221no—
516C42—1221no—
517C42—1221no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C23—612no—
522C23—1221no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23C23×#692311yescharacteristic
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—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—612no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—611no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4#1451985612yes—
579C2×C4D4×#692312yes—
580C8—1222no—
581C8—1222no—
582C8—2441no—
583C8—2441no—
584C8—2441no—
585C8—2441no—
586C8—2441no—
587C8—2441no—
588C22—1222no—
589C22—1222no—
590C22—1222no—
591C22—2441no—
592C22—2441no—
593C22—2441no—
594C22—2441no—
595C22C2×#1451985611yescharacteristic
596C22C2×D4×#692311yescharacteristic
597C22#1684083011yescharacteristic
598C22D61⋊#13111yescharacteristic
599C22D61⋊#13111yescharacteristic
600C22D61⋊#13111yescharacteristic
601C22#1684084111yescharacteristic
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—611no—
610C4—611no—
611C4—611no—
612C4—611no—
613C4—614no—
614C4—1222no—
615C4—2441no—
616C4—2441no—
617C4—2441no—
618C4—2441no—
619C2—2442no—
620C2#2146232711yescharacteristic
621C2D61⋊#43311yescharacteristic
622C2#2146251111yescharacteristic
623C2#2146254311yescharacteristic
624C2#2146256411yescharacteristic
625C2#2146272611yescharacteristic
626C2#2146272911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#382C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenter#382#6923
hypocenterC61#22600
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
3. #21461918C2semidirect
4. #21461918C2semidirect
5. #21461878C2semidirect
6. #21461877C2semidirect
7. #21461874C2semidirect
8. #21461873C2semidirect
12. #21461614C2semidirect
13. #21461611C2semidirect
301. Dic61#433semidirect
374. C61#22600semidirect

Maximal subgroups

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

subgroupindex
C2×#168407872
#214619182
#214618782
#214618772
#214618742
#214618732
#214617132
#214616762
#214616142
#214616112
C4×#145198342
#214482252
#2260061