Finite Group Databasebeta

← #30617265

Subgroups — #30617265

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840813C211yesmaximal, characteristic, complemented
2C2×C4×#11758217C211yesmaximal, characteristic
3#21461993C211yesmaximal, characteristic, complemented
4#21461921C211yesmaximal, characteristic, complemented
5#21461875C211yesmaximal, characteristic, complemented
6#21461872C211yesmaximal, characteristic, complemented
7#21461766C211yesmaximal, characteristic
8#21461766C211yesmaximal, characteristic
9#21461675C211yesmaximal, characteristic
10#21461619C211yesmaximal, characteristic, complemented
11#21461619C211yesmaximal, characteristic, complemented
12C4⋊C4×#92420C211yesmaximal, characteristic, complemented
13#21461561C211yesmaximal, characteristic
14#21461561C211yesmaximal, characteristic
15C22⋊Q8×Dic61C211yesmaximal, characteristic
16C22×#11758217C2211yescharacteristic
17C22×#11758217C2211yescharacteristic
18#16840813—24no—
19C2×#14519844C2211yescharacteristic
20C2×#14519844C2211yescharacteristic
21C2×#14519844C2211yescharacteristic
22C2×#14519844C2211yescharacteristic
23C2×C4⋊#92420C2211yescharacteristic
24C2×C4⋊#92420C2211yescharacteristic
25C2×C22⋊#92420C2211yescharacteristic
26C2×C22⋊#92420C2211yescharacteristic
27C4×#11758217—24no—
28C2×#14519832C2211yescharacteristic
29C2×#14519832C2211yescharacteristic
30C2×#14519832C2211yescharacteristic
31C2×#14519832C2211yescharacteristic
32C2×C4×#92420C2211yescharacteristic
33C2×C4×#92420C2211yescharacteristic
34C2×C4×#92420C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840628C2211yescharacteristic
37#16840628C2211yescharacteristic
38#16840628C2211yescharacteristic
39#16840628C2211yescharacteristic
40#16840628C2211yescharacteristic
41#16840628C2211yescharacteristic
42#16840628C2211yescharacteristic
43#16840628C2211yescharacteristic
44#16840628C2211yescharacteristic
45#16840628C2211yescharacteristic
46C2×Q8×Dic61C2211yescharacteristic
47C22×C4×Dic61C2211yescharacteristic
48#16839669C411yescharacteristic
49#16839661C411yescharacteristic
50#16839648C411yescharacteristic
51#16839648C411yescharacteristic
52#16839637C411yescharacteristic
53#16839637C411yescharacteristic
54#16839622C411yescharacteristic
55C4⋊C4×Dic61C2211yescharacteristic
56C4⋊C4×Dic61C2211yescharacteristic
57C4⋊C4×Dic61C2211yescharacteristic
58C22⋊C4×Dic61C2211yescharacteristic
59C22⋊C4×Dic61C2211yescharacteristic
60C2×C22⋊Q8×C61C411yescharacteristic
61C2×#11758217—22no—
62C2×#11758217—22no—
63C2×#11758217—22no—
64C2×#11758217—22no—
65C2×#11758217—22no—
66C2×#11758217Q814yescomplemented
67C22×#92420C2311yescharacteristic
68C22×#92420C2311yescharacteristic
69C22×#92420C2311yescharacteristic
70C22×#92420C2311yescharacteristic
71C22×#92420C2311yescharacteristic
72C22×#92420C2311yescharacteristic
73C22×#92420C2311yescharacteristic
74C22×#92420C2311yescharacteristic
75#14519844—22no—
76#14519844—22no—
77#14519844—22no—
78#14519844—22no—
79C4⋊#92420—22no—
80C4⋊#92420—22no—
81C22⋊#92420—22no—
82C22⋊#92420—22no—
83#14519832—22no—
84#14519832—22no—
85#14519832—22no—
86#14519832D414yes—
87C4×#92420—22no—
88C4×#92420—22no—
89C4×#92420—22no—
90C4×#92420—22no—
91C23×Dic61C2×C411yescharacteristic
92C23×Dic61C2311yescharacteristic
93Q8×Dic61—22no—
94Q8×Dic61—22no—
95C2×C4⋊Dic61C2×C411yescharacteristic
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×C22⋊Dic61C2×C411yescharacteristic
102C2×C22⋊Dic61C2×C411yescharacteristic
103C2×C4×Dic61—22no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113#14519684C2×C411yescharacteristic
114#14519684C2×C411yescharacteristic
115#14519684C2×C411yescharacteristic
116#14519684C2×C411yescharacteristic
117#14519684C2×C411yescharacteristic
118#14519684C2×C411yescharacteristic
119#14519684C2×C411yescharacteristic
120#14519684C2×C411yescharacteristic
121#14519684C2×C411yescharacteristic
122#14519684C2×C411yescharacteristic
123C22×Q8×C61C2×C411yescharacteristic
124C23×C4×C61C2×C411yescharacteristic
125C22⋊Q8×C61—24no—
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C22⋊C4×C61C2×C411yescharacteristic
130C2×C22⋊C4×C61C2×C411yescharacteristic
131C24×C61C22×C411yescharacteristic
132C2×Q8×C61—22no—
133C2×Q8×C61—22no—
134C2×Q8×C61—41no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—41no—
138C22×C4×C61C22×C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—41no—
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C22×Dic61C22×C411yescharacteristic
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C2×Q812yes—
159C22×Dic61#4112yes—
160C22×Dic61C2411yescharacteristic
161#11758217—28no—
162#11758217—28no—
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×#92420—21no—
176C2×#92420—21no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—21no—
180C2×#92420—21no—
181C2×#92420C2×D412yes—
182C2×#92420C2×D412yes—
183C2×#92420C2×Q812yes—
184C2×#92420C2×Q812yes—
185C2×#92420#4112yes—
186C2×#92420#4112yes—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61M1614yescomplemented
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊Dic61—41no—
193C22⋊Dic61—41no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—41no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61C2×D412yes—
213C4×Dic61#4112yes—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61C4×Q812yes—
221C23×C61C23×C411yescharacteristic
222Q8×C61—44no—
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—44no—
238C2×C4×C61C4×D412yes—
239C2×C4×C61C2×M1612yes—
240C2×C4×C61C2×M1612yes—
241C2×C4×C61C2×M1612yes—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—44no—
265C2×Dic61C4×D412yes—
266C2×Dic61C4×Q812yes—
267C2×Dic61#13112yes—
268C2×Dic61C22×D411yescharacteristic
269C2×Dic61C22×Q811yescharacteristic
270C2×Dic61C2×#4111yescharacteristic
271C2×Dic61C2×#4111yescharacteristic
272C2×Dic61C2×#4111yescharacteristic
273C2×Dic61C2×#4111yescharacteristic
274#92420—22no—
275#92420—22no—
276#92420—22no—
277#92420—41no—
278#92420—41no—
279#92420—41no—
280#92420—41no—
281#92420C22⋊Q814yescomplemented
282#21600—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61C2×C4×D411yescharacteristic
291C22×C61C2×C4×Q811yescharacteristic
292C22×C61C4×#4111yescharacteristic
293C22×C61C22×M1611yescharacteristic
294C22×C61C2×#13111yescharacteristic
295C22×C61#56811yescharacteristic
296C22×C61#56811yescharacteristic
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—41no—
302C4×C61—41no—
303Dic61—22no—
304Dic61—22no—
305Dic61—22no—
306Dic61—41no—
307Dic61—41no—
308Dic61—41no—
309Dic61—41no—
310Dic61C2×C22⋊Q811yescharacteristic
311Dic61#52511yescharacteristic
312Dic61#52911yescharacteristic
313Dic61#53311yescharacteristic
314C2×#446—611no—
315C2×C4×M16—611no—
316C4×C22⋊Q8—611no—
317#1985—611no—
318#1940—611no—
319#1916—611no—
320#1916—611no—
321#1851—611no—
322#1851—611no—
323#1797—611no—
324#1771—611no—
325#1771—611no—
326C8×C4⋊C4—611no—
327#1753—611no—
328#1753—611no—
329C2×C61—42no—
330C2×C61C4×C22⋊Q811yescharacteristic
331C2×C61#293311yescharacteristic
332C2×C61D4×M1611yescharacteristic
333C2×C61#296211yescharacteristic
334C2×C61Q8×M1611yescharacteristic
335C2×C61#296511yescharacteristic
336C2×C61#296711yescharacteristic
337C22×M16—611no—
338C22×M16—611no—
339C2×C22⋊Q8#692311yescharacteristic
340C2×C4×Q8—611no—
341C22×C42—611no—
342#446—1224no—
343C2×C4⋊C8—611no—
344C2×C4⋊C8—611no—
345C2×C4⋊C8—611no—
346C2×C4⋊C8—611no—
347C2×C4⋊C8—611no—
348C2×C4⋊C8—611no—
349C2×C22⋊C8—611no—
350C2×C22⋊C8—611no—
351C4×M16—1224no—
352C2×#97—611no—
353C2×#97—611no—
354C2×#97—611no—
355C2×#97—611no—
356C2×C4×C8—611no—
357C2×C4×C8—611no—
358C2×C4×C8—611no—
359C2×C4×C8—611no—
360#391—611no—
361#389—611no—
362#387—611no—
363#387—611no—
364#385—611no—
365#385—611no—
366#380—611no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C22⋊C4—611no—
371C4×C22⋊C4—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376#336—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382C61#2160011yescharacteristic, complemented, Sylow
383C22×Q8C2×#692311yescharacteristic
384C23×C4—611no—
385C23×C4—611no—
386C23×C4C2×#692311yescharacteristic
387C2×M16—614no—
388C2×M16—1222no—
389C2×M16—1222no—
390C2×M16—1222no—
391C2×M16—1222no—
392C2×M16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—611no—
400C22×C8—611no—
401C22⋊Q8—24no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4C2×#692311yescharacteristic
411C2×C4⋊C4C2×#692311yescharacteristic
412C2×C4⋊C4C2×#692311yescharacteristic
413C2×C22⋊C4—611no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4C2×#692311yescharacteristic
416C2×C22⋊C4C2×#692311yescharacteristic
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—1222no—
426C2×C42—1222no—
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1222no—
435#97—614no—
436#97—1222no—
437#97—1222no—
438#97—1222no—
439C4×C8—1222no—
440C4×C8—1222no—
441C4×C8—1222no—
442C4×C8—1222no—
443#95—611no—
444#95—611no—
445#95—611no—
446#95—611no—
447#95—611no—
448#95—611no—
449#95—611no—
450#95—611no—
451#95—611no—
452#95—611no—
453C24C22×#692311yescharacteristic
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—41no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—611no—
461C22×C4—612no—
462C22×C4—612no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—2441no—
477C22×C4C22×#692311yescharacteristic
478C22×C4C22×#692311yescharacteristic
479C22×C4C22×#692311yescharacteristic
480C22×C4C22×#692311yescharacteristic
481C22×C4C22×#692311yescharacteristic
482C22×C4C22×#692311yescharacteristic
483M16—1228no—
484M16—1228no—
485C2×C8—612no—
486C2×C8—612no—
487C2×C8—612no—
488C2×C8—612no—
489C2×C8—612no—
490C2×C8—612no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1221no—
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—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—2441no—
517C4⋊C4#1175821714yescomplemented
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—2441no—
521C22⋊C4—2441no—
522C42—612no—
523C42—612no—
524C42—1221no—
525C42—1221no—
526C42—1221no—
527C42—1221no—
528C42—1221no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1222no—
533C42—1221no—
534C42—1221no—
535C42—1222no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23Q8×#692312yes—
543C23C23×#692311yescharacteristic
544Q8—44no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—44no—
560C2×C4—611no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—2444no—
592C2×C4D4×#692312yes—
593C2×C4C2×#1175821712yes—
594C2×C4C2×#1175821712yes—
595C2×C4C2×#1175821712yes—
596C8—614no—
597C8—1222no—
598C8—1222no—
599C8—1222no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C8—2441no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×D4×#692311yescharacteristic
612C22C2×Q8×#692311yescharacteristic
613C22#41×#692311yescharacteristic
614C22C22×#1175821711yescharacteristic
615C22D61⋊#13111yescharacteristic
616C22#1684084011yescharacteristic
617C22#1684084211yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—611no—
625C4—611no—
626C4—611no—
627C4—611no—
628C4—1222no—
629C4—1222no—
630C4—1222no—
631C4—2441no—
632C4—2441no—
633C4—2441no—
634C4—2441no—
635C2—42no—
636C2C22⋊Q8×#692311yescharacteristic
637C2#2146255411yescharacteristic
638C2D4×#1175821711yescharacteristic
639C2#2146261611yescharacteristic
640C2Q8×#1175821711yescharacteristic
641C2#2146262811yescharacteristic
642C2#2146264011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C22⋊Q8×C61C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#21600
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840813C2semidirect
3. #21461993C2semidirect
4. #21461921C2semidirect
5. #21461875C2semidirect
6. #21461872C2semidirect
10. #21461619C2semidirect
11. #21461619C2semidirect
12. C4⋊C4×#92420C2semidirect
66. C2×#11758217Q8semidirect
189. C4⋊C4×C61M16semidirect
281. #92420C22⋊Q8semidirect
382. C61#21600semidirect
517. C4⋊C4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168408132
C2×C4×#117582172
#214619932
#214619212
#214618752
#214618722
#214617662
#214616752
#214616192
C4⋊C4×#924202
#214615612
C22⋊Q8×Dic612
#2160061