Finite Group Databasebeta

← #29697115

Subgroups — #29697115

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#103×D61C211yesmaximal, characteristic
2#21448692C211yesmaximal, characteristic
3Q8×#11758198C211yesmaximal, characteristic
4#21446509C211yesmaximal, characteristic, complemented
5#21446507C211yesmaximal, characteristic, complemented
6#21446453C211yesmaximal, characteristic, complemented
7#21446451C211yesmaximal, characteristic, complemented
8#21446087C211yesmaximal, characteristic, complemented
9#21446085C211yesmaximal, characteristic, complemented
10#21445908C211yesmaximal, characteristic
11#21445843C211yesmaximal, characteristic
12#21445752C211yesmaximal, characteristic
13#21445651C211yesmaximal, characteristic, complemented
14#21445596C211yesmaximal, characteristic
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28#103×D61C418yescomplemented
29C2×#14519689C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31#16839668—21no—
32#16839668C2211yescharacteristic
33#16839667—21no—
34#16839667C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839646—21no—
38#16839644—21no—
39#16839621—21no—
40#16839619—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758198—21no—
45C4×#11758198—21no—
46C4×#11758198C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C2×Q8×D61—22no—
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—41no—
66C2×Q8×D61—41no—
67C2×Q8×D61C2×C414yes—
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C2×C8×D61—41no—
74C2×C8×D61—41no—
75C2×C8×D61C2×C414yes—
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61C2×C414yes—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519708—41no—
95#14519708—41no—
96#14519702—41no—
97#14519702C2×C414yes—
98#14519701—22no—
99#14519701C2×C414yes—
100#14519700—41no—
101#14519700C2×C414yes—
102#14519689—22no—
103#14519687—41no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4×Q8×C61—22no—
119C4×Q8×C61—41no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#103×C61—22no—
128#103×C61C2×C414yes—
129#97×C61—41no—
130#95×C61—21no—
131C23×D61C2×D411yescharacteristic
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61C22×C412yes—
136C2×Q8×C61#4112yes—
137C2×Q8×C61#4112yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152Q8×D61—28no—
153Q8×D61—44no—
154Q8×D61—44no—
155Q8×D61—44no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—81no—
165C2×C4×D61—81no—
166C2×C4×D61C22×C412yes—
167C2×C4×D61#4112yes—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—24no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122C22×C412yes—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C22×C412yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—42no—
189C4⋊C4×C61C22×C412yes—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—42no—
193C42×C61—81no—
194#11758198—22no—
195#11758198—22no—
196#11758198—42no—
197#11758198—42no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204C4⋊Dic61C22×C412yes—
205#11758196—24no—
206#11758196—81no—
207#11758196—81no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—42no—
211C4×Dic61—81no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C22×C412yes—
218C8×D61—44no—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61C4×D412yes—
225C22×D61#13712yes—
226C22×D61#13712yes—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61#11714yes—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
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—41no—
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×C61C23×C411yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
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×Dic61C4×D412yes—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—48no—
281C4×D61—84no—
282Dic122—22no—
283Dic122—22no—
284Dic122—48no—
285Dic122—41no—
286Dic122—41no—
287C8×C61—22no—
288C8×C61—41no—
289#92418—22no—
290#92418—41no—
291#22567—611nomaximal, Sylow
292C22×C61C2×C4×D411yescharacteristic
293C22×C61#54611yescharacteristic
294C22×C61#54611yescharacteristic
295C22×C61C2×#13711yescharacteristic
296C22×C61C2×#13711yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57711yescharacteristic
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61#44114yes—
306C4×C61—24no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C2×#11711yescharacteristic
313C4×C61#51811yescharacteristic
314C4×C61#55111yescharacteristic
315C4×C61#55111yescharacteristic
316Dic61—22no—
317Dic61—42no—
318Dic61—42no—
319Dic61—41no—
320Dic61—41no—
321Dic61—81no—
322Dic61—81no—
323C22×#103—611no—
324#2345—611no—
325Q8×C22⋊C4—611no—
326#1935—611no—
327#1935—611no—
328#1927—611no—
329#1927—611no—
330#1865—611no—
331#1865—611no—
332#1824—611no—
333#1807—611no—
334#1787—611no—
335#1763—611no—
336#1763D6111yescharacteristic, complemented
337#1752—611no—
338C2×C61#234211yescharacteristic
339C2×C61C2×#44111yescharacteristic
340C2×C61#294911yescharacteristic
341C2×C61#331611yescharacteristic
342C2×C61#331611yescharacteristic
343C2×C61#332011yescharacteristic
344C2×C61#332011yescharacteristic
345D61—24no—
346C23×Q8—611no—
347C23×C8—611no—
348C2×C4×Q8—611no—
349C2×C4×Q8C2×D6111yescharacteristic
350C22×C4⋊C4—611no—
351C2×#106—611no—
352C2×#106—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—618no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103C2×D6111yescharacteristic
361C2×#103C2×D6111yescharacteristic
362C2×C22⋊C8—611no—
363C2×#97—611no—
364C2×#97C2×D6111yescharacteristic
365#391—611no—
366#391—611no—
367#391—1221no—
368#391—1221no—
369#389—611no—
370#389—611no—
371#387—1221no—
372#387—1221no—
373#380—611no—
374#380—611no—
375#380—1221no—
376#380—1221no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4C2×D6111yescharacteristic
379C4×C22⋊C4—611no—
380C4×C22⋊C4—1221no—
381C4×C22⋊C4—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340C2×D6111yescharacteristic
387#340C2×D6111yescharacteristic
388#336—611no—
389#336—611no—
390C61#2256711yescharacteristic, complemented, Sylow
391C22×Q8—614no—
392C22×Q8—611no—
393C22×Q8—1221no—
394C22×Q8—1222no—
395C22×Q8—1222no—
396C22×Q8—1222no—
397C22×Q8—2441no—
398C22×Q8—2441no—
399C22×Q8C22×D6111yescharacteristic
400C23×C4—611no—
401C23×C4—1221no—
402C23×C4—1221no—
403C22×C8—611no—
404C22×C8—614no—
405C22×C8—611no—
406C22×C8—2441no—
407C22×C8—2441no—
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410C4×Q8—22no—
411C4×Q8—41no—
412C4×Q8—1222no—
413C4×Q8—2441no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—614no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—21no—
432C2×C42—21no—
433C2×C42—611no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437#106—2441no—
438#106—2441no—
439#103—22no—
440#103—614no—
441#103—614no—
442#103—614no—
443#103—1222no—
444#103—2441no—
445#103—2441no—
446#103C4×D6114yes—
447C22⋊C8—1222no—
448#97—41no—
449#97—2441no—
450#95—21no—
451#95—611no—
452#95—611no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—611no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—612no—
467C2×Q8—1224no—
468C2×Q8—1221no—
469C2×Q8—1228no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—2442no—
475C2×Q8—2442no—
476C2×Q8—2444no—
477C2×Q8—2444no—
478C2×Q8—2444no—
479C2×Q8C2×C4×D6112yes—
480C2×Q8#1175822212yes—
481C2×Q8#1175822212yes—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—611no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1224no—
492C22×C4—1224no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—4881no—
502C22×C4—4881no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4D4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4C23×D6111yescharacteristic
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—612no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—2444no—
520C2×C8C2×C4×D6112yes—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—612no—
528C4⋊C4—1224no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—4881no—
535C4⋊C4—4881no—
536C4⋊C4—2442no—
537C4⋊C4C2×C4×D6112yes—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—2442no—
541C22⋊C4—2442no—
542C42—21no—
543C42—21no—
544C42—42no—
545C42—81no—
546C42—1221no—
547C42—1221no—
548C42—4881no—
549C42—2442no—
550C23—612no—
551C23—612no—
552C23—612no—
553C23—1221no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23C2×D4×D6111yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—22no—
561Q8—1222no—
562Q8—1222no—
563Q8—2448no—
564Q8—2441no—
565Q8—2441no—
566Q8#1451973414yes—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—4884no—
601C2×C4—2448no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4C2×#1175822211yescharacteristic
615C2×C4C2×#1175822211yescharacteristic
616C2×C4#1451982211yescharacteristic
617C2×C4#1451982311yescharacteristic
618C8—22no—
619C8—41no—
620C8—1222no—
621C8—2441no—
622C22—614no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—1222no—
628C22—1222no—
629C22C4×D4×D6111yescharacteristic
630C22#1684040211yescharacteristic
631C22#1684040811yescharacteristic
632C22#137×D6111yescharacteristic
633C22#137×D6111yescharacteristic
634C22#1684057111yescharacteristic
635C22#1684057211yescharacteristic
636C4—24no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—81no—
642C4—1222no—
643C4—4881no—
644C4—4881no—
645C4—2441no—
646C4—2441no—
647C4—2442no—
648C4—2442no—
649C4C2×#1451973411yescharacteristic
650C4#1684023411yescharacteristic
651C4#1684043611yescharacteristic
652C4#1684044111yescharacteristic
653C2—1224no—
654C2#2144866211yescharacteristic
655C2#441×D6111yescharacteristic
656C2#2145467511yescharacteristic
657C2#2145793111yescharacteristic
658C2#2145793511yescharacteristic
659C2#2145799111yescharacteristic
660C2#2145799511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#22567
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. #21446509C2semidirect
5. #21446507C2semidirect
6. #21446453C2semidirect
7. #21446451C2semidirect
8. #21446087C2semidirect
9. #21446085C2semidirect
13. #21445651C2semidirect
15. C61×#1763C2semidirect
28. #103×D61C4semidirect
336. #1763D61semidirect
390. C61#22567semidirect

Maximal subgroups

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

subgroupindex
C2×#103×D612
#214486922
Q8×#117581982
#214465092
#214465072
#214464532
#214464512
#214460872
#214460852
#214459082
#214458432
#214457522
#214456512
#214455962
C61×#17632
#2256761