Finite Group Databasebeta

← #29697341

Subgroups — #29697341

Order 15616.

934 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×#16839769C211yesmaximal, characteristic
2#21448700C211yesmaximal, characteristic
3Q8×#11758198C211yesmaximal, characteristic
4#21446509C211yesmaximal, characteristic, complemented
5#21446493C211yesmaximal, characteristic, complemented
6#21446460C211yesmaximal, characteristic, complemented
7#21446451C211yesmaximal, characteristic, complemented
8#21446087C211yesmaximal, characteristic, complemented
9#21446071C211yesmaximal, characteristic, complemented
10#21445913C211yesmaximal, characteristic
11#21445851C211yesmaximal, characteristic
12#21445752C211yesmaximal, characteristic
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445604C211yesmaximal, characteristic
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22#16839769C418yescomplemented
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839668—21no—
32#16839668C2211yescharacteristic
33#16839667—21no—
34#16839667C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839653—21no—
38#16839651—21no—
39#16839626—21no—
40#16839619—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758198—21no—
45C4×#11758198C2211yescharacteristic
46#16839608—21no—
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×D61C2311yescharacteristic
70C2×#11758189—41no—
71C2×#11758189—41no—
72C2×#11758189C2×C414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75Q8×Dic61—22no—
76Q8×Dic61—41no—
77#14519735—22no—
78#14519735C2×C414yes—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198D411yescharacteristic
89C2×#11758198D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94#14519710—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—
103C8×Dic61—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—42no—
162C2×C4×D61—42no—
163C2×C4×D61C22×C412yes—
164C2×C4×D61#4112yes—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—24no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122C22×C412yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C412yes—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61C22×C412yes—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—42no—
190C42×C61—42no—
191#11758198—24no—
192#11758198—22no—
193#11758198—41no—
194#11758198—42no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—42no—
202C4⋊Dic61C22×C412yes—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—24no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C22×C412yes—
216#11758189—44no—
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61C4×D412yes—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61#11714yes—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
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×C61C23×C411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C4×D412yes—
273C2×Dic61#13712yes—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—48no—
279Dic122—22no—
280Dic122—22no—
281Dic122—48no—
282Dic122—41no—
283Dic122—41no—
284C8×C61—22no—
285C8×C61—41no—
286#92418—22no—
287#92418—41no—
288#22585—611nomaximal, Sylow
289C22×C61C2×C4×D411yescharacteristic
290C22×C61#54611yescharacteristic
291C22×C61#54611yescharacteristic
292C22×C61C2×#13711yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57811yescharacteristic
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61—41no—
303C4×C61—24no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61C2×#11711yescharacteristic
310C4×C61#51811yescharacteristic
311C4×C61#55111yescharacteristic
312C4×C61#55111yescharacteristic
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316Dic61—42no—
317Dic61—41no—
318Dic61—81no—
319Dic61—81no—
320C2×#419—611no—
321#2346—611no—
322Q8×C22⋊C4—611no—
323#1935—611no—
324#1933—611no—
325#1928—611no—
326#1927—611no—
327#1865—611no—
328#1863—611no—
329#1825—611no—
330#1809—611no—
331#1787—611no—
332#1763D6111yescharacteristic, complemented
333C4×#103—611no—
334#1753—611no—
335C2×C61#234211yescharacteristic
336C2×C61C4×#13711yescharacteristic
337C2×C61#294711yescharacteristic
338C2×C61#331211yescharacteristic
339C2×C61#332111yescharacteristic
340C2×C61#332711yescharacteristic
341C2×C61#332911yescharacteristic
342D61—42no—
343C23×Q8—611no—
344C22×M16—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8C2×D6111yescharacteristic
347C2×#117—611no—
348C2×#107—611no—
349C2×#106—611no—
350#419—618no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103C2×D6111yescharacteristic
358C2×#103C2×D6111yescharacteristic
359C2×C22⋊C8—611no—
360C2×#97C2×D6111yescharacteristic
361C2×C4×C8—611no—
362#391—611no—
363#391—611no—
364#391—1221no—
365#391—1221no—
366#389—611no—
367#389—611no—
368#388—1221no—
369#388—1221no—
370#382—1221no—
371#380—611no—
372#380—611no—
373#380—1221no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4C2×D6111yescharacteristic
376C4×C22⋊C4—611no—
377C4×C22⋊C4—1221no—
378#376—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340C2×D6111yescharacteristic
384#340C2×D6111yescharacteristic
385#336—611no—
386#336—611no—
387C61#2258511yescharacteristic, complemented, Sylow
388C22×Q8—614no—
389C22×Q8—611no—
390C22×Q8—1221no—
391C22×Q8—1222no—
392C22×Q8—1222no—
393C22×Q8—1222no—
394C22×Q8—2441no—
395C22×Q8—2441no—
396C22×Q8C22×D6111yescharacteristic
397C23×C4—611no—
398C23×C4—1221no—
399C2×M16—614no—
400C2×M16—2441no—
401C2×M16—2441no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4×Q8—22no—
407C4×Q8—41no—
408C4×Q8—1222no—
409C4×Q8—2441no—
410#117—614no—
411#117—1222no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—611no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—2441no—
435#106—2441no—
436#103—22no—
437#103—614no—
438#103—614no—
439#103—614no—
440#103—1222no—
441#103—2441no—
442#103—2441no—
443#103C4×D6114yes—
444C22⋊C8—1222no—
445#97—41no—
446C4×C8—2441no—
447#95—21no—
448#95—611no—
449#95—611no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459C24—611no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—612no—
464C2×Q8—1224no—
465C2×Q8—1221no—
466C2×Q8—1228no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—2442no—
472C2×Q8—2442no—
473C2×Q8—2444no—
474C2×Q8—2444no—
475C2×Q8—2444no—
476C2×Q8C2×C4×D6112yes—
477C2×Q8#1175822212yes—
478C2×Q8#1175822212yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—611no—
488C22×C4—1224no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502M16—2444no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8C2×C4×D6112yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—42no—
521C4⋊C4—612no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—2442no—
530C4⋊C4C2×C4×D6112yes—
531C22⋊C4—1222no—
532C22⋊C4—1224no—
533C22⋊C4—4881no—
534C22⋊C4—2441no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—42no—
540C42—1224no—
541C42—1221no—
542C42—1221no—
543C42—2442no—
544C42—2442no—
545C23—612no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23C2×D4×D6111yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—1222no—
557Q8—1222no—
558Q8—2448no—
559Q8—2441no—
560Q8—2441no—
561Q8#1451973414yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—2448no—
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—2441no—
605C2×C4—2441no—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982311yescharacteristic
612C8—22no—
613C8—41no—
614C8—1222no—
615C8—2441no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—2441no—
620C22—2441no—
621C22—2441no—
622C22—2441no—
623C22C4×D4×D6111yescharacteristic
624C22#1684040211yescharacteristic
625C22#1684040811yescharacteristic
626C22#137×D6111yescharacteristic
627C22#1684055811yescharacteristic
628C22#1684057111yescharacteristic
629C22#1684057811yescharacteristic
630C4—24no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—1222no—
637C4—1222no—
638C4—4881no—
639C4—4881no—
640C4—2442no—
641C4—2441no—
642C4—2442no—
643C4C2×#1451973411yescharacteristic
644C4#1684023411yescharacteristic
645C4#1684043611yescharacteristic
646C4#1684044111yescharacteristic
647C2—2442no—
648C2#2144866211yescharacteristic
649C2#2145464111yescharacteristic
650C2#2145466111yescharacteristic
651C2#2145787511yescharacteristic
652C2#2145801011yescharacteristic
653C2#2145808811yescharacteristic
654C2#2145810911yescharacteristic
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#22585
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. #21446493C2semidirect
6. #21446460C2semidirect
7. #21446451C2semidirect
8. #21446087C2semidirect
9. #21446071C2semidirect
13. #103×Dic61C2semidirect
15. C61×#1763C2semidirect
22. #16839769C4semidirect
332. #1763D61semidirect
387. C61#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#168397692
#214487002
Q8×#117581982
#214465092
#214464932
#214464602
#214464512
#214460872
#214460712
#214459132
#214458512
#214457522
#103×Dic612
#214456042
C61×#17632
#2258561