Finite Group Databasebeta

← #29697287

Subgroups — #29697287

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
1#21448712C211yesmaximal, characteristic
2Q8×C22⋊Dic61C211yesmaximal, characteristic
3#21446511C211yesmaximal, characteristic, complemented
4#21446497C211yesmaximal, characteristic, complemented
5#21446462C211yesmaximal, characteristic, complemented
6#21446455C211yesmaximal, characteristic, complemented
7#21446089C211yesmaximal, characteristic, complemented
8#21446075C211yesmaximal, characteristic, complemented
9#21445914C211yesmaximal, characteristic
10#21445852C211yesmaximal, characteristic
11#21445754C211yesmaximal, characteristic
12#21445651C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445600C211yesmaximal, characteristic
15C2×C61×#419C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25C2×#14519687C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839669—21no—
28#16839669—21no—
29#16839669C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839655—21no—
34#16839655—21no—
35#16839630—21no—
36#16839622—21no—
37#16839622C2211yescharacteristic
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C4⋊Dic61—21no—
42C4×C22⋊Dic61—21no—
43C4×C22⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C23×Q8×C61C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C61×#419C418yescomplemented
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64Q8×Dic61—41no—
65Q8×Dic61—41no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C22⋊Dic61—21no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61D411yescharacteristic
77C2×C22⋊Dic61D411yescharacteristic
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519711—41no—
85#14519709—41no—
86#14519701—22no—
87#14519701—22no—
88#14519701—41no—
89#14519701—41no—
90C22⋊#92418—22no—
91#14519687—41no—
92C8×Dic61—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×Q8×C61—21no—
106C22×Q8×C61—22no—
107C22×Q8×C61—22no—
108C22×Q8×C61—22no—
109C22×Q8×C61—41no—
110C22×Q8×C61—41no—
111C22×Q8×C61C2×C414yes—
112C22×Q8×C61C2311yescharacteristic
113C22×Q8×C61C2311yescharacteristic
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C2×M16×C61—41no—
117C2×M16×C61—41no—
118C2×M16×C61C2×C414yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121#117×C61—22no—
122#117×C61C2×C414yes—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126C2×C42×C61—21no—
127#103×C61C2×C414yes—
128#103×C61C2×C414yes—
129#103×C61C2×C414yes—
130#103×C61C2×C414yes—
131C24×C61C2×D411yescharacteristic
132C2×Q8×C61—24no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—28no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—42no—
143C2×Q8×C61—42no—
144C2×Q8×C61—44no—
145C2×Q8×C61—44no—
146C2×Q8×C61—44no—
147C2×Q8×C61C22×C412yes—
148C2×Q8×C61C22×C412yes—
149C2×Q8×C61#4112yes—
150C2×Q8×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—24no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—42no—
161C22×C4×C61—42no—
162C22×C4×C61C22×C412yes—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174M16×C61—44no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C412yes—
180C2×C8×C61C22×C412yes—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C412yes—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—24no—
186C42×C61—24no—
187C22⋊Dic61—22no—
188C22⋊Dic61—41no—
189C22⋊Dic61—42no—
190C22⋊Dic61—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61—81no—
210C4×Dic61—81no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61C4×D412yes—
224C23×C61C22×D411yescharacteristic
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61—48no—
231Q8×C61—41no—
232Q8×C61—41no—
233Q8×C61#11714yes—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—48no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61#13712yes—
284C8×C61—22no—
285C8×C61—22no—
286#92418—41no—
287#92418—41no—
288#22581—611nomaximal, Sylow
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61C2×C4×D411yescharacteristic
297C22×C61#54611yescharacteristic
298C22×C61#54611yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57811yescharacteristic
303C4×C61—22no—
304C4×C61—24no—
305C4×C61—42no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61C2×#11711yescharacteristic
309C4×C61#51811yescharacteristic
310C4×C61#55111yescharacteristic
311C4×C61#55111yescharacteristic
312Dic61—22no—
313Dic61—41no—
314Dic61—41no—
315Dic61—41no—
316Dic61—81no—
317Dic61—81no—
318Dic61—81no—
319Dic61—81no—
320C2×#419D6111yescharacteristic
321#2347—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#1763—611no—
333C4×#103—611no—
334#1753—611no—
335C2×C61—42no—
336C2×C61#234211yescharacteristic
337C2×C61C4×#13711yescharacteristic
338C2×C61#294711yescharacteristic
339C2×C61#331311yescharacteristic
340C2×C61#332011yescharacteristic
341C2×C61#332611yescharacteristic
342C2×C61#332811yescharacteristic
343C23×Q8C2×D6111yescharacteristic
344C22×M16C2×D6111yescharacteristic
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×#117C2×D6111yescharacteristic
348C2×#107—611no—
349C2×#106—611no—
350#419Dic6118yescomplemented
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103C2×D6111yescharacteristic
356C2×#103C2×D6111yescharacteristic
357C2×#103C2×D6111yescharacteristic
358C2×#103C2×D6111yescharacteristic
359C2×C22⋊C8—611no—
360C2×#97—611no—
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#383—1221no—
371#380—611no—
372#380—611no—
373#380—1221no—
374C4×C4⋊C4—611no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—1221no—
377C4×C22⋊C4—611no—
378C4×C22⋊C4—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#336—611no—
386#336—611no—
387C61#2258111yescharacteristic, complemented, Sylow
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C2×Dic6114yes—
395C22×Q8C22×D6111yescharacteristic
396C22×Q8C22×D6111yescharacteristic
397C23×C4—21no—
398C23×C4C22×D6111yescharacteristic
399C2×M16—41no—
400C2×M16—41no—
401C2×M16C2×Dic6114yes—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×Q8—2441no—
410#117—22no—
411#117C2×Dic6114yes—
412C2×C4⋊C4—611no—
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⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4—611no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C42—21no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434#107—2441no—
435#106—2441no—
436#103—1222no—
437#103—1222no—
438#103—2441no—
439#103—2441no—
440#103C2×Dic6114yes—
441#103C2×Dic6114yes—
442#103C2×Dic6114yes—
443#103C2×Dic6114yes—
444C22⋊C8—1222no—
445#97—2441no—
446C4×C8—2441no—
447#95—611no—
448#95—611no—
449#95—1221no—
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—
459C24D4×D6111yescharacteristic
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—24no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8#1175822212yes—
476C2×Q8#1175822212yes—
477C2×Q8C22×Dic6112yes—
478C2×Q8C22×Dic6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—24no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4D4×D6111yescharacteristic
499C22×C4#1175822412yes—
500C22×C4C22×Dic6112yes—
501C22×C4C23×D6111yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8C22×Dic6112yes—
514C2×C8C22×Dic6112yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4C22×Dic6112yes—
530C4⋊C4C22×Dic6112yes—
531C22⋊C4—24no—
532C22⋊C4—1222no—
533C22⋊C4—4881no—
534C22⋊C4—2441no—
535C22⋊C4—2442no—
536C42—24no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—4881no—
542C42—4881no—
543C42—2442no—
544C42—2442no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23D4×Dic6112yes—
552C23C2×D4×D6111yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1451973614yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—612no—
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—1221no—
592C2×C4—1221no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
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×C4D4×Dic6112yes—
606C2×C4C23×Dic6111yescharacteristic
607C2×C4C2×#1175822411yescharacteristic
608C2×C4C2×#1175822211yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982211yescharacteristic
612C8—22no—
613C8—22no—
614C8—2441no—
615C8—2441no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×D4×Dic6111yescharacteristic
624C22#1684040211yescharacteristic
625C22#1684040211yescharacteristic
626C22#137×D6111yescharacteristic
627C22#1684056111yescharacteristic
628C22#1684057111yescharacteristic
629C22#1684057811yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—1222no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—4881no—
640C4—2441no—
641C4—2441no—
642C4—2441no—
643C4C2×#1451973611yescharacteristic
644C4#1684023711yescharacteristic
645C4#1684043811yescharacteristic
646C4#1684043811yescharacteristic
647C2—42no—
648C2#2144866811yescharacteristic
649C2#137×Dic6111yescharacteristic
650C2#2145466311yescharacteristic
651C2#2145788611yescharacteristic
652C2#2145799111yescharacteristic
653C2#2145806811yescharacteristic
654C2#2145809811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#22581
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. #21446511C2semidirect
4. #21446497C2semidirect
5. #21446462C2semidirect
6. #21446455C2semidirect
7. #21446089C2semidirect
8. #21446075C2semidirect
12. #21445651C2semidirect
13. #103×Dic61C2semidirect
55. C61×#419C4semidirect
350. #419Dic61semidirect
387. C61#22581semidirect

Maximal subgroups

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

subgroupindex
#214487122
Q8×C22⋊Dic612
#214465112
#214464972
#214464622
#214464552
#214460892
#214460752
#214459142
#214458522
#214457542
#214456512
#103×Dic612
#214456002
C2×C61×#4192
#2258161