Finite Group Databasebeta

← #29697352

Subgroups — #29697352

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×#16839772C211yesmaximal, characteristic
2#21448702C211yesmaximal, characteristic
3C22⋊C4×Dic122C211yesmaximal, characteristic
4#21446509C211yesmaximal, characteristic, complemented
5#21446495C211yesmaximal, characteristic, complemented
6#21446461C211yesmaximal, characteristic, complemented
7#21446454C211yesmaximal, characteristic, complemented
8#21446090C211yesmaximal, characteristic, complemented
9#21446076C211yesmaximal, characteristic, complemented
10#21445915C211yesmaximal, characteristic
11#21445852C211yesmaximal, characteristic
12#21445755C211yesmaximal, characteristic
13#21445650C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1753C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22#16839772C418yescomplemented
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839666—21no—
34#16839666—21no—
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839650—21no—
38#16839650—21no—
39#16839626—21no—
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839620—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C22⋊C4×Dic61—21no—
46#16839607—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C22⋊C4×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—22no—
61C22×Dic122—22no—
62C22×Dic122—21no—
63C22×Dic122—22no—
64C22×Dic122—41no—
65C22×Dic122—41no—
66C22×Dic122C2×C414yes—
67C22×Dic122C2311yescharacteristic
68C22×Dic122C2311yescharacteristic
69C23×Dic61—21no—
70C4×Dic122—22no—
71C4×Dic122—22no—
72C4×Dic122—41no—
73C4×Dic122—41no—
74#14519736—22no—
75#14519736C2×C414yes—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94#14519711—41no—
95#14519709—41no—
96#14519702—22no—
97#14519702—22no—
98#14519702—41no—
99#14519702—41no—
100#14519702C2×C414yes—
101#14519702C2×C414yes—
102#14519702C2×C414yes—
103#14519702C2×C414yes—
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#14519684—21no—
114C23×C4×C61C2311yescharacteristic
115C2×M16×C61—41no—
116C2×M16×C61—41no—
117C2×M16×C61C2×C414yes—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C22⋊C4×C61D411yescharacteristic
123C2×C22⋊C4×C61D411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C22⋊C8×C61—22no—
127#97×C61—41no—
128C4×C8×C61—41no—
129#95×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131C24×C61C2×D411yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—24no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—28no—
163C2×Dic122—24no—
164C2×Dic122—21no—
165C2×Dic122—44no—
166C2×Dic122—44no—
167C2×Dic122—44no—
168C2×Dic122—42no—
169C2×Dic122—42no—
170C2×Dic122C22×C412yes—
171C2×Dic122C22×C412yes—
172C2×Dic122#4112yes—
173C2×Dic122#4112yes—
174M16×C61—44no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C2×C8×C61C22×C412yes—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C22⋊Dic61—24no—
194C22⋊Dic61—42no—
195C22⋊Dic61—81no—
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—21no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208#11758196—42no—
209#11758196—42no—
210#11758196—81no—
211#11758196—81no—
212C4×Dic61—24no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61C4×D412yes—
224C23×C61C22×D411yescharacteristic
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
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—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61C4×D412yes—
243C2×C4×C61#13712yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—48no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
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×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275Dic122—22no—
276Dic122—22no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122—48no—
281Dic122—41no—
282Dic122—41no—
283Dic122#11714yes—
284C8×C61—22no—
285C8×C61—22no—
286C8×C61—41no—
287C8×C61—41no—
288#22585—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—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61C2×#11711yescharacteristic
309C4×C61#51811yescharacteristic
310C4×C61#55111yescharacteristic
311C4×C61#55111yescharacteristic
312Dic61—24no—
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316Dic61—81no—
317Dic61—81no—
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#1763—611no—
333C4×#103—611no—
334#1753D6111yescharacteristic
335C2×C61—42no—
336C2×C61#234211yescharacteristic
337C2×C61C4×#13711yescharacteristic
338C2×C61#294711yescharacteristic
339C2×C61#331211yescharacteristic
340C2×C61#332111yescharacteristic
341C2×C61#332711yescharacteristic
342C2×C61#332911yescharacteristic
343C23×Q8—611no—
344C22×M16C2×D6111yescharacteristic
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
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×#103—611no—
358C2×#103—611no—
359C2×C22⋊C8C2×D6111yescharacteristic
360C2×#97C2×D6111yescharacteristic
361C2×C4×C8C2×D6111yescharacteristic
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⋊C4—611no—
376C4×C22⋊C4—1221no—
377C4×C22⋊C4C2×D6111yescharacteristic
378#376—1221no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#336C2×D6111yescharacteristic
386#336C2×D6111yescharacteristic
387C61#2258511yescharacteristic, complemented, Sylow
388C22×Q8—614no—
389C22×Q8—611no—
390C22×Q8—611no—
391C22×Q8—1222no—
392C22×Q8—1222no—
393C22×Q8—1221no—
394C22×Q8—1222no—
395C22×Q8—2441no—
396C22×Q8—2441no—
397C23×C4—1221no—
398C23×C4C22×D6111yescharacteristic
399C2×M16—41no—
400C2×M16—41no—
401C2×M16C4×D6114yes—
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×Q8—2441no—
410#117—614no—
411#117—1222no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4D24411yescharacteristic
426C2×C22⋊C4D24411yescharacteristic
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42C22×D6111yescharacteristic
433C2×C42C22×D6111yescharacteristic
434#107—2441no—
435#106—2441no—
436#103—614no—
437#103—614no—
438#103—614no—
439#103—614no—
440#103—1222no—
441#103—1222no—
442#103—2441no—
443#103—2441no—
444C22⋊C8—22no—
445#97—41no—
446C4×C8—41no—
447#95—1221no—
448#95—1221no—
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#95D24411yescharacteristic
458#95D24411yescharacteristic
459C24C2×D24411yescharacteristic
460C2×Q8—612no—
461C2×Q8—612no—
462C2×Q8—612no—
463C2×Q8—612no—
464C2×Q8—1228no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1224no—
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—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1224no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×C4×D6112yes—
495C22×C4C2×D24411yescharacteristic
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24411yescharacteristic
499C22×C4C2×D24411yescharacteristic
500C22×C4#1175822212yes—
501C22×C4C23×D6111yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8C2×C4×D6112yes—
514C2×C8C2×C4×D6112yes—
515C4⋊C4—612no—
516C4⋊C4—612no—
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—1221no—
526C4⋊C4—1221no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—2442no—
530C4⋊C4—2442no—
531C22⋊C4—22no—
532C22⋊C4—41no—
533C22⋊C4—1224no—
534C22⋊C4—4881no—
535C22⋊C4—2442no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—1224no—
541C42—2442no—
542C42—2442no—
543C42—2442no—
544C42—2442no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×D24412yes—
552C23C22×D24411yescharacteristic
553Q8—614no—
554Q8—1222no—
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558Q8—1222no—
559Q8—2448no—
560Q8—2441no—
561Q8—2441no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—2448no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
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×C4C4×D24412yes—
605C2×C4#1451981012yes—
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4#41×D6111yescharacteristic
609C2×C4C2×#1175822211yescharacteristic
610C2×C4#1451982211yescharacteristic
611C2×C4#1451982211yescharacteristic
612C8—22no—
613C8—22no—
614C8—41no—
615C8—41no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×D24411yescharacteristic
624C22#1684040711yescharacteristic
625C22#1684040711yescharacteristic
626C22C2×#1451981011yescharacteristic
627C22#1684056211yescharacteristic
628C22#1684057411yescharacteristic
629C22#1684057911yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—1224no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—4881no—
640C4—2442no—
641C4—2442no—
642C4—2442no—
643C4#117×D6111yescharacteristic
644C4#1684023511yescharacteristic
645C4#1684044111yescharacteristic
646C4#1684044111yescharacteristic
647C2—42no—
648C2#2144867111yescharacteristic
649C2C4×#1451981011yescharacteristic
650C2#2145466411yescharacteristic
651C2#2145787211yescharacteristic
652C2#2145800711yescharacteristic
653C2#2145808911yescharacteristic
654C2#2145811411yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1753C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1753D61
hypocenterC61#22585
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #21446509C2semidirect
5. #21446495C2semidirect
6. #21446461C2semidirect
7. #21446454C2semidirect
8. #21446090C2semidirect
9. #21446076C2semidirect
13. #21445650C2semidirect
14. C4×#14519702C2semidirect
22. #16839772C4semidirect
387. C61#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#168397722
#214487022
C22⋊C4×Dic1222
#214465092
#214464952
#214464612
#214464542
#214460902
#214460762
#214459152
#214458522
#214457552
#214456502
C4×#145197022
C61×#17532
#2258561