Finite Group Databasebeta

← #29698382

Subgroups — #29698382

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2C2×#16839859C211yesmaximal, characteristic
3C2×#16839853C211yesmaximal, characteristic
4#21448731C211yesmaximal, characteristic, complemented
5#21448729C211yesmaximal, characteristic
6#21446556C211yesmaximal, characteristic, complemented
7#21446551C211yesmaximal, characteristic, complemented
8#21446088C211yesmaximal, characteristic, complemented
9#21446078C211yesmaximal, characteristic
10#21446074C211yesmaximal, characteristic, complemented
11#21446064C211yesmaximal, characteristic
12#21445985C211yesmaximal, characteristic, complemented
13#21445835C211yesmaximal, characteristic, complemented
14#21445696C211yesmaximal, characteristic, complemented
15C61×#1802C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16839862C418yescomplemented
23#16839859—24no—
24#16839853—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839657—21no—
42#16839657—21no—
43#16839637—21no—
44#16839637—21no—
45#16839631—21no—
46#16839628C2211yescharacteristic
47#16839622—21no—
48C4⋊C4×Dic61—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50C2×#14519684—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#384C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C2×#11758200C2×C414yes—
66C2×#11758200D414yes—
67C2×#11758200Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C4×Dic122—22no—
71C4×Dic122C2×C414yescomplemented
72#14519742—22no—
73#14519742—22no—
74#14519742—41no—
75#14519742—41no—
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—41no—
79D4×Dic61—41no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519710—22no—
100#14519710—41no—
101#14519710C2×C414yescomplemented
102#14519708—22no—
103#14519708—41no—
104#14519702—22no—
105#14519702—22no—
106#14519700C2×C414yescomplemented
107#14519698C2×C414yes—
108#14519688C2×C414yescomplemented
109#14519687—22no—
110C8×Dic61—22no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×D4×C61C2311yescharacteristic
120C23×C4×C61—21no—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×D4×C61—22no—
124C4×D4×C61C2×C414yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—22no—
133#102×C61—22no—
134#102×C61—22no—
135C24×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61C22×C412yes—
140C2×D4×C61C2×D412yes—
141C2×D4×C61C2×Q812yes—
142C22×C4×C61—21no—
143C22×C4×C61—24no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—81no—
147C22×C4×C61—81no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—81no—
160C22×Dic61—81no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C2×Dic122C2×D412yes—
170C2×Dic122C2×Q812yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—24no—
186C42×C61C22×C412yes—
187C42×C61#4112yes—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190#11758200C4⋊C4116yescomplemented
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—41no—
198C4⋊Dic61—81no—
199C4⋊Dic61—81no—
200C4⋊Dic61C22×C412yes—
201#11758196—24no—
202#11758196—42no—
203#11758196—42no—
204#11758196—81no—
205#11758196—81no—
206C4×Dic61—24no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C22×C412yes—
216C2×#92418C22×C412yes—
217C2×#92418C2×D412yes—
218C2×#92418C2×Q812yes—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61C22×D411yescharacteristic
227D4×C61—22no—
228D4×C61—22no—
229D4×C61—22no—
230D4×C61C2×C4⋊C414yes—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—84no—
245C2×C4×C61C4×D412yes—
246C2×C4×C61C4×D412yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×Q811yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—21no—
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—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—84no—
277C2×Dic61—84no—
278C2×Dic61#13712yes—
279Dic122—22no—
280Dic122—22no—
281Dic122—22no—
282Dic122C2×C4⋊C414yes—
283C8×C61—41no—
284C8×C61—41no—
285#92418—22no—
286#92418C2×C4⋊C414yes—
287#22678—611nomaximal, Sylow
288C22×C61—24no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61C2×C4×D411yescharacteristic
296C22×C61D4211yescharacteristic
297C22×C61D4×Q811yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61#57511yescharacteristic
300C22×C61#57611yescharacteristic
301C22×C61#57711yescharacteristic
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61C22×C4⋊C411yescharacteristic
308C4×C61#52011yescharacteristic
309C4×C61#54711yescharacteristic
310C4×C61#55411yescharacteristic
311Dic61—24no—
312Dic61—22no—
313Dic61—42no—
314Dic61—41no—
315Dic61—41no—
316Dic61—41no—
317Dic61—81no—
318Dic61—81no—
319C2×#440—611no—
320C2×#440—611no—
321C2×C4×SD16—611no—
322Q8×C4⋊C4—611no—
323#2349—611no—
324#1943—611no—
325#1942—611no—
326#1865—611no—
327#1864—611no—
328#1863—611no—
329#1862—611no—
330#1847—611no—
331#1805—611no—
332#1802D6111yescharacteristic
333#1771—611no—
334C2×C61—44no—
335C2×C61D4×C4⋊C411yescharacteristic
336C2×C61C4×#13711yescharacteristic
337C2×C61#294611yescharacteristic
338C2×C61#326611yescharacteristic
339C2×C61#326811yescharacteristic
340C2×C61#338511yescharacteristic
341C2×C61#338811yescharacteristic
342C22×SD16—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×D4—611no—
347C2×C4×D4—611no—
348C2×C4×D4C2×D6111yescharacteristic
349#440—618no—
350#440—1224no—
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#107—611no—
354C2×#107—611no—
355C2×#106—611no—
356C2×#106—611no—
357C2×C4⋊C8C2×D6111yescharacteristic
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102C2×D6111yescharacteristic
363C2×#102C2×D6111yescharacteristic
364C2×#97—611no—
365C2×#97—611no—
366C2×C4×C8—611no—
367#389—611no—
368#389—611no—
369#389—611no—
370#389—611no—
371#389—1221no—
372#389—1221no—
373#385—1221no—
374#385—1221no—
375#384—1221no—
376#384C2×D6111yescharacteristic
377#382—611no—
378#380—1221no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—1221no—
381C2×#95—1221no—
382#340—611no—
383#340—611no—
384#340C2×D6111yescharacteristic
385#340C2×D6111yescharacteristic
386#336—611no—
387#336—611no—
388C61#2267811yescharacteristic, complemented, Sylow
389C22×Q8—611no—
390C22×D4C22×D6111yescharacteristic
391C23×C4—21no—
392C23×C4—1221no—
393C23×C4—1221no—
394C2×SD16—614no—
395C2×SD16—614no—
396C2×SD16—614no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8C22×D6111yescharacteristic
400C22×C8C22×D6111yescharacteristic
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—2441no—
406C4×Q8—2441no—
407C4×D4—22no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—2441no—
411C4×D4—2441no—
412C4×D4C4×D6114yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439#107—614no—
440#107—1222no—
441#107—2441no—
442#106—1222no—
443#106—2441no—
444C4⋊C8—22no—
445#103—614no—
446#103—1222no—
447#103—1222no—
448#102—22no—
449#102—22no—
450#102—614no—
451#97—614no—
452#97—1222no—
453C4×C8—1222no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—21no—
463C2×Q8—612no—
464C2×Q8—612no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D6112yes—
473C2×D4D4×D6112yes—
474C2×D4Q8×D6112yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—4881no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4D4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498SD16—6116no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—612no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—612no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4—1224no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—4881no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—2441no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C4⋊C4C2×C4×D6112yes—
534C22⋊C4—24no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—1224no—
542C42—2442no—
543C42—2442no—
544C42C2×C4×D6112yes—
545C42#1175822412yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D6111yescharacteristic
554Q8—614no—
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C4⋊C4×D6114yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—612no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
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—4884no—
593C2×C4—4884no—
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#1451973712yes—
603C2×C4#1451973712yes—
604C2×C4C22×C4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×Q8×D6111yescharacteristic
607C2×C4C2×#1175822411yescharacteristic
608C2×C4#1451982311yescharacteristic
609C2×C4#1451982411yescharacteristic
610C8—41no—
611C8—41no—
612C8—614no—
613C8—1222no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×#1451973711yescharacteristic
622C22#1684039611yescharacteristic
623C22#1684042511yescharacteristic
624C22#137×D6111yescharacteristic
625C22#1684055811yescharacteristic
626C22#1684056311yescharacteristic
627C22#1684056811yescharacteristic
628C4—22no—
629C4—22no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—1224no—
634C4—1222no—
635C4—4881no—
636C4—4881no—
637C4—2442no—
638C4—2441no—
639C4—2441no—
640C4—2441no—
641C4C2×C4⋊C4×D6111yescharacteristic
642C4#1684024211yescharacteristic
643C4#1684041211yescharacteristic
644C4#1684047011yescharacteristic
645C2—44no—
646C2#2144871511yescharacteristic
647C2#2145464311yescharacteristic
648C2#2145464511yescharacteristic
649C2#2145733511yescharacteristic
650C2#2145736511yescharacteristic
651C2#2145872411yescharacteristic
652C2#2145875711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1802C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1802D61
hypocenterC61#22678
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
4. #21448731C2semidirect
6. #21446556C2semidirect
7. #21446551C2semidirect
8. #21446088C2semidirect
10. #21446074C2semidirect
12. #21445985C2semidirect
13. #21445835C2semidirect
14. #21445696C2semidirect
22. #16839862C4semidirect
71. C4×Dic122C2×C4semidirect
101. #14519710C2×C4semidirect
106. #14519700C2×C4semidirect
108. #14519688C2×C4semidirect
190. #11758200C4⋊C4semidirect
388. C61#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
C2×#168398592
C2×#168398532
#214487312
#214487292
#214465562
#214465512
#214460882
#214460782
#214460742
#214460642
#214459852
#214458352
#214456962
C61×#18022
#2267861