Finite Group Databasebeta

← #29683402

Subgroups — #29683402

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic
2C2×#16839801C211yesmaximal, characteristic
3C2×#16839714C211yesmaximal, characteristic
4#21448300C211yesmaximal, characteristic
5#389×D61C211yesmaximal, characteristic
6#21446559C211yesmaximal, characteristic, complemented
7#21446557C211yesmaximal, characteristic, complemented
8#21446554C211yesmaximal, characteristic, complemented
9#21446551C211yesmaximal, characteristic, complemented
10#21446550C211yesmaximal, characteristic, complemented
11#21446548C211yesmaximal, characteristic, complemented
12#21446546C211yesmaximal, characteristic, complemented
13#21445537C211yesmaximal, characteristic
14#21445537C211yesmaximal, characteristic
15C61×#1944C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839803—24no—
25#16839801C418yescomplemented
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32#16839714—24no—
33C2×#14519688C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839657—21no—
44#16839636—21no—
45#16839633—21no—
46C4⋊C4×Dic61—21no—
47#16839613—21no—
48#95×D61—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C2×#11758189—22no—
68C2×#11758189—22no—
69C2×#11758189C2×C414yes—
70C2×#11758189D414yes—
71C2×#11758189Q814yes—
72C2×#11758189Q814yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519735—22no—
76#14519735—22no—
77#14519735—22no—
78#14519735C2×C414yes—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—22no—
93C4⋊C4×D61—22no—
94C4⋊C4×D61—44no—
95C4⋊C4×D61C2×C414yes—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61D411yescharacteristic
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103C42×D61—22no—
104C42×D61—22no—
105#14519711C2×C414yescomplemented
106#14519710—22no—
107#14519710C2×C414yescomplemented
108#14519709—22no—
109#14519708—22no—
110#14519708C2×C414yescomplemented
111#14519688—22no—
112C8×Dic61—22no—
113C8×Dic61—22no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#107×C61—22no—
130#106×C61C2×C414yescomplemented
131#97×C61—22no—
132#95×C61—21no—
133#95×C61—21no—
134C23×D61C2×D411yescharacteristic
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×C4×D61—24no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×Q812yes—
164C2×C4×D61C2×Q812yes—
165C2×C4×D61#4112yes—
166C2×C4×D61#4112yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C22×C412yes—
170C2×C8×C61C2×D412yes—
171C2×C8×C61C2×Q812yes—
172C2×C8×C61C2×Q812yes—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61C22×C412yes—
182C4⋊C4×C61C22×C412yes—
183C42×C61—21no—
184C42×C61—21no—
185#11758198—24no—
186#11758198—42no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197#11758196—24no—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201#11758196—42no—
202#11758196—42no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—24no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C22×C412yes—
212C2×#92418C2×D412yes—
213C2×#92418C2×Q812yes—
214C2×#92418C2×Q812yes—
215#11758189—28no—
216#11758189C4⋊C4116yescomplemented
217C23×C61C22×D411yescharacteristic
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61C4×D412yes—
223C22×D61#12112yes—
224C22×D61C4⋊Q812yes—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
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×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×Q811yescharacteristic
246C2×C4×C61C22×Q811yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—41no—
256C2×Dic61—41no—
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×Dic61C4×D412yes—
266C2×Dic61#12112yes—
267C2×Dic61C4⋊Q812yes—
268C2×Dic61AGL(1,ℤ/8)12yes—
269C2×Dic61#13712yes—
270C4×D61—44no—
271C4×D61—44no—
272C4×D61—48no—
273C4×D61—84no—
274C4×D61C2×C4⋊C414yes—
275C4×D61C4×Q814yes—
276C4×D61C22⋊Q814yes—
277C4×D61#12514yes—
278C8×C61—22no—
279C8×C61C2×C4⋊C414yescomplemented
280#92418—22no—
281#92418C2×C4⋊C414yescomplemented
282#21159—611nomaximal, Sylow
283C22×C61C2×C4×D411yescharacteristic
284C22×C61C2×#12111yescharacteristic
285C22×C61C2×C4⋊Q811yescharacteristic
286C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C61C2×#13711yescharacteristic
288C22×C61#57511yescharacteristic
289C22×C61#57511yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61#38914yes—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61C22×C4⋊C411yescharacteristic
304C4×C61C2×C4×Q811yescharacteristic
305C4×C61C2×C22⋊Q811yescharacteristic
306C4×C61C2×#12511yescharacteristic
307Dic61—22no—
308Dic61—22no—
309Dic61—42no—
310Dic61—42no—
311Dic61—81no—
312Dic61—81no—
313Dic61#38914yes—
314C2×#428—611no—
315C2×#428—611no—
316C2×C4×M16—611no—
317#2296—611no—
318C2×#389—611no—
319#1944—611no—
320#1944D6111yescharacteristic, complemented
321#1943—611no—
322#1943—611no—
323#1942—611no—
324#1942—611no—
325#1941—611no—
326#1941—611no—
327#1736—611no—
328#1736—611no—
329C2×C61C2×#38911yescharacteristic
330C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
331C2×C61C4×#13711yescharacteristic
332C2×C61#305511yescharacteristic
333C2×C61#305611yescharacteristic
334C2×C61#316511yescharacteristic
335C2×C61#316611yescharacteristic
336D61—24no—
337C22×M16—611no—
338C22×M16—611no—
339C2×#117—611no—
340C2×#117—611no—
341C22×C4⋊C4—611no—
342C22×C4⋊C4—611no—
343C22×C4⋊C4—1221no—
344C22×C42—611no—
345#428—618no—
346#428—1224no—
347C2×#107—611no—
348C2×#107—611no—
349C2×#107—611no—
350C2×#107C2×D6111yescharacteristic
351C2×#106—611no—
352C2×#106—611no—
353C2×#106—611no—
354C2×#106C2×D6111yescharacteristic
355C4×M16—1224no—
356C2×#97—611no—
357C2×#97C2×D6111yescharacteristic
358C2×C4×C8—611no—
359C2×C4×C8—611no—
360#389—611no—
361#389—611no—
362#389—611no—
363#389—611no—
364#389—611no—
365#389—611no—
366#389—1221no—
367#389—1221no—
368#389C2×D6111yescharacteristic
369#389C2×D6111yescharacteristic
370#385—1221no—
371#385—1221no—
372C4×C4⋊C4—1221no—
373C4×C4⋊C4—1221no—
374C2×#95—1221no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#340C2×D6111yescharacteristic
383C61#2115911yescharacteristic, complemented, Sylow
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—1221no—
388C23×C4—1221no—
389C2×M16—614no—
390C2×M16—614no—
391C2×M16—614no—
392C2×M16—614no—
393C2×M16—1222no—
394C2×M16—1222no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#117—614no—
400#117—1222no—
401#117—1222no—
402#117—1222no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—614no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—2444no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1222no—
432C2×C42—1222no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42C22×D6111yescharacteristic
436#107—22no—
437#107—614no—
438#107—614no—
439#107—1222no—
440#106—614no—
441#106—1222no—
442#106—1222no—
443#106C4×D6114yescomplemented
444#97—22no—
445#97—1222no—
446C4×C8—1222no—
447C4×C8—1222no—
448#95—21no—
449#95—21no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—611no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—611no—
462C22×C4—612no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1224no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—2441no—
480C22×C4—2441no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4—2442no—
486C22×C4D4×D6111yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489M16—6116no—
490M16—1228no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—612no—
494C2×C8—612no—
495C2×C8—612no—
496C2×C8—612no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8C2×C4×D6112yes—
500C2×C8D4×D6112yes—
501C2×C8Q8×D6112yes—
502C2×C8Q8×D6112yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4C2×C4×D6112yes—
528C4⋊C4C2×C4×D6112yes—
529C22⋊C4—1224no—
530C22⋊C4—2442no—
531C42—21no—
532C42—21no—
533C42—1222no—
534C42—1222no—
535C42—1224no—
536C42—1221no—
537C42—1221no—
538C42—2442no—
539C23—612no—
540C23—612no—
541C23—612no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23C2×D4×D6111yescharacteristic
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—614no—
569C2×C4—614no—
570C2×C4—614no—
571C2×C4—614no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—4884no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2448no—
584C2×C4—2441no—
585C2×C4—2441no—
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×C4C22×C4×D6111yescharacteristic
595C2×C4C2×D4×D6111yescharacteristic
596C2×C4C2×Q8×D6111yescharacteristic
597C2×C4C2×Q8×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C8—22no—
601C8—614no—
602C8—1222no—
603C8C4⋊C4×D6114yescomplemented
604C22—614no—
605C22—1222no—
606C22—1222no—
607C22—1222no—
608C22—1222no—
609C22—1222no—
610C22—1222no—
611C22C4×D4×D6111yescharacteristic
612C22#121×D6111yescharacteristic
613C22C4⋊Q8×D6111yescharacteristic
614C22AGL(1,ℤ/8)×D6111yescharacteristic
615C22#137×D6111yescharacteristic
616C22#1684055811yescharacteristic
617C22#1684056111yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—614no—
625C4—1222no—
626C4—1222no—
627C4—4881no—
628C4—4881no—
629C4—2442no—
630C4—2442no—
631C4C2×C4⋊C4×D6111yescharacteristic
632C4C4×Q8×D6111yescharacteristic
633C4C22⋊Q8×D6111yescharacteristic
634C4#125×D6111yescharacteristic
635C2—1224no—
636C2#389×D6111yescharacteristic
637C2#2145462411yescharacteristic
638C2#2145464111yescharacteristic
639C2#2145543711yescharacteristic
640C2#2145544811yescharacteristic
641C2#2145636411yescharacteristic
642C2#2145637011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1944C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1944D61
hypocenterC61#21159
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
6. #21446559C2semidirect
7. #21446557C2semidirect
8. #21446554C2semidirect
9. #21446551C2semidirect
10. #21446550C2semidirect
11. #21446548C2semidirect
12. #21446546C2semidirect
15. C61×#1944C2semidirect
25. #16839801C4semidirect
105. #14519711C2×C4semidirect
107. #14519710C2×C4semidirect
110. #14519708C2×C4semidirect
130. #106×C61C2×C4semidirect
216. #11758189C4⋊C4semidirect
279. C8×C61C2×C4⋊C4semidirect
281. #92418C2×C4⋊C4semidirect
320. #1944D61semidirect
383. C61#21159semidirect
443. #106C4×D61semidirect
603. C8C4⋊C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
C2×#168398012
C2×#168397142
#214483002
#389×D612
#214465592
#214465572
#214465542
#214465512
#214465502
#214465482
#214465462
#214455372
C61×#19442
#2115961