Finite Group Databasebeta

← #29698375

Subgroups — #29698375

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×#16839864C211yesmaximal, characteristic
2C2×#16839861C211yesmaximal, characteristic
3C2×#16839849C211yesmaximal, characteristic
4#21448731C211yesmaximal, characteristic, complemented
5#21448727C211yesmaximal, characteristic
6#21446555C211yesmaximal, characteristic, complemented
7#21446550C211yesmaximal, characteristic, complemented
8#21446084C211yesmaximal, characteristic, complemented
9#21446077C211yesmaximal, characteristic
10#21446070C211yesmaximal, characteristic, complemented
11#21446063C211yesmaximal, characteristic
12#21445831C211yesmaximal, characteristic, complemented
13#21445816C211yesmaximal, characteristic
14#21445691C211yesmaximal, characteristic, complemented
15C61×#1847C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839864C418yescomplemented
24#16839861—24no—
25#16839849—24no—
26C2×#14519711C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839658C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839657—21no—
41#16839657—21no—
42#16839636—21no—
43#16839636—21no—
44#16839631—21no—
45#16839621—21no—
46C4⋊C4×Dic61—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#95×D61—21no—
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C61×#384C2211yescharacteristic
61C61×#382C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190C2×C414yes—
68C2×#11758190D414yes—
69C2×#11758190Q814yes—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×Dic122—22no—
73C4×Dic122C2×C414yescomplemented
74#14519742—22no—
75#14519742—22no—
76#14519742—41no—
77#14519742—41no—
78C4×D244—22no—
79C4×D244C2×C414yes—
80#14519739—22no—
81#14519739—22no—
82#14519739—41no—
83#14519739—41no—
84C2×C4⋊Dic61D411yescharacteristic
85C2×C4⋊Dic61D411yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519711C2×C414yescomplemented
104C4⋊#92418—22no—
105#14519702C2×C414yescomplemented
106#14519700—22no—
107#14519700—22no—
108#14519699C2×C414yes—
109#14519697—22no—
110#14519697—22no—
111#14519687—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#107×C61—22no—
129#107×C61—41no—
130#106×C61—22no—
131#106×C61—41no—
132#97×C61C2×C414yescomplemented
133#95×C61—21no—
134#95×C61—21no—
135C23×D61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×D244C2×D412yes—
155C2×D244C2×Q812yes—
156C2×C4×D61—24no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
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×C61C22×C412yes—
174C2×C8×C61C22×C412yes—
175C2×C8×C61C2×D412yes—
176C2×C8×C61C2×Q812yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
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×C61—81no—
187C42×C61C22×C412yes—
188C42×C61#4112yes—
189#11758198—24no—
190#11758198—42no—
191#11758198—42no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—22no—
195C4⋊Dic61—41no—
196C4⋊Dic61C22×C412yes—
197C4⋊Dic61C22×C412yes—
198#11758196—24no—
199#11758196—42no—
200#11758196—42no—
201#11758196—81no—
202#11758196—81no—
203#11758196—81no—
204#11758196—81no—
205C4×Dic61—24no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218#11758190C4⋊C4116yescomplemented
219C23×C61C22×D411yescharacteristic
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
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×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—21no—
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—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61#13712yes—
272D244—22no—
273D244—22no—
274D244—22no—
275D244C2×C4⋊C414yes—
276C4×D61—84no—
277C4×D61—84no—
278C4×D61—84no—
279Dic122—22no—
280Dic122—22no—
281Dic122—22no—
282Dic122C2×C4⋊C414yes—
283C8×C61—22no—
284C8×C61C2×C4⋊C414yes—
285#92418—41no—
286#92418—41no—
287#22678—611nomaximal, Sylow
288C22×C61C2×C4×D411yescharacteristic
289C22×C61D4211yescharacteristic
290C22×C61D4×Q811yescharacteristic
291C22×C61C2×#13711yescharacteristic
292C22×C61#57511yescharacteristic
293C22×C61#57611yescharacteristic
294C22×C61#57711yescharacteristic
295C2×D61—24no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61—81no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C22×C4⋊C411yescharacteristic
309C4×C61#52011yescharacteristic
310C4×C61#54711yescharacteristic
311C4×C61#55411yescharacteristic
312Dic61—24no—
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
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#1847D6111yescharacteristic, complemented
331#1805—611no—
332#1802—611no—
333#1771—611no—
334C2×C61D4×C4⋊C411yescharacteristic
335C2×C61C4×#13711yescharacteristic
336C2×C61#294611yescharacteristic
337C2×C61#326611yescharacteristic
338C2×C61#326811yescharacteristic
339C2×C61#338511yescharacteristic
340C2×C61#338811yescharacteristic
341D61—44no—
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×D4—611no—
349#440—618no—
350#440—1224no—
351C4×SD16—1224no—
352C2×#107—611no—
353C2×#107C2×D6111yescharacteristic
354C2×#107C2×D6111yescharacteristic
355C2×#106C2×D6111yescharacteristic
356C2×#106C2×D6111yescharacteristic
357C2×C4⋊C8—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#97—611no—
365C2×#97C2×D6111yescharacteristic
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#382C2×D6111yescharacteristic
378#380—1221no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—1221no—
381C2×#95—1221no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#336—611no—
387#336—611no—
388C61#2267811yescharacteristic, complemented, Sylow
389C22×Q8—611no—
390C22×D4—611no—
391C23×C4—1221no—
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—614no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—2441no—
412C4×D4—2441no—
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—1221no—
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⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—1221no—
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—22no—
440#107—41no—
441#107—614no—
442#106—22no—
443#106—41no—
444C4⋊C8—1222no—
445#103—614no—
446#103—1222no—
447#103—1222no—
448#102—614no—
449#102—1222no—
450#102—1222no—
451#97—1222no—
452#97C4×D6114yescomplemented
453C4×C8—1222no—
454#95—21no—
455#95—21no—
456#95—611no—
457#95—611no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—1221no—
463C2×Q8—612no—
464C2×Q8—612no—
465C2×Q8—612no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×D4—612no—
470C2×D4—612no—
471C2×D4—612no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1224no—
487C22×C4—4881no—
488C22×C4—4881no—
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—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8C2×C4×D6112yes—
508C2×C8C2×C4×D6112yes—
509C2×C8D4×D6112yes—
510C2×C8Q8×D6112yes—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1224no—
527C4⋊C4—4881no—
528C4⋊C4—4881no—
529C4⋊C4—4881no—
530C4⋊C4—4881no—
531C4⋊C4—2441no—
532C4⋊C4—2442no—
533C4⋊C4—2442no—
534C22⋊C4—1224no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—1221no—
538C42—1221no—
539C42—1221no—
540C42—1221no—
541C42—1224no—
542C42—2442no—
543C42—2442no—
544C42C2×C4×D6112yes—
545C42#1175822612yes—
546C23—1222no—
547C23—1222no—
548C23—1222no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23C2×D4×D6111yescharacteristic
554Q8—614no—
555Q8—1222no—
556Q8—1222no—
557Q8—1222no—
558D4—614no—
559D4—1222no—
560D4—1222no—
561D4—1222no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
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—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—612no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—4884no—
596C2×C4—4884no—
597C2×C4—4884no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4#1451973912yes—
603C2×C4#1451973912yes—
604C2×C4C22×C4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C2×C4C2×Q8×D6111yescharacteristic
607C2×C4C2×#1175822611yescharacteristic
608C2×C4#1451982211yescharacteristic
609C2×C4#1451982511yescharacteristic
610C8—22no—
611C8—2441no—
612C8—2441no—
613C8C4⋊C4×D6114yes—
614C22—1224no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22C2×#1451973911yescharacteristic
622C22#1684039711yescharacteristic
623C22#1684042711yescharacteristic
624C22#137×D6111yescharacteristic
625C22#1684055811yescharacteristic
626C22#1684056611yescharacteristic
627C22#1684057111yescharacteristic
628C4—22no—
629C4—22no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—1224no—
635C4—1222no—
636C4—2442no—
637C4—2442no—
638C4—2441no—
639C4—2441no—
640C4—2441no—
641C4C2×C4⋊C4×D6111yescharacteristic
642C4#1684024311yescharacteristic
643C4#1684041511yescharacteristic
644C4#1684047311yescharacteristic
645C2—2444no—
646C2#2144871911yescharacteristic
647C2#2145463711yescharacteristic
648C2#2145464911yescharacteristic
649C2#2145732811yescharacteristic
650C2#2145735811yescharacteristic
651C2#2145872611yescharacteristic
652C2#2145875911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#22678
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #21448731C2semidirect
6. #21446555C2semidirect
7. #21446550C2semidirect
8. #21446084C2semidirect
10. #21446070C2semidirect
12. #21445831C2semidirect
14. #21445691C2semidirect
15. C61×#1847C2semidirect
23. #16839864C4semidirect
73. C4×Dic122C2×C4semidirect
103. #14519711C2×C4semidirect
105. #14519702C2×C4semidirect
132. #97×C61C2×C4semidirect
218. #11758190C4⋊C4semidirect
330. #1847D61semidirect
388. C61#22678semidirect
452. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
C2×#168398612
C2×#168398492
#214487312
#214487272
#214465552
#214465502
#214460842
#214460772
#214460702
#214460632
#214458312
#214458162
#214456912
C61×#18472
#2267861