Finite Group Databasebeta

← #29686384

Subgroups — #29686384

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839985C211yesmaximal, characteristic
2C2×#16839864C211yesmaximal, characteristic
3C2×C4×Dic244C211yesmaximal, characteristic, complemented
4#21448530C211yesmaximal, characteristic
5#21448182C211yesmaximal, characteristic, complemented
6#21446749C211yesmaximal, characteristic
7#21446578C211yesmaximal, characteristic, complemented
8#21446461C211yesmaximal, characteristic, complemented
9#21446440C211yesmaximal, characteristic
10#21446173C211yesmaximal, characteristic
11#21446159C211yesmaximal, characteristic, complemented
12#21445825C211yesmaximal, characteristic, complemented
13#21445650C211yesmaximal, characteristic, complemented
14C4×#14519699C211yesmaximal, characteristic
15C61×#1768C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24#16839985C418yescomplemented
25#16839864—24no—
26C4×Dic244—24no—
27C2×#14519711C2211yescharacteristic
28C2×#14519711C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839654—21no—
40#16839654—21no—
41#16839636—21no—
42#16839636—21no—
43#16839632C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C4⋊Dic61C2211yescharacteristic
46C4×C4⋊Dic61C2211yescharacteristic
47#16839613—21no—
48#16839613—21no—
49#16839613—21no—
50#16839607—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54#16839525C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#376C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×Dic244C2×C414yescomplemented
69C2×#11758190—22no—
70C2×#11758190—22no—
71C2×#11758190C2×C414yes—
72#14519765—22no—
73#14519765C2×C414yes—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122C2×C414yescomplemented
81C4×Dic122D414yes—
82C4×D244—22no—
83C4×D244—22no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107#14519711—22no—
108#14519711—22no—
109#14519702—22no—
110#14519702—22no—
111#14519702—22no—
112#14519702—22no—
113#14519702C2×C414yescomplemented
114#14519699—22no—
115#14519699—22no—
116#14519699C2×C414yes—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—22no—
130C4⋊C8×C61C2×C414yescomplemented
131#97×C61—22no—
132C4×C8×C61—22no—
133#95×C61D411yescharacteristic
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136#95×C61D411yescharacteristic
137C23×D61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C22×C412yes—
158C2×C4×D61—42no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C412yes—
171C2×Dic122C2×D412yes—
172C2×Dic122#4112yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
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×C61C22×C412yes—
184C2×C8×C61C22×C412yes—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61C22×C412yes—
188C42×C61C2×D412yes—
189C42×C61#4112yes—
190C42×C61#4112yes—
191#11758198—24no—
192#11758198—24no—
193#11758198—42no—
194#11758198—81no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—81no—
204C4⋊Dic61—81no—
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61#4112yes—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—42no—
212C4×Dic61—24no—
213C4×Dic61—24no—
214C4×Dic61—24no—
215C4×Dic61—42no—
216C4×Dic61—81no—
217C4×Dic61—81no—
218C4×Dic61—81no—
219Dic244—28no—
220#11758190—28no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
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×C61—21no—
243C2×C4×C61C4×D412yes—
244C2×C4×C61C4×D412yes—
245C2×C4×C61#12112yes—
246C2×C4×C61#12112yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
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—
274D244—22no—
275D244—22no—
276D244—41no—
277D244—41no—
278C4×D61—84no—
279Dic122—22no—
280Dic122—22no—
281Dic122—22no—
282Dic122—22no—
283Dic122—22no—
284Dic122—41no—
285Dic122—41no—
286Dic122C4×D414yes—
287C8×C61—22no—
288C8×C61—22no—
289C8×C61—41no—
290C8×C61—41no—
291#21512—611nomaximal, Sylow
292C22×C61C2×C4×D411yescharacteristic
293C22×C61C2×#12111yescharacteristic
294C22×C61#52511yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57711yescharacteristic
297C22×C61#57811yescharacteristic
298C22×C61#57811yescharacteristic
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—81no—
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—41no—
311C4×C61—41no—
312C4×C61C2×C4×D411yescharacteristic
313C4×C61C4×#4111yescharacteristic
314C4×C61#52911yescharacteristic
315C4×C61#53211yescharacteristic
316Dic61—24no—
317Dic61—42no—
318Dic61—42no—
319Dic61—42no—
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323C2×#464—611no—
324C2×#440—611no—
325C2×C4×Q16—611no—
326#2326—611no—
327#2276—611no—
328#1983—611no—
329#1947—611no—
330#1928—611no—
331#1925—611no—
332#1880—611no—
333#1878—611no—
334#1803—611no—
335#1768D6111yescharacteristic, complemented
336#1763—611no—
337C4×#102—611no—
338C2×C61C4×#12111yescharacteristic
339C2×C61#294711yescharacteristic
340C2×C61#295111yescharacteristic
341C2×C61#304411yescharacteristic
342C2×C61#304511yescharacteristic
343C2×C61#312111yescharacteristic
344C2×C61#312411yescharacteristic
345D61—82no—
346C22×Q16—611no—
347C22×SD16—611no—
348C2×#124—611no—
349C2×C4×Q8—611no—
350C2×C4×Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8—611no—
353C2×C4×D4—611no—
354#464—618no—
355#440—1224no—
356C4×Q16—1224no—
357C2×#107—611no—
358C2×#107—611no—
359C2×C4⋊C8C2×D6111yescharacteristic
360C2×C4⋊C8C2×D6111yescharacteristic
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#97C2×D6111yescharacteristic
370C2×C4×C8C2×D6111yescharacteristic
371#391—611no—
372#390—611no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#385—1221no—
377#384—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C4⋊C4—1221no—
383C4×C4⋊C4—1221no—
384#376—1221no—
385#376C2×D6111yescharacteristic
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#336C2×D6111yescharacteristic
391#336C2×D6111yescharacteristic
392C61#2151211yescharacteristic, complemented, Sylow
393C22×Q8—611no—
394C22×Q8—611no—
395C22×D4—611no—
396C23×C4—1221no—
397C2×Q16—614no—
398C2×Q16—1222no—
399C2×Q16—1222no—
400C2×SD16—614no—
401C2×SD16—1222no—
402C2×SD16—1222no—
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407#124—614no—
408#124—1222no—
409C4×Q8—614no—
410C4×Q8—614no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#107—1222no—
447C4⋊C8—22no—
448C4⋊C8C4×D6114yescomplemented
449#103—614no—
450#103—1222no—
451#103—1222no—
452#103—1222no—
453#103—1222no—
454#102—614no—
455#102—1222no—
456#102—1222no—
457#97—22no—
458C4×C8—22no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95D24411yescharacteristic
465#95D24411yescharacteristic
466#95D24411yescharacteristic
467#95D24411yescharacteristic
468C24—1221no—
469C2×Q8—612no—
470C2×Q8—612no—
471C2×Q8—612no—
472C2×Q8—612no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×D4—612no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—4881no—
496C22×C4—2442no—
497C22×C4C2×D24411yescharacteristic
498C22×C4C2×D24411yescharacteristic
499C22×C4C2×D24411yescharacteristic
500C22×C4C2×D24411yescharacteristic
501C22×C4C2×D24411yescharacteristic
502C22×C4C2×D24411yescharacteristic
503C22×C4C23×D6111yescharacteristic
504Q16—1228no—
505SD16—1228no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8C2×C4×D6112yes—
517C2×C8C2×C4×D6112yes—
518C4⋊C4—612no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—1224no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1224no—
526C4⋊C4—1224no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—4881no—
532C4⋊C4—4881no—
533C4⋊C4—2442no—
534C22⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—4881no—
537C22⋊C4—4881no—
538C22⋊C4—2442no—
539C42—21no—
540C42—21no—
541C42—1224no—
542C42—1224no—
543C42—1224no—
544C42—4881no—
545C42—4881no—
546C42—4881no—
547C42—2442no—
548C42C2×C4×D6112yes—
549C42#1175822212yes—
550C42D4×D6112yes—
551C42#1175822612yes—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C22×D24411yescharacteristic
560Q8—614no—
561Q8—1222no—
562Q8—1222no—
563Q8—1222no—
564Q8—1222no—
565Q8—1222no—
566Q8—2441no—
567Q8—2441no—
568D4—1222no—
569D4—1222no—
570D4—2441no—
571D4—2441no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
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—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4C4×D24412yes—
609C2×C4C4×D24412yes—
610C2×C4#1451975212yes—
611C2×C4#1451975212yes—
612C2×C4C22×C4×D6111yescharacteristic
613C2×C4C2×D4×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4C2×#1175822211yescharacteristic
617C2×C4C2×#1175822611yescharacteristic
618C8—22no—
619C8—22no—
620C8—41no—
621C8—41no—
622C22—4881no—
623C22—4881no—
624C22—4881no—
625C22—4881no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22C2×C4×D24411yescharacteristic
630C22C2×#1451975211yescharacteristic
631C22#1684027411yescharacteristic
632C22#1684056711yescharacteristic
633C22#1684057411yescharacteristic
634C22#1684057911yescharacteristic
635C22#1684057911yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—1224no—
643C4—4881no—
644C4—4881no—
645C4—2442no—
646C4—2442no—
647C4—2442no—
648C4—2442no—
649C4C4×D4×D6111yescharacteristic
650C4C4×#1175822611yescharacteristic
651C4#1684029911yescharacteristic
652C4#1684030911yescharacteristic
653C2—4882no—
654C2C4×#1451975211yescharacteristic
655C2#2145466411yescharacteristic
656C2#2145469611yescharacteristic
657C2#2145535011yescharacteristic
658C2#2145536011yescharacteristic
659C2#2145605011yescharacteristic
660C2#2145608311yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1768C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1768D61
hypocenterC61#21512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
3. C2×C4×Dic244C2semidirect
5. #21448182C2semidirect
7. #21446578C2semidirect
8. #21446461C2semidirect
11. #21446159C2semidirect
12. #21445825C2semidirect
13. #21445650C2semidirect
15. C61×#1768C2semidirect
24. #16839985C4semidirect
68. C2×Dic244C2×C4semidirect
80. C4×Dic122C2×C4semidirect
113. #14519702C2×C4semidirect
130. C4⋊C8×C61C2×C4semidirect
335. #1768D61semidirect
392. C61#21512semidirect
448. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399852
C2×#168398642
C2×C4×Dic2442
#214485302
#214481822
#214467492
#214465782
#214464612
#214464402
#214461732
#214461592
#214458252
#214456502
C4×#145196992
C61×#17682
#2151261