Finite Group Databasebeta

← #29688479

Subgroups — #29688479

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840133C211yesmaximal, characteristic
2C2×#16839865C211yesmaximal, characteristic, complemented
3C2×#16839863C211yesmaximal, characteristic, complemented
4C2×#16839861C211yesmaximal, characteristic, complemented
5C2×#16839859C211yesmaximal, characteristic, complemented
6#21448388C211yesmaximal, characteristic, complemented
7#21448383C211yesmaximal, characteristic
8#21446703C211yesmaximal, characteristic, complemented
9#21446700C211yesmaximal, characteristic, complemented
10#21446699C211yesmaximal, characteristic, complemented
11#21446623C211yesmaximal, characteristic
12#21446619C211yesmaximal, characteristic, complemented
13#21445977C211yesmaximal, characteristic, complemented
14#97×Dic61C211yesmaximal, characteristic, complemented
15C61×#1973C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519773C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24C2×#14519739C2211yescharacteristic
25C2×#14519737—22no—
26#16840133C418yescomplemented
27#16839865—24no—
28#16839863—24no—
29#16839861—24no—
30#16839859—24no—
31C2×#14519711C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519710C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519698C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519688C2211yescharacteristic
41C2×#14519687C2211yescharacteristic
42C2×#14519687C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44C2×C8×Dic61C2211yescharacteristic
45#16839668C2211yescharacteristic
46#16839664C2211yescharacteristic
47#16839662—22no—
48#16839659—22no—
49#16839631C2211yescharacteristic
50#16839631C2211yescharacteristic
51#16839613—22no—
52C42×Dic61C2211yescharacteristic
53C22×SD16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—22no—
63C23×Dic61—21no—
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190C2×C414yescomplemented
68C2×#11758200—22no—
69C2×#11758200—22no—
70C2×#11758200C2×C414yescomplemented
71C2×#11758201—22no—
72C2×#11758201—22no—
73C2×#11758201C2×C414yescomplemented
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519775—22no—
77#14519775C2×C414yescomplemented
78#14519773—22no—
79#14519773C2×C414yes—
80Q8×Dic61—22no—
81Q8×Dic61—22no—
82#14519742—22no—
83#14519742—22no—
84D4×Dic61—22no—
85D4×Dic61—22no—
86#14519739—22no—
87#14519739—22no—
88#14519737—48no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—22no—
93C2×#11758196—22no—
94C2×#11758196—21no—
95C2×#11758196—22no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—22no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61D412yes—
102C2×C4×Dic61D412yes—
103C2×C4×Dic61C2311yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105#14519711—22no—
106#14519710—22no—
107#14519710—22no—
108#14519702—22no—
109#14519701—22no—
110#14519700—22no—
111#14519699—22no—
112#14519698—22no—
113#14519697—22no—
114#14519688—22no—
115#14519687D414yescomplemented
116#14519687D414yescomplemented
117C8×Dic61—22no—
118C8×Dic61C2×C414yescomplemented
119#14519684—22no—
120C22×Q8×C61C2311yescharacteristic
121C22×D4×C61C2311yescharacteristic
122C2×SD16×C61—22no—
123C2×SD16×C61—22no—
124C2×SD16×C61C2×C414yescomplemented
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C22⋊C4×C61—22no—
130C2×C42×C61C2311yescharacteristic
131#107×C61—22no—
132#103×C61—22no—
133#102×C61—22no—
134#97×C61—22no—
135#95×C61—22no—
136C24×C61—21no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D412yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—28no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—44no—
162C22×Dic61—42no—
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D412yes—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244C22×C412yes—
172C2×C4×D61—44no—
173C2×C4×D61—42no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22×C412yes—
180SD16×C61—28no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C22×C412yes—
184C2×C8×C61C2×D412yes—
185C2×C8×C61C2×D412yes—
186C2×C8×C61#4112yes—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C22⋊C4×C61—44no—
192C42×C61—21no—
193C42×C61—21no—
194C22⋊Dic61—42no—
195#11758201—28no—
196#11758200—28no—
197#11758198—44no—
198#11758198—42no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—42no—
204#11758196—28no—
205#11758196—42no—
206#11758196—44no—
207#11758196—44no—
208C4×Dic61—24no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61C22×C412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C22×C412yes—
219C2×#92418C2×D412yes—
220C2×#92418C2×D412yes—
221C2×#92418#4112yes—
222#11758190—28no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—42no—
227C23×C61—42no—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—42no—
233C22×D61—42no—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—41no—
241D4×C61—41no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C22⋊D61—416no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—84no—
269C2×Dic61C4×D414yes—
270C2×Dic61#12712yes—
271C2×Dic61#12712yes—
272C2×Dic61AGL(1,ℤ/8)14yes—
273C2×Dic61#13714yes—
274D244—22no—
275D244—22no—
276D244—41no—
277D244—41no—
278C4×D61—84no—
279Dic122—22no—
280Dic122—22no—
281Dic122—41no—
282Dic122—41no—
283C8×C61—22no—
284C8×C61C4×D414yescomplemented
285#92418—22no—
286#92418C4×D414yescomplemented
287#21692—611nomaximal, Sylow
288C22×C61—42no—
289C22×C61—44no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61—82no—
293C22×C61C2×C4×D411yescharacteristic
294C22×C61C2×#12711yescharacteristic
295C22×C61#53211yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)12yes—
297C22×C61C2×#13712yes—
298C2×D61—42no—
299C2×D61—44no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—82no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61C2×C4×D411yescharacteristic
308C4×C61C2×C4×D411yescharacteristic
309C4×C61#53211yescharacteristic
310C4×C61#53211yescharacteristic
311Dic61—22no—
312Dic61—22no—
313Dic61—42no—
314Dic61—81no—
315Dic61—81no—
316Dic61#44018yescomplemented
317C2×#492—611no—
318C2×#440—611no—
319C2×#440—611no—
320C2×#440—611no—
321C2×#440—611no—
322C4×C4⋊Q8—611no—
323C4×#127—611no—
324#1973—611no—
325#1973—611no—
326#1973—611no—
327#1973D6111yescharacteristic, complemented
328#1957—611no—
329#1956—611no—
330#1845—611no—
331C4×#97—611no—
332C2×C61—82no—
333C2×C61C4×#12711yescharacteristic
334C2×C61C2×#44012yes—
335C2×C61#315111yescharacteristic
336C2×C61#315211yescharacteristic
337C2×C61#318212yes—
338D61—82no—
339C22×SD16—611no—
340C22×SD16—611no—
341C22×SD16—611no—
342C22×SD16C2×D6111yescharacteristic
343C2×C4⋊Q8—611no—
344C2×#127—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4—1222no—
350#492—618no—
351#440—1224no—
352#440—1224no—
353#440—1224no—
354#440—1224no—
355C2×#107—611no—
356C2×#107—611no—
357C2×#107—611no—
358C2×#107C2×D6111yescharacteristic
359C2×#103—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103C2×D6111yescharacteristic
363C2×#102—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102C2×D6111yescharacteristic
367C2×#97—611no—
368C2×#97—611no—
369C2×#97—611no—
370C2×#97C2×D6111yescharacteristic
371C2×C4×C8—611no—
372C2×C4×C8—611no—
373#391—611no—
374#391C2×D6111yescharacteristic
375#390—611no—
376#390—1222no—
377#390C2×D6111yescharacteristic
378#389—1222no—
379#384—611no—
380#384—611no—
381C4×C4⋊C4—1222no—
382C43—611no—
383C61#2169211yescharacteristic, complemented, Sylow
384C22×Q8—611no—
385C22×Q8C22×D6111yescharacteristic
386C22×D4—611no—
387C22×D4—1222no—
388C22×D4C22×D6111yescharacteristic
389C23×C4—1221no—
390C23×C4—1221no—
391C2×SD16—22no—
392C2×SD16—22no—
393C2×SD16—614no—
394C2×SD16—614no—
395C2×SD16—614no—
396C2×SD16—1222no—
397C2×SD16—1222no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C2×SD16—1222no—
401C2×SD16—1222no—
402C2×SD16C4×D6114yescomplemented
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C4⋊Q8—614no—
408C4⋊Q8—1222no—
409#127—614no—
410#127—1222no—
411C4×Q8—1222no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×D4—1222no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—2448no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C22⋊C4—22no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1222no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—612no—
436C2×C42—612no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42C22×D6111yescharacteristic
440#107—22no—
441#107—1222no—
442#107—1222no—
443#107—1222no—
444#103—22no—
445#103—1222no—
446#103—1222no—
447#103—1222no—
448#102—22no—
449#102—1222no—
450#102—1222no—
451#102—1222no—
452#97—22no—
453#97—614no—
454#97—614no—
455#97—1222no—
456C4×C8—614no—
457C4×C8—1222no—
458#95—22no—
459#95—1222no—
460C24—21no—
461C24—1221no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—612no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8C2×C4×D6112yes—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—612no—
480C2×D4—1228no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—2444no—
487C2×D4—2444no—
488C2×D4C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—612no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—2442no—
499C22×C4—2444no—
500C22×C4—2442no—
501C22×C4—2444no—
502C22×C4D4×D6112yes—
503C22×C4C23×D6111yescharacteristic
504SD16—28no—
505SD16—1228no—
506SD16—1228no—
507SD16—1228no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—612no—
511C2×C8—612no—
512C2×C8—612no—
513C2×C8—612no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8C2×C4×D6112yes—
517C2×C8D4×D6112yes—
518C2×C8D4×D6112yes—
519C2×C8#1175822412yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—1228no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—2444no—
530C4⋊C4—2444no—
531C4⋊C4—2442no—
532C4⋊C4—2442no—
533C22⋊C4—44no—
534C22⋊C4—2444no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—21no—
538C42—21no—
539C42—612no—
540C42—612no—
541C42—612no—
542C42—612no—
543C42—1224no—
544C42—1224no—
545C42—2442no—
546C42—2442no—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—1222no—
553C23—2442no—
554C23—2442no—
555C23—2441no—
556C23—2441no—
557C23C2×D4×D6111yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—1222no—
563Q8—1222no—
564Q8—2441no—
565Q8—2441no—
566D4—22no—
567D4—22no—
568D4—41no—
569D4—41no—
570D4—1222no—
571D4—1222no—
572D4—24416no—
573D4—2441no—
574D4—2441no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—614no—
586C2×C4—614no—
587C2×C4—614no—
588C2×C4—612no—
589C2×C4—612no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—4884no—
594C2×C4—4884no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4C2×D4×D6111yescharacteristic
604C2×C4#41×D6111yescharacteristic
605C2×C4#41×D6111yescharacteristic
606C2×C4C2×#1175822411yescharacteristic
607C8—22no—
608C8—614no—
609C8—1222no—
610C8#1451973714yescomplemented
611C22—44no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—82no—
616C22—4881no—
617C22—4881no—
618C22—4882no—
619C22—2444no—
620C22—2442no—
621C22C4×D4×D6111yescharacteristic
622C22#127×D6111yescharacteristic
623C22#1684030711yescharacteristic
624C22AGL(1,ℤ/8)×D6112yes—
625C22#137×D6112yes—
626C4—42no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—618no—
631C4—1222no—
632C4—1222no—
633C4—4881no—
634C4—4881no—
635C4—2442no—
636C4C4×D4×D6111yescharacteristic
637C4C2×#1451973711yescharacteristic
638C4#1684030711yescharacteristic
639C4#1684031111yescharacteristic
640C2—82no—
641C2—4882no—
642C2#2144838311yescharacteristic
643C2#440×D6112yes—
644C2#2145626811yescharacteristic
645C2#2145627511yescharacteristic
646C2#2145653212yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1973C2
socleC23×C61C22×D4
prime radicalC2×#14519773C22
cosocleC22×C4×C61C24
hypercenter#1973D61
hypocenterC61#21692
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. C2×#16839865C2semidirect
3. C2×#16839863C2semidirect
4. C2×#16839861C2semidirect
5. C2×#16839859C2semidirect
6. #21448388C2semidirect
8. #21446703C2semidirect
9. #21446700C2semidirect
10. #21446699C2semidirect
12. #21446619C2semidirect
13. #21445977C2semidirect
14. #97×Dic61C2semidirect
15. C61×#1973C2semidirect
26. #16840133C4semidirect
67. C2×#11758190C2×C4semidirect
70. C2×#11758200C2×C4semidirect
73. C2×#11758201C2×C4semidirect
77. #14519775C2×C4semidirect
115. #14519687D4semidirect
116. #14519687D4semidirect
118. C8×Dic61C2×C4semidirect
124. C2×SD16×C61C2×C4semidirect
284. C8×C61C4×D4semidirect
286. #92418C4×D4semidirect
316. Dic61#440semidirect
327. #1973D61semidirect
383. C61#21692semidirect
402. C2×SD16C4×D61semidirect
610. C8#14519737semidirect

Maximal subgroups

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

subgroupindex
C2×#168401332
C2×#168398652
C2×#168398632
C2×#168398612
C2×#168398592
#214483882
#214483832
#214467032
#214467002
#214466992
#214466232
#214466192
#214459772
#97×Dic612
C61×#19732
#2169261