Finite Group Databasebeta

← #29736923

Subgroups — #29736923

Order 15616.

1028 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840119C211yesmaximal, characteristic
2C2×#16839984C211yesmaximal, characteristic
3#21451307C211yesmaximal, characteristic
4#21446967C211yesmaximal, characteristic, complemented
5#21446962C211yesmaximal, characteristic
6#21446957C211yesmaximal, characteristic, complemented
7#21446952C211yesmaximal, characteristic
8#21446777C211yesmaximal, characteristic, complemented
9#21446756C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446174C211yesmaximal, characteristic
12#21446160C211yesmaximal, characteristic, complemented
13#21446160C211yesmaximal, characteristic, complemented
14C4×#14519701C211yesmaximal, characteristic, complemented
15C2×C61×#557C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19#16840119—24no—
20#16839984—24no—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839688—21no—
32#16839673—21no—
33#16839668C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839664C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839652—21no—
38#16839652—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C61×#557—24no—
51C61×#557—24no—
52C2×C4⋊Q8×C61—21no—
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#125×C61—21no—
55C2×#125×C61—21no—
56C2×#125×C61C2211yescharacteristic
57C2×#125×C61C2211yescharacteristic
58C2×C4×Q8×C61—21no—
59C2×C4×Q8×C61C2211yescharacteristic
60C2×C4×Q8×C61C2211yescharacteristic
61C2×C4×Q8×C61C2211yescharacteristic
62C2×C4×Q8×C61C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202—22no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201—22no—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519765—22no—
76#14519765—22no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C4⋊#92418—22no—
87#14519701—22no—
88#14519701—22no—
89#14519700—22no—
90#14519700—22no—
91#14519700—22no—
92#14519697—22no—
93#14519697—22no—
94#14519697—22no—
95C4×#92418—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101C22×Q8×C61—21no—
102C22×Q8×C61D411yescharacteristic
103C22×Q8×C61C2311yescharacteristic
104C4⋊Q8×C61—28no—
105C4⋊Q8×C61—22no—
106C4⋊Q8×C61—22no—
107#125×C61—28no—
108#125×C61—28no—
109#125×C61—22no—
110#125×C61—22no—
111#125×C61—22no—
112#125×C61D414yes—
113C4×Q8×C61—28no—
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×Q8×C61—22no—
117C4×Q8×C61—22no—
118C4×Q8×C61—22no—
119C4×Q8×C61—22no—
120C4×Q8×C61—22no—
121C4×Q8×C61D414yes—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61D411yescharacteristic
131C2×C4⋊C4×C61D411yescharacteristic
132C2×C4⋊C4×C61D411yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61—21no—
138C2×C42×C61—21no—
139C2×C42×C61C2311yescharacteristic
140C2×C42×C61C2311yescharacteristic
141C2×C42×C61C2311yescharacteristic
142C23×D61—21no—
143C2×Q8×C61—24no—
144C2×Q8×C61—22no—
145C2×Q8×C61—22no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—41no—
150C2×Q8×C61—42no—
151C2×Q8×C61—42no—
152C2×Q8×C61C2×D412yes—
153C2×Q8×C61#4112yes—
154C2×Q8×C61#4112yes—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—24no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61—24no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61C2×D412yes—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61C2×D412yes—
198C42×C61—24no—
199C42×C61—24no—
200C42×C61—24no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61C2×D412yes—
206C42×C61C2×D412yes—
207#11758202—44no—
208#11758201—44no—
209#11758198—42no—
210#11758198—42no—
211#11758198—42no—
212#11758198—42no—
213#11758198—81no—
214#11758198—81no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4×Dic61—81no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C23×C61C22×D411yescharacteristic
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61—44no—
244Q8×C61—48no—
245Q8×C61#12414yes—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—22no—
258C2×C4×C61—21no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61#12012yes—
274C2×C4×C61#12012yes—
275C2×C4×C61#13512yes—
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61C2×#4111yescharacteristic
280C2×C4×C612−1+411yescharacteristic
281C2×C4×C612−1+411yescharacteristic
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291D244—41no—
292D244—41no—
293D244—41no—
294D244—41no—
295Dic122—41no—
296Dic122—41no—
297Dic122—41no—
298Dic122—41no—
299#92418—41no—
300#92418—41no—
301#92418—41no—
302#92418—41no—
303#25937—611nomaximal, Sylow
304C22×C61C2×#12011yescharacteristic
305C22×C61#54811yescharacteristic
306C22×C61#54811yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61#57511yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C4×C61—24no—
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—22no—
322C4×C61—41no—
323C4×C61—42no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—42no—
328C4×C61C2×#12411yescharacteristic
329C4×C61#53611yescharacteristic
330C4×C61#54811yescharacteristic
331C4×C61#54811yescharacteristic
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335C2×#557D6111yescharacteristic, complemented
336C2×#488—611no—
337C2×#464—611no—
338#2592—611no—
339#2026—611no—
340#2025—611no—
341#2024—611no—
342#2023—611no—
343#1989—611no—
344#1984—611no—
345#1880—611no—
346#1880—611no—
347#1878—611no—
348#1878—611no—
349C4×#103—611no—
350C2×C61#243611yescharacteristic
351C2×C61#300211yescharacteristic
352C2×C61#301311yescharacteristic
353C2×C61#336411yescharacteristic
354C2×C61#336511yescharacteristic
355C2×C61#337111yescharacteristic
356C2×C61#337711yescharacteristic
357D61—161no—
358C22×Q16—611no—
359C22×SD16—611no—
360#557—24no—
361#557—24no—
362C2×C4⋊Q8—21no—
363C2×C4⋊Q8C2×D6111yescharacteristic
364C2×#125—21no—
365C2×#125—21no—
366C2×#125C2×D6111yescharacteristic
367C2×#125C2×D6111yescharacteristic
368C2×#124—611no—
369C2×C4×Q8—21no—
370C2×C4×Q8C2×D6111yescharacteristic
371C2×C4×Q8C2×D6111yescharacteristic
372C2×C4×Q8C2×D6111yescharacteristic
373C2×C4×Q8C2×D6111yescharacteristic
374#488—1224no—
375#464—1224no—
376C2×C4⋊C8—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#102—611no—
383C2×#102—611no—
384C2×#102—611no—
385C2×C4×C8—611no—
386#397—1221no—
387#393—1221no—
388#391—611no—
389#391—611no—
390#390—611no—
391#390—611no—
392#388—1221no—
393#388—1221no—
394C4×C4⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#336—611no—
402#336—611no—
403#336—611no—
404#336—611no—
405C61#2593711yescharacteristic, complemented, Sylow
406C22×Q8—21no—
407C22×Q8—611no—
408C22×Q8C22⋊D6111yescharacteristic
409C22×Q8C22×D6111yescharacteristic
410C22×D4—611no—
411C2×Q16—1222no—
412C2×Q16—1222no—
413C2×Q16—1222no—
414C2×SD16—1222no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8—611no—
421C4⋊Q8—28no—
422C4⋊Q8—22no—
423C4⋊Q8—22no—
424#125—28no—
425#125—28no—
426#125—22no—
427#125—22no—
428#125—22no—
429#125C22⋊D6114yes—
430#124—1222no—
431#124—1222no—
432C4×Q8—28no—
433C4×Q8—22no—
434C4×Q8—22no—
435C4×Q8—22no—
436C4×Q8—22no—
437C4×Q8—22no—
438C4×Q8—22no—
439C4×Q8—22no—
440C4×Q8C22⋊D6114yes—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4—21no—
447C2×C4⋊C4—21no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4C22⋊D6111yescharacteristic
452C2×C4⋊C4C22⋊D6111yescharacteristic
453C2×C4⋊C4C22⋊D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—21no—
464C2×C42—21no—
465C2×C42—21no—
466C2×C42—1221no—
467C2×C42C22×D6111yescharacteristic
468C2×C42C22×D6111yescharacteristic
469C2×C42C22×D6111yescharacteristic
470C4⋊C8—1222no—
471#103—1222no—
472#103—1222no—
473#103—1222no—
474#103—1222no—
475#103—1222no—
476#102—1222no—
477#102—1222no—
478#102—1222no—
479C4×C8—1222no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485C24—1221no—
486C2×Q8—24no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—22no—
491C2×Q8—22no—
492C2×Q8—41no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8#1175822612yes—
502C2×Q8#1175822612yes—
503C2×Q8C2×C22⋊D6112yes—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—1221no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C2×C22⋊D6111yescharacteristic
519C22×C4C2×C22⋊D6111yescharacteristic
520C22×C4C2×C22⋊D6111yescharacteristic
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C23×D6111yescharacteristic
524Q16—2444no—
525SD16—2444no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—24no—
547C4⋊C4—24no—
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4C2×C22⋊D6112yes—
557C4⋊C4C2×C22⋊D6112yes—
558C4⋊C4C2×C22⋊D6112yes—
559C22⋊C4—4881no—
560C22⋊C4—4881no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C42—24no—
566C42—24no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—24no—
572C42—4881no—
573C42C2×C22⋊D6112yes—
574C42C2×C22⋊D6112yes—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23C22×C22⋊D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—44no—
587Q8—48no—
588Q8—2441no—
589Q8—2441no—
590Q8—2441no—
591Q8—2441no—
592Q8#1451976614yes—
593D4—2441no—
594D4—2441no—
595D4—2441no—
596D4—2441no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4#1451974612yes—
634C2×C4#1451974612yes—
635C2×C4#1451979712yes—
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4C22×C22⋊D6111yescharacteristic
638C2×C4C2×#1175822611yescharacteristic
639C2×C4C2×#1175822611yescharacteristic
640C2×C4#1451982511yescharacteristic
641C2×C4#1451982511yescharacteristic
642C8—2441no—
643C8—2441no—
644C8—2441no—
645C8—2441no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22C2×#1451974611yescharacteristic
654C22#1684042311yescharacteristic
655C22#1684042311yescharacteristic
656C22C2×#1451979711yescharacteristic
657C22#1684056011yescharacteristic
658C22#1684057011yescharacteristic
659C22#1684057711yescharacteristic
660C4—24no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—42no—
670C4—4881no—
671C4—4881no—
672C4—4881no—
673C4C2×#1451976611yescharacteristic
674C4#1684033611yescharacteristic
675C4#1684042311yescharacteristic
676C4#1684042311yescharacteristic
677C2—9761no—
678C2#2144947311yescharacteristic
679C2#2145503711yescharacteristic
680C2#2145513411yescharacteristic
681C2#2145850411yescharacteristic
682C2#2145851511yescharacteristic
683C2#2145858711yescharacteristic
684C2#2145866111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#557C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#557D61
hypocenterC61#25937
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #21446967C2semidirect
6. #21446957C2semidirect
8. #21446777C2semidirect
9. #21446756C2semidirect
12. #21446160C2semidirect
13. #21446160C2semidirect
14. C4×#14519701C2semidirect
15. C2×C61×#557C2semidirect
335. C2×#557D61semidirect
405. C61#25937semidirect

Maximal subgroups

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

subgroupindex
C2×#168401192
C2×#168399842
#214513072
#214469672
#214469622
#214469572
#214469522
#214467772
#214467562
#214461742
#214461602
C4×#145197012
C2×C61×#5572
#2593761