Finite Group Databasebeta

← #29718832

Subgroups — #29718832

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839918C211yesmaximal, characteristic
2C2×#16839820C211yesmaximal, characteristic
3#21451618C211yesmaximal, characteristic
4#21448405C211yesmaximal, characteristic
5#21447210C211yesmaximal, characteristic, complemented
6#21447196C211yesmaximal, characteristic, complemented
7#21447140C211yesmaximal, characteristic, complemented
8#21447140C211yesmaximal, characteristic, complemented
9#21446687C211yesmaximal, characteristic, complemented
10#21446677C211yesmaximal, characteristic, complemented
11#21446511C211yesmaximal, characteristic, complemented
12#21446497C211yesmaximal, characteristic, complemented
13#21446129C211yesmaximal, characteristic
14#21446016C211yesmaximal, characteristic
15C61×#1787C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839918—24no—
22#16839820—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839702—21no—
35#16839694—21no—
36#16839687—21no—
37#16839687—21no—
38#16839669—21no—
39#16839669—21no—
40#16839669C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839669C2211yescharacteristic
43#16839669C2211yescharacteristic
44#16839667C2211yescharacteristic
45#16839667C2211yescharacteristic
46#16839661C2211yescharacteristic
47#16839661C2211yescharacteristic
48#16839625—21no—
49#16839625—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C23×Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C61×#380C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—41no—
67C2×#11758202—41no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519756—22no—
71#14519756—22no—
72#14519736—22no—
73#14519736—22no—
74#14519736—41no—
75#14519736—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61D411yescharacteristic
87C2×C22⋊Dic61D411yescharacteristic
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519710—41no—
92#14519708—41no—
93#14519706—22no—
94#14519706—22no—
95#14519702—41no—
96#14519702—41no—
97#14519701—22no—
98#14519701—22no—
99C22⋊#92418D414yes—
100#14519687—22no—
101C8×Dic61—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C22×Q8×C61—22no—
117C22×Q8×C61—22no—
118C22×Q8×C61—21no—
119C22×Q8×C61—22no—
120C22×Q8×C61—22no—
121C22×Q8×C61—41no—
122C22×Q8×C61—41no—
123C22×Q8×C61C2311yescharacteristic
124C22×Q8×C61C2311yescharacteristic
125C23×C4×C61—21no—
126C23×C4×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132#103×C61—22no—
133#103×C61—22no—
134#103×C61—41no—
135#103×C61—41no—
136C22⋊C8×C61—22no—
137#95×C61—21no—
138C24×C61C2×D411yescharacteristic
139C2×Q8×C61—24no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—44no—
151C2×Q8×C61—44no—
152C2×Q8×C61—44no—
153C2×Q8×C61—44no—
154C2×Q8×C61—42no—
155C2×Q8×C61—42no—
156C2×Q8×C61#4112yes—
157C2×Q8×C61#4112yes—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—24no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—42no—
168C22×C4×C61—42no—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61#4112yes—
172C22×C4×C61C2411yescharacteristic
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D411yescharacteristic
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C22⋊C4×C61—81no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—42no—
201C22⋊Dic61—42no—
202#11758202—44no—
203#11758202—44no—
204C4⋊Dic61—22no—
205C4⋊Dic61—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—81no—
213C4⋊Dic61—81no—
214#11758196—42no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—42no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418C2×D412yes—
225C2×#92418C2×D412yes—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61#12112yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—48no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—48no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61#12112yes—
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—41no—
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—
291C2×Dic61—41no—
292C2×Dic61—41no—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—41no—
298C8×C61—41no—
299#92418—22no—
300#92418—22no—
301#24514—611nomaximal, Sylow
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61C2×#12111yescharacteristic
310C22×C61#52511yescharacteristic
311C22×C61#53511yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57811yescharacteristic
314C22×C61#57811yescharacteristic
315C22×C61#57811yescharacteristic
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61#54611yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#55111yescharacteristic
324C4×C61#55111yescharacteristic
325Dic61—41no—
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333C2×#451—611no—
334C2×#433—611no—
335#2625—611no—
336#2310—611no—
337#2068—611no—
338#2066—611no—
339#2056—611no—
340#2056—611no—
341#1971—611no—
342#1969—611no—
343#1935—611no—
344#1933—611no—
345#1873—611no—
346#1854—611no—
347#1787D6111yescharacteristic
348C2×C61—42no—
349C2×C61#241111yescharacteristic
350C2×C61#305911yescharacteristic
351C2×C61#306311yescharacteristic
352C2×C61#312111yescharacteristic
353C2×C61#312711yescharacteristic
354C2×C61#322311yescharacteristic
355C2×C61#322511yescharacteristic
356C23×Q8C2×D6111yescharacteristic
357C22×Q16—611no—
358C22×Q16—611no—
359C2×C22⋊Q8—611no—
360C2×#117—611no—
361C2×#117—611no—
362#451—1224no—
363#433—1224no—
364C2×#107—611no—
365C2×#106—611no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103C2×D6111yescharacteristic
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×#103C2×D6111yescharacteristic
376C2×C22⋊C8—611no—
377C2×C22⋊C8C2×D6111yescharacteristic
378C2×#97—611no—
379C2×C4×C8—611no—
380#400—1221no—
381#398—1221no—
382#397—1221no—
383#397—1221no—
384#391—611no—
385#391—611no—
386#391—611no—
387#391—611no—
388#391—611no—
389#391—611no—
390#391—1221no—
391#391—1221no—
392#389—611no—
393#389—611no—
394#381—1221no—
395#381—1221no—
396#380C2×D6111yescharacteristic
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401C61#2451411yescharacteristic, complemented, Sylow
402C22×Q8—22no—
403C22×Q8—22no—
404C22×Q8—21no—
405C22×Q8—22no—
406C22×Q8—22no—
407C22×Q8—41no—
408C22×Q8—41no—
409C22×Q8—611no—
410C22×Q8C22×D6111yescharacteristic
411C22×Q8C22×D6111yescharacteristic
412C23×C4—21no—
413C23×C4C22×D6111yescharacteristic
414C2×Q16—1222no—
415C2×Q16—1222no—
416C2×Q16—1222no—
417C2×Q16—1222no—
418C2×Q16—2441no—
419C2×Q16—2441no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426#117—1222no—
427#117—1222no—
428#117—2441no—
429#117—2441no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—611no—
442C2×C22⋊C4—611no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—1221no—
448#107—2441no—
449#106—2441no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#103—22no—
453#103—22no—
454#103—41no—
455#103—41no—
456#103—1222no—
457#103—1222no—
458#103—2441no—
459#103—2441no—
460C22⋊C8—22no—
461C22⋊C8—614no—
462#97—1222no—
463C4×C8—1222no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24D4×D6111yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8#1175822412yes—
504C2×Q8#1175822412yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4#1175822212yes—
525C22×C4D4×D6112yes—
526C22×C4D4×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528Q16—2444no—
529Q16—2444no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—612no—
537C2×C8—612no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C22⋊C4—81no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—1221no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—2442no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1451975012yes—
574C23C2×D4×D6111yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—22no—
579Q8—48no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584Q8—2441no—
585Q8—2441no—
586Q8—2441no—
587Q8—2441no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4#1451975012yes—
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C2×C4C2×#1175822411yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C8—41no—
639C8—41no—
640C8—1222no—
641C8—1222no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1451975011yescharacteristic
650C22#1684026811yescharacteristic
651C22#1684032711yescharacteristic
652C22#1684056811yescharacteristic
653C22#1684057511yescharacteristic
654C22#1684057811yescharacteristic
655C22#1684057811yescharacteristic
656C4—22no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—2441no—
666C4—2441no—
667C4—2441no—
668C4—2441no—
669C4#1684039911yescharacteristic
670C4#1684039911yescharacteristic
671C4#1684043811yescharacteristic
672C4#1684043811yescharacteristic
673C2—42no—
674C2#2144920611yescharacteristic
675C2#2145547311yescharacteristic
676C2#2145550811yescharacteristic
677C2#2145604111yescharacteristic
678C2#2145611311yescharacteristic
679C2#2145689811yescharacteristic
680C2#2145692611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1787D61
hypocenterC61#24514
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
5. #21447210C2semidirect
6. #21447196C2semidirect
7. #21447140C2semidirect
8. #21447140C2semidirect
9. #21446687C2semidirect
10. #21446677C2semidirect
11. #21446511C2semidirect
12. #21446497C2semidirect
401. C61#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#168399182
C2×#168398202
#214516182
#214484052
#214472102
#214471962
#214471402
#214466872
#214466772
#214465112
#214464972
#214461292
#214460162
C61×#17872
#2451461