Finite Group Databasebeta

← #29718834

Subgroups — #29718834

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×#16839917C211yesmaximal, characteristic
2C2×#16839820C211yesmaximal, characteristic
3#21451617C211yesmaximal, characteristic
4#21448403C211yesmaximal, characteristic
5#21447215C211yesmaximal, characteristic, complemented
6#21447201C211yesmaximal, characteristic, complemented
7#21447140C211yesmaximal, characteristic, complemented
8#21447140C211yesmaximal, characteristic, complemented
9#21446687C211yesmaximal, characteristic, complemented
10#21446677C211yesmaximal, characteristic, complemented
11#21446507C211yesmaximal, characteristic, complemented
12#21446493C211yesmaximal, characteristic, complemented
13#21446018C211yesmaximal, characteristic
14#21445755C211yesmaximal, characteristic
15C61×#1873C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16839917—24no—
21#16839820—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839701—21no—
36#16839692—21no—
37#16839690—21no—
38#16839690—21no—
39#16839669C2211yescharacteristic
40#16839669C2211yescharacteristic
41#16839668—21no—
42#16839668—21no—
43#16839667C2211yescharacteristic
44#16839667C2211yescharacteristic
45#16839667C2211yescharacteristic
46#16839667C2211yescharacteristic
47#16839658C2211yescharacteristic
48#16839658C2211yescharacteristic
49#16839623—21no—
50#16839623—21no—
51#16839622C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#117×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—22no—
65C22×Dic122—22no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202—22no—
74C2×#11758202—22no—
75C2×#11758202—41no—
76C2×#11758202—41no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79#14519736—22no—
80#14519736—22no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×C22⋊Dic61D411yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519710—41no—
93#14519708—41no—
94#14519706—22no—
95#14519706—22no—
96#14519702—22no—
97#14519702—22no—
98#14519702—41no—
99#14519702—41no—
100#14519700—22no—
101#14519700—22no—
102C22⋊#92418D414yes—
103#14519687—22no—
104C8×Dic61—22no—
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—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120C22×Q8×C61C2311yescharacteristic
121C23×C4×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—22no—
125C22⋊Q8×C61—22no—
126#117×C61—41no—
127#117×C61—41no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61—21no—
133C2×C22⋊C4×C61—21no—
134C2×C42×C61—21no—
135#103×C61—41no—
136#103×C61—41no—
137C22⋊C8×C61—22no—
138C24×C61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61#4112yes—
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—24no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—24no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—44no—
180C2×Dic122—44no—
181C2×Dic122—44no—
182C2×Dic122—44no—
183C2×Dic122—42no—
184C2×Dic122—42no—
185C2×Dic122#4112yes—
186C2×Dic122#4112yes—
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—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—42no—
200C22⋊C4×C61—42no—
201C22⋊C4×C61—42no—
202C42×C61—42no—
203C22⋊Dic61—22no—
204C22⋊Dic61—22no—
205C22⋊Dic61—81no—
206#11758202—44no—
207#11758202—44no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214#11758196—81no—
215#11758196—81no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
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—41no—
235Q8×C61—41no—
236Q8×C61—41no—
237Q8×C61—41no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#12112yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
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—48no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288Dic122—22no—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122—48no—
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—81no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61#54611yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#55111yescharacteristic
324C4×C61#55111yescharacteristic
325Dic61—42no—
326Dic61—42no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
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#1873D6111yescharacteristic
346#1854—611no—
347#1787—611no—
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×Q8—611no—
357C22×Q16—611no—
358C22×Q16—611no—
359C2×C22⋊Q8C2×D6111yescharacteristic
360C2×#117—611no—
361C2×#117C2×D6111yescharacteristic
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×#103—611no—
373C2×#103—611no—
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#380—611no—
397#340—611no—
398#340—611no—
399#340C2×D6111yescharacteristic
400#340C2×D6111yescharacteristic
401C61#2451411yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—611no—
404C22×Q8—1222no—
405C22×Q8—1222no—
406C22×Q8—1221no—
407C22×Q8—1222no—
408C22×Q8—1222no—
409C22×Q8—2441no—
410C22×Q8—2441no—
411C22×Q8C22×D6111yescharacteristic
412C23×C4—1221no—
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—22no—
425C22⋊Q8—22no—
426#117—41no—
427#117—41no—
428#117—1222no—
429#117—1222no—
430C2×C4⋊C4—21no—
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⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—611no—
443C2×C22⋊C4—611no—
444C2×C22⋊C4—1221no—
445C2×C42—21no—
446C2×C42—611no—
447C2×C42—611no—
448#107—2441no—
449#106—2441no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452#103—41no—
453#103—41no—
454#103—1222no—
455#103—1222no—
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—1221no—
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—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—1224no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—2444no—
500C2×Q8—2444no—
501C2×Q8—2444no—
502C2×Q8—2444no—
503C2×Q8—2442no—
504C2×Q8—2442no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1224no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—2442no—
523C22×C4—2442no—
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—21no—
547C4⋊C4—21no—
548C4⋊C4—42no—
549C4⋊C4—1222no—
550C4⋊C4—1222no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—4881no—
556C4⋊C4—4881no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—4881no—
562C42—42no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—1221no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1451975012yes—
574C23C2×D4×D6111yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579Q8—1222no—
580Q8—1222no—
581Q8—1222no—
582Q8—1222no—
583Q8—2448no—
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—41no—
595C2×C4—41no—
596C2×C4—41no—
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—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2448no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4#1451975012yes—
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
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#1684027011yescharacteristic
651C22#1684032711yescharacteristic
652C22#1684056911yescharacteristic
653C22#1684057611yescharacteristic
654C22#1684057811yescharacteristic
655C22#1684057811yescharacteristic
656C4—22no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—4881no—
662C4—4881no—
663C4—2441no—
664C4—2441no—
665C4—2441no—
666C4—2441no—
667C4—2442no—
668C4—2442no—
669C4#1684040111yescharacteristic
670C4#1684040111yescharacteristic
671C4#1684044111yescharacteristic
672C4#1684044111yescharacteristic
673C2—42no—
674C2#2144920711yescharacteristic
675C2#2145547311yescharacteristic
676C2#2145551311yescharacteristic
677C2#2145605111yescharacteristic
678C2#2145611511yescharacteristic
679C2#2145690011yescharacteristic
680C2#2145693211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1873C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1873D61
hypocenterC61#24514
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
5. #21447215C2semidirect
6. #21447201C2semidirect
7. #21447140C2semidirect
8. #21447140C2semidirect
9. #21446687C2semidirect
10. #21446677C2semidirect
11. #21446507C2semidirect
12. #21446493C2semidirect
401. C61#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
C2×#168398202
#214516172
#214484032
#214472152
#214472012
#214471402
#214466872
#214466772
#214465072
#214464932
#214460182
#214457552
C61×#18732
#2451461