Finite Group Databasebeta

← #29718822

Subgroups — #29718822

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×#16839920C211yesmaximal, characteristic
2#21451616C211yesmaximal, characteristic
3#21448403C211yesmaximal, characteristic
4#21447213C211yesmaximal, characteristic, complemented
5#21447199C211yesmaximal, characteristic, complemented
6#21447142C211yesmaximal, characteristic, complemented
7#21447142C211yesmaximal, characteristic, complemented
8#21446690C211yesmaximal, characteristic, complemented
9#21446680C211yesmaximal, characteristic, complemented
10#21446509C211yesmaximal, characteristic, complemented
11#21446495C211yesmaximal, characteristic, complemented
12#21446129C211yesmaximal, characteristic
13#21446020C211yesmaximal, characteristic
14#21445755C211yesmaximal, characteristic
15C2×C61×#433C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×Dic244C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839920—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32#16839700—21no—
33#16839693—21no—
34#16839690—21no—
35#16839690—21no—
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839668C2211yescharacteristic
42#16839667—21no—
43#16839667—21no—
44#16839660C2211yescharacteristic
45#16839660C2211yescharacteristic
46#16839623—21no—
47#16839623—21no—
48#16839622C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C61×#433—24no—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
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
70C22×Dic122C2311yescharacteristic
71C23×Dic61—21no—
72C2×Dic244—22no—
73C2×Dic244—22no—
74C2×Dic244—22no—
75C2×Dic244—22no—
76C2×Dic244—41no—
77C2×Dic244—41no—
78#14519756—22no—
79#14519756—22no—
80#14519736—41no—
81#14519736—41no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C4×Dic61—21no—
94#14519711—41no—
95#14519709—41no—
96#14519702—22no—
97#14519702—22no—
98#14519702—22no—
99#14519702—22no—
100#14519702—41no—
101#14519702—41no—
102#14519702—41no—
103#14519702—41no—
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—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C23×C4×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#117×C61—22no—
125#117×C61—22no—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D411yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—22no—
133C4⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61D414yes—
136#97×C61—22no—
137C4×C8×C61—22no—
138C24×C61C2×D411yescharacteristic
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61#4112yes—
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—24no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—24no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
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—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61C2×D412yes—
198C2×C8×C61C2×D412yes—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—22no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61—21no—
206C42×C61—21no—
207C22⋊Dic61—42no—
208C22⋊Dic61—42no—
209C22⋊Dic61—81no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220#11758196—42no—
221#11758196—81no—
222#11758196—81no—
223C4×Dic61—42no—
224Dic244—44no—
225Dic244—44no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61#12112yes—
233C23×C61C22×D411yescharacteristic
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61#12112yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—48no—
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—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284Dic122—22no—
285Dic122—22no—
286Dic122—22no—
287Dic122—22no—
288Dic122—48no—
289Dic122—41no—
290Dic122—41no—
291Dic122—41no—
292Dic122—41no—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—22no—
298C8×C61—22no—
299C8×C61—41no—
300C8×C61—41no—
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—41no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61#54611yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#55111yescharacteristic
324C4×C61#55111yescharacteristic
325Dic61—42no—
326Dic61—42no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333C2×#451—611no—
334C2×#433D6111yescharacteristic
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#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⋊Q8—611no—
360C2×#117—611no—
361C2×#117C2×D6111yescharacteristic
362#451—1224no—
363#433—24no—
364C2×#107—611no—
365C2×#106—611no—
366C2×C4⋊C8C2×D6111yescharacteristic
367C2×C4⋊C8C2×D6111yescharacteristic
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×C22⋊C8C2×D6111yescharacteristic
377C2×C22⋊C8C2×D6111yescharacteristic
378C2×#97C2×D6111yescharacteristic
379C2×C4×C8C2×D6111yescharacteristic
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#340—611no—
400#340—611no—
401C61#2451411yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—611no—
404C22×Q8—611no—
405C22×Q8—1222no—
406C22×Q8—1222no—
407C22×Q8—1221no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—2441no—
411C22×Q8—2441no—
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×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426#117—22no—
427#117—22no—
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⋊C4D24411yescharacteristic
439C2×C4⋊C4D24411yescharacteristic
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4D24411yescharacteristic
444C2×C22⋊C4D24411yescharacteristic
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447C2×C42C22×D6111yescharacteristic
448#107—2441no—
449#106—2441no—
450C4⋊C8—22no—
451C4⋊C8—22no—
452#103—1222no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—2441no—
457#103—2441no—
458#103—2441no—
459#103—2441no—
460C22⋊C8—22no—
461C22⋊C8C22⋊D6114yes—
462#97—22no—
463C4×C8—22no—
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—
479C24C2×D24411yescharacteristic
480C2×Q8—612no—
481C2×Q8—612no—
482C2×Q8—1224no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
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—1224no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4C2×D24411yescharacteristic
521C22×C4C2×D24411yescharacteristic
522C22×C4C2×D24411yescharacteristic
523C22×C4C2×D24411yescharacteristic
524C22×C4C2×D24411yescharacteristic
525C22×C4#1175822212yes—
526C22×C4C2×C22⋊D6112yes—
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—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8C2×C22⋊D6112yes—
541C2×C8C2×C22⋊D6112yes—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
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—22no—
558C22⋊C4—22no—
559C22⋊C4—4881no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—2442no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1451975112yes—
574C23C22×D24411yescharacteristic
575Q8—1222no—
576Q8—1222no—
577Q8—1222no—
578Q8—1222no—
579Q8—2448no—
580Q8—2441no—
581Q8—2441no—
582Q8—2441no—
583Q8—2441no—
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—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
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#1451975112yes—
632C2×C4C22×C22⋊D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4#1451982311yescharacteristic
637C2×C4#1451982311yescharacteristic
638C8—22no—
639C8—22no—
640C8—41no—
641C8—41no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1451975111yescharacteristic
650C22#1684027411yescharacteristic
651C22#1684033111yescharacteristic
652C22#1684057411yescharacteristic
653C22#1684057911yescharacteristic
654C22#1684057911yescharacteristic
655C22#1684057911yescharacteristic
656C4—22no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—2442no—
668C4—2442no—
669C4#1684040011yescharacteristic
670C4#1684040011yescharacteristic
671C4#1684043611yescharacteristic
672C4#1684043611yescharacteristic
673C2—42no—
674C2#2144919811yescharacteristic
675C2#2145547411yescharacteristic
676C2#2145551011yescharacteristic
677C2#2145605011yescharacteristic
678C2#2145612011yescharacteristic
679C2#2145690411yescharacteristic
680C2#2145693011yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#433C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#433D61
hypocenterC61#24514
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
4. #21447213C2semidirect
5. #21447199C2semidirect
6. #21447142C2semidirect
7. #21447142C2semidirect
8. #21446690C2semidirect
9. #21446680C2semidirect
10. #21446509C2semidirect
11. #21446495C2semidirect
401. C61#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#168399202
#214516162
#214484032
#214472132
#214471992
#214471422
#214466902
#214466802
#214465092
#214464952
#214461292
#214460202
#214457552
C2×C61×#4332
#2451461