Finite Group Databasebeta

← #29707391

Subgroups — #29707391

Order 15616.

1214 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic, complemented
2#21454338C211yesmaximal, characteristic
3#21448411C211yesmaximal, characteristic
4#21446978C214yesmaximal, complemented
5#21446973C214yesmaximal, complemented
6C2×#16839530C211yesmaximal, characteristic, complemented
7#21445530C211yesmaximal, characteristic, complemented
8#21445528C211yesmaximal, characteristic, complemented
9C61×#2896C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11#16840390—22no—
12#16840390—22no—
13#16840390—22no—
14#16840390C2214yes—
15C2×#14519777—21no—
16C2×#14519776—21no—
17C2×#14519769—21no—
18C2×#14519768—21no—
19C2×#14519754—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22C22×C4⋊Dic61C2211yescharacteristic
23#16839771C2212yes—
24#16839767C2212yes—
25#16839763—21no—
26#16839763—21no—
27#16839758—22no—
28#16839669C2212yes—
29#16839665C2212yes—
30#16839642—22no—
31#16839622—22no—
32#16839530C2214yes—
33#16839523C2214yes—
34#16839521C2214yes—
35C22×#41×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#419C2212yes—
39C61×#418C2212yes—
40C61×#417—21no—
41C61×#417—21no—
42C61×#416—22no—
43C22×Dic122—22no—
44C23×Dic61D412yes—
45C2×#11758194—21no—
46C2×#11758194—22no—
47C2×#11758194—22no—
48C2×#11758194C2312yes—
49C22×#92418—21no—
50#14519777—24no—
51#14519777—24no—
52#14519776—24no—
53#14519776—24no—
54#14519769—24no—
55#14519769—24no—
56#14519768—24no—
57#14519768—24no—
58#14519754—28no—
59#14519754—28no—
60#14519736—21no—
61#14519736—21no—
62#14519736—22no—
63#14519736—22no—
64#14519736D412yes—
65#14519736C2312yes—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61D412yes—
73C2×C4⋊Dic61D411yescharacteristic
74C2×C4⋊Dic61D411yescharacteristic
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—22no—
77C2×#11758196—22no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—24no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2312yes—
83#14519701—22no—
84#14519701—22no—
85#14519698—22no—
86#14519698—22no—
87#14519684—24no—
88C2×#41×C61—22no—
89C2×#41×C61—21no—
90C2×#41×C61—21no—
91C2×#41×C61—21no—
92C2×#41×C61—21no—
93C2×#41×C61—22no—
94C2×#41×C61—22no—
95C2×#41×C61Q814yescomplemented
96C22×Q8×C61C2311yescharacteristic
97C22×D4×C61—21no—
98C22×D4×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C2×M16×C61—21no—
102C2×M16×C61—22no—
103C2×M16×C61—22no—
104C2×M16×C61C2312yes—
105C22×C8×C61—21no—
106#117×C61—21no—
107#117×C61—21no—
108#117×C61D412yes—
109#117×C61C2312yes—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#103×C61—22no—
115#103×C61—22no—
116#102×C61—22no—
117#102×C61—22no—
118C24×C61—21no—
119C24×C61C2×D411yescharacteristic
120#41×C61—24no—
121#41×C61—24no—
122#41×C61—24no—
123#41×C61—24no—
124#41×C61—42no—
125#41×C61—42no—
126#41×C61—42no—
127#41×C61—42no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61C2×Q812yescomplemented
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—41no—
141C2×D4×C61—41no—
142C2×D4×C61—42no—
143C2×D4×C61C2×Q812yescomplemented
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—24no—
148C22×C4×C61—41no—
149C22×C4×C61—41no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×Q812yes—
157C22×C4×C61#4112yes—
158C22×C4×C61#4112yes—
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—22no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—82no—
167C22×Dic61C2×D414yes—
168C2×Dic122—28no—
169C2×Dic122—84no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—42no—
173C2×C8×C61—22no—
174C2×C8×C61—42no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—42no—
186C22⋊Dic61—84no—
187C4⋊Dic61—28no—
188C4⋊Dic61—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—42no—
195C4⋊Dic61C2×D412yes—
196C4⋊Dic61C2×D412yes—
197#11758196—28no—
198#11758196—28no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203#11758194—22no—
204#11758194—22no—
205#11758194—42no—
206C2×#92418—22no—
207C2×#92418—42no—
208C23×C61—22no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61#12011yescharacteristic
216C23×C61#12011yescharacteristic
217C23×C61C22⋊Q812yes—
218C23×C61#12412yes—
219C23×C61C22×D411yescharacteristic
220Q8×C61—22no—
221Q8×C61—22no—
222Q8×C61—41no—
223Q8×C61—41no—
224D4×C61—22no—
225D4×C61—22no—
226D4×C61—44no—
227D4×C61—44no—
228D4×C61—41no—
229D4×C61—41no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—44no—
238C2×C4×C61—44no—
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61#12011yescharacteristic
241C2×C4×C61C22⋊Q812yes—
242C2×C4×C61C22⋊Q812yes—
243C2×C4×C61C22⋊Q812yes—
244C2×C4×C61C22⋊Q812yes—
245C2×C4×C61C22⋊Q812yes—
246C2×C4×C61#12412yes—
247C2×C4×C61#12412yes—
248C2×C4×C61#12412yes—
249C2×C4×C61#12412yes—
250C2×C4×C61#12412yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—24no—
260C2×Dic61—42no—
261C2×Dic61—44no—
262C2×Dic61—42no—
263C2×Dic61—44no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—82no—
268Dic122—88no—
269C8×C61—42no—
270#92418—42no—
271#23462—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—44no—
281C22×C61—42no—
282C22×C61—82no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61#39312yes—
286C22×C61#39312yes—
287C22×C61C2×#12011yescharacteristic
288C22×C61C2×C22⋊Q811yescharacteristic
289C22×C61C2×#12411yescharacteristic
290C4×C61—42no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61#39312yes—
294C4×C61#39312yes—
295C4×C61C2×C22⋊Q811yescharacteristic
296C4×C61C2×C22⋊Q811yescharacteristic
297C4×C61C2×#12411yescharacteristic
298C4×C61C2×#12411yescharacteristic
299Dic61—42no—
300Dic61—44no—
301Dic61—42no—
302Dic61—82no—
303C2×#544—611no—
304#2896—611no—
305#2896D6111yescharacteristic
306#2311—611no—
307#2028—614no—
308#2027—614no—
309C2×#343—611no—
310#1733—611no—
311#1733—611no—
312C2×C61—21no—
313C2×C61—21no—
314C2×C61—22no—
315C2×C61—22no—
316C2×C61—82no—
317C2×C61C2×#39311yescharacteristic
318C2×C61#301911yescharacteristic
319C2×C61#302011yescharacteristic
320C22×#41C2×D6111yescharacteristic
321C22×M16—611no—
322C22×M16C2×D6111yescharacteristic
323#544—614no—
324#544—1222no—
325#544—1222no—
326#544—1222no—
327C2×C4⋊Q8—1221no—
328C2×C4⋊Q8—1221no—
329C2×#125—1221no—
330C2×#125—1221no—
331C2×C22⋊Q8—1222no—
332C2×#117—611no—
333C2×#117—611no—
334C2×#117C2×D6111yescharacteristic
335C22×C4⋊C4—611no—
336#419—612no—
337#419C2×D6112yes—
338#418—612no—
339#418C2×D6112yes—
340#417—21no—
341#417—21no—
342#417—1221no—
343#417—1221no—
344#416—22no—
345#416—1222no—
346#391—612no—
347#390—612no—
348#386—1222no—
349#380—1222no—
350#343—614no—
351#339—614no—
352#339—614no—
353C61#2346211yescharacteristic, complemented, Sylow
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—22no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41Dic12214yescomplemented
362C22×Q8—1222no—
363C22×Q8C22×D6111yescharacteristic
364C22×D4—21no—
365C22×D4C22×D6111yescharacteristic
366C23×C4—21no—
367C23×C4—612no—
368C23×C4C22×D6111yescharacteristic
369C2×M16—21no—
370C2×M16—22no—
371C2×M16—22no—
372C2×M16—612no—
373C2×M16—1221no—
374C2×M16—1222no—
375C2×M16—1222no—
376C2×M16C22×D6112yes—
377C22×C8—21no—
378C22×C8—1221no—
379C4⋊Q8—1224no—
380C4⋊Q8—1224no—
381C4⋊Q8—1224no—
382C4⋊Q8—1224no—
383#125—1224no—
384#125—1224no—
385#125—1224no—
386#125—1224no—
387C22⋊Q8—1228no—
388C22⋊Q8—1228no—
389#117—21no—
390#117—21no—
391#117—612no—
392#117—612no—
393#117—1221no—
394#117—1221no—
395#117—1222no—
396#117—1222no—
397#117C22⋊D6112yes—
398#117C22×D6112yes—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4C22⋊D6111yescharacteristic
412C2×C4⋊C4C22⋊D6111yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1224no—
416C2×C22⋊C4—1221no—
417C2×C42—21no—
418C2×C42—612no—
419C2×C42—1221no—
420#103—22no—
421#103—22no—
422#103—1222no—
423#103—1222no—
424#102—22no—
425#102—22no—
426#102—1222no—
427#102—1222no—
428#95—1224no—
429C24—21no—
430C24D4×D6111yescharacteristic
431#41—24no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—42no—
436#41—42no—
437#41—42no—
438#41—42no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—1228no—
445C2×Q8—4884no—
446C2×Q8C2×Dic12212yescomplemented
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4—41no—
455C2×D4—42no—
456C2×D4C2×Dic12212yescomplemented
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—614no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—4882no—
471C22×C4C2×Dic12212yes—
472C22×C4#1175822212yes—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4#1175822412yes—
477C22×C4#1175822412yes—
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C23×D6111yescharacteristic
481M16—22no—
482M16—22no—
483M16—42no—
484M16—1222no—
485M16—1222no—
486M16—2442no—
487C2×C8—22no—
488C2×C8—42no—
489C2×C8—1222no—
490C2×C8—2442no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—612no—
494C4⋊C4—612no—
495C4⋊C4—1228no—
496C4⋊C4—1228no—
497C4⋊C4—1228no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—2442no—
505C4⋊C4C2×C22⋊D6112yes—
506C4⋊C4C2×C22⋊D6112yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—4884no—
512C22⋊C4—2442no—
513C42—22no—
514C42—22no—
515C42—1222no—
516C42—1222no—
517C42—1222no—
518C42—1222no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23#1451974511yescharacteristic
527C23#1451974511yescharacteristic
528C23#1451975412yes—
529C23#1451976312yes—
530C23C2×D4×D6111yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—4888no—
536D4—22no—
537D4—22no—
538D4—44no—
539D4—44no—
540D4—41no—
541D4—41no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1224no—
554C2×C4—4882no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4—2444no—
560C2×C4—2442no—
561C2×C4—2444no—
562C2×C4#1451974511yescharacteristic
563C2×C4#1451974511yescharacteristic
564C2×C4#1451975412yes—
565C2×C4#1451975412yes—
566C2×C4#1451975412yes—
567C2×C4#1451975612yes—
568C2×C4#1451975612yes—
569C2×C4#1451976312yes—
570C2×C4#1451976312yes—
571C2×C4#1451976312yes—
572C2×C4#1451976412yes—
573C2×C4#1451976412yes—
574C2×C4C2×D4×D6111yescharacteristic
575C2×C4C22×C22⋊D6111yescharacteristic
576C2×C4C22×Dic12211yescharacteristic
577C2×C4C2×#1175822411yescharacteristic
578C2×C4C2×#1175822411yescharacteristic
579C2×C4C2×#1175822211yescharacteristic
580C8—42no—
581C8—2442no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—42no—
592C22—82no—
593C22—81no—
594C22—81no—
595C22#1683967412yes—
596C22#1683967412yes—
597C22C2×#1451974511yescharacteristic
598C22C2×#1451975411yescharacteristic
599C22C2×#1451976311yescharacteristic
600C4—42no—
601C4—42no—
602C4—42no—
603C4—4882no—
604C4—2444no—
605C4—2442no—
606C4—2442no—
607C4#1683967412yes—
608C4#1683967412yes—
609C4C2×#1451975411yescharacteristic
610C4C2×#1451975611yescharacteristic
611C4C2×#1451976311yescharacteristic
612C4C2×#1451976411yescharacteristic
613C2—21no—
614C2—21no—
615C2—22no—
616C2—22no—
617C2—82no—
618C2C2×#1683967411yescharacteristic
619C2#2145517411yescharacteristic
620C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2896C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
hypercenter#2896D61
hypocenterC61#23462
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
1. C2×#16840390C2semidirect
4. #21446978C2semidirect
5. #21446973C2semidirect
6. C2×#16839530C2semidirect
7. #21445530C2semidirect
8. #21445528C2semidirect
95. C2×#41×C61Q8semidirect
133. C2×Q8×C61C2×Q8semidirect
143. C2×D4×C61C2×Q8semidirect
353. C61#23462semidirect
361. C2×#41Dic122semidirect
446. C2×Q8C2×Dic122semidirect
456. C2×D4C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
#214543382
#214484112
#214469782
#214469732
C2×#168395302
#214455302
#214455282
C61×#28962
#2346261