Finite Group Databasebeta

← #29681539

Subgroups — #29681539

Order 15616.

1702 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459022C211yesmaximal, characteristic
2C2×#16839779C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4#21448405C211yesmaximal, characteristic, complemented
5#21446190C218yesmaximal, complemented
6#21445557C211yesmaximal, characteristic, complemented
7#21445530C211yesmaximal, characteristic, complemented
8C22×#137×C61C211yesmaximal, characteristic
9C22×#11758194C2211yescharacteristic
10#16840245—22no—
11#16840245C2214yes—
12#16840240—21no—
13#16840237—22no—
14#16840237—21no—
15#41×Dic61—22no—
16#41×Dic61—22no—
17C2×#14519736—22no—
18C2×#14519736C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16839779—24no—
22C2×#14519704C2214yes—
23#16839771—22no—
24#16839771C2214yescomplemented
25C2×#14519701—22no—
26#16839669—22no—
27#16839669C2214yescomplemented
28#16839625—22no—
29#16839530C2214yescomplemented
30#16839523C2214yescomplemented
31C22×#41×C61C2211yescharacteristic
32C23×Q8×C61C2211yescharacteristic
33C2×#137×C61—24no—
34C2×#137×C61—22no—
35C2×#137×C61C418yescomplemented
36C2×#137×C61C2214yes—
37C22×Q16×C61—21no—
38C22×SD16×C61—21no—
39C22×M16×C61C2211yescharacteristic
40C2×#11758194—21no—
41C2×#11758194—22no—
42C2×#11758194—22no—
43C2×#11758194C2312yes—
44C22×#92418—21no—
45Q8×Dic61—24no—
46Q8×Dic61—24no—
47D4×Dic61—28no—
48D4×Dic61—24no—
49D4×Dic61—24no—
50D4×Dic61—28no—
51#14519736—24no—
52#14519736—24no—
53#14519736—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58#14519736—21no—
59#14519736—24no—
60#14519736—22no—
61#14519736—22no—
62#14519736D412yes—
63#14519736C2312yes—
64C2×C4⋊Dic61—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D412yes—
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—24no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—21no—
71C2×C22⋊Dic61D412yes—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—21no—
74C2×C4×Dic61D412yes—
75C2×C4×Dic61C2312yes—
76#14519704—28no—
77#14519701—28no—
78#14519684—28no—
79C2×#41×C61—21no—
80C2×#41×C61—21no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61C2×C414yes—
84C2×#41×C61D414yes—
85C2×#41×C61D412yes—
86C2×#41×C61C2312yes—
87C22×Q8×C61—21no—
88C22×Q8×C61—22no—
89C22×Q8×C61—21no—
90C22×Q8×C61—21no—
91C22×Q8×C61—22no—
92C22×Q8×C61—21no—
93C22×Q8×C61—21no—
94C22×Q8×C61—22no—
95C22×Q8×C61C2×C414yescomplemented
96C22×Q8×C61D411yescharacteristic
97C22×Q8×C61C2312yes—
98C22×D4×C61—21no—
99C22×D4×C61D411yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103#137×C61—216no—
104#137×C61—28no—
105#137×C61C2×C4116yescomplemented
106C2×Q16×C61—28no—
107C2×Q16×C61—24no—
108C2×SD16×C61—28no—
109C2×SD16×C61—24no—
110C2×M16×C61—22no—
111C2×M16×C61—21no—
112C2×M16×C61C2×C414yescomplemented
113C2×M16×C61C2312yes—
114C22×C8×C61—21no—
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117#41×C61—24no—
118#41×C61—24no—
119#41×C61—24no—
120#41×C61—24no—
121#41×C61—24no—
122#41×C61—22no—
123#41×C61—22no—
124#41×C61C22⋊C418yes—
125#41×C61C22×C414yes—
126#41×C61C2×D414yes—
127C2×Q8×C61—24no—
128C2×Q8×C61—21no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—24no—
132C2×Q8×C61—24no—
133C2×Q8×C61—24no—
134C2×Q8×C61—24no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—41no—
140C2×Q8×C61—41no—
141C2×Q8×C61—42no—
142C2×Q8×C61—42no—
143C2×Q8×C61—42no—
144C2×Q8×C61—42no—
145C2×Q8×C61—42no—
146C2×Q8×C61C22⋊C414yes—
147C2×Q8×C61C22×C414yes—
148C2×Q8×C61C2×D412yes—
149C2×D4×C61—24no—
150C2×D4×C61—22no—
151C2×D4×C61—24no—
152C2×D4×C61—22no—
153C2×D4×C61—21no—
154C2×D4×C61—22no—
155C2×D4×C61C22⋊C414yes—
156C2×D4×C61C2×D412yes—
157C22×C4×C61—21no—
158C22×C4×C61—24no—
159C22×C4×C61—22no—
160C22×C4×C61—24no—
161C22×C4×C61—22no—
162C22×C4×C61—24no—
163C22×C4×C61—22no—
164C22×C4×C61—41no—
165C22×C4×C61—41no—
166C22×C4×C61—42no—
167C22×C4×C61C22×C412yes—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61#4112yes—
175C22×C4×C61#4112yes—
176C22×C4×C61C2411yescharacteristic
177C22×Dic61—28no—
178C22×Dic61—22no—
179C22×Dic61—22no—
180C22×Dic61C2×D414yes—
181Q16×C61—216no—
182SD16×C61—216no—
183M16×C61—24no—
184M16×C61—22no—
185M16×C61C22×C414yes—
186C2×C8×C61—24no—
187C2×C8×C61—22no—
188C22⋊Dic61—28no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—42no—
194C4⋊Dic61—28no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—42no—
199C4⋊Dic61C2×D414yes—
200C4×Dic61—28no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207#11758194—22no—
208#11758194—22no—
209#11758194—42no—
210C2×#92418—22no—
211C2×#92418—42no—
212C23×C61—22no—
213C23×C61—24no—
214C23×C61—42no—
215C23×C61—82no—
216C23×C61C4×D412yes—
217C23×C61#12011yescharacteristic
218C23×C61#12011yescharacteristic
219C23×C61#12112yes—
220C23×C61C22×D411yescharacteristic
221Q8×C61—22no—
222Q8×C61—24no—
223Q8×C61—24no—
224Q8×C61—24no—
225Q8×C61—42no—
226Q8×C61—42no—
227Q8×C61—44no—
228Q8×C61—44no—
229Q8×C61C2×C22⋊C414yes—
230D4×C61—24no—
231D4×C61—24no—
232D4×C61—28no—
233D4×C61—22no—
234D4×C61—24no—
235D4×C61C2×C22⋊C414yes—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—28no—
241C2×C4×C61—24no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—44no—
245C2×C4×C61—44no—
246C2×C4×C61C2×C22⋊C412yes—
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C4×D412yes—
249C2×C4×C61C4×D412yes—
250C2×C4×C61C4×D412yes—
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12112yes—
254C2×C4×C61#12112yes—
255C2×C4×C61#12112yes—
256C2×C4×C61#12312yes—
257C2×C4×C61#12312yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—44no—
267C2×Dic61—44no—
268C2×Dic61—42no—
269C2×Dic61—82no—
270C2×Dic61—88no—
271C8×C61—24no—
272#92418—42no—
273#20944—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—24no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—44no—
281C22×C61—84no—
282C22×C61#38612yes—
283C22×C61#38612yes—
284C22×C61C2×C4×D411yescharacteristic
285C22×C61C2×#12011yescharacteristic
286C22×C61C2×#12111yescharacteristic
287C4×C61—24no—
288C4×C61—44no—
289C4×C61#38612yes—
290C4×C61#38612yes—
291C4×C61C22×C22⋊C411yescharacteristic
292C4×C61C2×C4×D411yescharacteristic
293C4×C61C2×#12111yescharacteristic
294C4×C61C2×#12311yescharacteristic
295Dic61—44no—
296Dic61—44no—
297Dic61—82no—
298C22×#137D6111yescharacteristic
299#3428—611no—
300C2×#421—611no—
301C2×#419—611no—
302#2310—611no—
303#1882—618no—
304#1742—611no—
305#1733—611no—
306C2×C61—21no—
307C2×C61—22no—
308C2×C61—21no—
309C2×C61—22no—
310C2×C61—44no—
311C2×C61C2×#38611yescharacteristic
312C2×C61#302212yes—
313C22×#41C2×D6111yescharacteristic
314C23×Q8C2×D6111yescharacteristic
315C2×#137—24no—
316C2×#137—22no—
317C2×#137Dic6118yescomplemented
318C2×#137C2×D6114yes—
319C22×Q16—21no—
320C22×SD16—21no—
321C22×M16—611no—
322C22×M16C2×D6111yescharacteristic
323#520—614no—
324#520—1222no—
325#519—1221no—
326#518—1222no—
327#518—1221no—
328C4×#41—1222no—
329C4×#41—1222no—
330C2×#117—611no—
331C2×#117—611no—
332C2×#117—611no—
333C2×#117—1222no—
334#421—1224no—
335C2×#104—614no—
336#419—614no—
337#419—1222no—
338C2×#103—1222no—
339#391—614no—
340#391—1222no—
341#381—1222no—
342#343—614no—
343#339—614no—
344C61#2094411yescharacteristic, complemented, Sylow
345C2×#41—21no—
346C2×#41—21no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41C2×Dic6114yes—
350C2×#41C22⋊D6114yes—
351C2×#41C22⋊D6112yes—
352C2×#41C22×D6112yes—
353C22×Q8—21no—
354C22×Q8—22no—
355C22×Q8—21no—
356C22×Q8—21no—
357C22×Q8—22no—
358C22×Q8—21no—
359C22×Q8—21no—
360C22×Q8—22no—
361C22×Q8C2×Dic6114yescomplemented
362C22×Q8C22⋊D6111yescharacteristic
363C22×Q8C22×D6112yes—
364C22×D4—21no—
365C22×D4C22⋊D6111yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4C22×D6111yescharacteristic
369#137—216no—
370#137—28no—
371#137C2×Dic61116yescomplemented
372C2×Q16—28no—
373C2×Q16—24no—
374C2×SD16—28no—
375C2×SD16—24no—
376C2×M16—21no—
377C2×M16—22no—
378C2×M16—612no—
379C2×M16—1221no—
380C2×M16—1222no—
381C2×M16—1222no—
382C2×M16C2×Dic6114yescomplemented
383C2×M16C22×D6112yes—
384C22×C8—21no—
385C22×C8—1221no—
386C4×Q8—1224no—
387C4×Q8—1224no—
388C4×D4—1228no—
389C4×D4—1228no—
390C4×D4—1224no—
391C4×D4—1224no—
392#117—612no—
393#117—612no—
394#117—1224no—
395#117—1224no—
396#117—1224no—
397#117—1221no—
398#117—1221no—
399#117—1221no—
400#117—1221no—
401#117—1221no—
402#117—1221no—
403#117—1222no—
404#117—1222no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4—1221no—
409C2×C22⋊C4—612no—
410C2×C22⋊C4—1224no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C42—612no—
414C2×C42—612no—
415C2×C42—1224no—
416C2×C42—1221no—
417#104—1228no—
418#103—1228no—
419#95—1228no—
420C24—21no—
421C24D4×D6111yescharacteristic
422#41—22no—
423#41—22no—
424#41—24no—
425#41—24no—
426#41—24no—
427#41—24no—
428#41—24no—
429#41C22⋊Dic6118yes—
430#41C22×Dic6114yes—
431#41C2×C22⋊D6114yes—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8C22⋊Dic6114yes—
452C2×Q8C22×Dic6114yes—
453C2×Q8C2×C22⋊D6112yes—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4C22⋊Dic6114yes—
461C2×D4C2×C22⋊D6112yes—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—614no—
473C22×C4—1228no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4D4×D6111yescharacteristic
477C22×C4D4×D6111yescharacteristic
478C22×C4D4×D6111yescharacteristic
479C22×C4#1175822412yes—
480C22×C4#1175822412yes—
481C22×C4C22×Dic6112yes—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6112yes—
485C22×C4C23×D6111yescharacteristic
486Q16—216no—
487SD16—216no—
488M16—22no—
489M16—24no—
490M16—1222no—
491M16—1222no—
492M16—2442no—
493M16C22×Dic6114yes—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—1222no—
497C2×C8—2442no—
498C4⋊C4—614no—
499C4⋊C4—1228no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—2442no—
504C22⋊C4—1228no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—2442no—
510C42—1228no—
511C42—1222no—
512C42—1222no—
513C42—1222no—
514C42—1222no—
515C42—1222no—
516C42—1222no—
517C23—22no—
518C23—24no—
519C23—42no—
520C23—82no—
521C23D4×Dic6112yes—
522C23#1451974511yescharacteristic
523C23#1451974511yescharacteristic
524C23#1451974912yes—
525C23C2×D4×D6111yescharacteristic
526Q8—22no—
527Q8—24no—
528Q8—24no—
529Q8—24no—
530Q8—42no—
531Q8—42no—
532Q8—44no—
533Q8—44no—
534Q8C2×C22⋊Dic6114yes—
535D4—24no—
536D4—24no—
537D4—28no—
538D4—22no—
539D4—24no—
540D4C2×C22⋊Dic6114yes—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—4882no—
554C2×C4—4888no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2444no—
558C2×C4C2×C22⋊Dic6112yes—
559C2×C4C2×C22⋊Dic6112yes—
560C2×C4D4×Dic6112yes—
561C2×C4D4×Dic6112yes—
562C2×C4D4×Dic6112yes—
563C2×C4#1451974511yescharacteristic
564C2×C4#1451974511yescharacteristic
565C2×C4#1451974912yes—
566C2×C4#1451974912yes—
567C2×C4#1451974912yes—
568C2×C4#1451975812yes—
569C2×C4#1451975812yes—
570C2×C4C23×Dic6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C22×C22⋊D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C8—24no—
577C8—2442no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—84no—
586C22#1683964212yes—
587C22#1683964212yes—
588C22C2×D4×Dic6111yescharacteristic
589C22C2×#1451974511yescharacteristic
590C22C2×#1451974911yescharacteristic
591C4—24no—
592C4—44no—
593C4—4882no—
594C4—2444no—
595C4—2444no—
596C4#1683964212yes—
597C4#1683964212yes—
598C4C22×C22⋊Dic6111yescharacteristic
599C4C2×D4×Dic6111yescharacteristic
600C4C2×#1451974911yescharacteristic
601C4C2×#1451975811yescharacteristic
602C2—21no—
603C2—22no—
604C2—21no—
605C2—22no—
606C2—44no—
607C2C2×#1683964211yescharacteristic
608C2#2145519512yes—
subgroupG/H
centerC22C2×#14519745
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC22×C61C2×#120
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC22×#137D61
hypocenterC61#20944
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
4. #21448405C2semidirect
5. #21446190C2semidirect
6. #21445557C2semidirect
7. #21445530C2semidirect
24. #16839771C22semidirect
27. #16839669C22semidirect
29. #16839530C22semidirect
30. #16839523C22semidirect
35. C2×#137×C61C4semidirect
95. C22×Q8×C61C2×C4semidirect
105. #137×C61C2×C4semidirect
112. C2×M16×C61C2×C4semidirect
317. C2×#137Dic61semidirect
344. C61#20944semidirect
361. C22×Q8C2×Dic61semidirect
371. #137C2×Dic61semidirect
382. C2×M16C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214590222
C2×#168397792
C2×#168397712
#214484052
#214461902
#214455572
#214455302
C22×#137×C612
#2094461