Finite Group Databasebeta

← #29682031

Subgroups — #29682031

Order 15616.

1748 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459022C211yesmaximal, characteristic, complemented
2C2×#16839736C211yesmaximal, characteristic, complemented
3C2×#16839736C212yesmaximal, complemented
4#21446362C218yesmaximal, complemented
5#21445551C212yesmaximal, complemented
6C61×#3449C211yesmaximal, characteristic
7#16840245—24no—
8#16840240—21no—
9#16840237—21no—
10#16840237C2214yes—
11#41×Dic61—24no—
12C2×#14519736—22no—
13C2×#14519736C2212yes—
14C2×#14519736C2212yes—
15#16839736—24no—
16#16839736—24no—
17#16839736—22no—
18#16839736—22no—
19C2×#14519692C2218yes—
20C2×#14519692C2214yes—
21#16839528C2218yes—
22C22×#41×C61C2211yescharacteristic
23C22×#41×C61C2211yescharacteristic
24C61×#537—22no—
25C61×#536—21no—
26C61×#536C412yescomplemented
27C61×#535—21no—
28C61×#535C412yescomplemented
29C61×#534C2214yes—
30C61×#532—22no—
31C61×#525—22no—
32C61×#525C414yescomplemented
33C2×#117×C61C2211yescharacteristic
34Q8×Dic61—28no—
35D4×Dic61—28no—
36D4×Dic61—216no—
37#14519736—24no—
38#14519736—24no—
39#14519736—22no—
40#14519736—24no—
41#14519736—22no—
42#14519736—22no—
43#14519736—22no—
44#14519736—22no—
45#14519736—22no—
46#14519736D414yescomplemented
47#14519736D414yescomplemented
48C2×C4⋊Dic61—24no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—22no—
51C2×C22⋊Dic61—24no—
52C2×C22⋊Dic61C2314yes—
53C2×C22⋊Dic61C2314yes—
54C2×C4×Dic61—24no—
55C2×C4×Dic61—22no—
56C2×C4×Dic61—22no—
57#14519692—28no—
58#14519692—216no—
59C2×#41×C61—22no—
60C2×#41×C61—22no—
61C2×#41×C61—21no—
62C2×#41×C61—21no—
63C2×#41×C61—21no—
64C2×#41×C61—22no—
65C2×#41×C61—22no—
66C2×#41×C61—21no—
67C2×#41×C61—22no—
68C2×#41×C61—22no—
69C2×#41×C61—22no—
70C2×#41×C61—22no—
71C2×#41×C61C2×C412yes—
72C2×#41×C61C2×C412yes—
73C2×#41×C61D412yes—
74C2×#41×C61D412yes—
75C22×Q8×C61C2×C411yescharacteristic, complemented
76C22×Q8×C61D411yescharacteristic
77C22×D4×C61C2×C411yescharacteristic, complemented
78C22×D4×C61D411yescharacteristic
79C22×D4×C61C2312yes—
80C22×D4×C61C2312yes—
81C23×C4×C61C2×C412yescomplemented
82C23×C4×C61D412yes—
83C23×C4×C61C2311yescharacteristic
84C4⋊Q8×C61—22no—
85#127×C61—22no—
86#124×C61—22no—
87#124×C61—22no—
88#123×C61—24no—
89#123×C61C2×C418yescomplemented
90C22⋊Q8×C61—24no—
91C22⋊Q8×C61—24no—
92#121×C61—24no—
93#121×C61—28no—
94#121×C61—24no—
95#120×C61—24no—
96#120×C61C2×C418yescomplemented
97C4×D4×C61—28no—
98#117×C61—21no—
99#117×C61—21no—
100#117×C61C2×C412yescomplemented
101#117×C61C2×C412yescomplemented
102C2×C4⋊C4×C61—21no—
103C2×C22⋊C4×C61C2312yes—
104C2×C42×C61—21no—
105C24×C61C22×C412yes—
106C24×C61C2×D412yes—
107C24×C61C2411yescharacteristic
108#41×C61—24no—
109#41×C61—24no—
110#41×C61—24no—
111#41×C61—24no—
112#41×C61—24no—
113#41×C61—24no—
114#41×C61—24no—
115#41×C61—24no—
116#41×C61—41no—
117#41×C61—42no—
118#41×C61—41no—
119#41×C61—42no—
120#41×C61—42no—
121#41×C61—42no—
122#41×C61—42no—
123#41×C61—41no—
124#41×C61—41no—
125#41×C61—42no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61C22⋊C412yes—
133C2×Q8×C61C22⋊C412yes—
134C2×D4×C61—24no—
135C2×D4×C61—24no—
136C2×D4×C61—22no—
137C2×D4×C61—22no—
138C2×D4×C61—24no—
139C2×D4×C61—24no—
140C2×D4×C61—28no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C2×D4×C61C22⋊C412yes—
148C2×D4×C61C22⋊C412yes—
149C2×D4×C61C22×C414yes—
150C2×D4×C61C2×D414yes—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—24no—
159C22×C4×C61—24no—
160C22×C4×C61C22⋊C414yes—
161C22×C4×C61C22⋊C414yes—
162C22×C4×C61C22×C412yes—
163C22×C4×C61C22×C411yescharacteristic
164C22×C4×C61C22×C411yescharacteristic
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2×D412yes—
172C22×C4×C61#4112yes—
173C22×C4×C61#4112yes—
174C22×Dic61—28no—
175C22×Dic61—24no—
176C22×Dic61—24no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—28no—
180C22⋊C4×C61—28no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—28no—
183C22⋊C4×C61C22×C414yes—
184C42×C61—22no—
185C42×C61—22no—
186C22⋊Dic61—28no—
187C22⋊Dic61—24no—
188C22⋊Dic61—24no—
189C22⋊Dic61—24no—
190C22⋊Dic61C2×D418yes—
191C4⋊Dic61—28no—
192C4⋊Dic61—24no—
193C4⋊Dic61—24no—
194C4⋊Dic61—42no—
195C4⋊Dic61—42no—
196C4×Dic61—28no—
197C4×Dic61—24no—
198C4×Dic61—24no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—24no—
205C23×C61—81no—
206C23×C61—82no—
207C23×C61—82no—
208C23×C61C2×C22⋊C414yes—
209C23×C61C23×C411yescharacteristic
210C23×C61C22×D411yescharacteristic
211C23×C61C22×D411yescharacteristic
212C23×C61C22×D412yes—
213C23×C61C2×#4112yes—
214Q8×C61—24no—
215Q8×C61—24no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—42no—
219D4×C61—28no—
220D4×C61—28no—
221D4×C61—24no—
222D4×C61—24no—
223D4×C61—42no—
224D4×C61—44no—
225D4×C61—42no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—42no—
229D4×C61—88no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—22no—
238C2×C4×C61—24no—
239C2×C4×C61—42no—
240C2×C4×C61—44no—
241C2×C4×C61—42no—
242C2×C4×C61—82no—
243C2×C4×C61C2×C22⋊C414yes—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C4×D412yes—
249C2×C4×C61C4×D412yes—
250C2×C4×C61C4×D412yes—
251C2×C4×C61C4×D412yes—
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61#12012yes—
254C2×C4×C61#12011yescharacteristic
255C2×C4×C61#12112yes—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#12311yescharacteristic
260C2×C4×C61#12311yescharacteristic
261C2×C4×C61#12312yes—
262C2×Dic61—28no—
263C2×Dic61—44no—
264C2×Dic61—84no—
265C2×Dic61—88no—
266C2×Dic61—84no—
267#21014—611nomaximal, Sylow
268C22×C61—24no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—44no—
277C22×C61—42no—
278C22×C61—82no—
279C22×C61—84no—
280C22×C61C22×C22⋊C411yescharacteristic
281C22×C61C2×C4×D412yes—
282C22×C61C2×#12011yescharacteristic
283C22×C61C2×#12112yes—
284C22×C61C2×#12311yescharacteristic
285C4×C61—24no—
286C4×C61—44no—
287C4×C61—42no—
288C4×C61#38611yescharacteristic
289C4×C61#38611yescharacteristic
290C4×C61#38611yescharacteristic
291C4×C61#38612yes—
292C4×C61#38612yes—
293C4×C61#38611yescharacteristic
294Dic61—48no—
295Dic61—84no—
296#3449D6111yescharacteristic
297#3428—611no—
298C2×#410—612no—
299C2×#410—611no—
300#1913—618no—
301#1740—612no—
302C2×C61—21no—
303C2×C61—22no—
304C2×C61—21no—
305C2×C61—22no—
306C2×C61—21no—
307C2×C61—44no—
308C2×C61—82no—
309C2×C61C2×#38611yescharacteristic
310C22×#41C2×D6111yescharacteristic
311C22×#41C2×D6111yescharacteristic
312#537—22no—
313#536—21no—
314#536Dic6112yescomplemented
315#535—21no—
316#535Dic6112yescomplemented
317#534C2×D6114yes—
318#532—22no—
319#525—22no—
320#525Dic6114yescomplemented
321#520—1224no—
322#519—1221no—
323#518—614no—
324#518—1221no—
325C4×#41—1224no—
326C2×#117—612no—
327C2×#117—612no—
328C2×#117—1222no—
329C2×#117C2×D6111yescharacteristic
330#410—1224no—
331#410—1224no—
332#410—1222no—
333#410—1222no—
334C2×#99—618no—
335C2×#99—614no—
336#342—618no—
337C61#2101411yescharacteristic, complemented, Sylow
338C2×#41—22no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41—21no—
349C2×#41—21no—
350C2×#41C2×Dic6112yes—
351C2×#41C2×Dic6112yes—
352C2×#41C22⋊D6112yes—
353C2×#41C22⋊D6112yes—
354C22×Q8C2×Dic6111yescharacteristic, complemented
355C22×Q8C22⋊D6111yescharacteristic
356C22×D4C2×Dic6111yescharacteristic, complemented
357C22×D4C22⋊D6111yescharacteristic
358C22×D4C22×D6112yes—
359C22×D4C22×D6112yes—
360C23×C4C2×Dic6112yescomplemented
361C23×C4C22⋊D6112yes—
362C23×C4C22×D6111yescharacteristic
363C4⋊Q8—22no—
364#127—22no—
365#124—22no—
366#124—22no—
367#123—24no—
368#123C2×Dic6118yescomplemented
369C22⋊Q8—24no—
370C22⋊Q8—24no—
371#121—24no—
372#121—28no—
373#121—24no—
374#120—24no—
375#120C2×Dic6118yescomplemented
376C4×Q8—1228no—
377C4×D4—28no—
378C4×D4—12216no—
379C4×D4—1228no—
380#117—21no—
381#117—21no—
382#117—614no—
383#117—614no—
384#117—1224no—
385#117—1224no—
386#117—1222no—
387#117—1222no—
388#117—1224no—
389#117—1222no—
390#117—1222no—
391#117—1222no—
392#117—1222no—
393#117C2×Dic6112yescomplemented
394#117C2×Dic6112yescomplemented
395C2×C4⋊C4—21no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1222no—
399C2×C22⋊C4—614no—
400C2×C22⋊C4—614no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4C22×D6112yes—
403C2×C42—21no—
404C2×C42—1224no—
405C2×C42—1222no—
406C2×C42—1222no—
407#99—12216no—
408#99—1228no—
409C24C22×Dic6112yes—
410C24C2×C22⋊D6112yes—
411C24C23×D6111yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—24no—
418#41—24no—
419#41—24no—
420#41—42no—
421#41—41no—
422#41—41no—
423#41—42no—
424#41—42no—
425#41—42no—
426#41—42no—
427#41—41no—
428#41—42no—
429#41—41no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8C22⋊Dic6112yes—
437C2×Q8C22⋊Dic6112yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4C22⋊Dic6112yes—
452C2×D4C22⋊Dic6112yes—
453C2×D4C22×Dic6114yes—
454C2×D4C2×C22⋊D6114yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—1228no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4C22⋊Dic6114yes—
468C22×C4C22⋊Dic6114yes—
469C22×C4D4×D6112yes—
470C22×C4D4×D6112yes—
471C22×C4#1175822412yes—
472C22×C4#1175822412yes—
473C22×C4C22×Dic6112yes—
474C22×C4C22×Dic6111yescharacteristic
475C22×C4C22×Dic6111yescharacteristic
476C22×C4C2×C22⋊D6112yes—
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C2×C22⋊D6111yescharacteristic
480C22×C4C2×C22⋊D6111yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—28no—
484C4⋊C4—1228no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C22⋊C4—22no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—618no—
493C22⋊C4—1228no—
494C22⋊C4—1224no—
495C22⋊C4—1224no—
496C22⋊C4—1224no—
497C22⋊C4C22×Dic6114yes—
498C42—22no—
499C42—22no—
500C42—1228no—
501C42—1224no—
502C42—1224no—
503C42—2442no—
504C42—2442no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—81no—
510C23—82no—
511C23—82no—
512C23C2×C22⋊Dic6114yes—
513C23C23×Dic6111yescharacteristic
514C23C2×D4×D6112yes—
515C23C22×C22⋊D6111yescharacteristic
516C23C22×C22⋊D6111yescharacteristic
517C23C2×#1175822412yes—
518Q8—24no—
519Q8—24no—
520Q8—41no—
521Q8—41no—
522Q8—42no—
523D4—28no—
524D4—28no—
525D4—24no—
526D4—24no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—41no—
531D4—41no—
532D4—42no—
533D4—88no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—1228no—
548C2×C4—4884no—
549C2×C4—4884no—
550C2×C4—4888no—
551C2×C4—2444no—
552C2×C4C2×C22⋊Dic6114yes—
553C2×C4C2×C22⋊Dic6111yescharacteristic
554C2×C4C2×C22⋊Dic6111yescharacteristic
555C2×C4C2×C22⋊Dic6111yescharacteristic
556C2×C4C2×C22⋊Dic6111yescharacteristic
557C2×C4D4×Dic6112yes—
558C2×C4D4×Dic6112yes—
559C2×C4D4×Dic6112yes—
560C2×C4D4×Dic6112yes—
561C2×C4#1451974512yes—
562C2×C4#1451974511yescharacteristic
563C2×C4#1451974511yescharacteristic
564C2×C4#1451974912yes—
565C2×C4#1451974912yes—
566C2×C4#1451974912yes—
567C2×C4#1451974912yes—
568C2×C4#1451975811yescharacteristic
569C2×C4#1451975812yes—
570C2×C4#1451975811yescharacteristic
571C22—24no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—82no—
582C22—84no—
583C22C22×C22⋊Dic6111yescharacteristic
584C22C2×D4×Dic6112yes—
585C22C2×#1451974511yescharacteristic
586C22C2×#1451974912yes—
587C22C2×#1451975811yescharacteristic
588C4—24no—
589C4—44no—
590C4—42no—
591C4—4884no—
592C4—2448no—
593C4#1683964211yescharacteristic
594C4#1683964211yescharacteristic
595C4#1683964211yescharacteristic
596C4#1683964212yes—
597C4#1683964212yes—
598C4#1683964211yescharacteristic
599C2—21no—
600C2—22no—
601C2—21no—
602C2—22no—
603C2—21no—
604C2—44no—
605C2—82no—
606C2C2×#1683964211yescharacteristic
subgroupG/H
centerC4#16839642
commutatorC23×C61C23×C4
FrattiniC24C23×D61
Fitting, prime radicalC61×#3449C2
socleC2×C61C2×#386
cosocleC24×C61C24
hypercenter#3449D61
hypocenterC61#21014
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21459022C2semidirect
2. C2×#16839736C2semidirect
3. C2×#16839736C2semidirect
4. #21446362C2semidirect
5. #21445551C2semidirect
26. C61×#536C4semidirect
28. C61×#535C4semidirect
32. C61×#525C4semidirect
46. #14519736D4semidirect
47. #14519736D4semidirect
75. C22×Q8×C61C2×C4semidirect
77. C22×D4×C61C2×C4semidirect
81. C23×C4×C61C2×C4semidirect
89. #123×C61C2×C4semidirect
96. #120×C61C2×C4semidirect
100. #117×C61C2×C4semidirect
101. #117×C61C2×C4semidirect
314. #536Dic61semidirect
316. #535Dic61semidirect
320. #525Dic61semidirect
337. C61#21014semidirect
354. C22×Q8C2×Dic61semidirect
356. C22×D4C2×Dic61semidirect
360. C23×C4C2×Dic61semidirect
368. #123C2×Dic61semidirect
375. #120C2×Dic61semidirect
393. #117C2×Dic61semidirect
394. #117C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214590222
C2×#168397362
#214463622
#214455512
C61×#34492
#2101461