Finite Group Databasebeta

← #29680784

Subgroups — #29680784

Order 15616.

1462 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840245C211yesmaximal, characteristic
2C2×#16839805C211yesmaximal, characteristic, complemented
3C2×#16839798C211yesmaximal, characteristic, complemented
4C2×#16839729C211yesmaximal, characteristic
5#21446016C212yesmaximal, complemented
6#21446016C212yesmaximal, complemented
7#21446011C212yesmaximal, complemented
8#21446006C212yesmaximal, complemented
9#21445538C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#2311C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840245—24no—
15#16840245—24no—
16C2×Q8×Dic61—21no—
17C2×D4×Dic61—22no—
18C2×D4×Dic61—21no—
19C2×#14519736C2212yes—
20C2×#14519736C2211yescharacteristic
21C22×C4⋊Dic61C2212yes—
22C22×C4⋊Dic61C2211yescharacteristic
23#16839805—24no—
24#16839798—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2212yes—
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2212yes—
31#16839729C418yescomplemented
32C2×C22⋊#92418C2212yes—
33C2×C22⋊#92418C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839524C2212yes—
38C22×#41×C61C2211yescharacteristic
39C2×#117×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C61×#391—21no—
42C61×#390—21no—
43C61×#386—22no—
44C61×#380C2212yes—
45C61×#380C2212yes—
46C23×Dic61—22no—
47C23×Dic61—21no—
48C2×#11758194—22no—
49C2×#11758194—22no—
50C2×#11758194C2×C414yescomplemented
51C22×#92418—22no—
52C22×#92418—22no—
53C22×#92418C2×C414yescomplemented
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56C22×#92418C2312yes—
57Q8×Dic61—28no—
58D4×Dic61—216no—
59D4×Dic61—28no—
60#14519736—22no—
61#14519736—24no—
62#14519736—24no—
63#14519736D414yes—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—24no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61D414yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—21no—
75C2×C4×Dic61—22no—
76C2×C4×Dic61—21no—
77#14519710—22no—
78#14519710—22no—
79#14519710—24no—
80#14519708—22no—
81#14519708—22no—
82#14519708—24no—
83C22⋊#92418C2×C418yescomplemented
84C22⋊#92418C2×C418yescomplemented
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61—22no—
89C2×#41×C61C2×C414yes—
90C2×#41×C61Q814yes—
91C22×Q8×C61D411yescharacteristic
92C22×D4×C61D412yes—
93C22×D4×C61D411yescharacteristic
94C23×C4×C61—21no—
95C23×C4×C61C2312yes—
96C23×C4×C61C2311yescharacteristic
97#117×C61—22no—
98#117×C61—22no—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—22no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61—22no—
108C2×C42×C61—21no—
109#95×C61—22no—
110#95×C61—22no—
111C24×C61C2×D412yes—
112C24×C61C2×D411yescharacteristic
113#41×C61—28no—
114#41×C61—28no—
115#41×C61—28no—
116#41×C61—28no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61C4⋊C414yes—
120C2×D4×C61—24no—
121C2×D4×C61—24no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61C4⋊C418yes—
125C2×D4×C61C4⋊C414yes—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—42no—
134C22×C4×C61—81no—
135C22×C4×C61—81no—
136C22×C4×C61C22×C414yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×Q814yes—
143C22×C4×C61C2×Q812yes—
144C22×C4×C61#4112yes—
145C22×C4×C61#4112yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—28no—
151C22×Dic61—24no—
152C22×Dic61—81no—
153C22×Dic61—82no—
154C22×Dic61—81no—
155C22×Dic61—82no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C22⋊C4×C61—82no—
165C22⋊C4×C61—82no—
166C42×C61—42no—
167C22⋊Dic61—28no—
168C22⋊Dic61—24no—
169C4⋊Dic61—28no—
170C4⋊Dic61—24no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61C2×D412yes—
175C4⋊Dic61C2×D414yes—
176C4⋊Dic61C2×D412yes—
177C4×Dic61—28no—
178C4×Dic61—24no—
179#11758194—28no—
180C2×#92418—28no—
181C2×#92418—21no—
182C2×#92418—22no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—22no—
187C2×#92418C22×C412yes—
188C2×#92418C22×C412yes—
189C2×#92418C22×C414yes—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61C2×C4⋊C414yes—
201C23×C61C2×C4⋊C412yes—
202C23×C61#12012yes—
203C23×C61#12312yes—
204C23×C61C22×D411yescharacteristic
205Q8×C61—28no—
206D4×C61—216no—
207D4×C61—28no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—42no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—41no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—84no—
224C2×C4×C61C2×C4⋊C414yes—
225C2×C4×C61C2×C4⋊C412yes—
226C2×C4×C61C4×D414yes—
227C2×C4×C61C4×D414yes—
228C2×C4×C61#12012yes—
229C2×C4×C61C22⋊Q814yes—
230C2×C4×C61C22⋊Q814yes—
231C2×C4×C61#12312yes—
232C2×C4×C61C23×C411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×Q811yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×Dic61—22no—
239C2×Dic61—24no—
240C2×Dic61—22no—
241C2×Dic61—42no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—84no—
248C2×Dic61—88no—
249#92418—22no—
250#92418—22no—
251#92418—24no—
252#20849—611nomaximal, Sylow
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—24no—
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—42no—
266C22×C61#38514yes—
267C22×C61C22×C4⋊C411yescharacteristic
268C22×C61C2×#12011yescharacteristic
269C22×C61C2×#12311yescharacteristic
270C22×C61#57611yescharacteristic
271C22×C61#57711yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57811yescharacteristic
274C4×C61—24no—
275C4×C61—82no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61#38514yes—
279C4×C61C2×C4×D411yescharacteristic
280C4×C61C2×C4×D411yescharacteristic
281C4×C61C2×C22⋊Q811yescharacteristic
282C4×C61C2×C22⋊Q811yescharacteristic
283Dic61—44no—
284Dic61—42no—
285Dic61—42no—
286C2×#520—611no—
287C2×#428—611no—
288C2×#427—611no—
289C2×#408—611no—
290#2311D6111yescharacteristic
291#1854—612no—
292#1854—612no—
293#1853—612no—
294#1852—612no—
295#1736—611no—
296C2×#340—611no—
297C2×C61—24no—
298C2×C61—44no—
299C2×C61C2×#38511yescharacteristic
300C2×C61#291211yescharacteristic
301C2×C61#291311yescharacteristic
302C2×C61#301211yescharacteristic
303C2×C61#301311yescharacteristic
304C2×C61#309611yescharacteristic
305C2×C61#309711yescharacteristic
306C22×#41C2×D6111yescharacteristic
307C22×M16—611no—
308C23×C8—611no—
309#520—1224no—
310#520—1224no—
311C2×C4×Q8—1221no—
312C2×C4×D4—1222no—
313C2×C4×D4—1221no—
314C2×#117—611no—
315C2×#117—612no—
316C2×#117C2×D6111yescharacteristic
317C22×C4⋊C4—611no—
318C22×C4⋊C4—612no—
319C22×C4⋊C4C2×D6111yescharacteristic
320#428—1224no—
321#427—1224no—
322C2×#107—611no—
323C2×#107—612no—
324C2×#107—611no—
325C2×#106—611no—
326C2×#106—612no—
327C2×#106—611no—
328#408—618no—
329C2×C22⋊C8—612no—
330C2×C22⋊C8—612no—
331#391—21no—
332#390—21no—
333#386—22no—
334#380C2×D6112yes—
335#380C2×D6112yes—
336#340—612no—
337#340—612no—
338#340—612no—
339#340—612no—
340C61#2084911yescharacteristic, complemented, Sylow
341C2×#41—22no—
342C2×#41—22no—
343C2×#41—22no—
344C2×#41—22no—
345C2×#41Dic12214yes—
346C2×#41C4×D6114yes—
347C22×Q8C22⋊D6111yescharacteristic
348C22×D4C22⋊D6112yes—
349C22×D4C22⋊D6111yescharacteristic
350C23×C4—21no—
351C23×C4—1222no—
352C23×C4—1221no—
353C23×C4C22×D6112yes—
354C23×C4C22×D6111yescharacteristic
355C2×M16—614no—
356C2×M16—1222no—
357C2×M16—1222no—
358C22×C8—614no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8—612no—
362C22×C8—1222no—
363C22×C8—1222no—
364C4×Q8—1228no—
365C4×D4—12216no—
366C4×D4—1228no—
367#117—22no—
368#117—22no—
369#117—614no—
370#117—1224no—
371#117—1224no—
372#117—1222no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—614no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1224no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4C22×D6112yes—
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C4⋊C4C22×D6111yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—1222no—
392C2×C22⋊C4—1221no—
393C2×C42—21no—
394C2×C42—1222no—
395C2×C42—1221no—
396#107—1222no—
397#107—1222no—
398#107—1224no—
399#106—1222no—
400#106—1222no—
401#106—1224no—
402C22⋊C8—618no—
403C22⋊C8—618no—
404#95—22no—
405#95—22no—
406C24C2×C22⋊D6112yes—
407C24C2×C22⋊D6111yescharacteristic
408#41—28no—
409#41—28no—
410#41—28no—
411#41—28no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8#1175819614yes—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4#1175819618yes—
420C2×D4#1175819614yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—1228no—
432C22×C4—1224no—
433C22×C4—1221no—
434C22×C4—1222no—
435C22×C4—1221no—
436C22×C4—4881no—
437C22×C4—4881no—
438C22×C4—4882no—
439C22×C4—4882no—
440C22×C4C2×Dic12214yes—
441C22×C4C2×Dic12212yes—
442C22×C4C2×C4×D6114yes—
443C22×C4C2×C4×D6112yes—
444C22×C4D4×D6112yes—
445C22×C4D4×D6112yes—
446C22×C4#1175822412yes—
447C22×C4#1175822412yes—
448C22×C4C2×C22⋊D6112yes—
449C22×C4C2×C22⋊D6111yescharacteristic
450C22×C4C23×D6111yescharacteristic
451M16—1228no—
452C2×C8—612no—
453C2×C8—612no—
454C2×C8—614no—
455C2×C8—1221no—
456C2×C8—1222no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1222no—
461C2×C8—1228no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—614no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—1228no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1221no—
476C4⋊C4—1224no—
477C22⋊C4—42no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—1228no—
481C22⋊C4—1224no—
482C42—42no—
483C42—1228no—
484C42—1224no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23C2×#1175819614yes—
496C23C2×#1175819612yes—
497C23#1451974512yes—
498C23#1451975812yes—
499C23C22×C22⋊D6111yescharacteristic
500Q8—28no—
501D4—216no—
502D4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—84no—
519C2×C4—1222no—
520C2×C4—1224no—
521C2×C4—1222no—
522C2×C4—4884no—
523C2×C4—4888no—
524C2×C4—2442no—
525C2×C4—2441no—
526C2×C4—2442no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C2×#1175819614yes—
531C2×C4C2×#1175819612yes—
532C2×C4#1451973714yes—
533C2×C4#1451973714yes—
534C2×C4#1451974512yes—
535C2×C4#1451975414yes—
536C2×C4#1451975414yes—
537C2×C4#1451975812yes—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4C2×D4×D6111yescharacteristic
540C2×C4C2×D4×D6111yescharacteristic
541C2×C4C22×Dic12211yescharacteristic
542C2×C4C2×#1175822411yescharacteristic
543C2×C4C2×#1175822411yescharacteristic
544C8—1222no—
545C8—1222no—
546C8—1224no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22#1683963414yes—
561C22C22×#1175819611yescharacteristic
562C22C2×#1451974511yescharacteristic
563C22C2×#1451975811yescharacteristic
564C22#1684056511yescharacteristic
565C22#1684057011yescharacteristic
566C22#1684057311yescharacteristic
567C22#1684057711yescharacteristic
568C4—24no—
569C4—81no—
570C4—81no—
571C4—82no—
572C4—2444no—
573C4—2442no—
574C4—2442no—
575C4#1683963414yes—
576C4C2×#1451973711yescharacteristic
577C4C2×#1451973711yescharacteristic
578C4C2×#1451975411yescharacteristic
579C4C2×#1451975411yescharacteristic
580C2—24no—
581C2—44no—
582C2C2×#1683963411yescharacteristic
583C2#2145441211yescharacteristic
584C2#2145441711yescharacteristic
585C2#2145512011yescharacteristic
586C2#2145513111yescharacteristic
587C2#2145577111yescharacteristic
588C2#2145578211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#2311D61
hypocenterC61#20849
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16839805C2semidirect
3. C2×#16839798C2semidirect
5. #21446016C2semidirect
6. #21446016C2semidirect
7. #21446011C2semidirect
8. #21446006C2semidirect
9. #21445538C2semidirect
10. C2×#16839524C2semidirect
31. #16839729C4semidirect
50. C2×#11758194C2×C4semidirect
53. C22×#92418C2×C4semidirect
83. C22⋊#92418C2×C4semidirect
84. C22⋊#92418C2×C4semidirect
340. C61#20849semidirect

Maximal subgroups

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

subgroupindex
C2×#168402452
C2×#168398052
C2×#168397982
C2×#168397292
#214460162
#214460112
#214460062
#214455382
C2×#168395242
C61×#23112
#2084961