Finite Group Databasebeta

← #29681823

Subgroups — #29681823

Order 15616.

1718 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840333C211yesmaximal, characteristic
2C2×#16840239C211yesmaximal, characteristic, complemented
3C2×#16839736C211yesmaximal, characteristic
4#21448403C211yesmaximal, characteristic, complemented
5#21446274C218yesmaximal, complemented
6#21445551C211yesmaximal, characteristic, complemented
7#21445551C211yesmaximal, characteristic, complemented
8C2×C61×#410C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10#16840333—22no—
11#16840333—22no—
12#16840333—22no—
13#16840333—22no—
14C2×#14519775C412yescomplemented
15C2×#14519764C412yescomplemented
16C2×#14519758C2214yes—
17C2×#14519756C414yescomplemented
18#16840239—24no—
19#16840239—24no—
20C2×C4×Dic122—22no—
21C2×#14519742—22no—
22C2×#14519736C2211yescharacteristic
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25C2×#14519733C2214yes—
26#16839736—24no—
27C2×#14519692C2214yes—
28#16839668—22no—
29#16839667—22no—
30#16839623C2214yes—
31#16839528C2214yes—
32#16839528C2214yes—
33C22×#41×C61C2211yescharacteristic
34C2×#117×C61C2211yescharacteristic
35C2×#117×C61C2211yescharacteristic
36C61×#410—24no—
37C2×#99×C61C2214yes—
38C22×Dic122—22no—
39C22×Dic122—22no—
40C22×Dic122—22no—
41C22×Dic122—22no—
42C22×Dic122C2×C412yes—
43C22×Dic122C2×C412yes—
44C22×Dic122D412yes—
45C22×Dic122D412yes—
46C22×Dic122D412yes—
47C22×Dic122D412yes—
48C23×Dic61C2312yes—
49#14519775—24no—
50#14519775—24no—
51#14519764—24no—
52#14519764—24no—
53#14519758—28no—
54#14519758—28no—
55#14519756—28no—
56#14519756—28no—
57C4×Dic122—216no—
58#14519742—216no—
59#14519736—22no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64#14519736—22no—
65#14519736—22no—
66#14519736D414yes—
67#14519733—28no—
68#14519733—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2×C412yes—
72C2×#11758196—24no—
73C2×#11758196C2×C418yes—
74C2×C22⋊Dic61C2×C418yes—
75C2×C22⋊Dic61C2312yes—
76C2×C22⋊Dic61C2312yes—
77C2×C22⋊Dic61C2312yes—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—24no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C412yes—
82#14519692—28no—
83#14519684—28no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×#41×C61—21no—
89C2×#41×C61—21no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C22×Q8×C61C2×C411yescharacteristic
93C22×D4×C61C2×C411yescharacteristic
94C22×D4×C61C2312yes—
95C23×C4×C61C2×C412yes—
96C23×C4×C61C2311yescharacteristic
97#117×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100#117×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C22⋊C4×C61C2312yes—
104C2×C22⋊C4×C61C2312yes—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107#99×C61—28no—
108C24×C61C22×C412yes—
109C24×C61C2411yescharacteristic
110#41×C61—42no—
111#41×C61—42no—
112#41×C61—42no—
113#41×C61—42no—
114#41×C61—42no—
115#41×C61—42no—
116#41×C61—42no—
117#41×C61—42no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—24no—
136C22×C4×C61—24no—
137C22×C4×C61—22no—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4112yes—
148C22×C4×C61#4112yes—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61C22×C414yes—
156C22×Dic61C22×C414yes—
157C22×Dic61C2×D414yes—
158C22×Dic61C2×D414yes—
159C2×Dic122—24no—
160C2×Dic122—24no—
161C2×Dic122—24no—
162C2×Dic122—24no—
163C2×Dic122—24no—
164C2×Dic122—24no—
165C2×Dic122—24no—
166C2×Dic122—24no—
167C2×Dic122—24no—
168C2×Dic122—24no—
169C2×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122C22⋊C414yes—
172C2×Dic122C22⋊C414yes—
173C2×Dic122C22⋊C414yes—
174C2×Dic122C22⋊C414yes—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—42no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61C2×D414yes—
181C42×C61—24no—
182C42×C61—42no—
183C22⋊Dic61—22no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—216no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61C2×D414yes—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—24no—
193C4⋊Dic61—42no—
194#11758196—216no—
195#11758196—216no—
196C4×Dic61—216no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—24no—
200C4×Dic61—42no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—42no—
205C23×C61—44no—
206C23×C61C23×C411yescharacteristic
207C23×C61C22×D411yescharacteristic
208C23×C61C22×D411yescharacteristic
209C23×C61C22×D411yescharacteristic
210C23×C61C22×D411yescharacteristic
211C23×C61C2×#4111yescharacteristic
212C23×C61C2×#4111yescharacteristic
213Q8×C61—42no—
214Q8×C61—42no—
215D4×C61—42no—
216D4×C61—42no—
217D4×C61—42no—
218D4×C61—42no—
219D4×C61—42no—
220D4×C61—42no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61—44no—
228C2×C4×C61—44no—
229C2×C4×C61—42no—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C4×D412yes—
235C2×C4×C61C4×D412yes—
236C2×C4×C61C4×D412yes—
237C2×C4×C61C4×D412yes—
238C2×C4×C61#12012yes—
239C2×C4×C61#12012yes—
240C2×C4×C61#12112yes—
241C2×C4×C61#12112yes—
242C2×C4×C61#12112yes—
243C2×C4×C61#12112yes—
244C2×C4×C61#12312yes—
245C2×C4×C61#12312yes—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—44no—
255C2×Dic61—44no—
256C2×Dic61—44no—
257C2×Dic61—42no—
258C2×Dic61C2×C22⋊C418yes—
259Dic122—28no—
260Dic122—28no—
261Dic122—28no—
262Dic122—28no—
263#20986—611nomaximal, Sylow
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—44no—
272C22×C61—42no—
273C22×C61—82no—
274C22×C61—82no—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61C2×C4×D411yescharacteristic
277C22×C61C2×C4×D411yescharacteristic
278C22×C61C2×#12011yescharacteristic
279C22×C61C2×#12111yescharacteristic
280C22×C61C2×#12111yescharacteristic
281C22×C61C2×#12311yescharacteristic
282C4×C61—44no—
283C4×C61—42no—
284C4×C61—82no—
285C4×C61#38612yes—
286C4×C61#38612yes—
287C4×C61#38612yes—
288C4×C61#38612yes—
289Dic61—28no—
290Dic61—44no—
291Dic61—44no—
292Dic61—82no—
293C2×#536—611no—
294C2×#519—611no—
295C2×#410—611no—
296C2×#410D6111yescharacteristic
297#2310—611no—
298#1899—618no—
299#1740—611no—
300#1740—611no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61—21no—
304C2×C61—21no—
305C2×C61—21no—
306C2×C61—21no—
307C2×C61—82no—
308C2×C61C2×#38611yescharacteristic
309C2×C61#288412yes—
310C22×#41C2×D6111yescharacteristic
311C23×Q8—611no—
312#536—1222no—
313#536—1222no—
314#536—1222no—
315#536—1222no—
316C2×C4⋊Q8—612no—
317C2×#124—612no—
318C2×#123—614no—
319C2×C22⋊Q8—614no—
320#519—1224no—
321#519—1224no—
322C2×C4×Q8—1222no—
323C2×C4×Q8—1222no—
324C2×#117—611no—
325C2×#117—611no—
326C2×#117—611no—
327C2×#117—614no—
328C2×#117C2×D6111yescharacteristic
329C2×#117C2×D6111yescharacteristic
330#410—24no—
331#410—1224no—
332C2×#99—614no—
333C2×#99C2×D6114yes—
334#391—1222no—
335#391—1222no—
336#381—614no—
337#342—614no—
338#342—614no—
339C61#2098611yescharacteristic, complemented, Sylow
340C2×#41—22no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—21no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C22×Q8—612no—
349C22×Q8—612no—
350C22×Q8—612no—
351C22×Q8—612no—
352C22×Q8—612no—
353C22×Q8—612no—
354C22×Q8—1222no—
355C22×Q8—1222no—
356C22×Q8—1222no—
357C22×Q8—1222no—
358C22×Q8C4×D6111yescharacteristic
359C22×D4C4×D6111yescharacteristic
360C22×D4C22×D6112yes—
361C23×C4—612no—
362C23×C4C4×D6112yes—
363C23×C4C22×D6111yescharacteristic
364C4⋊Q8—1224no—
365C4⋊Q8—1224no—
366#124—1224no—
367#124—1224no—
368#123—1228no—
369#123—1228no—
370C22⋊Q8—1228no—
371C22⋊Q8—1228no—
372C4×Q8—12216no—
373C4×Q8—12216no—
374#117—22no—
375#117—22no—
376#117—22no—
377#117—614no—
378#117—1228no—
379#117—1221no—
380#117—1221no—
381#117—1221no—
382#117—1221no—
383#117—1222no—
384#117—1228no—
385#117—1222no—
386#117—1222no—
387#117C22⋊D6114yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—612no—
391C2×C4⋊C4—618no—
392C2×C4⋊C4—1224no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C22⋊C4—612no—
396C2×C22⋊C4—618no—
397C2×C22⋊C4—612no—
398C2×C22⋊C4—612no—
399C2×C22⋊C4C22×D6112yes—
400C2×C22⋊C4C22×D6112yes—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—612no—
404C2×C42—1224no—
405C2×C42—1221no—
406C2×C42—1221no—
407#99—28no—
408#99—1228no—
409#95—1228no—
410C24C2×C4×D6112yes—
411C24C23×D6111yescharacteristic
412#41—42no—
413#41—42no—
414#41—42no—
415#41—42no—
416#41—42no—
417#41—42no—
418#41—42no—
419#41—42no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—614no—
425C2×Q8—614no—
426C2×Q8—614no—
427C2×Q8—614no—
428C2×Q8—1224no—
429C2×Q8—1224no—
430C2×Q8—1224no—
431C2×Q8—1224no—
432C2×Q8—1224no—
433C2×Q8—1224no—
434C2×Q8—1224no—
435C2×Q8—1224no—
436C2×Q8—1224no—
437C2×Q8—1224no—
438C2×Q8—1224no—
439C2×Q8—1224no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—614no—
457C22×C4—614no—
458C22×C4—614no—
459C22×C4—614no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4C2×C4×D6112yes—
467C22×C4C2×C4×D6111yescharacteristic
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4#1175822212yes—
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6112yes—
475C22×C4#1175822412yes—
476C22×C4C2×C22⋊D6112yes—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—12216no—
480C4⋊C4—12216no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1224no—
484C4⋊C4—2442no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—614no—
489C22⋊C4—12216no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4C2×C22⋊D6114yes—
496C42—24no—
497C42—42no—
498C42—12216no—
499C42—1222no—
500C42—1222no—
501C42—1224no—
502C42—2442no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—42no—
507C23—44no—
508C23C22×C4×D6111yescharacteristic
509C23C2×D4×D6111yescharacteristic
510C23C2×D4×D6111yescharacteristic
511C23C2×D4×D6111yescharacteristic
512C23C22×C22⋊D6111yescharacteristic
513C23C2×#1175822411yescharacteristic
514C23C2×#1175822211yescharacteristic
515Q8—42no—
516Q8—42no—
517Q8—1228no—
518Q8—1228no—
519Q8—1228no—
520Q8—1228no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—618no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—2444no—
546C2×C4—2442no—
547C2×C4—2444no—
548C2×C4—2444no—
549C2×C4C22⋊C4×D6111yescharacteristic
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22⋊C4×D6111yescharacteristic
552C2×C4C22⋊C4×D6111yescharacteristic
553C2×C4#1451973712yes—
554C2×C4#1451973712yes—
555C2×C4C4×C22⋊D6112yes—
556C2×C4C4×C22⋊D6112yes—
557C2×C4#1451974512yes—
558C2×C4#1451974512yes—
559C2×C4#1451974812yes—
560C2×C4#1451974812yes—
561C2×C4#1451975012yes—
562C2×C4#1451975012yes—
563C2×C4#1451975912yes—
564C2×C4#1451975912yes—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22C2×C22⋊C4×D6111yescharacteristic
577C22C2×#1451973711yescharacteristic
578C22C2×C4×C22⋊D6111yescharacteristic
579C22C2×#1451974511yescharacteristic
580C22C2×#1451974811yescharacteristic
581C22C2×#1451975011yescharacteristic
582C22C2×#1451975911yescharacteristic
583C4—44no—
584C4—42no—
585C4—82no—
586C4—1228no—
587C4—4882no—
588C4—2444no—
589C4—2444no—
590C4#1683963812yes—
591C4#1683963812yes—
592C4#1683963812yes—
593C4#1683963812yes—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—82no—
601C2C2×#1683963811yescharacteristic
602C2#2145428112yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC2×C61×#410C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#41×C61C22
cosocleC24×C61C24
hypercenterC2×#410D61
hypocenterC61#20986
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16840239C2semidirect
4. #21448403C2semidirect
5. #21446274C2semidirect
6. #21445551C2semidirect
7. #21445551C2semidirect
14. C2×#14519775C4semidirect
15. C2×#14519764C4semidirect
17. C2×#14519756C4semidirect
339. C61#20986semidirect

Maximal subgroups

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

subgroupindex
C2×#168403332
C2×#168402392
C2×#168397362
#214484032
#214462742
#214455512
C2×C61×#4102
#2098661