Finite Group Databasebeta

← #30611248

Subgroups — #30611248

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#432C211yesmaximal, characteristic
2#21462266C211yesmaximal, characteristic
3#21461259C212yesmaximal
4#21461258C212yesmaximal
5#21461241C214yesmaximal, complemented
6#21461212C214yesmaximal, complemented
7C2×#16840316C211yesmaximal, characteristic
8#16840785C2212yes—
9#16840784C2212yes—
10#16840783—21no—
11#16840782—21no—
12D61⋊(C4⋊C8)—21no—
13D61⋊(C22⋊C8)C2212yes—
14#16840721—21no—
15#16840707C2212yes—
16#16840706—21no—
17#16840705—21no—
18#16840704C2212yes—
19C4×C4⋊#6923—21no—
20C4×C22⋊#6923—21no—
21#16840698C2212yes—
22#16840616C2218yes—
23#16840615C2218yes—
24#16840316—21no—
25#16840316—21no—
26#16840316—21no—
27#16840316—21no—
28#16840316C412yescomplemented
29#16840316C412yescomplemented
30#16840316C2212yes—
31#16840316C2212yes—
32C2×#14519776C411yescharacteristic, complemented
33C2×#14519775C411yescharacteristic, complemented
34C2×#14519767—21no—
35C2×#14519757C2211yescharacteristic
36C2×#14519755—21no—
37C2×#14519753C2211yescharacteristic
38C2×C4×Dic122—21no—
39C2×C4×Dic122C411yescharacteristic, complemented
40C2×#14519734—21no—
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic, complemented
43C2×D61⋊C8—22no—
44C2×C4⋊#6923—22no—
45C2×C22⋊#6923—22no—
46C2×C4×#6923—22no—
47#14519844—22no—
48C4⋊#92420—22no—
49#14519834C2314yes—
50C22⋊#92420—22no—
51#14519832—22no—
52#14519832—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4C2314yes—
57C22×Dic122—21no—
58C22×Dic122C2×C411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519776—21no—
63#14519776—21no—
64#14519776C2×C412yes—
65#14519776C2×C412yes—
66#14519775—21no—
67#14519775—21no—
68#14519775C2×C412yes—
69#14519775C2×C412yes—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519757—21no—
75#14519757—21no—
76#14519757C2×C412yes—
77#14519757C2312yes—
78#14519755—22no—
79#14519755—22no—
80#14519755—22no—
81#14519755—22no—
82#14519753—21no—
83#14519753—21no—
84#14519753C2×C412yes—
85#14519753C2312yes—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C4×Dic122C2×C412yes—
93C4×Dic122C2×C412yes—
94#14519734—22no—
95#14519734—22no—
96#14519734—22no—
97#14519734—22no—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C22×Q8×C61C2×C411yescharacteristic
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61C2×C412yes—
123C4×Q8×C61C2×C412yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2×C411yescharacteristic
128C23×D61C2×D411yescharacteristic
129C22×#6923—24no—
130C2×Q8×C61—21no—
131C2×Q8×C61—42no—
132C2×Q8×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C22⋊C411yescharacteristic
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C2×C4×D61—24no—
144C2×C4×D61—42no—
145C2×C4×D61—42no—
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22×C411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—44no—
162C2×Dic122—42no—
163C2×Dic122C22×C412yes—
164C22⋊#6923—44no—
165C2×#92420—24no—
166C4⋊#6923—44no—
167C4×#6923—44no—
168D61⋊C8—44no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22×C412yes—
173C42×C61—22no—
174C42×C61—22no—
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22×C411yescharacteristic
178C42×C61C22×C411yescharacteristic
179#11758198—22no—
180#11758198—22no—
181#11758198—22no—
182#11758198—22no—
183C4⋊Dic61—21no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61C22×C412yes—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22×C411yescharacteristic
206C4×Dic61C22×C411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C23×C61C2×C22⋊C411yescharacteristic
212C22×D61—41no—
213C22×D61—41no—
214C22×D61C2×C22⋊C411yescharacteristic
215C22×D61#13714yes—
216C22×D61C22×D411yescharacteristic
217C2×#6923—44no—
218C2×#6923—44no—
219C2×#6923—44no—
220Q8×C61—42no—
221Q8×C61—42no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C22×D411yescharacteristic
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—41no—
260Dic122—44no—
261Dic122—42no—
262Dic122—44no—
263Dic122—42no—
264#92420—44no—
265#17596—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61#41912yes—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—44no—
275C2×D61#41912yes—
276C2×D61C2×#13712yes—
277#6923—84no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61#41012yes—
286Dic61—21no—
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
291Dic61—42no—
292Dic61—42no—
293Dic61#42112yes—
294C2×#533—611no—
295C2×#432—611no—
296#2283—611no—
297#1520—612no—
298#1520—612no—
299#1515—614no—
300#1506—614no—
301C2×C61—22no—
302C2×C61C2×#41011yescharacteristic
303C2×C61C2×#41911yescharacteristic
304C2×C61C2×#42111yescharacteristic
305D61—22no—
306D61—22no—
307#533—612no—
308#533—612no—
309#533—612no—
310#533—612no—
311#533—1221no—
312#533—1221no—
313#533—1221no—
314#533—1221no—
315C2×C4⋊Q8—611no—
316C2×C4⋊Q8—611no—
317C2×#125—1221no—
318C2×C22⋊Q8—611no—
319C2×C22⋊Q8—611no—
320C2×C22⋊Q8—1221no—
321C2×C4×Q8—611no—
322C2×C4×Q8—1221no—
323C2×C4×Q8#692311yescharacteristic, complemented
324C2×#117—1221no—
325C22×C42—611no—
326#432—612no—
327#432—612no—
328#432—1221no—
329#432—1221no—
330C2×C4⋊C8—1221no—
331C2×C22⋊C8—612no—
332C2×#97—1221no—
333#383—612no—
334#382—1221no—
335#381—1221no—
336#380—612no—
337C4×C4⋊C4—1221no—
338C4×C22⋊C4—1221no—
339#376—612no—
340#328—618no—
341#328—618no—
342C61#1759611yescharacteristic, complemented, Sylow
343C22×Q8—611no—
344C22×Q8—1221no—
345C22×Q8C2×#692311yescharacteristic
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C22×C8—1222no—
350C4⋊Q8—612no—
351C4⋊Q8—612no—
352C4⋊Q8—612no—
353C4⋊Q8—612no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358#125—1222no—
359#125—1222no—
360#125—1222no—
361#125—1222no—
362C22⋊Q8—612no—
363C22⋊Q8—612no—
364C22⋊Q8—612no—
365C22⋊Q8—612no—
366C22⋊Q8—1221no—
367C22⋊Q8—1221no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—612no—
377C4×Q8—612no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×Q8C2×#692312yes—
385C4×Q8C2×#692312yes—
386#117—1222no—
387#117—1222no—
388#117—1222no—
389#117—1222no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4C2×#692311yescharacteristic
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1222no—
403C2×C42—21no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—1221no—
416C2×C42—1222no—
417C2×C42C2×#692311yescharacteristic
418C4⋊C8—1222no—
419C4⋊C8—1222no—
420C22⋊C8—614no—
421C22⋊C8—1222no—
422#97—1222no—
423#97—1222no—
424#95—614no—
425#95—1222no—
426#95—1222no—
427#95—1222no—
428C24—611no—
429C2×Q8—21no—
430C2×Q8—42no—
431C2×Q8—612no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×Q8—1221no—
435C2×Q8—2442no—
436C2×Q8—2444no—
437C2×Q8C22×#692312yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1224no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1224no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4C22⋊#692311yescharacteristic
462C22×C4C22⋊#692311yescharacteristic
463C22×C4C22×#692311yescharacteristic
464C2×C8—1224no—
465C2×C8—2444no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—612no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1221no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—2444no—
484C4⋊C4C22×#692312yes—
485C22⋊C4—1222no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—2444no—
490C42—22no—
491C42—22no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—1222no—
505C42—1222no—
506C42—2444no—
507C42C22⋊#692311yescharacteristic
508C42C22⋊#692311yescharacteristic
509C42C22×#692311yescharacteristic
510C42C22×#692311yescharacteristic
511C23—611no—
512C23—611no—
513C23—614no—
514C23—2441no—
515C23—2441no—
516C23C2×C22⋊#692311yescharacteristic
517Q8—42no—
518Q8—42no—
519Q8—2442no—
520Q8—2444no—
521Q8—2444no—
522Q8—2442no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4C2×C22⋊#692311yescharacteristic
559C2×C4C2×C22⋊#692311yescharacteristic
560C2×C4C2×C22⋊#692311yescharacteristic
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4C2×C22⋊#692311yescharacteristic
563C2×C4C23×#692311yescharacteristic
564C8—2444no—
565C22—21no—
566C22—21no—
567C22—612no—
568C22—612no—
569C22—1221no—
570C22—1221no—
571C22—1221no—
572C22—1221no—
573C22—2444no—
574C22#1684075512yes—
575C22C22×C22⋊#692311yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—42no—
582C4—42no—
583C4—612no—
584C4—1221no—
585C4—1221no—
586C4—1221no—
587C4—1221no—
588C4—1221no—
589C4—4884no—
590C4—2442no—
591C4—2442no—
592C4#1684074112yes—
593C2—22no—
594C2—1222no—
595C2—1222no—
596C2C2×#1684074111yescharacteristic
597C2C2×#1684075511yescharacteristic
598C2D61⋊#42111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#17596
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
5. #21461241C2semidirect
6. #21461212C2semidirect
28. #16840316C4semidirect
29. #16840316C4semidirect
32. C2×#14519776C4semidirect
33. C2×#14519775C4semidirect
39. C2×C4×Dic122C4semidirect
42. C2×C4×Q8×C61C4semidirect
323. C2×C4×Q8#6923semidirect
342. C61#17596semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4322
#214622662
#214612592
#214612582
#214612412
#214612122
C2×#168403162
#1759661