Finite Group Databasebeta

← #30610978

Subgroups — #30610978

Order 15616.

1084 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C211yesmaximal, characteristic
2C4×#14519856C211yesmaximal, characteristic
3#21461219C212yesmaximal
4#21461218C212yesmaximal
5#21461212C214yesmaximal, complemented
6#21461201C214yesmaximal, complemented
7C2×#16840316C211yesmaximal, characteristic
8C2×#14519856—21no—
9#16840785C2212yes—
10#16840782—21no—
11#16840780C2212yes—
12#16840779—21no—
13#16840767—21no—
14#16840733C2212yes—
15C4×#11758213—21no—
16C4×C4⋊#6923—21no—
17C4×C4⋊#6923C2212yes—
18C4×C22⋊#6923—21no—
19C4×C22⋊#6923C2212yes—
20#16840698C2212yes—
21C42×#6923—21no—
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×#11758213—22no—
44#14519856—28no—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923—22no—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50#14519844—22no—
51#14519844—22no—
52#14519834C2314yes—
53C22⋊#92420—22no—
54#14519832—22no—
55C4×#92420—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4C2314yes—
58C22×Dic122—21no—
59C22×Dic122C2×C411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519776—21no—
64#14519776—21no—
65#14519776C2×C412yes—
66#14519776C2×C412yes—
67#14519775—21no—
68#14519775—21no—
69#14519775C2×C412yes—
70#14519775C2×C412yes—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519767—22no—
75#14519757—21no—
76#14519757—21no—
77#14519757C2×C412yes—
78#14519757C2312yes—
79#14519755—22no—
80#14519755—22no—
81#14519755—22no—
82#14519755—22no—
83#14519753—21no—
84#14519753—21no—
85#14519753C2×C412yes—
86#14519753C2312yes—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C4×Dic122C2×C412yes—
94C4×Dic122C2×C412yes—
95#14519734—22no—
96#14519734—22no—
97#14519734—22no—
98#14519734—22no—
99C2×C4⋊Dic61—21no—
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×#11758196—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61C2311yescharacteristic
119C42×D61C2311yescharacteristic
120C22×Q8×C61C2×C411yescharacteristic
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61C2×C412yes—
124C4×Q8×C61C2×C412yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C22×#6923—24no—
131C2×Q8×C61—21no—
132C2×Q8×C61—42no—
133C2×Q8×C61C22×C412yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×C4×D61—24no—
145C2×C4×D61—24no—
146C2×C4×D61—42no—
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22×C411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—44no—
163C2×Dic122—42no—
164C2×Dic122C22×C412yes—
165C22⋊#6923—28no—
166C2×#92420—24no—
167C4⋊#6923—28no—
168C4×#6923—28no—
169C4×#6923—44no—
170C4×#6923—44no—
171#11758213—44no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C22×C412yes—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22×C411yescharacteristic
181C42×C61C22×C411yescharacteristic
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22×C411yescharacteristic
209C4×Dic61C22×C411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C23×C61C2×C22⋊C411yescharacteristic
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61C2×C22⋊C411yescharacteristic
219C22×D61C22×D411yescharacteristic
220C2×#6923—28no—
221C2×#6923—44no—
222C2×#6923—44no—
223Q8×C61—42no—
224Q8×C61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61#10414yes—
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C22×D411yescharacteristic
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61#13714yes—
263Dic122—42no—
264Dic122—44no—
265Dic122—44no—
266Dic122—42no—
267#92420—44no—
268#17545—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C2×#10412yes—
272C22×C61C22×C22⋊C411yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—44no—
279C2×D61C2×#13712yes—
280#6923—48no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61#41012yes—
288C4×C61#41912yes—
289Dic61—21no—
290Dic61—21no—
291Dic61—21no—
292Dic61—42no—
293Dic61—42no—
294Dic61#41912yes—
295Dic61C2×#10412yes—
296Dic61C2×#10412yes—
297C2×#533—611no—
298#2920—611no—
299C4×#117—611no—
300#1508—612no—
301#1508—612no—
302#1506—614no—
303#1503—614no—
304C2×C61—22no—
305C2×C61C2×#41011yescharacteristic
306C2×C61C2×#41911yescharacteristic
307C2×C61C22×#10411yescharacteristic
308D61—22no—
309D61—41no—
310#533—612no—
311#533—612no—
312#533—612no—
313#533—612no—
314#533—1221no—
315#533—1221no—
316#533—1221no—
317#533—1221no—
318C2×C4⋊Q8—611no—
319C2×C4⋊Q8—611no—
320C2×#125—1221no—
321C2×C22⋊Q8—611no—
322C2×C22⋊Q8—611no—
323C2×C22⋊Q8—1221no—
324C2×C4×Q8—611no—
325C2×C4×Q8—1221no—
326C2×C4×Q8#692311yescharacteristic, complemented
327C2×#117—1221no—
328C2×#117—1221no—
329C22×C42—611no—
330#432—612no—
331#432—1221no—
332#431—612no—
333#431—1221no—
334#423—1221no—
335#407—612no—
336C4×M16—1221no—
337C4×C4⋊C4—612no—
338C4×C4⋊C4—1221no—
339C4×C22⋊C4—612no—
340C4×C22⋊C4—1221no—
341#376—612no—
342C43—1221no—
343#328—618no—
344#328—618no—
345C61#1754511yescharacteristic, complemented, Sylow
346C22×Q8—611no—
347C22×Q8—1221no—
348C22×Q8C2×#692311yescharacteristic
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C2×M16—1222no—
353C4⋊Q8—612no—
354C4⋊Q8—612no—
355C4⋊Q8—612no—
356C4⋊Q8—612no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361#125—1222no—
362#125—1222no—
363#125—1222no—
364#125—1222no—
365C22⋊Q8—612no—
366C22⋊Q8—612no—
367C22⋊Q8—612no—
368C22⋊Q8—612no—
369C22⋊Q8—1221no—
370C22⋊Q8—1221no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—612no—
380C4×Q8—612no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1221no—
386C4×Q8—1221no—
387C4×Q8C2×#692312yes—
388C4×Q8C2×#692312yes—
389#117—1222no—
390#117—1222no—
391#117—1222no—
392#117—1222no—
393#117—1228no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—611no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4C2×#692311yescharacteristic
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1222no—
407C2×C42—21no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—1222no—
420C2×C42—1221no—
421C2×C42—1222no—
422C2×C42—1222no—
423C2×C42C2×#692311yescharacteristic
424C4⋊C8—1222no—
425C4⋊C8—1222no—
426C22⋊C8—614no—
427C22⋊C8—1222no—
428#97—1222no—
429C4×C8—1222no—
430#95—614no—
431#95—1222no—
432C24—611no—
433C2×Q8—21no—
434C2×Q8—42no—
435C2×Q8—612no—
436C2×Q8—1222no—
437C2×Q8—1222no—
438C2×Q8—1221no—
439C2×Q8—2442no—
440C2×Q8—2444no—
441C2×Q8C22×#692312yes—
442C22×C4—21no—
443C22×C4—21no—
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—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1224no—
460C22×C4—1224no—
461C22×C4—1224no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—2442no—
465C22×C4C22⋊#692311yescharacteristic
466C22×C4C22⋊#692311yescharacteristic
467C22×C4C22×#692311yescharacteristic
468M16—2444no—
469C2×C8—1224no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1228no—
485C4⋊C4—1221no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4C22×#692312yes—
489C22⋊C4—1222no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1228no—
494C42—22no—
495C42—22no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—1228no—
509C42—1222no—
510C42—1222no—
511C42—2444no—
512C42—2444no—
513C42C22⋊#692311yescharacteristic
514C42C22⋊#692311yescharacteristic
515C42C22×#692311yescharacteristic
516C42C22×#692311yescharacteristic
517C23—611no—
518C23—611no—
519C23—1222no—
520C23—1222no—
521C23—2441no—
522C23C2×C22⋊#692311yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—2442no—
526Q8—2444no—
527Q8—2444no—
528Q8—2442no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—614no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—614no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1228no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2442no—
561C2×C4—2444no—
562C2×C4—2444no—
563C2×C4—2442no—
564C2×C4C2×C22⋊#692311yescharacteristic
565C2×C4C2×C22⋊#692311yescharacteristic
566C2×C4C2×C22⋊#692311yescharacteristic
567C2×C4C2×C22⋊#692311yescharacteristic
568C2×C4C2×C22⋊#692311yescharacteristic
569C2×C4C23×#692311yescharacteristic
570C8—2444no—
571C22—21no—
572C22—21no—
573C22—612no—
574C22—1221no—
575C22—1221no—
576C22—1221no—
577C22—1221no—
578C22—1221no—
579C22—2444no—
580C22D61⋊#10412yes—
581C22C22×C22⋊#692311yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—612no—
589C4—612no—
590C4—612no—
591C4—1221no—
592C4—1221no—
593C4—1221no—
594C4—2448no—
595C4—2442no—
596C4—2442no—
597C4#1684074112yes—
598C4#1684075512yes—
599C2—22no—
600C2—1222no—
601C2—2441no—
602C2C2×#1684074111yescharacteristic
603C2C2×#1684075511yescharacteristic
604C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×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#17545
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
5. #21461212C2semidirect
6. #21461201C2semidirect
28. #16840316C4semidirect
29. #16840316C4semidirect
32. C2×#14519776C4semidirect
33. C2×#14519775C4semidirect
39. C2×C4×Dic122C4semidirect
42. C2×C4×Q8×C61C4semidirect
326. C2×C4×Q8#6923semidirect
345. C61#17545semidirect

Maximal subgroups

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

subgroupindex
#214624982
C4×#145198562
#214612192
#214612182
#214612122
#214612012
C2×#168403162
#1754561