Finite Group Databasebeta

← #30617881

Subgroups — #30617881

Order 15616.

1050 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462266C211yesmaximal, characteristic
2#21462266C211yesmaximal, characteristic
3#21462010C214yesmaximal
4#21462009C212yesmaximal
5#21462008C212yesmaximal
6#21461716C214yesmaximal
7C2×#16840316C211yesmaximal, characteristic
8#16840707—21no—
9#16840707—21no—
10#16840706—21no—
11#16840706—21no—
12#16840705C2212yes—
13#16840705C2212yes—
14#16840704C2212yes—
15#16840704C2212yes—
16C4×C4⋊#6923—21no—
17C4×C4⋊#6923—21no—
18C4×C22⋊#6923C2212yes—
19C4×C22⋊#6923C2212yes—
20#16840698—21no—
21#16840698—21no—
22#16840640C2218yes—
23#16840640C2218yes—
24#16840316—21no—
25#16840316—21no—
26#16840316—21no—
27#16840316—21no—
28#16840316—21no—
29#16840316—21no—
30#16840316—21no—
31#16840316—21no—
32C2×#14519776C411yescharacteristic, complemented
33C2×#14519775C411yescharacteristic, complemented
34C2×#14519767C412yescomplemented
35C2×#14519757C2211yescharacteristic
36C2×#14519755C2212yes—
37C2×#14519753C2211yescharacteristic
38C2×C4×Dic122C412yescomplemented
39C2×C4×Dic122C411yescharacteristic, complemented
40C2×#14519734C2212yes—
41C2×C42×D61C2211yescharacteristic
42C2×C4×Q8×C61C411yescharacteristic, complemented
43C2×C4⋊#6923—22no—
44C2×C4⋊#6923—22no—
45C2×C22⋊#6923C2314yes—
46C2×C22⋊#6923C2314yes—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57C22×Dic122C2×C412yes—
58C22×Dic122C2×C411yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519776—21no—
63#14519776—21no—
64#14519776—21no—
65#14519776—21no—
66#14519775—21no—
67#14519775—21no—
68#14519775—21no—
69#14519775—21no—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519757—21no—
75#14519757—21no—
76#14519757—21no—
77#14519757—21no—
78#14519755—22no—
79#14519755—22no—
80#14519755—22no—
81#14519755—22no—
82#14519753—21no—
83#14519753—21no—
84#14519753—21no—
85#14519753—21no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C4×Dic122—22no—
94#14519734—22no—
95#14519734—22no—
96#14519734—22no—
97#14519734—22no—
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×C4⋊Dic61C2×C412yes—
100C2×#11758196C2×C412yes—
101C2×#11758196C2×C412yes—
102C2×#11758196C2×C412yes—
103C2×#11758196C2×C412yes—
104C2×#11758196C2×C412yes—
105C2×#11758198C2312yes—
106C2×#11758198C2312yes—
107C2×C4×Dic61C2×C412yes—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C22×Q8×C61C2×C411yescharacteristic
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61—21no—
123C4×Q8×C61—21no—
124C2×C4⋊C4×C61C2×C412yes—
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C42×C61C2×C412yes—
127C2×C42×C61C2×C411yescharacteristic
128C23×D61C2411yescharacteristic
129C22×#6923—24no—
130C22×#6923—24no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—42no—
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C411yescharacteristic
139C22×Dic61C22×C411yescharacteristic
140C22×Dic61C22×C411yescharacteristic
141C22×Dic61C22×C412yes—
142C22×Dic61C22×C412yes—
143C22×Dic61C22×C411yescharacteristic
144C2×C4×D61—24no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
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×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122—44no—
164C2×Dic122—42no—
165C22⋊#6923—28no—
166C22⋊#6923—28no—
167C4⋊#6923—44no—
168C4⋊#6923—44no—
169C4×#6923—44no—
170C4×#6923—44no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181#11758198—22no—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—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—
197#11758196—22no—
198#11758196—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61C22⋊C411yescharacteristic
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×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C23×C61C23×C411yescharacteristic
214C22×D61—41no—
215C22×D61—41no—
216C22×D61C22×D411yescharacteristic
217C22×D61C22×D411yescharacteristic
218C22×D612−1+414yes—
219C2×#6923—44no—
220C2×#6923—44no—
221C2×#6923—44no—
222C2×#6923—44no—
223C2×#6923—44no—
224C2×#6923—44no—
225Q8×C61—42no—
226Q8×C61—42no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265Dic122—44no—
266Dic122—42no—
267Dic122—42no—
268Dic122—44no—
269#21826—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61C22×C22⋊C411yescharacteristic
273C22×C61#51912yes—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—44no—
279C2×D61#53612yes—
280C2×D61#53612yes—
281#6923—84no—
282#6923—84no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61#41012yes—
291Dic61—21no—
292Dic61—21no—
293Dic61—21no—
294Dic61—21no—
295Dic61—21no—
296Dic61—42no—
297Dic61—42no—
298Dic61#41012yes—
299C2×#533—611no—
300#2283—611no—
301#2283—611no—
302#1996—612no—
303#1996—612no—
304#1996—614no—
305#1820—614no—
306C2×C61—22no—
307C2×C61#231011yescharacteristic
308C2×C61C2×#41011yescharacteristic
309C2×C61C2×#41011yescharacteristic
310D61—22no—
311D61—22no—
312#533—1221no—
313#533—1221no—
314#533—1221no—
315#533—1221no—
316#533—1221no—
317#533—1221no—
318#533—1221no—
319#533—1221no—
320C2×C4⋊Q8—611no—
321C2×C4⋊Q8—611no—
322C2×#125—612no—
323C2×C22⋊Q8—611no—
324C2×C22⋊Q8—612no—
325C2×C22⋊Q8—611no—
326C2×C4×Q8—612no—
327C2×C4×Q8—611no—
328C2×C4×Q8#692311yescharacteristic, complemented
329C2×#117—612no—
330C22×C42—611no—
331#383—1221no—
332#383—1221no—
333#382—1221no—
334#382—1221no—
335#381—612no—
336#381—612no—
337#380—612no—
338#380—612no—
339C4×C4⋊C4—1221no—
340C4×C4⋊C4—1221no—
341C4×C22⋊C4—612no—
342C4×C22⋊C4—612no—
343#376—1221no—
344#376—1221no—
345#342—618no—
346#342—618no—
347C61#2182611yescharacteristic, complemented, Sylow
348C22×Q8—612no—
349C22×Q8—611no—
350C22×Q8C2×#692311yescharacteristic
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362#125—1222no—
363#125—1222no—
364#125—1222no—
365#125—1222no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C22⋊Q8—1221no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390#117—1222no—
391#117—1222no—
392#117—1222no—
393#117—1222no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—612no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4C2×#692312yes—
404C2×C4⋊C4C2×#692311yescharacteristic
405C2×C22⋊C4—612no—
406C2×C22⋊C4—612no—
407C2×C22⋊C4—614no—
408C2×C22⋊C4—614no—
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—611no—
420C2×C42—612no—
421C2×C42—1222no—
422C2×C42—1222no—
423C2×C42C2×#692312yes—
424C2×C42C2×#692311yescharacteristic
425#95—1222no—
426#95—1222no—
427#95—1222no—
428#95—1222no—
429#95—1222no—
430#95—1222no—
431#95—1222no—
432#95—1222no—
433C24—611no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—42no—
437C2×Q8—1222no—
438C2×Q8—1222no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—2442no—
442C2×Q8—2444no—
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—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—1224no—
461C22×C4—1224no—
462C22×C4—1224no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4C22×#692311yescharacteristic
466C22×C4C22×#692311yescharacteristic
467C22×C4C22×#692312yes—
468C22×C4C22×#692312yes—
469C22×C4C22×#692311yescharacteristic
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
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—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—2444no—
489C4⋊C4—2444no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1228no—
495C22⋊C4—1228no—
496C42—22no—
497C42—22no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—1222no—
511C42—1222no—
512C42—2444no—
513C42—2444no—
514C42C22⋊#692311yescharacteristic
515C42C22⋊#692311yescharacteristic
516C42C22⋊#692311yescharacteristic
517C42C22⋊#692311yescharacteristic
518C23—611no—
519C23—611no—
520C23—614no—
521C23—2441no—
522C23—2441no—
523C23C23×#692311yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—2442no—
527Q8—2444no—
528Q8—2444no—
529Q8—2442no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—1222no—
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—1222no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2444no—
561C2×C4—2444no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4—2444no—
565C2×C4—2444no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4C2×C22⋊#692311yescharacteristic
569C2×C4C2×C22⋊#692311yescharacteristic
570C2×C4C2×C22⋊#692311yescharacteristic
571C2×C4C2×C22⋊#692311yescharacteristic
572C2×C4C2×C22⋊#692311yescharacteristic
573C2×C4C2×C22⋊#692311yescharacteristic
574C22—21no—
575C22—21no—
576C22—612no—
577C22—612no—
578C22—1221no—
579C22—1221no—
580C22—1221no—
581C22—1221no—
582C22—2444no—
583C22C22×C22⋊#692311yescharacteristic
584C22#1684083012yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—42no—
591C4—42no—
592C4—612no—
593C4—1221no—
594C4—1221no—
595C4—1221no—
596C4—1221no—
597C4—1221no—
598C4—4884no—
599C4—4884no—
600C4—2442no—
601C4—2442no—
602C4#1684074112yes—
603C2—22no—
604C2—1222no—
605C2—1222no—
606C2#2146229311yescharacteristic
607C2C2×#1684074111yescharacteristic
608C2D61⋊#41011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radical, cosocleC23×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#21826
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
32. C2×#14519776C4semidirect
33. C2×#14519775C4semidirect
34. C2×#14519767C4semidirect
38. C2×C4×Dic122C4semidirect
39. C2×C4×Dic122C4semidirect
42. C2×C4×Q8×C61C4semidirect
328. C2×C4×Q8#6923semidirect
347. C61#21826semidirect

Maximal subgroups

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

subgroupindex
#214622662
#214620102
#214620092
#214620082
#214617162
C2×#168403162
#2182661