Finite Group Databasebeta

← #30617884

Subgroups — #30617884

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462295C211yesmaximal, characteristic
2#21462295C211yesmaximal, characteristic
3#21462010C212yesmaximal, complemented
4#21462010C212yesmaximal, complemented
5#21462009C212yesmaximal, complemented
6#21462008C212yesmaximal, complemented
7#21461715C212yesmaximal
8#21461714C212yesmaximal
9C2×#16840387C211yesmaximal, characteristic
10#16840712—24no—
11#16840712—24no—
12#16840704C2212yes—
13#16840704C2212yes—
14#16840703C2212yes—
15#16840703C2212yes—
16C4×C22⋊#6923C2212yes—
17C4×C22⋊#6923C2212yes—
18#16840640C2214yes—
19#16840640C2214yes—
20#16840640C2214yes—
21#16840640C2214yes—
22#16840387—24no—
23#16840387—24no—
24C2×#14519757C411yescharacteristic, complemented
25C2×#14519757C2211yescharacteristic
26C2×#14519756C411yescharacteristic, complemented
27C2×#14519755C411yescharacteristic, complemented
28C2×#14519755C411yescharacteristic, complemented
29C2×#14519755C2211yescharacteristic
30C2×#14519755C2211yescharacteristic
31C2×#14519754C411yescharacteristic, complemented
32C2×#14519754C411yescharacteristic, complemented
33C2×#14519753C411yescharacteristic, complemented
34C2×#14519753C2211yescharacteristic
35C22×#11758198C2211yescharacteristic
36C2×C22⋊C4×D61C2211yescharacteristic
37C2×C22⋊C4×D61C2211yescharacteristic
38C2×C22⋊Q8×C61C411yescharacteristic, complemented
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—22no—
41C2×C22⋊#6923—22no—
42C2×C22⋊#6923—22no—
43C2×C22⋊#6923—22no—
44C2×C22⋊#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—
53C24×D61C2311yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C22×Dic122C2×C411yescharacteristic
56C22×Dic122C2×C411yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519757—22no—
63#14519757—22no—
64#14519757—24no—
65#14519756—24no—
66#14519755—22no—
67#14519755—22no—
68#14519755—22no—
69#14519755—22no—
70#14519755—24no—
71#14519755—24no—
72#14519754—24no—
73#14519754—24no—
74#14519753—22no—
75#14519753—22no—
76#14519753—24no—
77C2×C4⋊Dic61C2×C411yescharacteristic
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×C4⋊Dic61C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C411yescharacteristic
84C2×#11758196C2×C411yescharacteristic
85C2×#11758196C2×C411yescharacteristic
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×#11758198—22no—
89C2×#11758198—22no—
90C2×#11758198C2×C411yescharacteristic
91C2×#11758198C2×C411yescharacteristic
92C2×#11758198C2×C411yescharacteristic
93C2×#11758198C2×C411yescharacteristic
94C2×#11758198C2311yescharacteristic
95C2×#11758198C2311yescharacteristic
96C2×#11758198C2311yescharacteristic
97C2×#11758198C2311yescharacteristic
98C22⋊C4×D61—22no—
99C22⋊C4×D61—22no—
100C22⋊C4×D61D414yes—
101C22⋊C4×D61D414yes—
102C22×Q8×C61C2×C411yescharacteristic
103C23×C4×C61C2×C411yescharacteristic
104C22⋊Q8×C61—24no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C4⋊C4×C61C2×C411yescharacteristic
108C2×C22⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61C2×C411yescharacteristic
110C24×C61C22×C411yescharacteristic
111C23×D61—24no—
112C23×D61—81no—
113C23×D61—81no—
114C23×D61C22×C411yescharacteristic
115C23×D61C2×D412yes—
116C23×D61C2×D412yes—
117C23×D61C2411yescharacteristic
118C22×#6923—24no—
119C22×#6923—24no—
120C2×Q8×C61—22no—
121C2×Q8×C61—81no—
122C22×C4×C61—22no—
123C22×C4×C61—81no—
124C22×C4×C61C22×C411yescharacteristic
125C22×C4×C61C22×C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C22×C4×C61C22×C411yescharacteristic
130C22×Dic61—22no—
131C22×Dic61—81no—
132C22×Dic61C22×C411yescharacteristic
133C22×Dic61C22×C411yescharacteristic
134C22×Dic61C22×C411yescharacteristic
135C22×Dic61C22×C411yescharacteristic
136C22×Dic61C22×C411yescharacteristic
137C22×Dic61C22×C411yescharacteristic
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—81no—
141C2×C4×D61—81no—
142C2×C4×D61—81no—
143C2×C4×D61—81no—
144C2×C4×D61C2×D412yes—
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—22no—
151C2×Dic122—81no—
152C2×Dic122—81no—
153C2×Dic122—81no—
154C22⋊#6923—28no—
155C22⋊#6923—28no—
156C22⋊#6923—82no—
157C22⋊#6923—82no—
158C22⋊#6923—82no—
159C22⋊#6923—82no—
160C4⋊#6923—82no—
161C4⋊#6923—82no—
162C4×#6923—82no—
163C4×#6923—82no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61C22⋊C414yes—
169C22⋊Dic61—22no—
170C22⋊Dic61C22⋊C414yes—
171#11758198—22no—
172#11758198—22no—
173#11758198—21no—
174#11758198—21no—
175#11758198—21no—
176#11758198—21no—
177#11758198—21no—
178#11758198—21no—
179#11758198—21no—
180#11758198—21no—
181#11758198C22⋊C414yes—
182#11758198C22⋊C414yes—
183C4⋊Dic61—22no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192C23×C61—22no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C2×C22⋊C412yes—
196C23×C61C23×C411yescharacteristic
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—41no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—88no—
209C22×D61C2×C22⋊C412yes—
210C22×D61C22×D411yescharacteristic
211C22×D61C22×D411yescharacteristic
212C22×D612−1+414yes—
213C2×#6923—44no—
214C2×#6923—44no—
215C2×#6923—44no—
216C2×#6923—44no—
217C2×#6923—44no—
218C2×#6923—44no—
219Q8×C61—82no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—82no—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61—82no—
251C4×D61—82no—
252Dic122—82no—
253Dic122—82no—
254Dic122—82no—
255#21827—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—44no—
262C22×C61C22×C22⋊C411yescharacteristic
263C22×C61#51912yes—
264C2×D61—21no—
265C2×D61—21no—
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—84no—
274C2×D61—84no—
275C2×D61#53612yes—
276C2×D61#53612yes—
277#6923—84no—
278#6923—84no—
279C4×C61—22no—
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285Dic61—22no—
286Dic61—22no—
287Dic61—41no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291C2×#543—611no—
292#2312—611no—
293#2312—611no—
294#1996—612no—
295#1996—612no—
296#1996—612no—
297#1996—612no—
298#1819—612no—
299#1819—612no—
300C2×C61—22no—
301C2×C61—42no—
302C2×C61#231011yescharacteristic
303C2×C61#288311yescharacteristic
304C2×C61#288311yescharacteristic
305D61—22no—
306D61—22no—
307D61—42no—
308#543—1224no—
309#543—1224no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—611no—
312C2×C22⋊Q8—611no—
313C2×C22⋊Q8—611no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—611no—
316C2×C22⋊Q8—611no—
317C2×C22⋊Q8—611no—
318C2×C22⋊Q8—611no—
319C2×C22⋊Q8—611no—
320C2×C22⋊Q8—611no—
321C2×C22⋊Q8#692311yescharacteristic, complemented
322C22×C22⋊C4—611no—
323C22×C22⋊C4—611no—
324C22×C22⋊C4—611no—
325#388—1224no—
326#388—1224no—
327#380—612no—
328#380—612no—
329#379—612no—
330#379—612no—
331C4×C22⋊C4—612no—
332C4×C22⋊C4—612no—
333#342—614no—
334#342—614no—
335#342—614no—
336#342—614no—
337C61#2182711yescharacteristic, complemented, Sylow
338C25—611no—
339C22×Q8—611no—
340C22×Q8—611no—
341C22×Q8—611no—
342C22×Q8C2×#692311yescharacteristic
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4C2×#692311yescharacteristic
349C22⋊Q8—24no—
350C22⋊Q8—1224no—
351C22⋊Q8—1224no—
352C22⋊Q8—1224no—
353C22⋊Q8—1224no—
354C22⋊Q8—1224no—
355C22⋊Q8—1224no—
356C22⋊Q8—1222no—
357C22⋊Q8—1222no—
358C22⋊Q8—1222no—
359C22⋊Q8—1222no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1224no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4C2×#692311yescharacteristic
377C2×C4⋊C4C2×#692311yescharacteristic
378C2×C4⋊C4C2×#692311yescharacteristic
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—614no—
383C2×C22⋊C4—614no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—611no—
391C2×C22⋊C4—614no—
392C2×C22⋊C4—614no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1222no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1222no—
401C2×C22⋊C4C2×#692311yescharacteristic
402C2×C22⋊C4C2×#692311yescharacteristic
403C2×C42—1222no—
404C2×C42—1222no—
405#95—1222no—
406#95—1222no—
407#95—1222no—
408#95—1222no—
409C24—611no—
410C24—612no—
411C24—612no—
412C24—611no—
413C24—1224no—
414C24—4881no—
415C24—4881no—
416C24C22×#692311yescharacteristic
417C2×Q8—22no—
418C2×Q8—81no—
419C2×Q8—1222no—
420C2×Q8—1222no—
421C2×Q8—1222no—
422C2×Q8—4881no—
423C2×Q8—4881no—
424C2×Q8—4881no—
425C22×C4—22no—
426C22×C4—81no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—4881no—
443C22×C4—4881no—
444C22×C4—4881no—
445C22×C4—4881no—
446C22×C4—4881no—
447C22×C4C22×#692311yescharacteristic
448C22×C4C22×#692311yescharacteristic
449C22×C4C22×#692311yescharacteristic
450C22×C4C22×#692311yescharacteristic
451C22×C4C22×#692311yescharacteristic
452C22×C4C22×#692311yescharacteristic
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—4882no—
466C4⋊C4—4882no—
467C22⋊C4—22no—
468C22⋊C4—614no—
469C22⋊C4—614no—
470C22⋊C4—614no—
471C22⋊C4—1221no—
472C22⋊C4—1221no—
473C22⋊C4—1221no—
474C22⋊C4—1221no—
475C22⋊C4—1221no—
476C22⋊C4—1221no—
477C22⋊C4—1222no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C22⋊C4—1222no—
481C22⋊C4—1222no—
482C22⋊C4—1228no—
483C22⋊C4—1228no—
484C22⋊C4—4882no—
485C22⋊C4—4882no—
486C22⋊C4—4882no—
487C22⋊C4—4882no—
488C22⋊C4C22⋊#692314yes—
489C42—4882no—
490C42—4882no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—611no—
495C23—611no—
496C23—614no—
497C23—612no—
498C23—1224no—
499C23—1224no—
500C23—1222no—
501C23—4888no—
502C23—2442no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23C2×C22⋊#692312yes—
511C23C23×#692311yescharacteristic
512Q8—82no—
513Q8—4882no—
514Q8—4882no—
515Q8—4882no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2444no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C2×C22⋊#692312yes—
553C2×C4C2×C22⋊#692312yes—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—44no—
560C22—612no—
561C22—612no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1222no—
567C22—1222no—
568C22—1222no—
569C22—4884no—
570C22—4884no—
571C22—2444no—
572C22—2444no—
573C22C22×C22⋊#692311yescharacteristic
574C22#1684083012yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—1222no—
582C4—1222no—
583C4—4884no—
584C4—4884no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C2—22no—
590C2—42no—
591C2—1222no—
592C2—1222no—
593C2—2442no—
594C2#2146229311yescharacteristic
595C2#2146241811yescharacteristic
596C2#2146241911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#21827
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21462010C2semidirect
4. #21462010C2semidirect
5. #21462009C2semidirect
6. #21462008C2semidirect
24. C2×#14519757C4semidirect
26. C2×#14519756C4semidirect
27. C2×#14519755C4semidirect
28. C2×#14519755C4semidirect
31. C2×#14519754C4semidirect
32. C2×#14519754C4semidirect
33. C2×#14519753C4semidirect
38. C2×C22⋊Q8×C61C4semidirect
321. C2×C22⋊Q8#6923semidirect
337. C61#21827semidirect

Maximal subgroups

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

subgroupindex
#214622952
#214620102
#214620092
#214620082
#214617152
#214617142
C2×#168403872
#2182761