Finite Group Databasebeta

← #30620118

Subgroups — #30620118

Order 15616.

1512 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×#16840792C212yesmaximal, complemented
2C2×#16840737C212yesmaximal, complemented
3#21461945C212yesmaximal, complemented
4#21461799C212yesmaximal, complemented
5#21461795C212yesmaximal, complemented
6#21461676C211yesmaximal, characteristic, complemented
7#21461644C212yesmaximal, complemented
8#21461546C211yesmaximal, characteristic, complemented
9C2×#16840313C211yesmaximal, characteristic
10C22×#11758213C2212yes—
11C22×D61⋊C8C2212yes—
12#16840792—28no—
13C2×#14519844C2212yes—
14#16840737—28no—
15C2×#14519834C2212yes—
16C2×#14519834C2212yes—
17C2×#14519834C2212yes—
18C2×#14519834C2212yes—
19C2×#14519834C2211yescharacteristic
20C2×#14519834C2212yes—
21C2×#14519834C2212yes—
22C2×#14519834C2212yes—
23C2×#14519834C2211yescharacteristic
24C2×#14519832C2212yes—
25#16840628C2211yescharacteristic
26#16840628C2212yes—
27#16840628C2211yescharacteristic
28C22×#11758226C2212yes—
29#16840313—28no—
30C2×#14519774C411yescharacteristic
31C2×#14519766C412yes—
32C2×#14519752C412yes—
33C2×#14519752C411yescharacteristic
34C2×#14519752C411yescharacteristic
35C2×#14519739C2212yes—
36C2×#14519739C2211yescharacteristic
37C2×#14519739C2211yescharacteristic
38C2×C42×D61C2211yescharacteristic
39C2×C4⋊Q8×C61C411yescharacteristic
40C2×#11758213—24no—
41C2×#11758213—24no—
42C2×#11758213—42no—
43C2×D61⋊C8—24no—
44C2×D61⋊C8—24no—
45C2×D61⋊C8—42no—
46C22×#92420C2312yes—
47C22×#92420C2312yes—
48C22×#92420C2312yes—
49C22×#92420C2312yes—
50#14519844—24no—
51#14519834—24no—
52#14519834—24no—
53#14519834—24no—
54#14519834—22no—
55#14519834—24no—
56#14519834—24no—
57#14519834—42no—
58#14519834—42no—
59#14519834D414yescomplemented
60#14519832—24no—
61C2×#11758226—24no—
62C2×#11758226—24no—
63C2×#11758226—24no—
64C2×#11758226—24no—
65C2×#11758226—42no—
66C2×#11758226—42no—
67C22×D244C2×C412yes—
68C22×D244C2×C412yes—
69C22×D244C2×C412yes—
70C22×D244C2×C412yes—
71C22×D244C2×C411yescharacteristic
72C22×D244C2×C411yescharacteristic
73C22×C4×D61C2×C411yescharacteristic
74C22×C4×D61C2×C411yescharacteristic
75C22×C4×D61C2312yes—
76C22×C4×D61C2312yes—
77C22×C4×D61C2311yescharacteristic
78#14519774—24no—
79#14519766—28no—
80#14519752—24no—
81#14519752—24no—
82#14519752—28no—
83#14519739—24no—
84#14519739—24no—
85#14519739—24no—
86#14519739D418yes—
87C2×#11758198C2×C412yes—
88C2×#11758198C2×C412yes—
89C2×#11758198C2×C412yes—
90C2×#11758198C2×C412yes—
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94C42×D61—24no—
95C22×Q8×C61C2×C412yes—
96C4⋊Q8×C61—24no—
97C2×C4⋊C4×C61C2×C411yescharacteristic
98C2×C4⋊C4×C61C2×C412yes—
99C2×C4⋊C4×C61C2×C411yescharacteristic
100C2×C42×C61C2×C411yescharacteristic
101C23×D61C22×C412yes—
102C23×D61C22×C412yes—
103C23×D61C22×C411yescharacteristic
104C2×Q8×C61—24no—
105C2×Q8×C61—24no—
106C2×Q8×C61—42no—
107C22×C4×C61C22×C412yes—
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C412yes—
110C22×C4×C61C22×C411yescharacteristic
111C22×C4×C61C22×C411yescharacteristic
112C22×Dic61C22×C411yescharacteristic
113C22×Dic61C22×C411yescharacteristic
114C22×Dic61C2411yescharacteristic
115#11758226—48no—
116#11758226—48no—
117#11758226—48no—
118#11758226—48no—
119C2×D244—24no—
120C2×D244—24no—
121C2×D244—24no—
122C2×D244—24no—
123C2×D244—24no—
124C2×D244—24no—
125C2×D244—22no—
126C2×D244—24no—
127C2×D244—24no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—42no—
131C2×D244—41no—
132C2×D244—41no—
133C2×D244—42no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244C22⋊C414yes—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61C2×D414yes—
156C2×C4×D61C2×D414yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61#4112yes—
159C2×#92420—22no—
160C2×#92420—22no—
161C2×#92420—22no—
162C2×#92420—22no—
163C2×#92420—22no—
164C2×#92420—22no—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420C2×D414yes—
171#11758213—48no—
172D61⋊C8—48no—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C22⋊C414yes—
176C42×C61—22no—
177#11758198—24no—
178#11758198—24no—
179#11758198—24no—
180#11758198C22⋊C418yes—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61C2×D414yes—
184C23×C61C23×C411yescharacteristic
185C22×D61—21no—
186C22×D61—22no—
187C22×D61—21no—
188C22×D61—22no—
189C22×D61—21no—
190C22×D61—22no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—22no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61C2×C22⋊C414yes—
205C22×D61C4×D412yes—
206Q8×C61—48no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61C2×C22⋊C412yes—
219C2×C4×C61C2×C22⋊C414yes—
220C2×C4×C61C4×D412yes—
221C2×C4×C61#13112yes—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61C2×C22⋊C412yes—
228C2×Dic61C22×D411yescharacteristic
229C2×Dic61C22×D412yes—
230C2×Dic61C2×#4111yescharacteristic
231C2×Dic61H(5,𝔽2)12yes—
232D244—44no—
233D244—48no—
234D244—48no—
235D244—48no—
236D244—44no—
237D244—48no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—24no—
243C4×D61—24no—
244C4×D61—44no—
245C4×D61—44no—
246C4×D61—42no—
247C4×D61—42no—
248C4×D61—42no—
249C4×D61—42no—
250#92420—24no—
251#92420—42no—
252#92420—42no—
253#92420—42no—
254#22403—611nomaximal, Sylow
255C22×C61C22×C22⋊C411yescharacteristic
256C22×C61C2×C4×D411yescharacteristic
257C22×C61#51811yescharacteristic
258C22×C61C2×#13111yescharacteristic
259C22×C61#56811yescharacteristic
260C22×C61#56811yescharacteristic
261C22×C61#56811yescharacteristic
262C2×D61—22no—
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—24no—
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—41no—
273C2×D61—42no—
274C2×D61—41no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C4×C61—22no—
284C4×C61—24no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—42no—
288Dic61—22no—
289Dic61—41no—
290Dic61D4212yes—
291Dic61#54612yes—
292C2×#532—611no—
293C2×#435—612no—
294C2×#408—612no—
295#1955—612no—
296#1870—612no—
297#1869—612no—
298#1797—611no—
299#1780—612no—
300#1749—611no—
301C2×C61D4×C22⋊C411yescharacteristic
302C2×C61#287911yescharacteristic
303C2×C61#288011yescharacteristic
304C2×C61#293211yescharacteristic
305C2×C61#293511yescharacteristic
306C2×C61#298011yescharacteristic
307C2×C61#298111yescharacteristic
308D61—42no—
309D61—44no—
310D61—82no—
311C22×#41—612no—
312C22×M16—612no—
313C23×C8—612no—
314#532—1228no—
315C2×C4⋊Q8#692311yescharacteristic
316C2×#127—611no—
317C2×#124—612no—
318C2×#121—612no—
319C2×#121—611no—
320C2×#121—611no—
321C2×C4×D4—611no—
322C2×C4×D4—612no—
323C2×C4×D4—611no—
324C22×C42—611no—
325#435—1228no—
326C2×C4⋊C8—612no—
327#408—1228no—
328C2×C22⋊C8—611no—
329C2×C22⋊C8—612no—
330C2×C22⋊C8—612no—
331C2×C22⋊C8—612no—
332C2×C22⋊C8—612no—
333C2×C22⋊C8—612no—
334C2×C22⋊C8—611no—
335C2×C22⋊C8—612no—
336C2×C22⋊C8—612no—
337C2×#97—612no—
338#336—611no—
339#336—611no—
340#336—612no—
341C61#2240311yescharacteristic, complemented, Sylow
342C2×#41—1224no—
343C2×#41—1224no—
344C2×#41—1224no—
345C2×#41—1224no—
346C2×#41—2442no—
347C2×#41—2442no—
348C22×Q8C2×#692312yes—
349C22×D4—611no—
350C22×D4—612no—
351C22×D4—612no—
352C22×D4—612no—
353C22×D4—612no—
354C22×D4—611no—
355C23×C4—612no—
356C23×C4—611no—
357C23×C4—612no—
358C23×C4—611no—
359C23×C4—611no—
360C2×M16—1224no—
361C2×M16—1224no—
362C2×M16—2442no—
363C22×C8—612no—
364C22×C8—612no—
365C22×C8—612no—
366C22×C8—612no—
367C22×C8—1224no—
368C22×C8—1224no—
369C22×C8—2442no—
370C4⋊Q8—24no—
371#127—1224no—
372#124—1228no—
373#121—1228no—
374#121—1224no—
375#121—1224no—
376C4×D4—618no—
377C4×D4—1224no—
378C4×D4—1224no—
379C4×D4—1224no—
380C2×C4⋊C4C2×#692311yescharacteristic
381C2×C4⋊C4C2×#692312yes—
382C2×C4⋊C4C2×#692311yescharacteristic
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—1224no—
391C2×C42C2×#692311yescharacteristic
392C4⋊C8—1224no—
393C22⋊C8—614no—
394C22⋊C8—1224no—
395C22⋊C8—1224no—
396C22⋊C8—1222no—
397C22⋊C8—1224no—
398C22⋊C8—1224no—
399C22⋊C8—1224no—
400C22⋊C8—2442no—
401C22⋊C8—2442no—
402#97—1224no—
403C24—612no—
404C24—611no—
405C24—612no—
406#41—2448no—
407#41—2448no—
408#41—2448no—
409#41—2448no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—42no—
413C2×D4—614no—
414C2×D4—1224no—
415C2×D4—1224no—
416C2×D4—1224no—
417C2×D4—1224no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1224no—
421C2×D4—1224no—
422C2×D4—1224no—
423C2×D4—2442no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4—2441no—
427C2×D4—2441no—
428C2×D4—2442no—
429C2×D4—2441no—
430C2×D4—2441no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—614no—
434C22×C4—614no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—611no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—2441no—
454C22×C4—2441no—
455C22×C4—2441no—
456C22×C4C22×#692312yes—
457C22×C4C22×#692311yescharacteristic
458C22×C4C22×#692312yes—
459C22×C4C22×#692311yescharacteristic
460C22×C4C22×#692311yescharacteristic
461M16—2448no—
462C2×C8—614no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—2448no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4C22⋊#692314yes—
478C22⋊C4—618no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C42—22no—
483C42—614no—
484C42—1222no—
485C42—1222no—
486C23—614no—
487C23—612no—
488C23—1221no—
489C23—1222no—
490C23—1221no—
491C23—1222no—
492C23—1221no—
493C23—1222no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1222no—
498C23—1222no—
499C23—1222no—
500C23—1222no—
501C23—1222no—
502C23—1222no—
503C23—1222no—
504C23—1222no—
505C23—1222no—
506C23—1222no—
507C23C23×#692311yescharacteristic
508Q8—48no—
509D4—2444no—
510D4—2448no—
511D4—2448no—
512D4—2448no—
513D4—2444no—
514D4—2448no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—612no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—1224no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4C2×C22⋊#692312yes—
549C2×C4C2×C22⋊#692314yes—
550C2×C4D4×#692312yes—
551C2×C4#1451986512yes—
552C8—1224no—
553C8—2442no—
554C8—2442no—
555C8—2442no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—1224no—
560C22—1224no—
561C22—1224no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22—2442no—
571C22—2441no—
572C22—2441no—
573C22—2441no—
574C22—2442no—
575C22—2441no—
576C22—2442no—
577C22C22×C22⋊#692311yescharacteristic
578C22C2×D4×#692311yescharacteristic
579C22#1684082911yescharacteristic
580C22C2×#1451986511yescharacteristic
581C22#1684084211yescharacteristic
582C22#1684084211yescharacteristic
583C22#1684084211yescharacteristic
584C4—24no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—612no—
590C4—612no—
591C4—1222no—
592C4—2441no—
593C2—4882no—
594C2—2442no—
595C2—2444no—
596C2D4×C22⋊#692311yescharacteristic
597C2#2146240611yescharacteristic
598C2#2146240811yescharacteristic
599C2#2146254611yescharacteristic
600C2#2146256511yescharacteristic
601C2#2146270611yescharacteristic
602C2#2146271111yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×C4⋊Q8×C61C4
prime radicalC2×#16840313C2
cosocleC22×Dic61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#22403
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840792C2semidirect
2. C2×#16840737C2semidirect
3. #21461945C2semidirect
4. #21461799C2semidirect
5. #21461795C2semidirect
6. #21461676C2semidirect
7. #21461644C2semidirect
8. #21461546C2semidirect
59. #14519834D4semidirect
341. C61#22403semidirect

Maximal subgroups

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

subgroupindex
C2×#168407922
C2×#168407372
#214619452
#214617992
#214617952
#214616762
#214616442
#214615462
C2×#168403132
#2240361