Finite Group Databasebeta

← #30620477

Subgroups — #30620477

Order 15616.

1412 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
1Q8×C22⋊#6923C212yesmaximal, complemented
2#21461960C214yesmaximal, complemented
3#21461959C212yesmaximal, complemented
4#21461682C212yesmaximal, complemented
5#21461679C211yesmaximal, characteristic, complemented
6#21461564C212yesmaximal, complemented
7C4×#14519835C211yesmaximal, characteristic, complemented
8C2×#16840422C211yesmaximal, characteristic
9C2×Q8×#6923—22no—
10C2×#14519836C2214yes—
11C2×#14519836C2214yes—
12C2×#14519835C2212yes—
13C2×#14519835C2212yes—
14(C2×#6923)⋊Q8—24no—
15(C2×#6923)⋊Q8—22no—
16#16840704C2212yes—
17#16840704C2214yes—
18C4×C22⋊#6923C2212yes—
19C4×C22⋊#6923C2214yes—
20#16840640C2211yescharacteristic
21#16840640C2211yescharacteristic
22#16840640C2212yes—
23C22×#11758222C411yescharacteristic, complemented
24C22×Q8×D61C2211yescharacteristic
25#16840422—22no—
26#16840422—24no—
27#16840422—22no—
28C2×#14519766C411yescharacteristic, complemented
29C2×#14519763C412yescomplemented
30C2×#14519755C412yescomplemented
31C2×#14519752C2211yescharacteristic
32C2×#14519750C2212yes—
33C2×#14519742C411yescharacteristic, complemented
34C2×#14519739C2212yes—
35C2×C4×C22⋊D61C2211yescharacteristic
36C2×C22⋊Q8×C61C411yescharacteristic, complemented
37Q8×#6923—48no—
38C2×C4⋊#6923—22no—
39C2×C4⋊#6923—24no—
40C2×C22⋊#6923C2312yes—
41C2×C22⋊#6923C2314yes—
42C2×C22⋊#6923C2312yes—
43C2×C4×#6923—22no—
44C2×C4×#6923—24no—
45#14519836—28no—
46#14519836—44no—
47#14519835—24no—
48#14519835—42no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—22no—
52(C2×#6923)⋊C4—24no—
53C2×#11758222—21no—
54C2×#11758222—22no—
55C2×#11758222—21no—
56C2×#11758222—44no—
57C22×Dic122C2×C411yescharacteristic
58C22×Dic122C2×C412yes—
59C2×Q8×D61—44no—
60C2×Q8×D61D412yes—
61C2×Q8×D61D414yes—
62C2×Q8×D61D412yes—
63C22×D244C2×C411yescharacteristic
64C22×D244C2312yes—
65C22×C22⋊D61C2×C412yes—
66C22×C22⋊D61C2311yescharacteristic
67C22×C4×D61C2×C412yes—
68C22×C4×D61C2311yescharacteristic
69C22×C4×D61C2312yes—
70#14519766—22no—
71#14519766—22no—
72#14519763—24no—
73#14519763—24no—
74#14519755—24no—
75#14519755—24no—
76#14519752—22no—
77#14519752—22no—
78#14519750—24no—
79#14519750—24no—
80#14519742—22no—
81#14519742—21no—
82#14519742—21no—
83#14519739—24no—
84#14519739—24no—
85C4×C22⋊D61—22no—
86C4×C22⋊D61—22no—
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C412yes—
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×#11758198C2×C412yes—
91C2×#11758198C2×C412yes—
92C2×#11758198C2×C412yes—
93C2×#11758198C2311yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C412yes—
96C22×Q8×C61C2×C411yescharacteristic
97C23×C4×C61C2×C411yescharacteristic
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C2×C4⋊C4×C61C2×C412yes—
101C2×C4⋊C4×C61C2×C411yescharacteristic
102C2×C22⋊C4×C61C2×C412yes—
103C24×C61C22×C411yescharacteristic
104C23×D61C22×C412yes—
105C23×D61C2411yescharacteristic
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—24no—
109C2×Q8×C61—42no—
110C2×Q8×C61C22⋊C412yes—
111C2×Q8×C61C22⋊C412yes—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—42no—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C412yes—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—21no—
122C2×C22⋊D61—21no—
123C2×C22⋊D61—22no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—44no—
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C412yes—
128C22×Dic61C22×C411yescharacteristic
129C22×Dic61C22×C411yescharacteristic
130Q8×D61—28no—
131Q8×D61—48no—
132Q8×D61—44no—
133#11758222—44no—
134#11758222—48no—
135#11758222—44no—
136C2×D244—24no—
137C2×D244—24no—
138C2×D244—22no—
139C2×D244—22no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—42no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—44no—
146C2×C4×D61—42no—
147C2×C4×D61—44no—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D414yes—
150C2×C4×D61C2×D412yes—
151C2×C4×D61C2×D414yes—
152C2×Dic122—24no—
153C2×Dic122—44no—
154C2×Dic122C22⋊C411yescharacteristic
155C2×Dic122C22⋊C412yes—
156C2×Dic122C22⋊C414yes—
157C2×Dic122C22⋊C414yes—
158C2×Dic122C22⋊C411yescharacteristic
159C22⋊#6923—22no—
160C22⋊#6923—22no—
161C22⋊#6923—24no—
162C22⋊#6923—24no—
163C22⋊#6923—22no—
164C22⋊#6923#4114yescomplemented
165C4⋊#6923—44no—
166C4⋊#6923—48no—
167C4×#6923—44no—
168C4×#6923—48no—
169C4⋊C4×C61—24no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C22⋊C4×C61—24no—
173C22⋊Dic61—22no—
174#11758198—22no—
175#11758198—24no—
176#11758198—24no—
177#11758198—24no—
178#11758196—22no—
179#11758196—22no—
180#11758196—22no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61C23×C411yescharacteristic
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61C22×D412yes—
199C22×D61C2×#4112yes—
200C22×D61H(5,𝔽2)11yescharacteristic
201C22×D612−1+411yescharacteristic
202C2×#6923—24no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923—42no—
206C2×#6923—44no—
207C2×#6923—42no—
208C2×#6923—44no—
209C2×#6923—42no—
210C2×#6923—44no—
211Q8×C61—24no—
212Q8×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—41no—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C414yes—
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C22⋊D61—44no—
226C22⋊D61—42no—
227C22⋊D61—42no—
228C22⋊D61—44no—
229C2×Dic61—24no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238C2×Dic61C2×C22⋊C412yes—
239D244—42no—
240D244—44no—
241D244—44no—
242D244—42no—
243C4×D61—24no—
244C4×D61—28no—
245C4×D61—42no—
246C4×D61—44no—
247C4×D61—44no—
248C4×D61—44no—
249Dic122—22no—
250Dic122—24no—
251Dic122—28no—
252Dic122—44no—
253Dic122#9914yescomplemented
254#22534—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C22×C22⋊C411yescharacteristic
263C22×C61C4×#4111yescharacteristic
264C22×C61#52011yescharacteristic
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—42no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—82no—
274C2×D61—82no—
275C2×D61#54812yes—
276C2×D61#55012yes—
277#6923—44no—
278#6923—82no—
279#6923—84no—
280C4×C61—21no—
281C4×C61—24no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61C2×#9912yes—
285Dic61—22no—
286Dic61—22no—
287Dic61—41no—
288Dic61—41no—
289Dic61C2×#9914yes—
290C2×#548—611no—
291Q8×C22⋊C4—612no—
292#1964—612no—
293#1964—614no—
294#1799—611no—
295#1799—612no—
296C4×#99—611no—
297C4×#99—612no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—81no—
301C2×C61#234111yescharacteristic
302C2×C61C22×#9911yescharacteristic
303C2×C61#288411yescharacteristic
304D61—22no—
305D61—22no—
306D61—82no—
307C22×#41—611no—
308C23×Q8—611no—
309#548—1222no—
310#548—1224no—
311#548—1222no—
312C2×#124—611no—
313C2×#124—612no—
314C2×C22⋊Q8—612no—
315C2×C22⋊Q8#692311yescharacteristic, complemented
316C2×#121—611no—
317C2×#121—612no—
318C2×C4×Q8—611no—
319C2×C4×Q8—1222no—
320C2×C4×D4—611no—
321C2×C4×D4—612no—
322C2×#99—612no—
323C2×#99—614no—
324C2×#99—612no—
325C2×#99—614no—
326#391—1224no—
327#391—1222no—
328#380—612no—
329#380—614no—
330C4×C22⋊C4—612no—
331C4×C22⋊C4—614no—
332#342—612no—
333#342—611no—
334#342—611no—
335C61#2253411yescharacteristic, complemented, Sylow
336C2×#41—1221no—
337C2×#41—1222no—
338C2×#41—1221no—
339C2×#41—2444no—
340C22×Q8—612no—
341C22×Q8—614no—
342C22×Q8—612no—
343C22×Q8—611no—
344C22×Q8—612no—
345C22×Q8—2444no—
346C22×Q8C2×#692311yescharacteristic
347C22×D4—611no—
348C22×D4—612no—
349C22×D4—612no—
350C22×D4—611no—
351C23×C4—612no—
352C23×C4—611no—
353C23×C4—612no—
354C23×C4C2×#692311yescharacteristic
355#124—1224no—
356#124—1224no—
357#124—1222no—
358#124—1222no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—1224no—
362C22⋊Q8—1224no—
363#121—1224no—
364#121—1224no—
365#121—1222no—
366#121—1222no—
367C4×Q8—1221no—
368C4×Q8—1222no—
369C4×Q8—1221no—
370C4×Q8—2448no—
371C4×D4—1224no—
372C4×D4—1224no—
373C4×D4—1222no—
374C4×D4—1222no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4C2×#692312yes—
380C2×C4⋊C4C2×#692311yescharacteristic
381C2×C22⋊C4—612no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4—612no—
387C2×C22⋊C4—612no—
388C2×C22⋊C4—614no—
389C2×C22⋊C4C2×#692312yes—
390C2×C42—612no—
391C2×C42—611no—
392C2×C42—1224no—
393C2×C42—1222no—
394#99—1224no—
395#99—1228no—
396#99—2442no—
397#99—2444no—
398#95—1224no—
399#95—1222no—
400#95—1224no—
401#95—1222no—
402C24—611no—
403C24—612no—
404C24C22×#692311yescharacteristic
405#41—2444no—
406#41—2448no—
407#41—2444no—
408C2×Q8—42no—
409C2×Q8—611no—
410C2×Q8—612no—
411C2×Q8—611no—
412C2×Q8—614no—
413C2×Q8—614no—
414C2×Q8—1224no—
415C2×Q8—1228no—
416C2×Q8—2444no—
417C2×Q8—2444no—
418C2×Q8—2448no—
419C2×Q8C22⋊#692312yes—
420C2×Q8C22⋊#692312yes—
421C2×D4—1224no—
422C2×D4—1222no—
423C2×D4—1224no—
424C2×D4—1222no—
425C2×D4—1222no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1221no—
429C2×D4—1222no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—2442no—
434C2×D4—2444no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—612no—
439C22×C4—611no—
440C22×C4—614no—
441C22×C4—612no—
442C22×C4—614no—
443C22×C4—612no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1224no—
451C22×C4—2442no—
452C22×C4—2444no—
453C22×C4—2444no—
454C22×C4C22×#692312yes—
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692312yes—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—1222no—
464C4⋊C4—2448no—
465C4⋊C4—2444no—
466C22⋊C4—24no—
467C22⋊C4—614no—
468C22⋊C4—1224no—
469C22⋊C4—1222no—
470C22⋊C4—1222no—
471C22⋊C4—1222no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—1222no—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—1222no—
478C42—1222no—
479C42—1222no—
480C42—1221no—
481C42—1221no—
482C42—2448no—
483C42—2444no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—41no—
488C23—41no—
489C23—612no—
490C23—612no—
491C23—611no—
492C23—611no—
493C23—1222no—
494C23—1222no—
495C23—1222no—
496C23—2441no—
497C23—2441no—
498C23—2442no—
499C23—2442no—
500C23C23×#692311yescharacteristic
501Q8—24no—
502Q8—42no—
503Q8—614no—
504Q8—1228no—
505Q8—1224no—
506Q8—1222no—
507Q8—2444no—
508D4—2442no—
509D4—2444no—
510D4—2444no—
511D4—2444no—
512D4—2444no—
513D4—2442no—
514D4—2442no—
515D4—2442no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—1228no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—1224no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—2442no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—2444no—
540C2×C4—2441no—
541C2×C4—2442no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2444no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C2×C22⋊#692314yes—
552C2×C4C2×C22⋊#692311yescharacteristic
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—81no—
559C22—81no—
560C22—612no—
561C22—612no—
562C22—1222no—
563C22—1222no—
564C22—4882no—
565C22—4882no—
566C22—2442no—
567C22—2441no—
568C22—2441no—
569C22—2442no—
570C22—2442no—
571C22—2442no—
572C22C22×C22⋊#692311yescharacteristic
573C22#41×#692311yescharacteristic
574C22#1684083111yescharacteristic
575C4—21no—
576C4—24no—
577C4—42no—
578C4—41no—
579C4—614no—
580C4—1222no—
581C4—1222no—
582C4—4882no—
583C4—4884no—
584C4—2444no—
585C4—2441no—
586C4—2441no—
587C4D61⋊#9912yes—
588C2—21no—
589C2—21no—
590C2—81no—
591C2—1222no—
592C2—1222no—
593C2—4882no—
594C2#2146232411yescharacteristic
595C2C2×D61⋊#9911yescharacteristic
596C2#2146242011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840422C2
cosocleC23×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#22534
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊#6923C2semidirect
2. #21461960C2semidirect
3. #21461959C2semidirect
4. #21461682C2semidirect
5. #21461679C2semidirect
6. #21461564C2semidirect
7. C4×#14519835C2semidirect
23. C22×#11758222C4semidirect
28. C2×#14519766C4semidirect
29. C2×#14519763C4semidirect
30. C2×#14519755C4semidirect
33. C2×#14519742C4semidirect
36. C2×C22⋊Q8×C61C4semidirect
164. C22⋊#6923#41semidirect
253. Dic122#99semidirect
315. C2×C22⋊Q8#6923semidirect
335. C61#22534semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊#69232
#214619602
#214619592
#214616822
#214616792
#214615642
C4×#145198352
C2×#168404222
#2253461