Finite Group Databasebeta

← #30602508

Subgroups — #30602508

Order 15616.

1866 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519836C212yesmaximal
2C2×#16840640C212yesmaximal, complemented
3#21460993C212yesmaximal, complemented
4C22×#14519739C211yesmaximal, characteristic
5C22×C22⋊#6923C2214yes—
6C2×#14519836—24no—
7C2×#14519836—24no—
8C2×#14519836C418yescomplemented
9C2×(C2×#6923)⋊C4—24no—
10#16840640—24no—
11C23×D244C2211yescharacteristic
12C23×C4×D61C411yescharacteristic, complemented
13C23×C4×D61C2211yescharacteristic
14C2×#14519739—24no—
15C2×#14519739—24no—
16C2×#14519739—24no—
17C22×#11758198C411yescharacteristic, complemented
18C22×#11758198C2211yescharacteristic
19C22×C4×Dic61C411yescharacteristic, complemented
20C22×C4⋊C4×C61C411yescharacteristic, complemented
21C23×#6923—24no—
22C2×C22⋊#6923—24no—
23C2×C22⋊#6923—24no—
24C2×C22⋊#6923—24no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—24no—
27C2×C22⋊#6923C2×C418yescomplemented
28#14519836—216no—
29(C2×#6923)⋊C4—48no—
30C24×D61C2×C411yescharacteristic
31C24×D61C2311yescharacteristic
32C22×D244—22no—
33C22×D244—22no—
34C22×D244—22no—
35C22×D244—22no—
36C22×D244C2×C414yes—
37C22×D244D412yes—
38C22×D244D414yes—
39C22×D244Q812yes—
40C23×Dic61C2×C411yescharacteristic
41C23×Dic61C2×C411yescharacteristic
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—44no—
48C22×C4×D61—44no—
49C22×C4×D61D414yes—
50#14519739—232no—
51C2×#11758198—22no—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×#11758198D414yes—
57C2×C4×Dic61—22no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—22no—
60C23×C4×C61C2×C411yescharacteristic
61C23×C4×C61C2×C411yescharacteristic
62C23×C4×C61C2×C411yescharacteristic
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61—22no—
65C2×C4⋊C4×C61—22no—
66C24×C61C22×C411yescharacteristic
67C23×D61—21no—
68C23×D61—21no—
69C23×D61—21no—
70C23×D61—21no—
71C23×D61—42no—
72C23×D61—42no—
73C23×D61—42no—
74C23×D61—42no—
75C23×D61C4212yes—
76C23×D61C22⋊C412yes—
77C23×D61C4⋊C412yes—
78C23×D61C22×C412yes—
79C23×D61C2×D412yes—
80C23×D61C2×D411yescharacteristic
81C23×D61C2×D412yes—
82C23×D61C2×D412yes—
83C23×D61C2×Q811yescharacteristic
84C23×D61#4112yes—
85C23×D61#4112yes—
86C22×#6923—28no—
87C22×#6923—48no—
88C22×#6923—48no—
89C22×#6923—44no—
90C22×#6923—44no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—42no—
100C22×C4×C61—42no—
101C22×C4×C61C4212yes—
102C22×C4×C61C22⋊C412yes—
103C22×C4×C61C22⋊C412yes—
104C22×C4×C61C22⋊C412yes—
105C22×C4×C61C4⋊C412yes—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—24no—
109C22×Dic61—21no—
110C22×Dic61—21no—
111C22×Dic61—42no—
112C22×Dic61C22⋊C412yes—
113C22×Dic61C22⋊C412yes—
114C2×D244—28no—
115C2×D244—28no—
116C2×D244—28no—
117C2×D244—24no—
118C2×D244—24no—
119C2×D244—24no—
120C2×D244—24no—
121C2×D244—28no—
122C2×D244C22⋊C418yes—
123C2×D244C4⋊C418yes—
124C2×C4×D61—28no—
125C2×C4×D61—28no—
126C2×C4×D61—28no—
127C2×C4×D61—44no—
128C2×C4×D61—48no—
129C2×C4×D61—48no—
130C2×C4×D61—48no—
131C2×C4×D61—48no—
132C2×C4×D61—44no—
133C2×C4×D61—44no—
134C22⋊#6923—28no—
135C22⋊#6923—28no—
136C22⋊#6923—44no—
137C22⋊#6923—44no—
138C4⋊C4×C61—28no—
139#11758198—28no—
140#11758198—28no—
141C4×Dic61—28no—
142C23×C61—22no—
143C23×C61—22no—
144C23×C61C2×C4211yescharacteristic
145C23×C61C2×C22⋊C411yescharacteristic
146C23×C61C2×C22⋊C411yescharacteristic
147C23×C61C2×C22⋊C411yescharacteristic
148C23×C61C2×C4⋊C411yescharacteristic
149C23×C61#11711yescharacteristic
150C23×C61#11711yescharacteristic
151C22×D61—24no—
152C22×D61—24no—
153C22×D61—22no—
154C22×D61—22no—
155C22×D61—22no—
156C22×D61—22no—
157C22×D61—22no—
158C22×D61—22no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61—44no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—44no—
167C22×D61—44no—
168C22×D61—42no—
169C22×D61—44no—
170C22×D61—44no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—44no—
176C22×D61—42no—
177C22×D61—44no—
178C22×D61#9514yescomplemented
179C22×D61C2×C22⋊C412yes—
180C22×D61C2×C4⋊C412yes—
181C22×D61C4×D414yes—
182C22×D61C4×D414yes—
183C22×D61#12014yes—
184C22×D61#12112yes—
185C22×D61#12112yes—
186C22×D61C22⋊Q812yes—
187C22×D61C22⋊Q812yes—
188C22×D61#12314yes—
189C2×#6923—416no—
190C2×#6923—48no—
191C2×#6923—88no—
192C2×#6923—84no—
193C2×#6923—84no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—24no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61#9514yescomplemented
209C2×Dic61—24no—
210C2×Dic61—24no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61#9914yescomplemented
218D244—216no—
219D244—216no—
220C4×D61—416no—
221C4×D61—416no—
222#10761—611nomaximal, Sylow
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61C2×#9511yescharacteristic
232C22×C61C4×C22⋊C411yescharacteristic
233C22×C61C4×C22⋊C411yescharacteristic
234C22×C61#37911yescharacteristic
235C22×C61#38011yescharacteristic
236C22×C61#38011yescharacteristic
237C22×C61#38111yescharacteristic
238C22×C61C2×#9912yes—
239C22×C61#41012yes—
240C2×D61—24no—
241C2×D61—24no—
242C2×D61—28no—
243C2×D61—24no—
244C2×D61—24no—
245C2×D61—24no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61—44no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—44no—
252C2×D61—44no—
253C2×D61—44no—
254C2×D61—44no—
255C2×D61—44no—
256C2×D61—44no—
257C2×D61—42no—
258C2×D61—42no—
259C2×D61—44no—
260C2×D61#38514yes—
261C2×D61#38614yes—
262#6923—88no—
263C4×C61—24no—
264C4×C61—42no—
265C4×C61—42no—
266Dic61—24no—
267Dic61—42no—
268C22×C4×D4—611no—
269C22×#99—612no—
270C2×#342—612no—
271C22⋊#95—612no—
272C2×C61—21no—
273C2×C61—21no—
274C2×C61—21no—
275C2×C61—21no—
276C2×C61C22⋊#9511yescharacteristic
277C2×C61C4×#9912yes—
278C2×C61#178111yescharacteristic
279C2×C61#178211yescharacteristic
280C2×C61#179912yes—
281D61—24no—
282D61—24no—
283D61—44no—
284C23×D4—611no—
285C24×C4—611no—
286C24×C4—611no—
287C2×C4×D4—1224no—
288C2×C4×D4—1224no—
289C2×C4×D4—1224no—
290C22×C4⋊C4#692311yescharacteristic, complemented
291C22×C22⋊C4—611no—
292C22×C22⋊C4—614no—
293C22×C22⋊C4—611no—
294C22×C42—611no—
295C2×#99—618no—
296C2×#99—1224no—
297C2×#99—1224no—
298C2×#95—1224no—
299#342—1224no—
300C61#1076111yescharacteristic, complemented, Sylow
301C25—611no—
302C25—611no—
303C22×D4—614no—
304C22×D4—612no—
305C22×D4—612no—
306C22×D4—614no—
307C22×D4—1222no—
308C22×D4—1222no—
309C22×D4—1222no—
310C22×D4—1222no—
311C23×C4—611no—
312C23×C4—611no—
313C23×C4—614no—
314C23×C4—1222no—
315C23×C4—1222no—
316C23×C4—1222no—
317C23×C4—1222no—
318C23×C4—1222no—
319C23×C4—1224no—
320C23×C4—2444no—
321C23×C4—2444no—
322C23×C4C2×#692311yescharacteristic
323C23×C4C2×#692311yescharacteristic
324C23×C4C2×#692311yescharacteristic
325C4×D4—12232no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C22⋊C4—614no—
330C2×C22⋊C4—618no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4—1222no—
333C2×C22⋊C4—1222no—
334C2×C22⋊C4—1222no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1224no—
337C2×C22⋊C4—1224no—
338C2×C22⋊C4—1224no—
339C2×C22⋊C4—1224no—
340C2×C22⋊C4—1224no—
341C2×C42—1222no—
342C2×C42—1222no—
343C2×C42—1222no—
344#99—12216no—
345#95—2448no—
346C24—612no—
347C24—612no—
348C24—612no—
349C24—612no—
350C24—611no—
351C24—611no—
352C24—612no—
353C24—612no—
354C24—612no—
355C24—612no—
356C24—612no—
357C24—1221no—
358C24—1221no—
359C24—1221no—
360C24—1221no—
361C24—2442no—
362C24—2442no—
363C24—2442no—
364C24—2442no—
365C24C22×#692311yescharacteristic
366C2×D4—618no—
367C2×D4—618no—
368C2×D4—1228no—
369C2×D4—1224no—
370C2×D4—1224no—
371C2×D4—1224no—
372C2×D4—1224no—
373C2×D4—1228no—
374C2×D4—1228no—
375C2×D4—1228no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—1228no—
389C22×C4—1228no—
390C22×C4—1228no—
391C22×C4—1224no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4—1228no—
397C22×C4—2444no—
398C22×C4—2444no—
399C22×C4—2448no—
400C22×C4—2448no—
401C22×C4—2448no—
402C22×C4—2444no—
403C22×C4—2448no—
404C22×C4—2444no—
405C22×C4—2444no—
406C22×C4—2448no—
407C22×C4—2448no—
408C22×C4—2442no—
409C22×C4C4×#692312yes—
410C22×C4C4⋊#692312yes—
411C22×C4C22⋊#692312yes—
412C22×C4C22⋊#692312yes—
413C22×C4C22⋊#692312yes—
414C4⋊C4—28no—
415C22⋊C4—1228no—
416C22⋊C4—1228no—
417C22⋊C4—1228no—
418C22⋊C4—1228no—
419C22⋊C4—2444no—
420C22⋊C4—2444no—
421C42—1228no—
422C23—22no—
423C23—22no—
424C23—612no—
425C23—612no—
426C23—614no—
427C23—614no—
428C23—612no—
429C23—612no—
430C23—614no—
431C23—612no—
432C23—612no—
433C23—614no—
434C23—614no—
435C23—1224no—
436C23—1224no—
437C23—1222no—
438C23—1222no—
439C23—1222no—
440C23—1222no—
441C23—1222no—
442C23—1222no—
443C23—2442no—
444C23—2442no—
445C23—2444no—
446C23—2444no—
447C23—2442no—
448C23—2442no—
449C23—2442no—
450C23—2442no—
451C23—2444no—
452C23—2442no—
453C23—2444no—
454C23—2442no—
455C23—2444no—
456C23—2442no—
457C23—2444no—
458C23—2442no—
459C23—2444no—
460C23—2442no—
461C23—2442no—
462C23C2×C4×#692311yescharacteristic
463C23C2×C22⋊#692311yescharacteristic
464C23C2×C22⋊#692311yescharacteristic
465C23C2×C22⋊#692311yescharacteristic
466C23C2×C4⋊#692311yescharacteristic
467C23#1451985611yescharacteristic
468C23#1451985611yescharacteristic
469D4—12216no—
470D4—12216no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—614no—
486C2×C4—1224no—
487C2×C4—1224no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—4884no—
492C2×C4—4884no—
493C2×C4—4888no—
494C2×C4—24416no—
495C2×C4—24416no—
496C2×C4—2448no—
497C2×C4—24416no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4(C2×#6923)⋊C414yescomplemented
502C22—22no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—614no—
511C22—614no—
512C22—1228no—
513C22—1224no—
514C22—1224no—
515C22—1224no—
516C22—1224no—
517C22—1224no—
518C22—2442no—
519C22—2442no—
520C22—2444no—
521C22—2442no—
522C22—2442no—
523C22—2444no—
524C22—2444no—
525C22—2444no—
526C22—2444no—
527C22—2444no—
528C22—2444no—
529C22—2442no—
530C22—2442no—
531C22—2444no—
532C22C2×(C2×#6923)⋊C411yescharacteristic
533C22C4×C22⋊#692311yescharacteristic
534C22C4×C22⋊#692311yescharacteristic
535C22#1684070311yescharacteristic
536C22#1684070411yescharacteristic
537C22#1684070411yescharacteristic
538C22#1684070511yescharacteristic
539C22D61⋊#9912yes—
540C22#1684074312yes—
541C4—24no—
542C4—42no—
543C4—42no—
544C4—1224no—
545C4—4888no—
546C4—2442no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2—21no—
551C2—1224no—
552C2—1224no—
553C2—2444no—
554C2#2146099311yescharacteristic
555C2#2146156412yes—
556C2#2146164611yescharacteristic
557C2#2146165011yescharacteristic
558C2#2146168212yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D244C22
cosocleC24×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#10761
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16840640C2semidirect
3. #21460993C2semidirect
8. C2×#14519836C4semidirect
12. C23×C4×D61C4semidirect
17. C22×#11758198C4semidirect
19. C22×C4×Dic61C4semidirect
20. C22×C4⋊C4×C61C4semidirect
27. C2×C22⋊#6923C2×C4semidirect
178. C22×D61#95semidirect
208. C2×C4×C61#95semidirect
217. C2×Dic61#99semidirect
290. C22×C4⋊C4#6923semidirect
300. C61#10761semidirect
501. C2×C4(C2×#6923)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145198362
C2×#168406402
#214609932
C22×#145197392
#1076161