Finite Group Databasebeta

← #30602626

Subgroups — #30602626

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840640C212yesmaximal
2C2×#16840640C212yesmaximal
3#21460993C212yesmaximal
4C2×C4×#11758198C211yesmaximal, characteristic
5C22×C22⋊#6923C2214yes—
6C2×(C2×#6923)⋊C4—22no—
7C2×(C2×#6923)⋊C4—22no—
8#16840640—24no—
9#16840640C418yescomplemented
10C23×C4×D61C2211yescharacteristic
11C22×#11758198C2211yescharacteristic
12C22×#11758198C2211yescharacteristic
13C22×C4×Dic61C411yescharacteristic, complemented
14C4×#11758198—22no—
15C4×#11758198—22no—
16C2×#14519684C411yescharacteristic, complemented
17C2×#14519684C411yescharacteristic, complemented
18C22×C42×C61C411yescharacteristic, complemented
19C23×#6923—24no—
20C2×C22⋊#6923—24no—
21C2×C22⋊#6923—24no—
22C2×C22⋊#6923—24no—
23C2×C22⋊#6923—24no—
24C2×C22⋊#6923—24no—
25C2×C22⋊#6923C2×C418yes—
26(C2×#6923)⋊C4—44no—
27(C2×#6923)⋊C4—44no—
28C24×D61C2311yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C23×Dic61C2×C411yescharacteristic
32C22×C4×D61—22no—
33C22×C4×D61—22no—
34C22×C4×D61—42no—
35C22×C4×D61—42no—
36C22×C4×D61D412yes—
37C22×C4×D61D412yes—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—22no—
42C2×#11758198C2×C414yes—
43C2×#11758198D412yes—
44C2×#11758198Q812yes—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—42no—
48#14519684—22no—
49#14519684—22no—
50C23×C4×C61C2×C411yescharacteristic
51C23×C4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53C2×C42×C61—21no—
54C2×C42×C61—21no—
55C2×C42×C61—42no—
56C24×C61C22×C411yescharacteristic
57C23×D61—41no—
58C23×D61—41no—
59C23×D61—41no—
60C23×D61—41no—
61C23×D61C22×C412yes—
62C23×D61C2×D412yes—
63C23×D61C2×D411yescharacteristic
64C23×D61C2×Q811yescharacteristic
65C23×D61#4112yes—
66C23×D61#4112yes—
67C23×D61#4112yes—
68C23×D61#4112yes—
69C22×#6923—28no—
70C22×#6923—44no—
71C22×#6923—44no—
72C22×#6923—44no—
73C22×#6923—44no—
74C22×#6923—44no—
75C22×#6923—44no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61C4212yes—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C4⋊C412yes—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100C22×Dic61—41no—
101C22×Dic61—41no—
102C22×Dic61—41no—
103C22×Dic61—41no—
104C22×Dic61C4212yes—
105C22×Dic61C22⋊C412yes—
106C22×Dic61C4⋊C412yes—
107C2×C4×D61—24no—
108C2×C4×D61—24no—
109C2×C4×D61—24no—
110C2×C4×D61—24no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—48no—
114C2×C4×D61—48no—
115C22⋊#6923—28no—
116C22⋊#6923—28no—
117C22⋊#6923—44no—
118C22⋊#6923—44no—
119C42×C61—42no—
120C42×C61—42no—
121#11758198—28no—
122#11758198—24no—
123#11758198—24no—
124C4×Dic61—42no—
125C4×Dic61—42no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61—21no—
130C23×C61C2×C4211yescharacteristic
131C23×C61C2×C22⋊C411yescharacteristic
132C23×C61C2×C4⋊C411yescharacteristic
133C23×C61#11711yescharacteristic
134C23×C61#11711yescharacteristic
135C23×C61#11711yescharacteristic
136C23×C61#11711yescharacteristic
137C22×D61—22no—
138C22×D61—22no—
139C22×D61—22no—
140C22×D61—22no—
141C22×D61—42no—
142C22×D61—42no—
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61—42no—
152C22×D61—42no—
153C22×D61—42no—
154C22×D61—42no—
155C22×D61#11714yes—
156C22×D61C4×D414yes—
157C22×D61C4×D412yes—
158C22×D61C4×Q812yes—
159C22×D61#12112yes—
160C22×D61C22⋊Q812yes—
161C22×D61#12314yes—
162C22×D61#12412yes—
163C22×D61#12512yes—
164C22×D61#12614yes—
165C2×#6923—48no—
166C2×#6923—48no—
167C2×#6923—48no—
168C2×#6923—84no—
169C2×#6923—84no—
170C2×#6923—84no—
171C2×#6923—84no—
172C2×C4×C61—24no—
173C2×C4×C61—24no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61#9912yes—
189C2×C4×C61#9912yes—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C4×D61—48no—
206C4×D61—48no—
207#10801—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61#37611yescharacteristic
221C22×C61C4×C22⋊C411yescharacteristic
222C22×C61C4×C4⋊C411yescharacteristic
223C22×C61#38011yescharacteristic
224C22×C61#38111yescharacteristic
225C22×C61#38211yescharacteristic
226C22×C61#38311yescharacteristic
227C22×C61C2×#9911yescharacteristic
228C22×C61C2×#9911yescharacteristic
229C22×C61#41011yescharacteristic
230C22×C61#41011yescharacteristic
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—24no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—24no—
239C2×D61—44no—
240C2×D61—42no—
241C2×D61—42no—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—44no—
245C2×D61#38714yes—
246C2×D61#38814yes—
247#6923—88no—
248C4×C61—22no—
249C4×C61—22no—
250C4×C61—42no—
251C4×C61—42no—
252Dic61—42no—
253Dic61—42no—
254Dic61—42no—
255C2×C4×C22⋊C4—611no—
256C2×#342—612no—
257C2×#342—612no—
258C22⋊#95—612no—
259C2×C61—21no—
260C2×C61—21no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61#143411yescharacteristic
264C2×C61C4×#9911yescharacteristic
265C2×C61C4×#9911yescharacteristic
266C2×C61#179911yescharacteristic
267C2×C61#179911yescharacteristic
268C2×C61#181911yescharacteristic
269C2×C61#182011yescharacteristic
270D61—24no—
271D61—24no—
272C24×C4—611no—
273C22×C22⋊C4—611no—
274C22×C22⋊C4—611no—
275C22×C22⋊C4—614no—
276C22×C42—611no—
277C22×C42#692311yescharacteristic, complemented
278C4×C22⋊C4—1222no—
279C4×C22⋊C4—1222no—
280C2×#95—611no—
281C2×#95—611no—
282C2×#95—1222no—
283C2×#95—1222no—
284#342—618no—
285#342—1224no—
286C61#1080111yescharacteristic, complemented, Sylow
287C25—611no—
288C23×C4—611no—
289C23×C4—611no—
290C23×C4—611no—
291C23×C4—612no—
292C23×C4—612no—
293C23×C4—1222no—
294C23×C4—1222no—
295C23×C4—1224no—
296C23×C4—2442no—
297C23×C4—2442no—
298C23×C4C2×#692311yescharacteristic
299C23×C4C2×#692311yescharacteristic
300C23×C4C2×#692311yescharacteristic
301C2×C22⋊C4—614no—
302C2×C22⋊C4—612no—
303C2×C22⋊C4—612no—
304C2×C22⋊C4—618no—
305C2×C22⋊C4—1222no—
306C2×C22⋊C4—1222no—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1224no—
310C2×C22⋊C4—1224no—
311C2×C22⋊C4—1224no—
312C2×C22⋊C4—1224no—
313C2×C22⋊C4—1224no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—42no—
317C2×C42—1221no—
318C2×C42—1221no—
319C2×C42—2442no—
320#95—1222no—
321#95—1222no—
322#95—2444no—
323#95—2444no—
324C24—612no—
325C24—612no—
326C24—611no—
327C24—611no—
328C24—612no—
329C24—612no—
330C24—612no—
331C24—612no—
332C24—2441no—
333C24—2441no—
334C24—2441no—
335C24—2441no—
336C24C22×#692311yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—612no—
350C22×C4—612no—
351C22×C4—612no—
352C22×C4—1224no—
353C22×C4—1224no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1224no—
361C22×C4—1224no—
362C22×C4—1228no—
363C22×C4—2442no—
364C22×C4—2442no—
365C22×C4—2448no—
366C22×C4—2448no—
367C22×C4—2441no—
368C22×C4—2441no—
369C22×C4—2444no—
370C22×C4—2444no—
371C22×C4—2444no—
372C22×C4—2444no—
373C22×C4—2444no—
374C22×C4—2444no—
375C22×C4—2441no—
376C22×C4—2441no—
377C22×C4—2441no—
378C22×C4—2441no—
379C22×C4C4×#692312yes—
380C22×C4C4⋊#692312yes—
381C22×C4C22⋊#692311yescharacteristic
382C22×C4C22⋊#692311yescharacteristic
383C22⋊C4—1224no—
384C22⋊C4—1224no—
385C22⋊C4—1228no—
386C22⋊C4—1228no—
387C22⋊C4—1228no—
388C22⋊C4—2444no—
389C22⋊C4—2444no—
390C42—42no—
391C42—42no—
392C42—2442no—
393C42—2442no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—612no—
399C23—612no—
400C23—614no—
401C23—614no—
402C23—612no—
403C23—612no—
404C23—614no—
405C23—614no—
406C23—612no—
407C23—612no—
408C23—1222no—
409C23—1222no—
410C23—1222no—
411C23—1222no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23—2442no—
417C23—2442no—
418C23—2441no—
419C23—2441no—
420C23—2442no—
421C23—2441no—
422C23—2441no—
423C23—2442no—
424C23—2442no—
425C23—2442no—
426C23C2×C4×#692311yescharacteristic
427C23C2×C22⋊#692311yescharacteristic
428C23C2×C4⋊#692311yescharacteristic
429C23#1451985611yescharacteristic
430C23#1451985611yescharacteristic
431C23#1451985611yescharacteristic
432C23#1451985611yescharacteristic
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—4884no—
455C2×C4—4884no—
456C2×C4—4884no—
457C2×C4—4884no—
458C2×C4—2448no—
459C2×C4—2448no—
460C2×C4—2448no—
461C2×C4—2448no—
462C2×C4—2448no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4#1451983512yes—
474C2×C4#1451983612yes—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—614no—
488C22—614no—
489C22—1224no—
490C22—1224no—
491C22—1222no—
492C22—1222no—
493C22—1222no—
494C22—1222no—
495C22—1224no—
496C22—1224no—
497C22—2442no—
498C22—2442no—
499C22—2444no—
500C22—2444no—
501C22—2442no—
502C22—2442no—
503C22#1684069811yescharacteristic
504C22C4×C22⋊#692311yescharacteristic
505C22C4×C4⋊#692311yescharacteristic
506C22#1684070411yescharacteristic
507C22#1684070511yescharacteristic
508C22#1684070611yescharacteristic
509C22#1684070711yescharacteristic
510C22C2×#1451983511yescharacteristic
511C22C2×#1451983611yescharacteristic
512C22#1684074111yescharacteristic
513C22#1684074111yescharacteristic
514C4—22no—
515C4—22no—
516C4—42no—
517C4—42no—
518C4—4888no—
519C4—2442no—
520C4—2442no—
521C4—2442no—
522C2—21no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—1224no—
527C2—1224no—
528C2#2146099511yescharacteristic
529C2C4×#1451983511yescharacteristic
530C2C4×#1451983611yescharacteristic
531C2#2146167911yescharacteristic
532C2#2146168011yescharacteristic
533C2#2146171411yescharacteristic
534C2#2146171611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC22×C42×C61C4
socleC23×C61C2×C22⋊C4
prime radical, cosocleC24×D61C23
hypercenterC22×C42#6923
hypocenterC61#10801
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
9. #16840640C4semidirect
13. C22×C4×Dic61C4semidirect
16. C2×#14519684C4semidirect
17. C2×#14519684C4semidirect
18. C22×C42×C61C4semidirect
277. C22×C42#6923semidirect
286. C61#10801semidirect

Maximal subgroups

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

subgroupindex
C2×#168406402
#214609932
C2×C4×#117581982
#1080161