Finite Group Databasebeta

← #30610335

Subgroups — #30610335

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462675C212yesmaximal, complemented
2#21461092C212yesmaximal, complemented
3#21461086C214yesmaximal, complemented
4#21461078C211yesmaximal, characteristic, complemented
5#21461075C212yesmaximal, complemented
6#21461058C211yesmaximal, characteristic, complemented
7#21461055C212yesmaximal, complemented
8C2×#16840464C211yesmaximal, characteristic
9#16840797—22no—
10#16840793—24no—
11D4×#92420—22no—
12#16840783C2212yes—
13#16840779C2212yes—
14#16840732C2214yes—
15C2×C22⋊#92420C2214yes—
16#16840607C2214yes—
17#16840607C2214yes—
18#16840606C2212yes—
19#16840606C2212yes—
20#16840605C2211yescharacteristic
21#16840605C2212yes—
22#16840605C2211yescharacteristic
23#16840464—22no—
24#16840464—24no—
25#16840464—22no—
26C2×#14519770C412yes—
27C2×#14519768C2211yescharacteristic
28C2×#14519761C2212yes—
29C2×#14519749C412yes—
30C2×#14519748C411yescharacteristic
31C2×D4×Dic61C2212yes—
32C2×D4×Dic61C2211yescharacteristic
33C2×C4×C22⋊D61C2211yescharacteristic
34C2×#14519735C411yescharacteristic
35C22×C4⋊Dic61C411yescharacteristic
36C2×#121×C61C411yescharacteristic
37C2×#11758217—24no—
38C22×#92420—24no—
39#14519844—22no—
40C4⋊#92420—22no—
41C22⋊#92420—24no—
42C22⋊#92420—24no—
43C22⋊#92420C2312yes—
44C22⋊#92420C2314yes—
45C22⋊#92420C2312yes—
46#14519832—22no—
47C4×#92420—22no—
48C22×C22⋊D61C2×C411yescharacteristic
49C22×C22⋊D61C2×C412yes—
50C23×Dic61C2×C412yes—
51C23×Dic61C2312yes—
52C22×C4×D61C2×C411yescharacteristic
53#14519770—24no—
54#14519770—24no—
55#14519768—22no—
56#14519768—22no—
57#14519761—24no—
58#14519761—24no—
59#14519749—24no—
60#14519749—24no—
61#14519748—22no—
62#14519748—22no—
63D4×Dic61—24no—
64D4×Dic61—24no—
65D4×Dic61D412yes—
66D4×Dic61D414yes—
67D4×Dic61D412yes—
68C4×C22⋊D61—22no—
69C4×C22⋊D61—22no—
70#14519735—22no—
71#14519735—21no—
72#14519735—21no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×C4⋊Dic61C2×C412yes—
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×C4⋊Dic61C2312yes—
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C412yes—
82C2×C22⋊Dic61C2×C412yes—
83C2×C22⋊Dic61C2×C412yes—
84C2×C22⋊Dic61C2311yescharacteristic
85C2×#11758198C2×C412yes—
86C2×#11758198C2311yescharacteristic
87C2×C4×Dic61C2×C412yes—
88C2×C4×Dic61C2311yescharacteristic
89C22×D4×C61C2×C412yes—
90C22×D4×C61C2×C411yescharacteristic
91C23×C4×C61C2×C411yescharacteristic
92#121×C61—22no—
93#121×C61—22no—
94C2×C4⋊C4×C61C2×C411yescharacteristic
95C2×C22⋊C4×C61C2×C412yes—
96C24×C61C22×C412yes—
97C24×C61C22×C411yescharacteristic
98C23×D61C22×C411yescharacteristic
99C2×D4×C61—24no—
100C2×D4×C61—24no—
101C2×D4×C61—42no—
102C2×D4×C61—42no—
103C2×D4×C61C22⋊C412yes—
104C2×D4×C61C22⋊C412yes—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—42no—
108C22×C4×C61C22×C412yes—
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C411yescharacteristic
111C2×C22⋊D61—22no—
112C2×C22⋊D61—24no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—41no—
115C2×C22⋊D61—82no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—82no—
119C22×Dic61—82no—
120C22×Dic61C22×C411yescharacteristic
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C412yes—
123C22×Dic61C22×C412yes—
124C22×Dic61C2×D414yes—
125C22×Dic61C2×D414yes—
126C22×Dic61C2411yescharacteristic
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—42no—
130#11758217—84no—
131C2×#92420—22no—
132C2×#92420—22no—
133C2×#92420—24no—
134C2×#92420—84no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C22⋊C4×C61—24no—
138C22⋊Dic61—24no—
139C22⋊Dic61—22no—
140C22⋊Dic61C22⋊C414yes—
141C22⋊Dic61C22⋊C414yes—
142#11758198—22no—
143#11758198—22no—
144#11758198—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—24no—
150C4⋊Dic61C22⋊C412yes—
151C4⋊Dic61C22⋊C411yescharacteristic
152C4⋊Dic61C22⋊C411yescharacteristic
153#11758196—22no—
154#11758196—24no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61C2×D412yes—
158C4×Dic61C2×D412yes—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—41no—
166C23×C61—41no—
167C23×C61C2×C22⋊C414yes—
168C23×C61#13112yes—
169C23×C61C23×C411yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—41no—
174C22×D61—41no—
175D4×C61—24no—
176D4×C61—42no—
177D4×C61—48no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C2×C4×C61C2×C22⋊C411yescharacteristic
188C22⋊D61—44no—
189C22⋊D61—84no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—22no—
194C2×Dic61—44no—
195C2×Dic61—44no—
196C2×Dic61—42no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—42no—
200C2×Dic61—82no—
201C2×Dic61—82no—
202C2×Dic61C2×C22⋊C412yes—
203C2×Dic61C2×C22⋊C412yes—
204C2×Dic61C2×C22⋊C412yes—
205C2×Dic61C22×D412yes—
206C4×D61—42no—
207C4×D61—42no—
208#92420—42no—
209#92420—44no—
210#92420—42no—
211#17424—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—22no—
215C22×C61—24no—
216C22×C61—24no—
217C22×C61—24no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—82no—
223C22×C61—82no—
224C22×C61C22×C22⋊C411yescharacteristic
225C22×C61C2×#13111yescharacteristic
226C22×C61#56811yescharacteristic
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—81no—
231C2×D61—81no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—41no—
235C4×C61#41312yes—
236Dic61—22no—
237Dic61—22no—
238Dic61—22no—
239Dic61—22no—
240Dic61—42no—
241Dic61—41no—
242C2×#553—611no—
243#2974—612no—
244#1470—612no—
245#1468—614no—
246#1466—611no—
247#1465—612no—
248#1460—611no—
249#1459—612no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61—44no—
253C2×C61—42no—
254C2×C61#288011yescharacteristic
255C2×C61#288311yescharacteristic
256C2×C61C2×#41311yescharacteristic
257D61—81no—
258#553—1222no—
259#553—1224no—
260#553—1222no—
261C2×#126—612no—
262C2×#125—611no—
263C2×#123—612no—
264C2×#121—612no—
265C2×#121—611no—
266C2×#121#692311yescharacteristic
267C2×C4×D4—611no—
268C2×C4×D4—612no—
269C2×C4×D4—611no—
270C2×#117—611no—
271C22×C4⋊C4—611no—
272#436—1222no—
273#435—1224no—
274C8×D4—1222no—
275#432—612no—
276#431—612no—
277#407—614no—
278C2×C22⋊C8—614no—
279#324—614no—
280#324—614no—
281#323—611no—
282#323—612no—
283#323—611no—
284#323—612no—
285#323—612no—
286C61#1742411yescharacteristic, complemented, Sylow
287C22×D4—611no—
288C22×D4—612no—
289C22×D4C2×#692312yes—
290C22×D4C2×#692311yescharacteristic
291C23×C4—612no—
292C23×C4—612no—
293C23×C4—611no—
294C23×C4C2×#692311yescharacteristic
295C2×M16—1224no—
296C22×C8—1224no—
297#126—1224no—
298#126—1224no—
299#125—1222no—
300#125—1222no—
301#123—1224no—
302#123—1224no—
303#121—22no—
304#121—22no—
305#121—1224no—
306#121—1224no—
307#121—1222no—
308#121—1222no—
309C4×D4—612no—
310C4×D4—614no—
311C4×D4—612no—
312C4×D4—1222no—
313C4×D4—1222no—
314C4×D4—1224no—
315C4×D4—1224no—
316#117—1222no—
317#117—1221no—
318#117—1221no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—1222no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4C2×#692311yescharacteristic
329C2×C22⋊C4—612no—
330C2×C22⋊C4—611no—
331C2×C22⋊C4—612no—
332C2×C22⋊C4—612no—
333C2×C22⋊C4—611no—
334C2×C22⋊C4C2×#692312yes—
335C2×C42—611no—
336C2×C42—612no—
337C4⋊C8—1222no—
338C4⋊C8—1222no—
339C22⋊C8—612no—
340C22⋊C8—614no—
341C22⋊C8—612no—
342C22⋊C8—1224no—
343C22⋊C8—1224no—
344#97—1222no—
345C4×C8—1222no—
346C24—611no—
347C24C22×#692312yes—
348C24C22×#692311yescharacteristic
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—1222no—
354C2×D4—1224no—
355C2×D4—1222no—
356C2×D4—4882no—
357C2×D4—2441no—
358C2×D4C22⋊#692312yes—
359C2×D4C22⋊#692312yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—42no—
363C22×C4—611no—
364C22×C4—614no—
365C22×C4—614no—
366C22×C4—611no—
367C22×C4—612no—
368C22×C4—611no—
369C22×C4—612no—
370C22×C4—1222no—
371C22×C4—1222no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—4882no—
375C22×C4—4882no—
376C22×C4—2442no—
377C22×C4C22×#692312yes—
378C22×C4C22×#692311yescharacteristic
379C22×C4C22×#692311yescharacteristic
380M16—4884no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8—1224no—
384C2×C8—4884no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—612no—
388C4⋊C4—611no—
389C4⋊C4—611no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4—1224no—
394C4⋊C4—1224no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C22⋊C4—24no—
398C22⋊C4—614no—
399C22⋊C4—614no—
400C22⋊C4—1222no—
401C22⋊C4—1222no—
402C22⋊C4—1222no—
403C22⋊C4—1222no—
404C22⋊C4—1224no—
405C42—612no—
406C42—612no—
407C42—1222no—
408C42—1222no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—1221no—
418C23—1221no—
419C23—1221no—
420C23—2441no—
421C23—2441no—
422C23C2×C22⋊#692314yes—
423C23#1451986412yes—
424C23C23×#692311yescharacteristic
425D4—24no—
426D4—42no—
427D4—48no—
428D4—4884no—
429D4—2444no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—612no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1222no—
446C2×C4—4882no—
447C2×C4—4882no—
448C2×C4—2442no—
449C2×C4—2444no—
450C2×C4—2444no—
451C2×C4—2441no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4C2×C22⋊#692311yescharacteristic
457C2×C4C2×C22⋊#692311yescharacteristic
458C8—2442no—
459C8—2444no—
460C8—2442no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—42no—
468C22—42no—
469C22—41no—
470C22—41no—
471C22—82no—
472C22—82no—
473C22—4881no—
474C22—4881no—
475C22—2441no—
476C22—2441no—
477C22—2441no—
478C22C22×C22⋊#692311yescharacteristic
479C22C2×#1451986411yescharacteristic
480C22#1684084011yescharacteristic
481C4—21no—
482C4—42no—
483C4—41no—
484C4—1222no—
485C4—1222no—
486C4—1222no—
487C4—1222no—
488C4—2442no—
489C4—2441no—
490C4#1684074912yes—
491C2—21no—
492C2—21no—
493C2—44no—
494C2—42no—
495C2—4881no—
496C2#2146240711yescharacteristic
497C2#2146241911yescharacteristic
498C2C2×#1684074911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#121×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840464C2
cosocleC22×Dic61C24
hypercenterC2×#121#6923
hypocenterC61#17424
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #21462675C2semidirect
2. #21461092C2semidirect
3. #21461086C2semidirect
4. #21461078C2semidirect
5. #21461075C2semidirect
6. #21461058C2semidirect
7. #21461055C2semidirect
286. C61#17424semidirect

Maximal subgroups

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

subgroupindex
#214626752
#214610922
#214610862
#214610782
#214610752
#214610582
#214610552
C2×#168404642
#1742461