Finite Group Databasebeta

← #30617880

Subgroups — #30617880

Order 15616.

1474 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462263C212yesmaximal
2#21462010C218yesmaximal, complemented
3#21461714C212yesmaximal
4#21461714C212yesmaximal
5C22×#14519755C211yesmaximal, characteristic
6C2×#14519856—22no—
7#16840705—22no—
8#16840704—22no—
9#16840704C2214yes—
10#16840703C2214yes—
11C4×C22⋊#6923—22no—
12C4×C22⋊#6923C2214yes—
13#16840640C2218yes—
14#16840640C2218yes—
15C23×Dic122C411yescharacteristic, complemented
16C23×C4×D61C2211yescharacteristic
17C2×#14519755—22no—
18C2×#14519755—22no—
19C2×#14519755—22no—
20C2×#14519755—22no—
21C2×#14519755C414yescomplemented
22C2×#14519755C2214yes—
23C22×#11758196C411yescharacteristic, complemented
24C22×#11758196C411yescharacteristic, complemented
25C22×#11758198C2211yescharacteristic
26C22×#11758198C2211yescharacteristic
27C22×C4⋊C4×C61C411yescharacteristic, complemented
28#14519856—48no—
29C2×C4⋊#6923—24no—
30C2×C22⋊#6923—24no—
31C2×C22⋊#6923—24no—
32C2×C22⋊#6923C2318yes—
33C2×C4×#6923—24no—
34(C2×#6923)⋊C4—24no—
35(C2×#6923)⋊C4—24no—
36C24×D61C2311yescharacteristic
37C22×Dic122—21no—
38C22×Dic122—22no—
39C22×Dic122—21no—
40C22×Dic122—44no—
41C22×Dic122C2×C414yes—
42C23×Dic61C2×C411yescharacteristic
43C23×Dic61C2×C411yescharacteristic
44C23×Dic61C2×C411yescharacteristic
45C22×C4×D61—44no—
46C22×C4×D61C2×C412yes—
47C22×C4×D61D412yes—
48C22×C4×D61D412yes—
49C22×C4×D61D412yes—
50C22×C4×D61D412yes—
51C22×C4×D61C2312yes—
52#14519755—24no—
53#14519755—24no—
54#14519755—24no—
55#14519755—24no—
56#14519755—24no—
57#14519755—24no—
58#14519755—24no—
59#14519755—24no—
60C2×#11758196—22no—
61C2×#11758196—21no—
62C2×#11758196—21no—
63C2×#11758196—21no—
64C2×#11758196—22no—
65C2×#11758196—21no—
66C2×#11758196C2×C414yes—
67C2×#11758196C2×C414yes—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198C2×C412yes—
73C2×#11758198C2×C412yes—
74C2×#11758198C2312yes—
75C2×#11758198C2312yes—
76C23×C4×C61C2×C411yescharacteristic
77C23×C4×C61C2×C411yescharacteristic
78C23×C4×C61C2×C411yescharacteristic
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2×C414yes—
83C24×C61C22×C411yescharacteristic
84C23×D61—24no—
85C23×D61—42no—
86C23×D61—42no—
87C23×D61C22×C411yescharacteristic
88C23×D61C2×D412yes—
89C23×D61C2×D412yes—
90C23×D61C2411yescharacteristic
91C22×#6923—28no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—42no—
97C22×C4×C61—42no—
98C22×C4×C61—42no—
99C22×C4×C61C22⋊C412yes—
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C22⋊C411yescharacteristic
102C22×C4×C61C22×C412yes—
103C22×C4×C61C22×C412yes—
104C22×C4×C61C22×C412yes—
105C22×Dic61—24no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—42no—
111C22×Dic61—42no—
112C22×Dic61C22⋊C411yescharacteristic
113C22×Dic61C22⋊C412yes—
114C22×Dic61C22⋊C411yescharacteristic
115C22×Dic61C22×C412yes—
116C22×Dic61C22×C412yes—
117C22×Dic61C22×C412yes—
118C2×C4×D61—28no—
119C2×C4×D61—44no—
120C2×C4×D61—44no—
121C2×C4×D61—44no—
122C2×C4×D61—44no—
123C2×C4×D61—44no—
124C2×C4×D61C22⋊C412yes—
125C2×C4×D61C22⋊C412yes—
126C2×C4×D61C22⋊C412yes—
127C2×C4×D61C22⋊C412yes—
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×D412yes—
131C2×C4×D61C2×D412yes—
132C2×Dic122—24no—
133C2×Dic122—24no—
134C2×Dic122—44no—
135C2×Dic122—48no—
136C2×Dic122—44no—
137C2×Dic122—48no—
138C22⋊#6923—216no—
139C22⋊#6923—48no—
140C22⋊#6923—48no—
141C4⋊#6923—48no—
142C4×#6923—48no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—24no—
145#11758198—24no—
146#11758198—24no—
147#11758198—24no—
148#11758198—24no—
149#11758196—24no—
150#11758196—24no—
151#11758196—24no—
152#11758196—24no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61C2×C22⋊C411yescharacteristic
157C23×C61C2×C22⋊C411yescharacteristic
158C23×C61C23×C411yescharacteristic
159C22×D61—22no—
160C22×D61—24no—
161C22×D61—24no—
162C22×D61—24no—
163C22×D61—24no—
164C22×D61—41no—
165C22×D61—42no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—42no—
169C22×D61—41no—
170C22×D61—416no—
171C22×D61C2×C22⋊C412yes—
172C22×D61C22×D412yes—
173C22×D612−1+414yes—
174C2×#6923—48no—
175C2×#6923—48no—
176C2×#6923—48no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61C2×C22⋊C412yes—
190C2×C4×C61C2×C22⋊C412yes—
191C2×Dic61—22no—
192C2×Dic61—24no—
193C2×Dic61—24no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61#9914yes—
203C2×Dic61C2×C22⋊C412yes—
204C2×Dic61C2×C22⋊C412yes—
205C4×D61—24no—
206C4×D61—24no—
207C4×D61—24no—
208C4×D61—24no—
209C4×D61—42no—
210C4×D61—42no—
211C4×D61—42no—
212C4×D61—42no—
213Dic122—48no—
214Dic122—48no—
215#21825—611nomaximal, Sylow
216C22×C61—28no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—21no—
223C22×C61C2×#9912yes—
224C22×C61#41012yes—
225C22×C61C22×C22⋊C411yescharacteristic
226C22×C61#51912yes—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—44no—
233C2×D61—48no—
234C2×D61—48no—
235C2×D61—44no—
236C2×D61#53614yes—
237#6923—88no—
238C4×C61—22no—
239C4×C61—22no—
240C4×C61—42no—
241C4×C61—42no—
242Dic61—22no—
243Dic61—42no—
244Dic61—42no—
245Dic61C2×#9914yes—
246C22×C22⋊Q8—611no—
247#2280—612no—
248#1996—618no—
249#1819—612no—
250#1819—612no—
251C2×C61—22no—
252C2×C61—22no—
253C2×C61—22no—
254C2×C61#231011yescharacteristic
255C2×C61C22×#9911yescharacteristic
256C2×C61C2×#41011yescharacteristic
257D61—24no—
258D61—42no—
259C23×Q8—611no—
260C24×C4—611no—
261C2×C22⋊Q8—614no—
262C2×C22⋊Q8—614no—
263C2×C22⋊Q8—1222no—
264C2×C22⋊Q8—1222no—
265C2×C22⋊Q8—1222no—
266C2×C22⋊Q8—1222no—
267C2×#117—1222no—
268C22×C4⋊C4—611no—
269C22×C4⋊C4—611no—
270C22×C4⋊C4#692311yescharacteristic, complemented
271C22×C22⋊C4—611no—
272C22×C22⋊C4—611no—
273#381—1222no—
274#380—614no—
275#380—1222no—
276#379—614no—
277C4×C22⋊C4—614no—
278C4×C22⋊C4—1222no—
279#342—618no—
280#342—618no—
281C61#2182511yescharacteristic, complemented, Sylow
282C25—611no—
283C22×Q8—614no—
284C22×Q8—1221no—
285C22×Q8—1222no—
286C22×Q8—1221no—
287C22×Q8—2444no—
288C23×C4—611no—
289C23×C4—612no—
290C23×C4—612no—
291C23×C4—612no—
292C23×C4—612no—
293C23×C4—612no—
294C23×C4—612no—
295C23×C4—611no—
296C23×C4—611no—
297C23×C4—2444no—
298C23×C4C2×#692311yescharacteristic
299C23×C4C2×#692311yescharacteristic
300C23×C4C2×#692311yescharacteristic
301C22⋊Q8—1224no—
302C22⋊Q8—1224no—
303C22⋊Q8—1224no—
304C22⋊Q8—1224no—
305C22⋊Q8—1224no—
306C22⋊Q8—1224no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309#117—2448no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—614no—
314C2×C4⋊C4—614no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1224no—
322C2×C4⋊C4C2×#692314yes—
323C2×C22⋊C4—612no—
324C2×C22⋊C4—612no—
325C2×C22⋊C4—612no—
326C2×C22⋊C4—612no—
327C2×C22⋊C4—618no—
328C2×C22⋊C4—1222no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1222no—
331C2×C22⋊C4—1222no—
332C2×C22⋊C4—1224no—
333C2×C22⋊C4—1224no—
334C2×C42—1224no—
335#95—1224no—
336#95—1224no—
337C24—611no—
338C24—612no—
339C24—612no—
340C24—611no—
341C24—1224no—
342C24—2442no—
343C24—2442no—
344C24C22×#692311yescharacteristic
345C2×Q8—1224no—
346C2×Q8—1224no—
347C2×Q8—2444no—
348C2×Q8—2448no—
349C2×Q8—2448no—
350C2×Q8—2444no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—611no—
359C22×C4—612no—
360C22×C4—611no—
361C22×C4—612no—
362C22×C4—612no—
363C22×C4—612no—
364C22×C4—612no—
365C22×C4—612no—
366C22×C4—612no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—1228no—
373C22×C4—1224no—
374C22×C4—1221no—
375C22×C4—1221no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1228no—
379C22×C4—2444no—
380C22×C4—2444no—
381C22×C4—2444no—
382C22×C4—2444no—
383C22×C4—2444no—
384C22×C4—2442no—
385C22×C4—2442no—
386C22×C4C22⋊#692312yes—
387C22×C4C22⋊#692311yescharacteristic
388C22×C4C22⋊#692311yescharacteristic
389C22×C4C22×#692312yes—
390C22×C4C22×#692312yes—
391C22×C4C22×#692312yes—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—1224no—
395C4⋊C4—1224no—
396C4⋊C4—1224no—
397C4⋊C4—1224no—
398C4⋊C4—2448no—
399C22⋊C4—1224no—
400C22⋊C4—1224no—
401C22⋊C4—1224no—
402C22⋊C4—1224no—
403C22⋊C4—12216no—
404C22⋊C4—2448no—
405C22⋊C4—2448no—
406C42—2448no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—612no—
411C23—614no—
412C23—612no—
413C23—1224no—
414C23—1224no—
415C23—1224no—
416C23—1224no—
417C23—1222no—
418C23—2442no—
419C23—2441no—
420C23—2441no—
421C23—2442no—
422C23—2441no—
423C23—2441no—
424C23—24416no—
425C23C2×C22⋊#692311yescharacteristic
426C23C2×C22⋊#692311yescharacteristic
427C23C23×#692311yescharacteristic
428Q8—2448no—
429Q8—2448no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—614no—
443C2×C4—612no—
444C2×C4—612no—
445C2×C4—1224no—
446C2×C4—1224no—
447C2×C4—1224no—
448C2×C4—1224no—
449C2×C4—1224no—
450C2×C4—1224no—
451C2×C4—1222no—
452C2×C4—1222no—
453C2×C4—1222no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2448no—
459C2×C4—2448no—
460C2×C4—2448no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4C2×C22⋊#692312yes—
468C2×C4C2×C22⋊#692312yes—
469C22—28no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—614no—
477C22—1222no—
478C22—1222no—
479C22—1222no—
480C22—1222no—
481C22—1222no—
482C22—2444no—
483C22—2448no—
484C22—2448no—
485C22—2444no—
486C22D61⋊#9912yes—
487C22#1684074112yes—
488C22C22×C22⋊#692311yescharacteristic
489C22#1684083012yes—
490C4—22no—
491C4—22no—
492C4—42no—
493C4—42no—
494C4—614no—
495C4—1222no—
496C4—4888no—
497C4—2442no—
498C4—2442no—
499C2—22no—
500C2—22no—
501C2—22no—
502C2—1224no—
503C2—2442no—
504C2#2146229311yescharacteristic
505C2C2×D61⋊#9911yescharacteristic
506C2C2×#1684074111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61#21825
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21462010C2semidirect
15. C23×Dic122C4semidirect
21. C2×#14519755C4semidirect
23. C22×#11758196C4semidirect
24. C22×#11758196C4semidirect
27. C22×C4⋊C4×C61C4semidirect
270. C22×C4⋊C4#6923semidirect
281. C61#21825semidirect

Maximal subgroups

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

subgroupindex
#214622632
#214620102
#214617142
C22×#145197552
#2182561