Finite Group Databasebeta

← #30620419

Subgroups — #30620419

Order 15616.

1114 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462328C212yesmaximal, complemented
2#21462010C214yesmaximal, complemented
3#21462008C212yesmaximal, complemented
4#21461680C211yesmaximal, characteristic
5#21461680C212yesmaximal
6C4×#14519836C212yesmaximal
7C4×#14519836C211yesmaximal, characteristic
8C2×#16840472C211yesmaximal, characteristic
9C2×#14519836C2212yes—
10C2×#14519836C2211yescharacteristic
11C2×#14519836C2211yescharacteristic
12#16840713—22no—
13#16840711—24no—
14#16840704C2212yes—
15#16840704C2214yes—
16C4⋊C4×#6923—22no—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2214yes—
19#16840640C2214yes—
20#16840640C2212yes—
21#16840640C2214yes—
22#16840640C2212yes—
23#16840472—24no—
24#16840472—22no—
25#16840472—22no—
26C2×#14519777C411yescharacteristic, complemented
27C2×#14519769C412yescomplemented
28C2×#14519755C412yescomplemented
29C2×#14519755C2212yes—
30C2×#14519753C411yescharacteristic, complemented
31C2×#14519753C2211yescharacteristic
32C2×C4×D244C2211yescharacteristic
33C2×C4×D244C2212yes—
34C2×C4⋊C4×D61C411yescharacteristic, complemented
35C2×C4⋊C4×D61C2211yescharacteristic
36C2×C4×Q8×C61C411yescharacteristic, complemented
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—24no—
39C2×C22⋊#6923C2314yes—
40C2×C22⋊#6923C2312yes—
41C2×C22⋊#6923C2312yes—
42C2×C4×#6923—22no—
43C2×C4×#6923—24no—
44#14519836—24no—
45#14519836—22no—
46#14519836Q814yescomplemented
47(C2×#6923)⋊C4—22no—
48(C2×#6923)⋊C4—24no—
49(C2×#6923)⋊C4—22no—
50(C2×#6923)⋊C4—24no—
51C22×Dic122C2×C412yes—
52C22×D244C2311yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C412yes—
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2312yes—
57#14519777—22no—
58#14519777—22no—
59#14519769—24no—
60#14519769—24no—
61#14519755—24no—
62#14519755—24no—
63#14519755—24no—
64#14519755—24no—
65#14519753—22no—
66#14519753—22no—
67#14519753—22no—
68#14519753—22no—
69C4×D244—22no—
70C4×D244—22no—
71C4×D244—24no—
72C4×D244—24no—
73C2×C4⋊Dic61C2×C411yescharacteristic
74C2×C4⋊Dic61C2×C411yescharacteristic
75C2×C4⋊Dic61C2×C412yes—
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×#11758196C2×C412yes—
78C2×#11758196C2×C412yes—
79C2×#11758196C2×C412yes—
80C2×#11758196C2×C412yes—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—21no—
83C4⋊C4×D61—21no—
84C4⋊C4×D61D414yes—
85C4⋊C4×D61D412yes—
86C4⋊C4×D61D412yes—
87C2×#11758198C2×C412yes—
88C2×#11758198C2×C411yescharacteristic
89C2×#11758198C2312yes—
90C2×#11758198C2311yescharacteristic
91C22×Q8×C61C2×C411yescharacteristic
92C4×Q8×C61—22no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C2×C4⋊C4×C61C2×C412yes—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C42×C61C2×C412yes—
98C2×C42×C61C2×C411yescharacteristic
99C23×D61C22×C411yescharacteristic
100C23×D61C2411yescharacteristic
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—24no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—42no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C412yes—
111C22×C4×C61C22×C412yes—
112C22×Dic61C22×C412yes—
113C22×Dic61C22×C411yescharacteristic
114C22×Dic61C22×C412yes—
115C22×Dic61C22×C411yescharacteristic
116C2×D244—22no—
117C2×D244—22no—
118C2×D244C2×Q812yes—
119C2×D244#4112yes—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—42no—
125C2×C4×D61—42no—
126C2×C4×D61—82no—
127C2×C4×D61—82no—
128C2×C4×D61C2×D412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×D414yes—
131C2×C4×D61C2×D414yes—
132C2×Dic122—24no—
133C2×Dic122—82no—
134C22⋊#6923—22no—
135C22⋊#6923—24no—
136C22⋊#6923—22no—
137C22⋊#6923—24no—
138C22⋊#6923—22no—
139C22⋊#6923C2×Q814yes—
140C4⋊#6923—44no—
141C4⋊#6923—84no—
142C4×#6923—44no—
143C4×#6923—84no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C22⋊C412yes—
147C4⋊C4×C61C22⋊C411yescharacteristic
148C4⋊C4×C61C22⋊C411yescharacteristic
149C42×C61—22no—
150C42×C61—22no—
151C42×C61—21no—
152C42×C61—21no—
153#11758198—24no—
154#11758198—22no—
155#11758198—24no—
156#11758198—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—24no—
161C4⋊Dic61C22⋊C412yes—
162C4⋊Dic61C22⋊C412yes—
163#11758196—22no—
164#11758196—22no—
165#11758196—24no—
166#11758196—24no—
167#11758196C22⋊C414yes—
168#11758196C22⋊C414yes—
169C23×C61C23×C411yescharacteristic
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61C4×Q812yes—
177C22×D61C22×D412yes—
178C22×D61C22×Q811yescharacteristic
179C22×D61C2×#4111yescharacteristic
180C22×D612−1+412yes—
181C2×#6923—24no—
182C2×#6923—42no—
183C2×#6923—42no—
184C2×#6923—42no—
185C2×#6923—44no—
186C2×#6923—44no—
187C2×#6923—42no—
188C2×#6923—42no—
189C2×#6923—44no—
190Q8×C61—42no—
191Q8×C61—42no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—42no—
198C2×C4×C61—41no—
199C2×C4×C61—42no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C4×Q812yes—
205C2×Dic61—21no—
206C2×Dic61—22no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61C2×C22⋊C412yes—
212C2×Dic61C2×C22⋊C414yes—
213D244—24no—
214D244—24no—
215C4×D61—24no—
216C4×D61—44no—
217C4×D61—44no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—42no—
221C4×D61—82no—
222C4×D61—82no—
223Dic122—84no—
224#22513—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—21no—
227C22×C61C22×C22⋊C411yescharacteristic
228C22×C61C2×C4×Q811yescharacteristic
229C22×C61#51911yescharacteristic
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—41no—
234C2×D61—41no—
235C2×D61—42no—
236C2×D61—42no—
237C2×D61—41no—
238C2×D61—41no—
239C2×D61—42no—
240C2×D61#54812yes—
241C2×D61D4×Q812yes—
242#6923—44no—
243#6923—82no—
244#6923—84no—
245C4×C61—22no—
246C4×C61—21no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—42no—
250C4×C61—41no—
251C4×C61#41012yes—
252Dic61—22no—
253Dic61—24no—
254Dic61—41no—
255Dic61—42no—
256C2×#554—611no—
257#2345—612no—
258#1996—612no—
259#1996—614no—
260#1799—612no—
261#1799—611no—
262C4×#99—612no—
263C4×#99—611no—
264C2×C61—21no—
265C2×C61—21no—
266C2×C61Q8×C22⋊C411yescharacteristic
267C2×C61C2×#41011yescharacteristic
268C2×C61#288311yescharacteristic
269D61—22no—
270D61—22no—
271D61—42no—
272#554—1222no—
273#554—1224no—
274#554—1222no—
275C2×C4⋊Q8—611no—
276C2×#125—612no—
277C2×C22⋊Q8—611no—
278C2×C22⋊Q8—611no—
279C2×C22⋊Q8—612no—
280C2×C22⋊Q8—612no—
281C2×C4×Q8#692311yescharacteristic, complemented
282C2×C4×D4—612no—
283C2×C4×D4—611no—
284C22×C4⋊C4—611no—
285C22×C4⋊C4—611no—
286C2×#99—611no—
287C2×#99—612no—
288C2×#99—611no—
289#389—1222no—
290#387—1224no—
291#380—614no—
292#380—612no—
293C4×C4⋊C4—1222no—
294C4×C22⋊C4—612no—
295C4×C22⋊C4—614no—
296#342—614no—
297#342—612no—
298#342—614no—
299#342—612no—
300C61#2251311yescharacteristic, complemented, Sylow
301C22×Q8—612no—
302C22×Q8C2×#692311yescharacteristic
303C22×D4—611no—
304C23×C4—612no—
305C23×C4—611no—
306C23×C4—612no—
307C23×C4—611no—
308C4⋊Q8—1222no—
309C4⋊Q8—1222no—
310#125—1224no—
311#125—1224no—
312C22⋊Q8—1224no—
313C22⋊Q8—1224no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—1222no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×D4—1224no—
324C4×D4—1224no—
325C4×D4—1222no—
326C4×D4—1222no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—614no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1224no—
342C2×C4⋊C4—1222no—
343C2×C4⋊C4C2×#692312yes—
344C2×C4⋊C4C2×#692311yescharacteristic
345C2×C22⋊C4—612no—
346C2×C22⋊C4—611no—
347C2×C22⋊C4—612no—
348C2×C22⋊C4—611no—
349C2×C22⋊C4—612no—
350C2×C22⋊C4—614no—
351C2×C22⋊C4—612no—
352C2×C42—1224no—
353C2×C42—1222no—
354C2×C42C2×#692312yes—
355C2×C42C2×#692311yescharacteristic
356#99—614no—
357#99—1224no—
358#99—1222no—
359#95—1222no—
360#95—1224no—
361#95—1224no—
362#95—1222no—
363C24—611no—
364C24—611no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—42no—
368C2×Q8—1224no—
369C2×Q8—4882no—
370C2×D4—612no—
371C2×D4—612no—
372C2×D4—1222no—
373C2×D4—1222no—
374C22×C4—611no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—614no—
379C22×C4—614no—
380C22×C4—611no—
381C22×C4—612no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—1224no—
388C22×C4—1222no—
389C22×C4—4882no—
390C22×C4—4882no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4C22×#692311yescharacteristic
394C22×C4C22×#692311yescharacteristic
395C22×C4C22×#692312yes—
396C22×C4C22×#692311yescharacteristic
397C22×C4C22×#692312yes—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—614no—
401C4⋊C4—614no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1222no—
407C4⋊C4—1224no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—4884no—
413C4⋊C4—2444no—
414C4⋊C4C22⋊#692312yes—
415C4⋊C4C22⋊#692311yescharacteristic
416C4⋊C4C22⋊#692311yescharacteristic
417C22⋊C4—614no—
418C22⋊C4—1222no—
419C22⋊C4—1224no—
420C22⋊C4—1222no—
421C22⋊C4—1224no—
422C22⋊C4—1222no—
423C22⋊C4—1224no—
424C22⋊C4—1222no—
425C22⋊C4—1224no—
426C22⋊C4—1222no—
427C42—22no—
428C42—22no—
429C42—21no—
430C42—21no—
431C42—4884no—
432C42—2444no—
433C23—612no—
434C23—611no—
435C23—611no—
436C23—612no—
437C23—612no—
438C23—1221no—
439C23—1221no—
440C23—2441no—
441C23—2441no—
442C23—2441no—
443C23—2441no—
444C23C23×#692311yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—4884no—
448D4—1224no—
449D4—1224no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—612no—
459C2×C4—614no—
460C2×C4—1224no—
461C2×C4—1224no—
462C2×C4—1221no—
463C2×C4—1222no—
464C2×C4—4882no—
465C2×C4—4882no—
466C2×C4—2444no—
467C2×C4—2444no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2441no—
472C2×C4—2444no—
473C2×C4—2442no—
474C2×C4—2444no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2444no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2442no—
482C2×C4—2441no—
483C2×C4C2×C22⋊#692312yes—
484C2×C4C2×C22⋊#692312yes—
485C2×C4C2×C22⋊#692311yescharacteristic
486C2×C4C2×C22⋊#692311yescharacteristic
487C2×C4Q8×#692312yes—
488C22—21no—
489C22—21no—
490C22—612no—
491C22—612no—
492C22—1222no—
493C22—1222no—
494C22—1222no—
495C22—2441no—
496C22—2441no—
497C22—2442no—
498C22—2442no—
499C22—2442no—
500C22—2441no—
501C22—2441no—
502C22C22×C22⋊#692311yescharacteristic
503C22C2×Q8×#692311yescharacteristic
504C22#1684083011yescharacteristic
505C4—22no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—42no—
510C4—41no—
511C4—1222no—
512C4—1224no—
513C4—4884no—
514C4—4882no—
515C4—2444no—
516C4—2441no—
517C4—2442no—
518C4#1684074112yes—
519C2—21no—
520C2—21no—
521C2—1222no—
522C2—1222no—
523C2—2442no—
524C2Q8×C22⋊#692311yescharacteristic
525C2C2×#1684074111yescharacteristic
526C2#2146241911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C23
cosocleC23×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#22513
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462328C2semidirect
2. #21462010C2semidirect
3. #21462008C2semidirect
26. C2×#14519777C4semidirect
27. C2×#14519769C4semidirect
28. C2×#14519755C4semidirect
30. C2×#14519753C4semidirect
34. C2×C4⋊C4×D61C4semidirect
36. C2×C4×Q8×C61C4semidirect
46. #14519836Q8semidirect
281. C2×C4×Q8#6923semidirect
300. C61#22513semidirect

Maximal subgroups

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

subgroupindex
#214623282
#214620102
#214620082
#214616802
C4×#145198362
C2×#168404722
#2251361