Finite Group Databasebeta

← #557×#6923 #30624725

Subgroups — #557×#6923 #30624725

Order 15616.

2220 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#21462353C216yesmaximal
2#21462348C213yesmaximal
3#21462338C213yesmaximal
4#21462334C212yesmaximal
5Q8×C4⋊#6923C211yesmaximal, characteristic
6C4⋊Q8×#6923C213yesmaximal
7#125×#6923C216yesmaximal
8C4×Q8×#6923C211yesmaximal, characteristic
9C4×Q8×#6923C212yesmaximal
10C4×Q8×#6923C213yesmaximal
11#557×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2213yes—
13C2×Q8×#6923C2212yes—
14C2×Q8×#6923C2211yescharacteristic
15C2×Q8×#6923C2213yes—
16C2×Q8×#6923C2211yescharacteristic
17(C2×#6923)⋊Q8C2216yes—
18#16840713C2216yes—
19#16840713C2216yes—
20#16840713C2216yes—
21#16840713C2216yes—
22#16840711C22112yes—
23#16840711C22112yes—
24#16840708C2213yes—
25#16840706C2216yes—
26C4⋊C4×#6923C2216yes—
27C4⋊C4×#6923C2216yes—
28C4⋊C4×#6923C2213yes—
29C4⋊C4×#6923C2216yes—
30C4⋊C4×#6923C2213yes—
31C4⋊C4×#6923C2216yes—
32C4⋊C4×#6923C2213yes—
33C4⋊C4×#6923C2216yes—
34C4⋊C4×#6923C2211yescharacteristic
35C4×C4⋊#6923C2213yes—
36C4×C4⋊#6923C2216yes—
37C4×C4⋊#6923C2213yes—
38C4×C4⋊#6923C2216yes—
39C42×#6923C2216yes—
40C42×#6923C2213yes—
41#16840491C411yescharacteristic, complemented
42#16840490C413yescomplemented
43#16840489C416yescomplemented
44#16840488C412yescomplemented
45#16840487C413yescomplemented
46C4⋊Q8×D61C2213yes—
47#125×D61C2216yes—
48C4×Q8×D61C2211yescharacteristic
49C4×Q8×D61C2212yes—
50C4×Q8×D61C2213yes—
51C61×#557C411yescharacteristic, complemented
52Q8×#6923—28no—
53Q8×#6923—24no—
54Q8×#6923—212no—
55Q8×#6923—212no—
56Q8×#6923Q818yescomplemented
57C2×C4⋊#6923C2313yes—
58C2×C4⋊#6923C2316yes—
59C2×C4⋊#6923C2313yes—
60C2×C4⋊#6923C2316yes—
61C2×C4⋊#6923C2316yes—
62C2×C4⋊#6923C2316yes—
63C2×C4⋊#6923C2313yes—
64C2×C4⋊#6923C2311yescharacteristic
65C2×C4×#6923C2316yes—
66C2×C4×#6923C2313yes—
67C2×C4×#6923C2316yes—
68C2×C4×#6923C2313yes—
69C2×C4×#6923C2316yes—
70C2×C4×#6923C2312yes—
71C2×C4×#6923C2313yes—
72C2×C4×#6923C2316yes—
73C2×C4×#6923C2313yes—
74C2×C4×#6923C2313yes—
75C2×C4×#6923C2311yescharacteristic
76(C2×#6923)⋊C4C2316yes—
77(C2×#6923)⋊C4C23112yes—
78(C2×#6923)⋊C4C2316yes—
79(C2×#6923)⋊C4C2316yes—
80(C2×#6923)⋊C4C23112yes—
81(C2×#6923)⋊C4C2312yes—
82C2×Q8×D61C2313yes—
83C2×Q8×D61C2311yescharacteristic
84#14519777C2×C413yes—
85#14519776C2×C416yes—
86#14519776C2×C416yes—
87#14519775C2×C416yes—
88#14519769C2×C416yes—
89#14519768C2×C416yes—
90#14519768C2×C416yes—
91#14519767C2×C4112yes—
92#14519767C2×C4112yes—
93Q8×Dic61C2×C413yes—
94Q8×Dic61C2×C411yescharacteristic
95Q8×Dic61C2×C412yes—
96C4×Dic122C2×C413yes—
97C4×Dic122C2×C413yes—
98C4×Dic122C2×C416yes—
99C4×Dic122C2×C416yes—
100#14519742C2×C416yes—
101#14519742C2×C413yes—
102#14519742C2×C416yes—
103#14519742C2×C413yes—
104C4⋊C4×D61C2316yes—
105C4⋊C4×D61C2316yes—
106C4⋊C4×D61C2313yes—
107C4⋊C4×D61C2316yes—
108C4⋊C4×D61C2311yescharacteristic
109C42×D61C2316yes—
110C42×D61C2313yes—
111C4⋊Q8×C61C2×C413yes—
112#125×C61C2×C416yes—
113C4×Q8×C61C2×C412yes—
114C4×Q8×C61C2×C413yes—
115C4×Q8×C61C2×C411yescharacteristic
116C22×#6923C2413yes—
117C22×#6923C2416yes—
118C22×#6923C2413yes—
119C22×#6923C2412yes—
120C22×#6923C2411yescharacteristic
121C22×#6923C2411yescharacteristic
122C2×Q8×C61C22×C413yes—
123C2×Q8×C61C22×C411yescharacteristic
124Q8×D61—26no—
125Q8×D61—26no—
126Q8×D61C2×Q814yes—
127Q8×D61#4114yes—
128C2×C4×D61C2416yes—
129C2×C4×D61C2412yes—
130C2×C4×D61C2413yes—
131C2×C4×D61C2413yes—
132C2×C4×D61C2411yescharacteristic
133C2×Dic122C22×C413yes—
134C2×Dic122C22×C416yes—
135C2×Dic122C22×C413yes—
136C4⋊#6923—212no—
137C4⋊#6923—212no—
138C4⋊#6923—26no—
139C4⋊#6923—212no—
140C4⋊#6923—26no—
141C4⋊#6923—46no—
142C4⋊#6923C2×Q8112yes—
143C4⋊#6923#4114yes—
144C4×#6923—212no—
145C4×#6923—212no—
146C4×#6923—24no—
147C4×#6923—26no—
148C4×#6923—26no—
149C4×#6923—212no—
150C4×#6923—26no—
151C4×#6923—46no—
152C4×#6923—43no—
153C4×#6923C2×Q8112yes—
154C4×#6923#4114yes—
155C4⋊C4×C61C22×C411yescharacteristic
156C4⋊C4×C61C22×C416yes—
157C4⋊C4×C61C22×C413yes—
158C4⋊C4×C61C22×C416yes—
159C4⋊C4×C61C22×C416yes—
160C42×C61C22×C416yes—
161C42×C61C22×C413yes—
162C4⋊Dic61C22×C411yescharacteristic
163C4⋊Dic61C22×C416yes—
164C4⋊Dic61C22×C413yes—
165C4⋊Dic61C22×C416yes—
166C4⋊Dic61C22×C416yes—
167#11758196C22×C412yes—
168#11758196C22×C416yes—
169#11758196C22×C416yes—
170#11758196C22×C416yes—
171#11758196C22×C4112yes—
172#11758196C22×C4112yes—
173C4×Dic61C22×C413yes—
174C4×Dic61C22×C416yes—
175C4×Dic61C22×C416yes—
176C4×Dic61C22×C416yes—
177C4×Dic61C22×C413yes—
178C4×Dic61C22×C413yes—
179C22×D61C2511yescharacteristic
180C2×#6923—23no—
181C2×#6923—21no—
182C2×#6923—22no—
183C2×#6923—22no—
184C2×#6923—23no—
185C2×#6923—21no—
186C2×#6923—26no—
187C2×#6923—26no—
188C2×#6923—23no—
189C2×#6923—23no—
190C2×#6923—23no—
191C2×#6923—26no—
192C2×#6923C22×Q816yes—
193C2×#6923C22×Q812yes—
194C2×#6923C2×#4114yes—
195C2×#6923C2×#4112yes—
196C2×#6923C2×#4112yes—
197C2×#69232−1+412yes—
198Q8×C61—26no—
199Q8×C61C4×Q814yescomplemented
200C2×C4×C61C23×C412yes—
201C2×C4×C61C23×C413yes—
202C2×C4×C61C23×C416yes—
203C2×C4×C61C23×C413yes—
204C2×C4×C61C23×C411yescharacteristic
205C2×Dic61C23×C412yes—
206C2×Dic61C23×C413yes—
207C2×Dic61C23×C416yes—
208C2×Dic61C23×C413yes—
209C2×Dic61C23×C411yescharacteristic
210C4×D61—26no—
211C4×D61—26no—
212C4×D61—22no—
213C4×D61—23no—
214C4×D61—22no—
215C4×D61—23no—
216C4×D61C22×Q816yes—
217C4×D61C2×#4116yes—
218C4×D61C2×#4112yes—
219C4×D61C2×#4112yes—
220Dic122—26no—
221Dic122—212no—
222Dic122C4×Q8112yescomplemented
223C4×#557—611nomaximal, Sylow
224C22×C61C24×C411yescharacteristic
225C2×D61C23×Q811yescharacteristic
226C2×D61C22×#4111yescharacteristic
227C2×D61C22×#4111yescharacteristic
228C2×D61C22×#4111yescharacteristic
229C2×D61C2×2−1+411yescharacteristic
230C2×D61#58511yescharacteristic
231#6923—26no—
232#6923—22no—
233#6923—24no—
234#6923—46no—
235#6923—43no—
236#6923#55714yesdirect product factor
237C4×C61—26no—
238C4×C61—22no—
239C4×C61—23no—
240C4×C61C2×C4×Q816yes—
241C4×C61C4×#4112yes—
242Dic61—26no—
243Dic61—22no—
244Dic61—23no—
245Dic61C2×C4×Q816yes—
246Dic61C4×#4112yes—
247C2×#557—611no—
248#2370—616no—
249#2365—613no—
250#2355—613no—
251#2351—612no—
252Q8×C4⋊C4—611no—
253C4×C4⋊Q8—613no—
254C4×#125—616no—
255C42×Q8—611no—
256C42×Q8—612no—
257C42×Q8—613no—
258C2×C61C22×C4×Q811yescharacteristic
259C2×C61C2×C4×#4111yescharacteristic
260C2×C61C4×2−1+411yescharacteristic
261D61C2×#55711yescharacteristic
262D61Q8×#4111yescharacteristic
263D61#348011yescharacteristic
264D61#348211yescharacteristic
265#557—613no—
266#557—616no—
267#557—612no—
268#557—613no—
269#557—611no—
270#557#692311yescharacteristic, direct product factor
271C2×C4⋊Q8—613no—
272C2×#125—616no—
273C2×C4×Q8—611no—
274C2×C4×Q8—611no—
275C2×C4×Q8—612no—
276C2×C4×Q8—613no—
277C2×C4×Q8—613no—
278C2×C4×Q8—612no—
279C2×C4×Q8—611no—
280C2×C4×Q8—613no—
281#391—616no—
282#389—616no—
283#389—616no—
284#389—616no—
285#389—616no—
286#387—6112no—
287#387—6112no—
288#384—613no—
289#382—616no—
290C4×C4⋊C4—616no—
291C4×C4⋊C4—616no—
292C4×C4⋊C4—613no—
293C4×C4⋊C4—616no—
294C4×C4⋊C4—613no—
295C4×C4⋊C4—616no—
296C4×C4⋊C4—613no—
297C4×C4⋊C4—613no—
298C4×C4⋊C4—616no—
299C4×C4⋊C4—616no—
300C4×C4⋊C4—616no—
301C4×C4⋊C4—613no—
302C4×C4⋊C4—611no—
303C43—616no—
304C43—613no—
305C61C4×#55711yescharacteristic, complemented, Sylow
306C22×Q8—611no—
307C22×Q8—613no—
308C4⋊Q8—616no—
309C4⋊Q8—616no—
310C4⋊Q8—616no—
311C4⋊Q8—613no—
312C4⋊Q8C2×#692313yes—
313#125—6112no—
314#125—6112no—
315#125—616no—
316#125—616no—
317#125—616no—
318#125C2×#692316yes—
319C4×Q8—618no—
320C4×Q8—612no—
321C4×Q8—613no—
322C4×Q8—611no—
323C4×Q8—616no—
324C4×Q8—616no—
325C4×Q8—613no—
326C4×Q8—613no—
327C4×Q8—616no—
328C4×Q8—613no—
329C4×Q8—616no—
330C4×Q8—613no—
331C4×Q8—12212no—
332C4×Q8—12212no—
333C4×Q8—1228no—
334C4×Q8—1224no—
335C4×Q8C2×#692312yes—
336C4×Q8C2×#692313yes—
337C4×Q8C2×#692311yescharacteristic
338C2×C4⋊C4—611no—
339C2×C4⋊C4—613no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—616no—
342C2×C4⋊C4—613no—
343C2×C4⋊C4—616no—
344C2×C4⋊C4—616no—
345C2×C4⋊C4—613no—
346C2×C4⋊C4—616no—
347C2×C4⋊C4—613no—
348C2×C4⋊C4—616no—
349C2×C4⋊C4—616no—
350C2×C4⋊C4—616no—
351C2×C42—611no—
352C2×C42—613no—
353C2×C42—616no—
354C2×C42—613no—
355C2×C42—616no—
356C2×C42—616no—
357C2×C42—616no—
358C2×C42—613no—
359C2×C42—613no—
360C2×C42—612no—
361C2×C42—613no—
362C2×C42—616no—
363C2×C42—613no—
364#95—612no—
365#95—616no—
366#95—616no—
367#95—616no—
368#95—6112no—
369#95—6112no—
370C2×Q8—614no—
371C2×Q8—614no—
372C2×Q8—613no—
373C2×Q8—616no—
374C2×Q8—613no—
375C2×Q8—1226no—
376C2×Q8—1226no—
377C2×Q8C22×#692313yes—
378C2×Q8C22×#692311yescharacteristic
379C22×C4—611no—
380C22×C4—613no—
381C22×C4—611no—
382C22×C4—613no—
383C22×C4—612no—
384C22×C4—611no—
385C22×C4—612no—
386C22×C4—613no—
387C22×C4—616no—
388C22×C4—613no—
389C22×C4—616no—
390C4⋊C4—6112no—
391C4⋊C4—614no—
392C4⋊C4—612no—
393C4⋊C4—616no—
394C4⋊C4—616no—
395C4⋊C4—616no—
396C4⋊C4—6112no—
397C4⋊C4—6112no—
398C4⋊C4—611no—
399C4⋊C4—616no—
400C4⋊C4—613no—
401C4⋊C4—616no—
402C4⋊C4—616no—
403C4⋊C4—1226no—
404C4⋊C4—12212no—
405C4⋊C4—1226no—
406C4⋊C4—12212no—
407C4⋊C4—12212no—
408C4⋊C4—2446no—
409C4⋊C4C22×#692311yescharacteristic
410C4⋊C4C22×#692316yes—
411C4⋊C4C22×#692313yes—
412C4⋊C4C22×#692316yes—
413C4⋊C4C22×#692316yes—
414C42—6112no—
415C42—614no—
416C42—616no—
417C42—616no—
418C42—616no—
419C42—613no—
420C42—613no—
421C42—613no—
422C42—12212no—
423C42—12212no—
424C42—1226no—
425C42—1226no—
426C42—12212no—
427C42—1224no—
428C42—1226no—
429C42—2446no—
430C42—2443no—
431C42C22×#692316yes—
432C42C22×#692313yes—
433C23—611no—
434Q8—26no—
435Q8—6112no—
436Q8—12212no—
437Q8—1226no—
438Q8Q8×#692314yescomplemented
439C2×C4—612no—
440C2×C4—612no—
441C2×C4—612no—
442C2×C4—613no—
443C2×C4—611no—
444C2×C4—616no—
445C2×C4—616no—
446C2×C4—612no—
447C2×C4—612no—
448C2×C4—616no—
449C2×C4—612no—
450C2×C4—614no—
451C2×C4—612no—
452C2×C4—613no—
453C2×C4—616no—
454C2×C4—1226no—
455C2×C4—1226no—
456C2×C4—1222no—
457C2×C4—1223no—
458C2×C4—1222no—
459C2×C4—1223no—
460C2×C4—1222no—
461C2×C4—1221no—
462C2×C4—1223no—
463C2×C4—1222no—
464C2×C4—1223no—
465C2×C4—1221no—
466C2×C4—1223no—
467C2×C4—1226no—
468C2×C4—1223no—
469C2×C4—1226no—
470C2×C4—1223no—
471C2×C4—1226no—
472C2×C4C23×#692312yes—
473C2×C4C23×#692313yes—
474C2×C4C23×#692316yes—
475C2×C4C23×#692313yes—
476C2×C4C23×#692311yescharacteristic
477C22—611no—
478C22—611no—
479C22—611no—
480C22—611no—
481C22—611no—
482C22—611no—
483C22C24×#692311yescharacteristic
484C4—26no—
485C4—22no—
486C4—23no—
487C4—614no—
488C4—616no—
489C4—612no—
490C4—1226no—
491C4—1224no—
492C4—1222no—
493C4—1226no—
494C4—1222no—
495C4—1223no—
496C4—2443no—
497C4—2446no—
498C4C2×Q8×#692316yes—
499C4#41×#692312yes—
500C2—611no—
501C2—611no—
502C2—611no—
503C2—611no—
504C2C22×Q8×#692311yescharacteristic
505C2C2×#41×#692311yescharacteristic
506C22−1+4×#692311yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#557C4
prime radical, cosocleC22×D61C25
hypercenter#557#6923
hypocenterC61C4×#557
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
41. #16840491C4semidirect
42. #16840490C4semidirect
43. #16840489C4semidirect
44. #16840488C4semidirect
45. #16840487C4semidirect
51. C61×#557C4semidirect
56. Q8×#6923Q8semidirect
199. Q8×C61C4×Q8semidirect
222. Dic122C4×Q8semidirect
236. #6923#557direct (N × H)
270. #557#6923direct (N × H)
305. C61C4×#557semidirect
438. Q8Q8×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623532
#214623482
#214623382
#214623342
Q8×C4⋊#69232
C4⋊Q8×#69232
#125×#69232
C4×Q8×#69232
#557×D612
C4×#55761