Finite Group Databasebeta

← #29729549

Subgroups — #29729549

Order 15616.

886 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
1C2×#16839962C211yesmaximal, characteristic
2C2×Q8×#92418C211yesmaximal, characteristic, complemented
3#21450818C211yesmaximal, characteristic
4#21447190C212yesmaximal, complemented
5#21447134C212yesmaximal, complemented
6#21446676C212yesmaximal, complemented
7#21446624C211yesmaximal, characteristic
8#21446580C211yesmaximal, characteristic, complemented
9#21446501C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C61×#2582C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519774C2211yescharacteristic
14C2×#14519766—21no—
15#16839962—24no—
16Q8×#92418—24no—
17C2×#14519708C2212yes—
18C2×C4⋊#92418C2212yes—
19C2×C4⋊#92418C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519697C2212yes—
23C2×#14519697C2212yes—
24C2×#14519697C2212yes—
25C2×C4×#92418C2211yescharacteristic
26C2×C4×#92418C2212yes—
27#16839679—22no—
28#16839664C2212yes—
29#16839664C2212yes—
30#16839663—21no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839524C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2212yes—
35C2×C4×Q8×C61C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#398—22no—
38C61×#391—21no—
39C61×#391C2212yes—
40C61×#389—21no—
41C61×#389C2212yes—
42C61×#384C2211yescharacteristic
43C22×D244—21no—
44C22×D244—21no—
45C22×D244C2312yes—
46C2×#11758201—24no—
47C2×#11758201—24no—
48C2×#11758201—42no—
49C22×#92418C2311yescharacteristic
50C22×#92418C2312yes—
51C22×#92418C2311yescharacteristic
52#14519774—22no—
53#14519774—22no—
54#14519766—44no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758198—22no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×C4×Dic61—21no—
62#14519708—42no—
63C4⋊#92418—24no—
64C4⋊#92418D414yes—
65#14519701—22no—
66#14519701—22no—
67#14519697—24no—
68#14519697—24no—
69#14519697—42no—
70C4×#92418—22no—
71C4×#92418—24no—
72#14519684—21no—
73C22×Q8×C61D411yescharacteristic
74C22×Q8×C61C2311yescharacteristic
75C4×Q8×C61—22no—
76C4×Q8×C61—22no—
77C4×Q8×C61—22no—
78C4×Q8×C61—22no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C4⋊C4×C61D412yes—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2312yes—
87C2×C42×C61—21no—
88C2×C42×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90#95×C61—22no—
91#95×C61—22no—
92#95×C61—22no—
93C23×D61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61SD1614yescomplemented
102C2×Q8×C61#4112yes—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—21no—
112C22×Dic61—21no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—42no—
123C2×D244—81no—
124C2×D244—81no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—42no—
132C4⋊C4×C61—42no—
133C4⋊C4×C61—81no—
134C4⋊C4×C61—82no—
135C42×C61—22no—
136C42×C61—22no—
137C42×C61—42no—
138C42×C61C2×D412yes—
139C42×C61#4112yes—
140#11758201—48no—
141#11758198—44no—
142#11758198—44no—
143#11758198—82no—
144#11758198—82no—
145C4⋊Dic61—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4×Dic61—42no—
150C2×#92418—21no—
151C2×#92418—22no—
152C2×#92418—22no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—21no—
156C2×#92418—21no—
157C2×#92418C2×D414yes—
158C2×#92418#4112yes—
159C23×C61C22×D411yescharacteristic
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167Q8×C61—28no—
168Q8×C61—22no—
169Q8×C61—22no—
170Q8×C61—41no—
171Q8×C61—41no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61#12112yes—
189C2×C4×C61#12112yes—
190C2×C4×C61C2×SD1614yes—
191C2×C4×C61C2×SD1612yes—
192C2×C4×C61#13512yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C61C2×#4111yescharacteristic
197C2×C4×C61H(5,𝔽2)11yescharacteristic
198C2×C4×C612−1+411yescharacteristic
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205D244—42no—
206D244—42no—
207D244—42no—
208D244—42no—
209D244—84no—
210D244—84no—
211#92418—24no—
212#92418—22no—
213#92418—41no—
214#25295—611nomaximal, Sylow
215C22×C61C2×#12111yescharacteristic
216C22×C61#54811yescharacteristic
217C22×C61#55011yescharacteristic
218C22×C61C22×SD1611yescharacteristic
219C22×C61C2×#13511yescharacteristic
220C22×C61#57611yescharacteristic
221C22×C61#57711yescharacteristic
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—82no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—82no—
229C4×C61—24no—
230C4×C61—22no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—82no—
234C4×C61—81no—
235C4×C61#46514yes—
236C4×C61#52911yescharacteristic
237C4×C61#54211yescharacteristic
238C4×C61#54611yescharacteristic
239C4×C61#54811yescharacteristic
240Dic61—81no—
241Dic61—81no—
242C2×#460—611no—
243C2×C8×Q8—611no—
244#2582D6111yescharacteristic, complemented
245#2548—611no—
246#2065—612no—
247#2055—612no—
248#1968—612no—
249#1957—611no—
250#1948—611no—
251#1934—612no—
252C4×#103—611no—
253C2×C61#242611yescharacteristic
254C2×C61C2×#46511yescharacteristic
255C2×C61#306211yescharacteristic
256C2×C61#336211yescharacteristic
257C2×C61#336511yescharacteristic
258C2×C61#339211yescharacteristic
259C2×C61#339611yescharacteristic
260D61—162no—
261C22×SD16—612no—
262C2×#127—611no—
263C2×#124—1221no—
264C2×C4×Q8C2×D6111yescharacteristic
265C2×C4×Q8C2×D6111yescharacteristic
266#460—1224no—
267C8×Q8—1224no—
268C2×#106—612no—
269C2×C4⋊C8—611no—
270C2×C4⋊C8—612no—
271C2×#103—611no—
272C2×#103—611no—
273C2×#102—612no—
274C2×#102—612no—
275C2×#102—612no—
276C2×C4×C8—612no—
277C2×C4×C8—611no—
278#398—22no—
279#394—1222no—
280#391—21no—
281#391C2×D6112yes—
282#390—612no—
283#390—612no—
284#390—1221no—
285#389—21no—
286#389C2×D6112yes—
287#384C2×D6111yescharacteristic
288C4×C4⋊C4—611no—
289#340—612no—
290#340—611no—
291#340—611no—
292C61#2529511yescharacteristic, complemented, Sylow
293C22×Q8C22⋊D6111yescharacteristic
294C22×Q8C22×D6111yescharacteristic
295C22×D4—612no—
296C22×D4—1221no—
297C22×D4—1221no—
298C2×SD16—1224no—
299C2×SD16—1224no—
300C2×SD16—2442no—
301C22×C8—612no—
302C22×C8—611no—
303C22×C8—611no—
304#127—1222no—
305#127—1222no—
306#124—2444no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4C22⋊D6111yescharacteristic
318C2×C4⋊C4C22⋊D6112yes—
319C2×C4⋊C4C22×D6111yescharacteristic
320C2×C4⋊C4C22×D6112yes—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1222no—
324C2×C22⋊C4—1222no—
325C2×C42—21no—
326C2×C42—1221no—
327C2×C42C22×D6112yes—
328C2×C42C22×D6111yescharacteristic
329#106—2442no—
330C4⋊C8—614no—
331C4⋊C8—1224no—
332#103—1222no—
333#103—1222no—
334#102—1224no—
335#102—1224no—
336#102—2442no—
337C4×C8—1224no—
338C4×C8—1222no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—1221no—
343C24—1222no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8#1175820114yescomplemented
352C2×Q8#1175822612yes—
353C2×D4—1222no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—1222no—
359C2×D4—4881no—
360C2×D4—4881no—
361C2×D4—2442no—
362C2×D4—2442no—
363C2×D4—2442no—
364C2×D4—2442no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4C2×C22⋊D6112yes—
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C2×C22⋊D6112yes—
373C22×C4C2×C22⋊D6111yescharacteristic
374C22×C4C23×D6111yescharacteristic
375SD16—2448no—
376C2×C8—614no—
377C2×C8—612no—
378C2×C8—1221no—
379C2×C8—1222no—
380C2×C8—1222no—
381C2×C8—1221no—
382C2×C8—1221no—
383C2×C8—1221no—
384C2×C8—1221no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—81no—
394C4⋊C4—82no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C22⋊C4—4882no—
400C22⋊C4—4882no—
401C22⋊C4—2444no—
402C22⋊C4—2444no—
403C42—22no—
404C42—22no—
405C42—42no—
406C42—2442no—
407C42D4×D6112yes—
408C42#1175822412yes—
409C23—2442no—
410C23—2442no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23—2442no—
415C23—2442no—
416C23C22×C22⋊D6111yescharacteristic
417Q8—28no—
418Q8—22no—
419Q8—22no—
420Q8—41no—
421Q8—41no—
422D4—4884no—
423D4—4884no—
424D4—2442no—
425D4—2442no—
426D4—2442no—
427D4—2442no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4#1451974912yes—
451C2×C4#1451974912yes—
452C2×C4C2×#1175820114yes—
453C2×C4C2×#1175820112yes—
454C2×C4#1451979712yes—
455C2×C4C2×D4×D6111yescharacteristic
456C2×C4#41×D6111yescharacteristic
457C2×C4C2×#1175822411yescharacteristic
458C2×C4C2×#1175822611yescharacteristic
459C2×C4#1451982311yescharacteristic
460C2×C4#1451982511yescharacteristic
461C8—1224no—
462C8—1222no—
463C8—2441no—
464C22—4882no—
465C22—4882no—
466C22—4882no—
467C22—4882no—
468C22—4882no—
469C22—4882no—
470C22—4882no—
471C22C2×#1451974911yescharacteristic
472C22#1684042311yescharacteristic
473C22#1684043111yescharacteristic
474C22C22×#1175820111yescharacteristic
475C22C2×#1451979711yescharacteristic
476C22#1684056511yescharacteristic
477C22#1684057011yescharacteristic
478C4—24no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—81no—
484C4—4881no—
485C4—4881no—
486C4#1683999314yes—
487C4#1684029811yescharacteristic
488C4#1684038511yescharacteristic
489C4#1684040311yescharacteristic
490C4#1684042211yescharacteristic
491C2—9762no—
492C2#2144938711yescharacteristic
493C2C2×#1683999311yescharacteristic
494C2#2145550111yescharacteristic
495C2#2145849011yescharacteristic
496C2#2145851511yescharacteristic
497C2#2145879611yescharacteristic
498C2#2145882711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2582C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2582D61
hypocenterC61#25295
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#92418C2semidirect
4. #21447190C2semidirect
5. #21447134C2semidirect
6. #21446676C2semidirect
8. #21446580C2semidirect
9. #21446501C2semidirect
10. C4×#14519701C2semidirect
11. C61×#2582C2semidirect
101. C2×Q8×C61SD16semidirect
244. #2582D61semidirect
292. C61#25295semidirect
351. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×#168399622
C2×Q8×#924182
#214508182
#214471902
#214471342
#214466762
#214466242
#214465802
#214465012
C4×#145197012
C61×#25822
#2529561