Finite Group Databasebeta

← #29730480

Subgroups — #29730480

Order 15616.

978 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839986C212yesmaximal
2C2×#16839964C211yesmaximal, characteristic
3#21450815C211yesmaximal, characteristic
4#21448879C211yesmaximal, characteristic, complemented
5#21447139C212yesmaximal, complemented
6#21446910C212yesmaximal
7#21446743C211yesmaximal, characteristic, complemented
8#21446537C211yesmaximal, characteristic, complemented
9#21446162C212yesmaximal, complemented
10C4×#14519699C211yesmaximal, characteristic
11C2×C61×#494C211yesmaximal, characteristic, complemented
12C22×#11758201C2212yes—
13C2×#14519766C2212yes—
14C2×Q8×Dic61C2212yes—
15C2×C4×D244C2211yescharacteristic
16#16839986—28no—
17#16839964—24no—
18C2×C4⋊#92418C2211yescharacteristic
19C2×#14519706C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25#16839688—22no—
26#16839669C2212yes—
27#16839669C2212yes—
28#16839661—21no—
29#16839648—22no—
30#16839619—21no—
31#16839619—21no—
32C4⋊C4×Dic61—21no—
33C4×C4⋊Dic61C2211yescharacteristic
34C4×C4⋊Dic61C2211yescharacteristic
35#16839525C2211yescharacteristic
36#16839525C2211yescharacteristic
37#16839516C2212yes—
38C22×Q16×C61C2212yes—
39C22×Q16×C61C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#494—24no—
43C2×C4×C8×C61C2211yescharacteristic
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46C2×#11758201—24no—
47C2×#11758201—24no—
48C2×#11758201—42no—
49C22×#92418C2312yes—
50#14519766—24no—
51#14519766—42no—
52Q8×Dic61—24no—
53Q8×Dic61—42no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61D411yescharacteristic
58C2×C4⋊Dic61D411yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2311yescharacteristic
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758198—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61C2312yes—
68C4⋊#92418—22no—
69#14519706—24no—
70#14519701—24no—
71#14519701—24no—
72#14519701—42no—
73#14519699—22no—
74#14519699D414yes—
75#14519684—21no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684D412yes—
81C22×Q8×C61C2312yes—
82C22×Q8×C61C2312yes—
83C2×Q16×C61—24no—
84C2×Q16×C61—24no—
85C2×Q16×C61—24no—
86C2×Q16×C61—24no—
87C2×Q16×C61—42no—
88C2×Q16×C61—42no—
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C4⋊Q8×C61—22no—
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4⋊Q8×C61D414yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4×C8×C61—22no—
101C23×D61—21no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C2×D414yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61C2×D412yes—
123C22×Dic61C2×D412yes—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244C2×D412yes—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133Q16×C61—48no—
134Q16×C61—48no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61C2×D412yes—
140C2×C8×C61#4112yes—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61—42no—
145C42×C61C2×D412yes—
146C42×C61#4112yes—
147#11758201—48no—
148#11758198—44no—
149#11758198—44no—
150#11758198—42no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—44no—
159C4⋊Dic61C2×D412yes—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C4×Dic61—44no—
163C4×Dic61—42no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61C22×D411yescharacteristic
168C22×D61—22no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175Q8×C61—24no—
176Q8×C61—24no—
177Q8×C61—42no—
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61#12112yes—
192C2×C4×C61#12112yes—
193C2×C4×C61#13512yes—
194C2×C4×C61#13712yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×C4×C61H(5,𝔽2)11yescharacteristic
200C2×C4×C61H(5,𝔽2)11yescharacteristic
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216D244—22no—
217D244—22no—
218D244—41no—
219D244—41no—
220C4×D61—84no—
221C8×C61—22no—
222C8×C61—22no—
223#92418—42no—
224#25365—611nomaximal, Sylow
225C22×C61C2×#12111yescharacteristic
226C22×C61D4211yescharacteristic
227C22×C61#55011yescharacteristic
228C22×C61C2×#13511yescharacteristic
229C22×C61C2×#13711yescharacteristic
230C22×C61#57811yescharacteristic
231C22×C61#57811yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—81no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C4×C61—22no—
240C4×C61—44no—
241C4×C61—82no—
242C4×C61#46414yescomplemented
243C4×C61#53211yescharacteristic
244C4×C61#54011yescharacteristic
245C4×C61D4211yescharacteristic
246C4×C61#54611yescharacteristic
247Dic61—42no—
248Dic61—42no—
249Dic61—42no—
250Dic61—81no—
251Dic61—81no—
252C2×#494D6111yescharacteristic, complemented
253C2×#464—612no—
254C2×#461—611no—
255#2547—611no—
256#2365—611no—
257#2056—612no—
258#2015—612no—
259#1981—611no—
260#1939—611no—
261#1878—612no—
262C4×#102—611no—
263C2×C61#242811yescharacteristic
264C2×C61C2×#46411yescharacteristic
265C2×C61#304411yescharacteristic
266C2×C61#327611yescharacteristic
267C2×C61#328311yescharacteristic
268C2×C61#339311yescharacteristic
269C2×C61#339811yescharacteristic
270D61—82no—
271C22×Q16C2×D6112yes—
272C22×Q16C2×D6112yes—
273C22×SD16—612no—
274C2×C4⋊Q8C2×D6111yescharacteristic
275C2×C4⋊Q8C2×D6111yescharacteristic
276C2×#124—612no—
277C2×C4×Q8—612no—
278C2×C4×D4—611no—
279#494—24no—
280#464—1228no—
281#461—1224no—
282C2×C4⋊C8—612no—
283C2×C4⋊C8—611no—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103—612no—
287C2×#102—611no—
288C2×#102—611no—
289C2×C4×C8C2×D6111yescharacteristic
290#397—1222no—
291#391—612no—
292#391—612no—
293#389—1221no—
294#387—1222no—
295#380—1221no—
296#380—1221no—
297C4×C4⋊C4—611no—
298C4×C4⋊C4—611no—
299C4×C4⋊C4—1221no—
300#340—611no—
301#340—611no—
302#336—612no—
303C61#2536511yescharacteristic, complemented, Sylow
304C22×Q8C22×D6112yes—
305C22×Q8C22×D6112yes—
306C22×D4—611no—
307C23×C4—1221no—
308C2×Q16—24no—
309C2×Q16—24no—
310C2×Q16—24no—
311C2×Q16—24no—
312C2×Q16—42no—
313C2×Q16—42no—
314C2×SD16—1224no—
315C2×SD16—1224no—
316C2×SD16—2442no—
317C22×C8—612no—
318C22×C8C22×D6111yescharacteristic
319C22×C8C22×D6111yescharacteristic
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—22no—
323C4⋊Q8C22⋊D6114yes—
324#124—1224no—
325#124—2442no—
326C4×Q8—1224no—
327C4×Q8—2442no—
328C4×D4—1222no—
329C4×D4—1222no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—611no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C42—612no—
344C2×C42—1222no—
345C2×C42—1221no—
346C2×C42C22×D6111yescharacteristic
347C4⋊C8—1224no—
348C4⋊C8—1222no—
349#103—1224no—
350#103—1224no—
351#103—2442no—
352#102—614no—
353#102—1222no—
354C4×C8—22no—
355#95—612no—
356#95—1222no—
357#95—1222no—
358#95—1222no—
359#95—1222no—
360#95—1221no—
361C24—1221no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8C2×C22⋊D6114yes—
374C2×D4—612no—
375C2×D4—1221no—
376C2×D4—1221no—
377C2×D4—1221no—
378C2×D4—1221no—
379C2×D4—1221no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—612no—
383C22×C4—612no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4D4×D6111yescharacteristic
391C22×C4D4×D6111yescharacteristic
392C22×C4C23×D6111yescharacteristic
393Q16—48no—
394Q16—48no—
395SD16—2448no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8D4×D6112yes—
404C2×C8#1175822412yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—612no—
410C4⋊C4—1222no—
411C4⋊C4—1221no—
412C4⋊C4—1222no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—1221no—
417C4⋊C4—2444no—
418C22⋊C4—2444no—
419C22⋊C4—2444no—
420C22⋊C4—2442no—
421C42—1222no—
422C42—1222no—
423C42—2444no—
424C42—2442no—
425C42#1175822412yes—
426C42C2×C22⋊D6112yes—
427C23—1222no—
428C23—2441no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23—2441no—
434C23C2×D4×D6111yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443D4—1222no—
444D4—1222no—
445D4—2441no—
446D4—2441no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—4884no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4#1451974912yes—
472C2×C4#1451974912yes—
473C2×C4#1451980012yes—
474C2×C4#1451980912yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C2×#1175822411yescharacteristic
478C2×C4C2×#1175822411yescharacteristic
479C2×C4#1451982211yescharacteristic
480C2×C4#1451982211yescharacteristic
481C8—22no—
482C8—22no—
483C8—2442no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—2442no—
489C22—2442no—
490C22—2442no—
491C22C2×#1451974911yescharacteristic
492C22#1684039711yescharacteristic
493C22#1684043211yescharacteristic
494C22C2×#1451980011yescharacteristic
495C22C2×#1451980911yescharacteristic
496C22#1684057511yescharacteristic
497C22#1684057511yescharacteristic
498C4—22no—
499C4—44no—
500C4—82no—
501C4—4881no—
502C4—4881no—
503C4—2442no—
504C4—2442no—
505C4—2442no—
506C4#1683998614yescomplemented
507C4#1684031011yescharacteristic
508C4#1684036611yescharacteristic
509C4D4×C22⋊D6111yescharacteristic
510C4#1684040611yescharacteristic
511C2—4882no—
512C2#2144941011yescharacteristic
513C2C2×#1683998611yescharacteristic
514C2#2145534611yescharacteristic
515C2#2145746411yescharacteristic
516C2#2145753911yescharacteristic
517C2#2145880911yescharacteristic
518C2#2145885511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#494C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#494D61
hypocenterC61#25365
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
4. #21448879C2semidirect
5. #21447139C2semidirect
7. #21446743C2semidirect
8. #21446537C2semidirect
9. #21446162C2semidirect
11. C2×C61×#494C2semidirect
242. C4×C61#464semidirect
252. C2×#494D61semidirect
303. C61#25365semidirect
506. C4#16839986semidirect

Maximal subgroups

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

subgroupindex
C2×#168399862
C2×#168399642
#214508152
#214488792
#214471392
#214469102
#214467432
#214465372
#214461622
C4×#145196992
C2×C61×#4942
#2536561