Finite Group Databasebeta

← #29665489

Subgroups — #29665489

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21455601C211yesmaximal, characteristic
3#21455307C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic
5#21445248C212yesmaximal, complemented
6#21445241C212yesmaximal, complemented
7#21445204C212yesmaximal, complemented
8#21445197C212yesmaximal, complemented
9#21444591C211yesmaximal, characteristic, complemented
10C2×#16839507C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13C2×#14519777C2211yescharacteristic
14C4×#11758222C2211yescharacteristic
15#16840158C2212yes—
16#16840112C2212yes—
17#16840090—22no—
18#16840066—21no—
19#16840059—21no—
20#16840054C2211yescharacteristic
21#16840049C2211yescharacteristic
22#16840047—21no—
23#16840047—21no—
24#16840044C2211yescharacteristic
25#16840039C2211yescharacteristic
26#16839973—21no—
27#16839973C2212yes—
28#16839968—21no—
29#16839968C2212yes—
30#16839886C2212yes—
31#16839827C2212yes—
32C8×C22⋊D61—22no—
33#16839815C2211yescharacteristic
34#16839814—21no—
35#16839798—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839772—21no—
39#16839756—21no—
40#16839512C2212yes—
41#16839512C2212yes—
42#16839507C2212yes—
43#16839507C2212yes—
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yes—
48C61×#332C2212yes—
49C2×#11758222D411yescharacteristic
50C22×Dic122—21no—
51#14519810—44no—
52C2×Dic244—22no—
53C2×#11758190—22no—
54C22×#92418—21no—
55C2×C8×D61—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777C2312yes—
60C4×Dic122D411yescharacteristic
61C4×Dic122C2311yescharacteristic
62C4×D244D411yescharacteristic
63C4×D244C2311yescharacteristic
64C4×C22⋊D61—21no—
65C4×C22⋊D61—21no—
66#14519736D411yescharacteristic
67#14519734—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C42×D61—21no—
71#14519710—21no—
72#14519710—21no—
73#14519708—21no—
74#14519708—21no—
75#14519708—22no—
76#14519708—22no—
77C4⋊#92418—21no—
78C4⋊#92418D411yescharacteristic
79C4⋊#92418D411yescharacteristic
80C4⋊#92418C2311yescharacteristic
81C4⋊#92418C2311yescharacteristic
82#14519706—22no—
83#14519702—22no—
84#14519702—22no—
85#14519701—22no—
86#14519701—22no—
87#14519700—21no—
88#14519700—21no—
89#14519697—21no—
90#14519697—21no—
91C22⋊#92418D412yes—
92#14519689—22no—
93C4×#92418C2312yes—
94C8×Dic61—22no—
95C2×M16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2311yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#125×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—21no—
116C22×Dic61—41no—
117#11758222—22no—
118#11758222SD1614yes—
119C2×D244C2×D411yescharacteristic
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×Dic122—22no—
124C2×Dic122—42no—
125C2×Dic122—42no—
126C2×Dic122—41no—
127C2×Dic122C2×D411yescharacteristic
128M16×C61—42no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61—81no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—21no—
145#11758198—21no—
146#11758198—21no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C4⋊Dic61C2×D411yescharacteristic
152C4⋊Dic61C2×D411yescharacteristic
153#11758196—21no—
154#11758196—21no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C4×Dic61—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—42no—
161C2×#92418C2×D412yes—
162C2×#92418C2×D412yes—
163Dic244—44no—
164#11758190—44no—
165C8×D61—82no—
166C23×C61#12111yescharacteristic
167C22×D61—21no—
168Q8×C61—41no—
169Q8×C61—41no—
170Q8×C61—81no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61C2×SD1612yes—
180C2×C4×C61#13512yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C22⋊D61—24no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—42no—
189C2×Dic61—82no—
190D244—21no—
191D244C2×SD1612yes—
192C4×D61—24no—
193C4×D61—41no—
194C4×D61—41no—
195C4×D61—81no—
196Dic122—21no—
197Dic122—42no—
198Dic122—42no—
199Dic122—82no—
200Dic122C2×SD1612yes—
201C8×C61—42no—
202C8×C61—42no—
203#92418—22no—
204#92418—22no—
205#92418—41no—
206#92418—41no—
207#19415—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61#46512yes—
211C22×C61C2×#12111yescharacteristic
212C2×D61—22no—
213C2×D61—41no—
214C2×D61—41no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—81no—
219C4×C61—81no—
220C4×C61#46512yes—
221C4×C61C22×SD1611yescharacteristic
222C4×C61C2×#13511yescharacteristic
223C4×C61#57711yescharacteristic
224C4×C61#57811yescharacteristic
225Dic61—22no—
226Dic61—41no—
227Dic61—41no—
228Dic61—82no—
229#3229—611no—
230#3076—611no—
231#3039—611no—
232#2988—611no—
233#1680—612no—
234#1679—612no—
235#1672—612no—
236#1671—612no—
237#1570—611no—
238#1550D6111yescharacteristic, complemented
239C2×#332—611no—
240C2×C61—22no—
241C2×C61#304611yescharacteristic
242C2×C61C2×#46511yescharacteristic
243C2×C61#306911yescharacteristic
244D61—42no—
245C2×#137—1221no—
246#533C2×D6111yescharacteristic
247C2×C4⋊Q8—611no—
248C4×#41—611no—
249#498—612no—
250#487—612no—
251#483—1222no—
252#479—1221no—
253#478—1221no—
254#477—611no—
255#476—611no—
256#475—611no—
257#475—1221no—
258#475—1221no—
259#474—611no—
260#462—612no—
261#462—1221no—
262#461—612no—
263#461—1221no—
264C8×Q8—612no—
265C8×D4—612no—
266C8×D4—1222no—
267#431—611no—
268#431—1221no—
269#427—1221no—
270#423C2×D6111yescharacteristic
271C2×C4⋊C8—611no—
272C2×C4⋊C8—1221no—
273C2×C4⋊C8C2×D6111yescharacteristic
274#419—1221no—
275#416—1221no—
276#334—612no—
277#334—612no—
278#333C2×D6112yes—
279#332—612no—
280#332—612no—
281#332C2×D6112yes—
282C61#1941511yescharacteristic, complemented, Sylow
283C2×#41—611no—
284C22×Q8—1221no—
285#137—2444no—
286C2×Q16—1222no—
287C2×SD16—1222no—
288C2×M16—21no—
289C22×C8—21no—
290C22×C8—1222no—
291C22×C8—1221no—
292C4⋊Q8—612no—
293C4⋊Q8—1221no—
294C4⋊Q8—1221no—
295C4⋊Q8—1221no—
296C4⋊Q8C22×D6111yescharacteristic
297C4⋊Q8C22×D6111yescharacteristic
298#125—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—611no—
304C4×Q8—611no—
305C4×D4—611no—
306C4×D4—611no—
307C4×D4—1221no—
308C4×D4—1221no—
309#117—21no—
310#117—611no—
311#117—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C42—1221no—
315C2×C42C22×D6111yescharacteristic
316#107—1221no—
317#107—1221no—
318#106—1221no—
319#106—1221no—
320#106—1222no—
321#106—1222no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—611no—
325C4⋊C8—611no—
326C4⋊C8—611no—
327C4⋊C8—611no—
328C4⋊C8—1222no—
329C4⋊C8—1221no—
330C4⋊C8C22×D6112yes—
331C4⋊C8C22×D6112yes—
332#103—1221no—
333#103—1221no—
334#103—1222no—
335#103—1222no—
336#103—1222no—
337#103—1222no—
338#102—1221no—
339#102—1221no—
340C22⋊C8—612no—
341C22⋊C8—1222no—
342C4×C8—612no—
343C4×C8—1222no—
344#41—614no—
345#41—1222no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—611no—
349C2×Q8—1222no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8—2441no—
353C2×D4—611no—
354C2×D4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—2441no—
359C22×C4#1175822211yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4D4×D6111yescharacteristic
362Q16—2444no—
363SD16—2444no—
364M16—42no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—81no—
368C2×C8—612no—
369C2×C8—612no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—4882no—
373C2×C8—2442no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—611no—
381C4⋊C4—611no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4—1222no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C22⋊C4—41no—
389C22⋊C4—1221no—
390C22⋊C4—1221no—
391C22⋊C4—1221no—
392C42—41no—
393C42—1221no—
394C42—1221no—
395C42—1221no—
396C42#1175822211yescharacteristic
397C42D4×D6111yescharacteristic
398C42D4×D6111yescharacteristic
399C42C23×D6111yescharacteristic
400C23—1221no—
401C23#1451975011yescharacteristic
402Q8—41no—
403Q8—41no—
404Q8—81no—
405Q8—612no—
406Q8—1221no—
407Q8—4882no—
408Q8—2442no—
409Q8—2442no—
410D4—612no—
411D4—1224no—
412D4—1221no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—1221no—
419C2×C4—1224no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—4881no—
423C2×C4—4882no—
424C2×C4—2442no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4#1451975011yescharacteristic
428C2×C4#1451975011yescharacteristic
429C2×C4#1451975011yescharacteristic
430C2×C4SD16×D6112yes—
431C2×C4#1451980012yes—
432C2×C4C2×D4×D6111yescharacteristic
433C2×C4C2×D4×D6111yescharacteristic
434C2×C4C2×#1175822211yescharacteristic
435C8—42no—
436C8—42no—
437C8—1222no—
438C8—1222no—
439C8—2441no—
440C8—2441no—
441C22—21no—
442C22—21no—
443C22—1222no—
444C22—2441no—
445C22—2441no—
446C22#1683999412yes—
447C22C2×#1451975011yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—81no—
452C4—81no—
453C4—1222no—
454C4—4882no—
455C4—2441no—
456C4—2441no—
457C4#1683999412yes—
458C4C2×SD16×D6111yescharacteristic
459C4C2×#1451980011yescharacteristic
460C4#1684057211yescharacteristic
461C4#1684057811yescharacteristic
462C2—22no—
463C2—2442no—
464C2#2145536811yescharacteristic
465C2C2×#1683999411yescharacteristic
466C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1550D61
hypocenterC61#19415
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
5. #21445248C2semidirect
6. #21445241C2semidirect
7. #21445204C2semidirect
8. #21445197C2semidirect
9. #21444591C2semidirect
10. C2×#16839507C2semidirect
11. C61×#1550C2semidirect
238. #1550D61semidirect
282. C61#19415semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214556012
#214553072
#214549282
#214452482
#214452412
#214452042
#214451972
#214445912
C2×#168395072
C61×#15502
#1941561