Finite Group Databasebeta

← #29663570

Subgroups — #29663570

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456741C211yesmaximal, characteristic, complemented
2#21456212C211yesmaximal, characteristic, complemented
3#21454563C211yesmaximal, characteristic, complemented
4#21445378C212yesmaximal, complemented
5#21445364C212yesmaximal, complemented
6#21445254C212yesmaximal, complemented
7#21445240C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444468C211yesmaximal, characteristic, complemented
10C2×#16839509C211yesmaximal, characteristic, complemented
11C61×#3245C211yesmaximal, characteristic
12C2×#14519784—21no—
13C2×#14519777C2211yescharacteristic
14#16840134—21no—
15#16840121—22no—
16#16840114—21no—
17#16840112C2212yes—
18#16840108C2212yes—
19#16839967C2212yes—
20#16839967C2212yes—
21#16839900—22no—
22#16839840C2212yes—
23#16839834—22no—
24D4×#92418C2212yes—
25#16839815C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839763—21no—
28#16839763—21no—
29#16839729—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yes—
32#16839512C2212yes—
33#16839509C2212yes—
34#16839509C2212yes—
35#16839508C2211yescharacteristic
36#16839508C2211yescharacteristic
37#16839506C2211yescharacteristic
38#16839506C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C61×#532C2211yescharacteristic
41C61×#474—21no—
42C61×#474C2212yes—
43C61×#466—22no—
44C61×#459—21no—
45C61×#459C2212yes—
46C2×C4⋊C8×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50#14519784—44no—
51C2×#11758194—22no—
52C2×#11758194D411yescharacteristic
53C2×#11758194D411yescharacteristic
54C22×#92418—21no—
55C22×#92418—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777C2312yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C4⋊#92418—21no—
63C4⋊#92418C2311yescharacteristic
64C4⋊#92418C2311yescharacteristic
65C4⋊#92418C2311yescharacteristic
66C4⋊#92418C2311yescharacteristic
67#14519701—22no—
68#14519701—22no—
69#14519700—22no—
70#14519700—22no—
71#14519698—22no—
72#14519698—22no—
73C22⋊#92418—21no—
74C22⋊#92418—22no—
75C22⋊#92418—22no—
76#14519688D412yes—
77C4×#92418—21no—
78C4×#92418C2312yes—
79C2×#41×C61—21no—
80C2×#41×C61—21no—
81C2×D8×C61—22no—
82C22×C8×C61—21no—
83C4⋊Q8×C61D411yescharacteristic
84C4⋊Q8×C61C2311yescharacteristic
85#127×C61D411yescharacteristic
86#127×C61C2311yescharacteristic
87#124×C61—21no—
88#124×C61—21no—
89#121×C61—21no—
90#121×C61—21no—
91#121×C61—21no—
92#121×C61D412yes—
93C4×D4×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98#107×C61—22no—
99C4⋊C8×C61—21no—
100C4⋊C8×C61C2312yes—
101#102×C61—22no—
102#102×C61—22no—
103#41×C61—42no—
104#41×C61—42no—
105#41×C61—42no—
106#41×C61—42no—
107C2×Q8×C61—21no—
108C2×Q8×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—41no—
116C2×D4×C61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61—41no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4111yescharacteristic
122C22×Dic61—41no—
123C2×Dic122—22no—
124C2×Dic122—41no—
125C2×Dic122—81no—
126C2×Dic122—81no—
127D8×C61—82no—
128C2×C8×C61—22no—
129C2×C8×C61—81no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61C2×D412yes—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61#4111yescharacteristic
141C42×C61C2411yescharacteristic
142#11758200—82no—
143#11758200—82no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—41no—
147C4⋊Dic61—41no—
148#11758194—22no—
149#11758194—22no—
150#11758194—44no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—21no—
155C2×#92418—44no—
156C2×#92418—42no—
157C2×#92418C2×D412yes—
158C2×#92418C2×D412yes—
159C23×C61—22no—
160C23×C61—41no—
161C23×C61#12111yescharacteristic
162Q8×C61—42no—
163Q8×C61—41no—
164Q8×C61—41no—
165D4×C61—22no—
166D4×C61—22no—
167D4×C61—44no—
168D4×C61—41no—
169D4×C61—41no—
170D4×C61—41no—
171D4×C61—41no—
172D4×C61—42no—
173D4×C61—81no—
174D4×C61—81no—
175D4×C61—82no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—82no—
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#13512yes—
188C2×C4×C61AGL(1,ℤ/8)12yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×Dic61—42no—
193C2×Dic61—82no—
194Dic122—42no—
195Dic122—42no—
196Dic122—82no—
197C8×C61—42no—
198#92418—22no—
199#92418—22no—
200#92418—41no—
201#92418—41no—
202#92418—41no—
203#92418—41no—
204#19237—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61#46312yes—
214C22×C61C2×#12111yescharacteristic
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—81no—
220C4×C61#46312yes—
221C4×C61C2×#13511yescharacteristic
222C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C4×C61#57611yescharacteristic
224C4×C61#57711yescharacteristic
225Dic61—82no—
226#3245D6111yescharacteristic
227#3206—611no—
228#3141—611no—
229#2935—611no—
230#1701—612no—
231#1699—612no—
232#1681—612no—
233#1679—612no—
234#1565—611no—
235#1540—611no—
236C2×#332—611no—
237C2×C61—22no—
238C2×C61—82no—
239C2×C61—161no—
240C2×C61C2×#46311yescharacteristic
241C2×C61#306211yescharacteristic
242C2×C61#306911yescharacteristic
243C2×#131—1221no—
244#532C2×D6111yescharacteristic
245#532C2×D6111yescharacteristic
246C2×C4⋊Q8—611no—
247#492—1221no—
248#489—1222no—
249#487—612no—
250#487—1221no—
251#486—612no—
252#474—21no—
253#474C2×D6112yes—
254#466—22no—
255#461—612no—
256#461—612no—
257#459—21no—
258#459C2×D6112yes—
259#448—1222no—
260#436—612no—
261#435—1222no—
262C8×D4—612no—
263#431—611no—
264C2×C4⋊C8—611no—
265C2×C4⋊C8C2×D6111yescharacteristic
266#417—1221no—
267#417—1221no—
268#408—1221no—
269C4×M16—611no—
270#335—612no—
271#334—612no—
272#332—611no—
273#332—611no—
274#332—612no—
275#332—612no—
276#331—611no—
277#331—611no—
278C61#1923711yescharacteristic, complemented, Sylow
279C2×#41—21no—
280C2×#41—21no—
281C22×Q8—1221no—
282C2×SD16—1222no—
283C2×SD16—1222no—
284C2×D8—22no—
285#131—2444no—
286C2×M16—611no—
287C2×M16—611no—
288C2×M16—1222no—
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#127C22⋊D6111yescharacteristic
299#127C22×D6111yescharacteristic
300#124—21no—
301#124—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#121C22⋊D6112yes—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4C22×D6112yes—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C42C22×D6111yescharacteristic
313#107—22no—
314C4⋊C8—21no—
315C4⋊C8—611no—
316C4⋊C8—611no—
317C4⋊C8—611no—
318C4⋊C8—611no—
319C4⋊C8—1221no—
320C4⋊C8C22×D6112yes—
321#103—1222no—
322#103—1222no—
323#103—1222no—
324#103—1222no—
325#102—22no—
326#102—22no—
327#102—1222no—
328#102—1222no—
329C22⋊C8—1222no—
330C22⋊C8—1222no—
331C22⋊C8—1221no—
332#97—612no—
333C4×C8—612no—
334C4×C8—1221no—
335#41—42no—
336#41—42no—
337#41—42no—
338#41—42no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—1222no—
342C2×Q8—4881no—
343C2×Q8—4881no—
344C2×Q8—2441no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4C2×C22⋊D6112yes—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—2441no—
356C22×C4D4×D6111yescharacteristic
357C22×C4#1175822411yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359SD16—4882no—
360SD16—4882no—
361D8—82no—
362M16—1222no—
363M16—1222no—
364M16—2444no—
365C2×C8—22no—
366C2×C8—81no—
367C2×C8—612no—
368C2×C8—612no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8—2444no—
374C2×C8—2442no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—1222no—
379C4⋊C4—1222no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C4⋊C4C2×C22⋊D6112yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C42D4×D6111yescharacteristic
388C42#1175822411yescharacteristic
389C42C2×C22⋊D6111yescharacteristic
390C42C23×D6111yescharacteristic
391C23—22no—
392C23—41no—
393C23#1451974911yescharacteristic
394Q8—42no—
395Q8—41no—
396Q8—41no—
397Q8—4882no—
398Q8—2442no—
399Q8—2442no—
400D4—22no—
401D4—22no—
402D4—44no—
403D4—42no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—41no—
408D4—81no—
409D4—81no—
410D4—82no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—82no—
419C2×C4—4882no—
420C2×C4—2442no—
421C2×C4#1451974911yescharacteristic
422C2×C4#1451974911yescharacteristic
423C2×C4#1451974911yescharacteristic
424C2×C4#1451979612yes—
425C2×C4#1451980112yes—
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4C22×C22⋊D6111yescharacteristic
428C2×C4C2×#1175822411yescharacteristic
429C8—42no—
430C8—1222no—
431C8—1222no—
432C8—2441no—
433C8—2441no—
434C8—2441no—
435C8—2441no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22#1683997512yes—
445C22C2×#1451974911yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—81no—
451C4—4882no—
452C4#1683997512yes—
453C4C2×#1451979611yescharacteristic
454C4C2×#1451980111yescharacteristic
455C4#1684056511yescharacteristic
456C4#1684057011yescharacteristic
457C2—22no—
458C2—82no—
459C2—161no—
460C2C2×#1683997511yescharacteristic
461C2#2145550111yescharacteristic
462C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3245C2
socleC22×C61C2×#121
hypercenter#3245D61
hypocenterC61#19237
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21456741C2semidirect
2. #21456212C2semidirect
3. #21454563C2semidirect
4. #21445378C2semidirect
5. #21445364C2semidirect
6. #21445254C2semidirect
7. #21445240C2semidirect
8. #21444569C2semidirect
9. #21444468C2semidirect
10. C2×#16839509C2semidirect
278. C61#19237semidirect

Maximal subgroups

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

subgroupindex
#214567412
#214562122
#214545632
#214453782
#214453642
#214452542
#214452402
#214445692
#214444682
C2×#168395092
C61×#32452
#1923761