Finite Group Databasebeta

← #29663880

Subgroups — #29663880

Order 15616.

650 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#21456394C211yesmaximal, characteristic
2#21455941C211yesmaximal, characteristic, complemented
3C2×Q8×#92418C211yesmaximal, characteristic, complemented
4#21445237C212yesmaximal, complemented
5#21445233C212yesmaximal, complemented
6#21445221C212yesmaximal, complemented
7#21445177C212yesmaximal, complemented
8#21444580C211yesmaximal, characteristic, complemented
9#21444515C211yesmaximal, characteristic, complemented
10#21444514C211yesmaximal, characteristic, complemented
11C61×#3112C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13#16840165C2211yescharacteristic
14#16840138C2211yescharacteristic
15#16840110C2211yescharacteristic
16#16840108C2211yescharacteristic
17#16840075—21no—
18#16840057C2212yes—
19#16839996—21no—
20#16839996—21no—
21#16839962C2212yes—
22#16839901—21no—
23Q8×#92418—21no—
24Q8×#92418—21no—
25Q8×#92418C2212yes—
26Q8×#92418C2212yes—
27#16839860—22no—
28C4×#11758199—21no—
29C4×#11758199—21no—
30#16839819C2211yescharacteristic
31C2×C4⋊#92418—21no—
32C2×C4⋊#92418C2211yescharacteristic
33C2×C4×#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yes—
36#16839512C2212yes—
37#16839511C2212yes—
38#16839510C2212yes—
39#16839506C2212yes—
40#16839505C2212yes—
41C61×#532C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#480—21no—
44C61×#477C2212yes—
45C61×#460C2212yes—
46C61×#448—21no—
47C61×#440—22no—
48C61×#432C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758201—22no—
51C2×#11758199—21no—
52C2×#11758199—21no—
53C22×#92418—21no—
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519765—21no—
59#14519751—21no—
60#14519751—21no—
61C4×D244—21no—
62C4×D244—21no—
63#14519710—22no—
64#14519710—21no—
65#14519710—21no—
66C4⋊#92418—21no—
67C4⋊#92418—22no—
68C4⋊#92418—22no—
69C4⋊#92418C2312yes—
70C4⋊#92418C2312yes—
71#14519701—22no—
72#14519698—21no—
73#14519698—21no—
74#14519697—22no—
75#14519697—21no—
76#14519697—21no—
77C22⋊#92418—21no—
78#14519688—21no—
79C4×#92418—22no—
80C4×#92418—22no—
81C4×#92418D411yescharacteristic
82C4×#92418D411yescharacteristic
83C4×#92418C2311yescharacteristic
84C4×#92418C2311yescharacteristic
85C2×#41×C61—21no—
86C22×Q8×C61D411yescharacteristic
87C2×SD16×C61—22no—
88C4⋊Q8×C61C2311yescharacteristic
89#127×C61C2311yescharacteristic
90#124×C61—21no—
91#121×C61—21no—
92#121×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C42×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109#41×C61—82no—
110C2×Q8×C61—24no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61C2×D412yes—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—41no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×C22⋊D61—41no—
125#11758222—82no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—21no—
129C2×C4×D61—41no—
130C2×Dic122—21no—
131SD16×C61—44no—
132C2×C8×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61C2×D412yes—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147#11758201—44no—
148#11758199—42no—
149#11758199—42no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—41no—
158C2×#92418—42no—
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163C23×C61—41no—
164C23×C61#12111yescharacteristic
165C22×D61—41no—
166Q8×C61—24no—
167Q8×C61—22no—
168Q8×C61—81no—
169Q8×C61#13514yes—
170D4×C61—42no—
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—81no—
174D4×C61—82no—
175C2×C4×C61—24no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—82no—
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C22⋊D61—82no—
192C2×Dic61—41no—
193D244—42no—
194D244—41no—
195D244—41no—
196D244—81no—
197C4×D61—82no—
198Dic122—81no—
199C8×C61—42no—
200#92418—21no—
201#92418—21no—
202#92418—21no—
203#92418—42no—
204#92418—42no—
205#92418#13712yes—
206#19264—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61#47212yes—
213C22×C61C2×#12111yescharacteristic
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—21no—
218C4×C61—24no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—42no—
223C4×C61—81no—
224C4×C61C2×#13512yes—
225C4×C61C2×#13711yescharacteristic
226C4×C61#57511yescharacteristic
227Dic61—81no—
228#3169—611no—
229#3112—611no—
230#3112D6111yescharacteristic, complemented
231C2×C8×Q8—611no—
232#1678—612no—
233#1678—612no—
234#1676—612no—
235#1668—612no—
236#1568—611no—
237#1551—611no—
238#1551—611no—
239C2×C61—22no—
240C2×C61—161no—
241C2×C61#303711yescharacteristic
242C2×C61#305011yescharacteristic
243C2×C61C2×#47211yescharacteristic
244D61—161no—
245#532—611no—
246#532C2×D6111yescharacteristic
247C2×C4×Q8C2×D6111yescharacteristic
248#500—611no—
249#493—611no—
250#486—611no—
251#486—611no—
252#480—21no—
253#480—1221no—
254#477—612no—
255#477C2×D6112yes—
256#466—1221no—
257#466—1221no—
258#460—612no—
259#460C2×D6112yes—
260#448—21no—
261#448—1221no—
262C8×Q8—612no—
263C8×Q8—612no—
264C8×Q8—1221no—
265C8×Q8—1221no—
266#440—22no—
267#440—1222no—
268C4×D8—1221no—
269C4×D8—1221no—
270#432—611no—
271#432C2×D6111yescharacteristic
272C2×C4⋊C8—611no—
273C2×C4⋊C8—1221no—
274C2×C4×C8—611no—
275C2×C4×C8—1221no—
276#335—612no—
277#334—612no—
278#333—612no—
279#333—612no—
280#331—612no—
281#331—612no—
282C61#1926411yescharacteristic, complemented, Sylow
283C2×#41—21no—
284C2×#41—1221no—
285C22×Q8C22⋊D6111yescharacteristic
286C2×SD16—22no—
287C2×SD16—1222no—
288C2×D8—1221no—
289C2×D8—1221no—
290C22×C8—611no—
291C22×C8—611no—
292C22×C8—1221no—
293C4⋊Q8—611no—
294C4⋊Q8C22×D6111yescharacteristic
295#127—611no—
296#127C22×D6111yescharacteristic
297#124—21no—
298#124—1221no—
299#121—21no—
300#121—21no—
301#121—1221no—
302#121—1221no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8C22⋊D6112yes—
306C4×Q8C22×D6112yes—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—1221no—
310C4×D4—1221no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4C22⋊D6111yescharacteristic
313C2×C42—21no—
314C2×C42C22×D6111yescharacteristic
315#107—22no—
316#107—1221no—
317#107—1221no—
318#107—1222no—
319C4⋊C8—612no—
320C4⋊C8—612no—
321C4⋊C8—1222no—
322C4⋊C8—1222no—
323C4⋊C8—1221no—
324C4⋊C8C22×D6112yes—
325#103—22no—
326#103—1222no—
327#102—22no—
328#102—1221no—
329#102—1221no—
330#102—1221no—
331#102—1221no—
332#102—1222no—
333C22⋊C8—21no—
334C22⋊C8—1221no—
335#97—21no—
336#97—1221no—
337C4×C8—611no—
338C4×C8—611no—
339C4×C8—611no—
340C4×C8—611no—
341C4×C8—1222no—
342C4×C8—1222no—
343#41—82no—
344#41—4882no—
345C2×Q8—24no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—1221no—
349C2×Q8C2×C22⋊D6112yes—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—2441no—
362C22×C4D4×D6111yescharacteristic
363C22×C4#1175822411yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365SD16—44no—
366SD16—2444no—
367D8—2442no—
368D8—2442no—
369C2×C8—22no—
370C2×C8—611no—
371C2×C8—611no—
372C2×C8—611no—
373C2×C8—611no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8—2441no—
378C2×C8—2442no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4C2×C22⋊D6112yes—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C42—22no—
392C42—22no—
393C42D4×D6111yescharacteristic
394C42#1175822411yescharacteristic
395C42C2×C22⋊D6111yescharacteristic
396C42C23×D6111yescharacteristic
397C23—41no—
398C23—2441no—
399C23#1451974911yescharacteristic
400Q8—24no—
401Q8—22no—
402Q8—81no—
403Q8—4881no—
404Q8#1451979714yes—
405D4—42no—
406D4—41no—
407D4—41no—
408D4—81no—
409D4—82no—
410D4—4881no—
411D4—4882no—
412D4—2442no—
413D4—2441no—
414D4—2441no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—82no—
425C2×C4—4882no—
426C2×C4—2441no—
427C2×C4#1451974911yescharacteristic
428C2×C4#1451974911yescharacteristic
429C2×C4#1451974911yescharacteristic
430C2×C4C2×D4×D6111yescharacteristic
431C2×C4C22×C22⋊D6111yescharacteristic
432C2×C4C2×#1175822411yescharacteristic
433C8—42no—
434C8—612no—
435C8—1221no—
436C8—1221no—
437C8—1221no—
438C8—2442no—
439C8—2442no—
440C22—21no—
441C22—21no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—4881no—
446C22—4881no—
447C22—4881no—
448C22#1684003012yes—
449C22C2×#1451974911yescharacteristic
450C4—24no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—4881no—
458C4C2×#1451979712yes—
459C4#137×D6111yescharacteristic
460C4#1684056111yescharacteristic
461C2—22no—
462C2—161no—
463C2—9761no—
464C2#2145528811yescharacteristic
465C2#2145539711yescharacteristic
466C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3112C2
socleC22×C61C2×#121
prime radical#21456394C2
hypercenter#3112D61
hypocenterC61#19264
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. #21455941C2semidirect
3. C2×Q8×#92418C2semidirect
4. #21445237C2semidirect
5. #21445233C2semidirect
6. #21445221C2semidirect
7. #21445177C2semidirect
8. #21444580C2semidirect
9. #21444515C2semidirect
10. #21444514C2semidirect
11. C61×#3112C2semidirect
230. #3112D61semidirect
282. C61#19264semidirect

Maximal subgroups

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

subgroupindex
#214563942
#214559412
C2×Q8×#924182
#214452372
#214452332
#214452212
#214451772
#214445802
#214445152
#214445142
C61×#31122
#1926461