Finite Group Databasebeta

← #29837764

Subgroups — #29837764

Order 15616.

474 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458807C211yesmaximal, characteristic
2Q16×Dic122C211yesmaximal, characteristic, complemented
3#21448057C211yesmaximal, characteristic
4#21448056C211yesmaximal, characteristic
5#21448045C211yesmaximal, characteristic, complemented
6#21447965C211yesmaximal, characteristic
7#21447961C211yesmaximal, characteristic, complemented
8#21447960C211yesmaximal, characteristic, complemented
9#21447883C211yesmaximal, characteristic
10#21447881C211yesmaximal, characteristic
11#21447867C211yesmaximal, characteristic, complemented
12#21447725C211yesmaximal, characteristic, complemented
13#21447687C211yesmaximal, characteristic, complemented
14#21447668C211yesmaximal, characteristic
15C61×#2176C211yesmaximal, characteristic, complemented
16Q8×Dic122—21no—
17#16840484—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519720C2211yescharacteristic
20#16840167C2211yescharacteristic
21#16840166C2211yescharacteristic
22#16840166C2211yescharacteristic
23#16840147C2211yescharacteristic
24#16840146C2211yescharacteristic
25#16840140C2211yescharacteristic
26#16840057—21no—
27#16840055—21no—
28#16840055—21no—
29#16839987—21no—
30#16839984—21no—
31#16839966—21no—
32C8×Dic122C2211yescharacteristic
33Q16×Dic61C2211yescharacteristic
34Q16×Dic61C2211yescharacteristic
35#16839856C2211yescharacteristic
36#16839844C2211yescharacteristic
37#16839589C2211yescharacteristic
38#16839589C2211yescharacteristic
39C4⋊#11758185C2211yescharacteristic
40#16839580C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839567C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839565C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×Q32×C61C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic
54C4×Q16×C61C2211yescharacteristic
55C61×#367C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#346C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758202—21no—
61C2×#11758190—21no—
62C2×#11758201—21no—
63C2×D488C2311yescharacteristic
64#14519777—21no—
65#14519777—41no—
66#14519776—21no—
67#14519776—21no—
68#14519776—41no—
69#14519776—41no—
70#14519775—21no—
71#14519772—41no—
72#14519771—41no—
73#14519766—21no—
74#14519765—21no—
75#14519755—41no—
76Q8×Dic61—21no—
77Q8×Dic61—21no—
78C4×Dic122—41no—
79C4×Dic122D411yescharacteristic
80#14519742—21no—
81#14519739—21no—
82#14519734—41no—
83#14519723—22no—
84#14519720—22no—
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87#14519711—21no—
88#14519711C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519710—21no—
91#14519710—21no—
92C4⋊#92418D411yescharacteristic
93#14519706D411yescharacteristic
94#14519706D411yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519700—21no—
98#14519697—21no—
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C2×Q16×C61C2311yescharacteristic
102C2×Q16×C61C2311yescharacteristic
103C4×Q8×C61—21no—
104C4×Q8×C61—21no—
105Q32×C61—22no—
106C2×C16×C61C2311yescharacteristic
107C2×C16×C61C2311yescharacteristic
108#107×C61C2311yescharacteristic
109#107×C61C2311yescharacteristic
110#103×C61—21no—
111#103×C61—21no—
112C4×C8×C61C2311yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×D244—21no—
116C2×C4×D61—41no—
117C2×Dic122—21no—
118C2×Dic122—41no—
119C2×Dic122—41no—
120C2×Dic122—41no—
121C2×Dic122D811yescharacteristic
122Q16×C61—21no—
123Q16×C61—21no—
124Q16×C61—21no—
125Q16×C61#4112yes—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2411yescharacteristic
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C42×C61—41no—
133C42×C61—41no—
134C42×C61C2×D411yescharacteristic
135#11758202—81no—
136#11758201—81no—
137#11758198—41no—
138#11758198—41no—
139#11758198—41no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4⋊Dic61D811yescharacteristic
148#11758196—41no—
149#11758196—41no—
150#11758196—41no—
151#11758196—41no—
152#11758196—41no—
153#11758196D811yescharacteristic
154#11758196D811yescharacteristic
155C4×Dic61—41no—
156C4×Dic61—41no—
157C4×Dic61—41no—
158C4×Dic61C2×D411yescharacteristic
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162Dic244—21no—
163Dic244—21no—
164D488—21no—
165D488—21no—
166#11758190—81no—
167C16×C61—21no—
168C16×C61—21no—
169#11758185—21no—
170#11758185—21no—
171C22×D61—41no—
172Q8×C61—22no—
173Q8×C61—41no—
174Q8×C61—41no—
175Q8×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61C2×D811yescharacteristic
181C2×C4×C61C2×D811yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61C2×D811yescharacteristic
187C2×Dic61C2×D811yescharacteristic
188C2×Dic61C2×D811yescharacteristic
189C2×Dic61C2×D811yescharacteristic
190D244—41no—
191D244—41no—
192C4×D61—82no—
193Dic122—22no—
194Dic122—41no—
195Dic122—41no—
196Dic122—161no—
197Dic122—161no—
198Dic122—161no—
199C8×C61—21no—
200C8×C61C2×#4111yescharacteristic
201C8×C612−1+411yescharacteristic
202#92418—21no—
203#92418—21no—
204#33936—611nomaximal, Sylow
205C22×C61C22×D811yescharacteristic
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—42no—
212C4×C61—81no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61#54811yescharacteristic
216C4×C61#57711yescharacteristic
217Dic61—21no—
218Dic61—21no—
219Dic61—21no—
220Dic61—81no—
221Dic61—81no—
222Dic61—81no—
223Dic61#51012yes—
224#3393—611no—
225Q8×Q16—611no—
226#2243—611no—
227#2243—611no—
228#2241—611no—
229#2226—611no—
230#2225—611no—
231#2225—611no—
232#2211—611no—
233#2211—611no—
234#2209—611no—
235#2183—611no—
236#2176—611no—
237#2176D6111yescharacteristic, complemented
238C4×SD32—611no—
239C2×C61#335811yescharacteristic
240C2×C61C2×#51011yescharacteristic
241C2×C61#341611yescharacteristic
242D61—161no—
243Q82—1221no—
244#556—1221no—
245C2×Q32—611no—
246C2×Q32C2×D6111yescharacteristic
247C2×SD32—611no—
248#500—611no—
249#500—611no—
250#500—611no—
251#495—611no—
252#495—611no—
253#494—611no—
254#477—1221no—
255#477—1221no—
256#477—1221no—
257#464—1221no—
258#464—1221no—
259#461—1221no—
260C8×Q8—611no—
261C4×Q16—611no—
262C4×Q16—611no—
263C4×Q16—611no—
264C4×Q16C2×D6111yescharacteristic
265C4×Q16C2×D6111yescharacteristic
266C4×D8—611no—
267#367—611no—
268#367—611no—
269#367C2×D6111yescharacteristic
270C4⋊C16—611no—
271C4⋊C16—611no—
272C4⋊C16—611no—
273#358—611no—
274#358—611no—
275#358—611no—
276#358—611no—
277#358—611no—
278#358C2×D6111yescharacteristic
279#358C2×D6111yescharacteristic
280#357—611no—
281#357—611no—
282#357—611no—
283#346—611no—
284#346C2×D6111yescharacteristic
285C4×C16—611no—
286C61#3393611yescharacteristic, complemented, Sylow
287C2×Q16—611no—
288C2×Q16—1221no—
289C2×Q16C22×D6111yescharacteristic
290C2×Q16C22×D6111yescharacteristic
291C2×SD16—1221no—
292C2×SD16—1221no—
293C2×D8—611no—
294C4⋊Q8—1221no—
295C4⋊Q8—1221no—
296C4⋊Q8—1221no—
297C4⋊Q8—1221no—
298C4⋊Q8—2441no—
299C4⋊Q8—2441no—
300C4⋊Q8—2441no—
301#126—2441no—
302#126—2441no—
303#124—1221no—
304#124—1221no—
305C22⋊Q8—2441no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—611no—
309C4×Q8—1221no—
310C4×Q8—1221no—
311C4×Q8—1221no—
312C4×Q8—2441no—
313C4×D4—1221no—
314#117—2441no—
315Q32—22no—
316Q32—1222no—
317SD32—1222no—
318C2×C16—611no—
319C2×C16—611no—
320C2×C16C22×D6111yescharacteristic
321C2×C16C22×D6111yescharacteristic
322#107—611no—
323#107—611no—
324#107—1221no—
325#107—1221no—
326#107—1221no—
327#107C22×D6111yescharacteristic
328#107C22×D6111yescharacteristic
329C4⋊C8—611no—
330C4⋊C8—611no—
331C4⋊C8—611no—
332#103—21no—
333#103—21no—
334#103—1221no—
335#103—1221no—
336#103—1221no—
337#102—1221no—
338C4×C8—611no—
339C4×C8—611no—
340C4×C8C22×D6111yescharacteristic
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—611no—
344C2×Q8—1221no—
345C2×Q8—2441no—
346C2×Q8—2441no—
347C2×Q8—2441no—
348C2×D4—1221no—
349C22×C4—2441no—
350Q16—21no—
351Q16—21no—
352Q16—21no—
353Q16—1221no—
354Q16—1221no—
355Q16—4881no—
356Q16#1175822212yes—
357SD16—4881no—
358SD16—4881no—
359D8—1221no—
360D8—1221no—
361C2×C8—611no—
362C2×C8—611no—
363C2×C8D4×D6111yescharacteristic
364C2×C8C23×D6111yescharacteristic
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—611no—
370C4⋊C4—611no—
371C4⋊C4—611no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—2441no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4—2441no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C22⋊C4—2441no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C42—41no—
388C42—41no—
389C42—611no—
390C42—611no—
391C42—2441no—
392C42—2441no—
393C42—2441no—
394C42D4×D6111yescharacteristic
395C16—21no—
396C16—21no—
397C16—1221no—
398C16—1221no—
399C23—2441no—
400Q8—22no—
401Q8—41no—
402Q8—41no—
403Q8—41no—
404Q8—1222no—
405Q8—9761no—
406Q8—9761no—
407Q8—9761no—
408Q8—2441no—
409Q8—2441no—
410D4—2441no—
411D4—2441no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—611no—
417C2×C4—611no—
418C2×C4—611no—
419C2×C4—611no—
420C2×C4—4882no—
421C2×C4—2441no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4D8×D6111yescharacteristic
425C2×C4D8×D6111yescharacteristic
426C2×C4C2×D4×D6111yescharacteristic
427C8—21no—
428C8—1221no—
429C8—1221no—
430C8C2×#1175822211yescharacteristic
431C8#1451982411yescharacteristic
432C22—4881no—
433C22—4881no—
434C22—4881no—
435C22C2×D8×D6111yescharacteristic
436C4—21no—
437C4—21no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—612no—
443C4—1221no—
444C4—1221no—
445C4—1221no—
446C4—4881no—
447C4—4881no—
448C4—4881no—
449C4#1684042111yescharacteristic
450C4#1684056911yescharacteristic
451C2—9761no—
452C2#2145845211yescharacteristic
453C2#510×D6111yescharacteristic
454C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2176C2
socleC22×C61C22×D8
prime radicalC2×D488C23
hypercenter#2176D61
hypocenterC61#33936
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
2. Q16×Dic122C2semidirect
5. #21448045C2semidirect
7. #21447961C2semidirect
8. #21447960C2semidirect
11. #21447867C2semidirect
12. #21447725C2semidirect
13. #21447687C2semidirect
15. C61×#2176C2semidirect
237. #2176D61semidirect
286. C61#33936semidirect

Maximal subgroups

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

subgroupindex
#214588072
Q16×Dic1222
#214480572
#214480562
#214480452
#214479652
#214479612
#214479602
#214478832
#214478812
#214478672
#214477252
#214476872
#214476682
C61×#21762
#3393661