Finite Group Databasebeta

← #29663217

Subgroups — #29663217

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic, complemented
2C2×#16840138C211yesmaximal, characteristic
3#21454754C211yesmaximal, characteristic, complemented
4#21445301C214yesmaximal, complemented
5#21445177C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#3208C211yesmaximal, characteristic, complemented
10C22×#11758199—21no—
11C22×#11758199—21no—
12C2×#14519774C2211yescharacteristic
13#16840138—21no—
14#16840138—21no—
15#16840138C2212yes—
16#16840138C2212yes—
17#16840136—21no—
18#16840110—21no—
19#16839962C2214yescomplemented
20#16839912—22no—
21#16839893—21no—
22#16839893C2212yescomplemented
23Q8×#92418—21no—
24Q8×#92418C2212yescomplemented
25#16839815C2211yescharacteristic
26#16839789—21no—
27#16839786C2211yescharacteristic
28#16839718—21no—
29C2×C4×#92418C2211yescharacteristic
30#16839514C2212yescomplemented
31#16839512C2212yescomplemented
32#16839506C2214yescomplemented
33#16839505C2214yescomplemented
34C61×#533C2211yescharacteristic
35C2×#127×C61C2211yescharacteristic
36C61×#492—21no—
37C61×#486—21no—
38C61×#460C2214yescomplemented
39C61×#450—22no—
40C61×#431C2211yescharacteristic
41C22×D244—21no—
42C22×D244—21no—
43C22×D244—21no—
44C2×#11758201—24no—
45C2×#11758199—22no—
46C2×#11758199—22no—
47C2×#11758199—22no—
48C2×#11758199—22no—
49C2×#11758199—44no—
50C2×#11758199—44no—
51C2×#11758194—21no—
52C22×#92418D411yescharacteristic
53C22×#92418D411yescharacteristic
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4⋊#92418—21no—
59C4⋊#92418—22no—
60C4⋊#92418—22no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519697—24no—
64C22⋊#92418—21no—
65#14519688—22no—
66C4×#92418—21no—
67C4×#92418—22no—
68C4×#92418D411yescharacteristic
69C4×#92418D411yescharacteristic
70C4×#92418C2311yescharacteristic
71C4×#92418C2311yescharacteristic
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61—21no—
75C2×SD16×C61—24no—
76C4⋊Q8×C61D411yescharacteristic
77C4⋊Q8×C61D411yescharacteristic
78#127×C61—21no—
79#127×C61—21no—
80#127×C61—21no—
81#127×C61C2312yes—
82#125×C61—21no—
83C22⋊Q8×C61—21no—
84C22⋊Q8×C61D412yes—
85C4×Q8×C61—21no—
86C4×Q8×C61C2312yes—
87#117×C61—21no—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61C2312yes—
90#102×C61—24no—
91C22⋊C8×C61—21no—
92C4×C8×C61—21no—
93C24×C61—41no—
94C23×D61—41no—
95C2×Q8×C61—21no—
96C2×Q8×C61C2×D412yes—
97C2×D4×C61—22no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C2×D4×C61—42no—
104C2×D4×C61—42no—
105C2×D4×C61—82no—
106C2×D4×C61—82no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—42no—
114C2×D244—42no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244—41no—
118C2×D244—82no—
119C2×D244—82no—
120SD16×C61—84no—
121C2×C8×C61—22no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61C2×D412yes—
128C22⋊C4×C61—22no—
129C42×C61—22no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C42×C61C2411yescharacteristic
134#11758201—84no—
135#11758199—44no—
136#11758199—44no—
137#11758199—44no—
138#11758199—44no—
139#11758194—42no—
140C2×#92418—22no—
141C2×#92418—22no—
142C2×#92418—22no—
143C2×#92418SD1614yescomplemented
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148C23×C61—42no—
149C23×C61—82no—
150C23×C61—82no—
151C23×C61—82no—
152C23×C61#12111yescharacteristic
153C22×D61—42no—
154C22×D61—82no—
155C22×D61—82no—
156C22×D61—82no—
157Q8×C61—24no—
158Q8×C61—42no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—44no—
162D4×C61—82no—
163D4×C61—82no—
164D4×C61—82no—
165D4×C61—82no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61C2×SD1612yes—
174C2×C4×C61AGL(1,ℤ/8)12yes—
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178D244—42no—
179D244—42no—
180D244—44no—
181D244—82no—
182D244—82no—
183D244—82no—
184D244—82no—
185C8×C61—42no—
186#92418—21no—
187#92418—21no—
188#92418—42no—
189#92418—42no—
190#92418C2×SD1612yes—
191#92418C2×SD1612yes—
192#19203—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—82no—
196C22×C61—82no—
197C22×C61—82no—
198C22×C61—164no—
199C22×C61#46012yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—42no—
208C4×C61—42no—
209C4×C61#46012yes—
210C4×C61#47212yes—
211C4×C61C22×SD1611yescharacteristic
212C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
213C4×C61#57612yes—
214#3208—611no—
215#3208D6111yescharacteristic, complemented
216C2×#493—611no—
217#2960—611no—
218#1688—614no—
219#1668—614no—
220#1564—611no—
221#1545—611no—
222#1545—611no—
223C2×C61—22no—
224C2×C61—162no—
225C2×C61C2×#46011yescharacteristic
226C2×C61#305811yescharacteristic
227C2×C61C2×#47211yescharacteristic
228D61—162no—
229C22×D8—1221no—
230C22×D8—1221no—
231#533C2×D6111yescharacteristic
232C2×#127—611no—
233C2×#127C2×D6111yescharacteristic
234#493—612no—
235#493—612no—
236#493—1221no—
237#493—1221no—
238#492—21no—
239#492—1221no—
240#486—21no—
241#486—1221no—
242#460—614no—
243#460C2×D6114yescomplemented
244#450—22no—
245#450—1222no—
246#446—612no—
247#446—1221no—
248C8×Q8—612no—
249C8×Q8—1221no—
250#431—611no—
251#431C2×D6111yescharacteristic
252#424—1221no—
253#423—611no—
254#405—1221no—
255C2×C4×C8—611no—
256#335—612no—
257#334—612no—
258#331—614no—
259#331—614no—
260C61#1920311yescharacteristic, complemented, Sylow
261C22×D4—21no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—1221no—
265C22×D4—1221no—
266C22×D4—1221no—
267C2×SD16—24no—
268C2×SD16—1224no—
269C2×D8—1222no—
270C2×D8—1222no—
271C2×D8—1222no—
272C2×D8—1222no—
273C2×D8—2444no—
274C2×D8—2444no—
275C2×M16—1221no—
276C22×C8—611no—
277C22×C8—611no—
278C4⋊Q8C22⋊D6111yescharacteristic
279C4⋊Q8C22⋊D6111yescharacteristic
280#127—21no—
281#127—21no—
282#127—21no—
283#127—612no—
284#127—1221no—
285#127—1221no—
286#127—1221no—
287#127C22×D6112yes—
288#125—21no—
289C22⋊Q8—21no—
290C22⋊Q8C22⋊D6112yes—
291C4×Q8—21no—
292C4×Q8C22×D6112yes—
293#117—21no—
294C2×C42C22×D6111yescharacteristic
295C4⋊C8—612no—
296C4⋊C8—612no—
297C4⋊C8—1222no—
298C4⋊C8—1222no—
299C4⋊C8—1221no—
300C4⋊C8C22×D6112yes—
301#102—24no—
302#102—1224no—
303C22⋊C8—21no—
304C22⋊C8—1221no—
305#97—1222no—
306C4×C8—21no—
307C4×C8—611no—
308C4×C8—611no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—1222no—
312C4×C8—1221no—
313C24—41no—
314C24—2441no—
315C2×Q8—21no—
316C2×Q8C2×C22⋊D6112yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—1222no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—4882no—
331C2×D4—4882no—
332C2×D4—2441no—
333C2×D4—2441no—
334C2×D4—2441no—
335C2×D4—2442no—
336C2×D4—2442no—
337C22×C4D4×D6111yescharacteristic
338C22×C4#1175822411yescharacteristic
339C22×C4C2×C22⋊D6111yescharacteristic
340SD16—84no—
341SD16—4884no—
342D8—2444no—
343D8—2444no—
344D8—2444no—
345D8—2444no—
346M16—2442no—
347C2×C8—22no—
348C2×C8—611no—
349C2×C8—611no—
350C2×C8—614no—
351C2×C8—611no—
352C2×C8—611no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8—1222no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4C2×C22⋊D6112yes—
362C22⋊C4—22no—
363C42—22no—
364C42D4×D6111yescharacteristic
365C42#1175822411yescharacteristic
366C42C2×C22⋊D6111yescharacteristic
367C42C23×D6111yescharacteristic
368C23—42no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—4882no—
373C23—4882no—
374C23—4882no—
375C23—2442no—
376C23#1451974911yescharacteristic
377Q8—24no—
378Q8—42no—
379D4—42no—
380D4—42no—
381D4—44no—
382D4—82no—
383D4—82no—
384D4—82no—
385D4—82no—
386D4—4882no—
387D4—4882no—
388D4—4882no—
389D4—4882no—
390D4—2442no—
391D4—2442no—
392D4—2444no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4#1451974911yescharacteristic
398C2×C4#1451974911yescharacteristic
399C2×C4#1451974911yescharacteristic
400C2×C4SD16×D6112yes—
401C2×C4#1451980412yes—
402C2×C4C2×D4×D6111yescharacteristic
403C2×C4C22×C22⋊D6111yescharacteristic
404C2×C4C2×#1175822411yescharacteristic
405C8—42no—
406C8—612no—
407C8—612no—
408C8—1221no—
409C8—1221no—
410C8—2442no—
411C8—2442no—
412C22—21no—
413C22—21no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—164no—
418C22—9764no—
419C22—4882no—
420C22—4882no—
421C22—4882no—
422C22#1683996012yes—
423C22C2×#1451974911yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—42no—
428C4#1683996012yes—
429C4#1684003012yes—
430C4C2×SD16×D6111yescharacteristic
431C4C2×#1451980411yescharacteristic
432C4#1684056512yes—
433C2—22no—
434C2—162no—
435C2—9762no—
436C2C2×#1683996011yescharacteristic
437C2#2145546411yescharacteristic
438C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3208C2
socleC22×C61C2×#121
prime radicalC2×#16840138C2
hypercenter#3208D61
hypocenterC61#19203
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
3. #21454754C2semidirect
4. #21445301C2semidirect
5. #21445177C2semidirect
6. #21444565C2semidirect
7. #21444487C2semidirect
8. #21444486C2semidirect
9. C61×#3208C2semidirect
19. #16839962C22semidirect
22. #16839893C22semidirect
24. Q8×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839506C22semidirect
33. #16839505C22semidirect
38. C61×#460C22semidirect
143. C2×#92418SD16semidirect
215. #3208D61semidirect
243. #460C2×D61semidirect
260. C61#19203semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168401382
#214547542
#214453012
#214451772
#214445652
#214444872
#214444862
C61×#32082
#1920361