Finite Group Databasebeta

← #29662352

Subgroups — #29662352

Order 15616.

714 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#21456246C211yesmaximal, characteristic
2#21455849C212yesmaximal, complemented
3C8×Q8×D61C211yesmaximal, characteristic, complemented
4#21445190C211yesmaximal, characteristic, complemented
5#21445189C211yesmaximal, characteristic, complemented
6#21445176C211yesmaximal, characteristic, complemented
7#21445166C212yesmaximal, complemented
8#21445165C212yesmaximal, complemented
9#21444566C211yesmaximal, characteristic, complemented
10#21444465C212yesmaximal, complemented
11C61×#1668C211yesmaximal, characteristic, complemented
12C2×#14519796—21no—
13C2×#14519796—21no—
14#16840310C2212yes—
15C4×Q8×D61C2211yescharacteristic
16#16840145—21no—
17#16840145—21no—
18#16840142C2211yescharacteristic
19#16840134C2212yes—
20#16840079—22no—
21#16840048C2212yes—
22#16840047C2212yes—
23#16839959C2212yes—
24#16839899—22no—
25Q8×#92418C2211yescharacteristic
26C8×Dic122—21no—
27C8×Dic122C2211yescharacteristic
28#16839886—21no—
29#16839886C2211yescharacteristic
30#16839853—22no—
31SD16×Dic61—22no—
32#16839814C2212yes—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839509C2212yes—
40#16839507C2212yes—
41#16839506C2212yes—
42C61×#493C2211yescharacteristic
43C61×#460C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#331C2212yes—
47C2×#11758224—22no—
48C2×Q8×D61D411yescharacteristic
49#14519796—44no—
50#14519796—44no—
51C2×Dic244—21no—
52C2×Dic244—21no—
53C2×#11758200—22no—
54C2×#11758200—22no—
55C2×#11758200—22no—
56C2×C8×D61—21no—
57C2×C8×D61D411yescharacteristic
58C2×C8×D61D411yescharacteristic
59#14519777C2312yes—
60#14519764—22no—
61#14519748—22no—
62#14519748—22no—
63Q8×Dic61D411yescharacteristic
64C4×Dic122—21no—
65C4×Dic122C2311yescharacteristic
66#14519742—21no—
67#14519742D411yescharacteristic
68D4×Dic61—22no—
69D4×Dic61—22no—
70C4⋊C4×D61—21no—
71C4⋊C4×D61D411yescharacteristic
72C42×D61—21no—
73C42×D61C2311yescharacteristic
74#14519709—22no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78C4⋊#92418C2311yescharacteristic
79#14519706—21no—
80#14519706—21no—
81#14519706—21no—
82#14519702—22no—
83#14519701—22no—
84#14519698—22no—
85#14519689—22no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61—22no—
90C8×Dic61—21no—
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C2×SD16×C61—22no—
94C2×D8×C61—21no—
95C2×D8×C61—21no—
96#127×C61C2312yes—
97C4×Q8×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2312yes—
101#102×C61—22no—
102C4×C8×C61—21no—
103C4×C8×C61C2311yescharacteristic
104C2×Q8×C61C2×D411yescharacteristic
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×C22⋊D61—42no—
109C22×Dic61—42no—
110Q8×D61—22no—
111Q8×D61—22no—
112#11758224—44no—
113#11758224—44no—
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61#4111yescharacteristic
119C2×Dic122—21no—
120C2×Dic122—22no—
121C2×Dic122C2×D411yescharacteristic
122SD16×C61—44no—
123D8×C61—42no—
124D8×C61—42no—
125C2×C8×C61—21no—
126C2×C8×C61—22no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C4⋊C4×C61—21no—
130C4⋊C4×C61C2×D411yescharacteristic
131C42×C61—21no—
132C42×C61C2411yescharacteristic
133C22⋊Dic61—42no—
134C22⋊Dic61—42no—
135#11758200—44no—
136#11758200—44no—
137#11758200—44no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61C2×D411yescharacteristic
142#11758196—21no—
143#11758196—21no—
144#11758196—21no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C4×Dic61C2×D411yescharacteristic
149C4×Dic61C2×D411yescharacteristic
150C4×Dic61#4111yescharacteristic
151C2×#92418—21no—
152C2×#92418—22no—
153C2×#92418C2×D411yescharacteristic
154C2×#92418C2×D411yescharacteristic
155Dic244—42no—
156Dic244—42no—
157C8×D61—22no—
158C8×D61—42no—
159C8×D61SD1614yescomplemented
160C23×C61—42no—
161C22×D61#12111yescharacteristic
162Q8×C61—21no—
163Q8×C61#13512yescomplemented
164D4×C61—42no—
165D4×C61—42no—
166D4×C61—42no—
167D4×C61—42no—
168D4×C61—82no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C22⋊D61—84no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—42no—
178C2×Dic61—84no—
179C2×Dic61#12111yescharacteristic
180C2×Dic61#12111yescharacteristic
181C2×Dic61#12111yescharacteristic
182C4×D61—24no—
183C4×D61—21no—
184C4×D61—21no—
185C4×D61—41no—
186C4×D61—42no—
187C4×D61C2×SD1612yes—
188C4×D61AGL(1,ℤ/8)12yes—
189Dic122—24no—
190Dic122—21no—
191Dic122—42no—
192Dic122—42no—
193Dic122#13512yescomplemented
194C8×C61—21no—
195C8×C61—42no—
196C8×C61—41no—
197C8×C61C2×SD1612yes—
198#92418—21no—
199#92418—42no—
200#92418—41no—
201#92418C2×SD1612yes—
202#19114—611nomaximal, Sylow
203C22×C61—82no—
204C22×C61—82no—
205C22×C61—82no—
206C22×C61C2×#12111yescharacteristic
207C2×D61—21no—
208C2×D61—21no—
209C2×D61#46012yes—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—41no—
213C4×C61C22×SD1611yescharacteristic
214C4×C61C2×#13511yescharacteristic
215C4×C61C2×#13511yescharacteristic
216C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
217Dic61—21no—
218Dic61—22no—
219Dic61—21no—
220Dic61—41no—
221Dic61—82no—
222Dic61#46012yes—
223#3147—611no—
224#3103—612no—
225C2×C8×Q8—611no—
226#1670—611no—
227#1670—611no—
228#1668—611no—
229#1668D6111yescharacteristic, complemented
230#1666—612no—
231#1666—612no—
232#1564—611no—
233#1540—612no—
234C2×C61—162no—
235C2×C61C2×#46011yescharacteristic
236C2×C61#305011yescharacteristic
237C2×C61#306711yescharacteristic
238D61—22no—
239C2×#135—1221no—
240C2×#135—1221no—
241#532—612no—
242C2×C4×Q8—611no—
243#495—1221no—
244#495—1221no—
245#494—611no—
246#493C2×D6111yescharacteristic
247#492—612no—
248#481—1222no—
249#475—612no—
250#475—612no—
251#460—612no—
252#460C2×D6112yes—
253#448—1222no—
254C8×Q8—611no—
255C8×Q8—611no—
256C8×Q8—611no—
257C8×Q8—1221no—
258C8×Q8—1221no—
259C8×Q8C2×D6111yescharacteristic
260C4×SD16—1222no—
261C4×SD16—1222no—
262#431—612no—
263C2×C4⋊C8—611no—
264C2×C4⋊C8—1221no—
265C2×C4×C8—611no—
266C2×C4×C8—1221no—
267#335—611no—
268#335C2×D6111yescharacteristic
269#334—612no—
270#332—612no—
271#332—612no—
272#331—612no—
273#331C2×D6112yes—
274C61#1911411yescharacteristic, complemented, Sylow
275C2×#41—1222no—
276C22×Q8—611no—
277#135—2444no—
278#135—2444no—
279C2×Q16—1221no—
280C2×Q16—1221no—
281C2×SD16—22no—
282C2×SD16—1222no—
283C2×SD16—1222no—
284C2×SD16—1222no—
285C2×D8—21no—
286C2×D8—21no—
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—1221no—
290C4⋊Q8—612no—
291#127C22×D6112yes—
292#124—1222no—
293#121—1222no—
294#121—1222no—
295C4×Q8—611no—
296C4×Q8—611no—
297C4×Q8—611no—
298C4×Q8—1221no—
299C4×Q8—1221no—
300C4×Q8C22×D6111yescharacteristic
301C4×D4—1222no—
302C4×D4—1222no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—1221no—
305C2×C42—611no—
306C2×C42—1221no—
307#106—1222no—
308#106—1222no—
309C4⋊C8—21no—
310C4⋊C8—611no—
311C4⋊C8—612no—
312C4⋊C8—1221no—
313C4⋊C8—1221no—
314C4⋊C8—1221no—
315C4⋊C8—1221no—
316C4⋊C8C22×D6111yescharacteristic
317C4⋊C8C22×D6112yes—
318#103—1222no—
319#103—1222no—
320#102—22no—
321#102—1222no—
322C22⋊C8—1222no—
323C4×C8—21no—
324C4×C8—611no—
325C4×C8—611no—
326C4×C8—611no—
327C4×C8—1221no—
328C4×C8—1222no—
329C4×C8—1221no—
330C4×C8—1221no—
331C4×C8C22×D6111yescharacteristic
332#41—2444no—
333#41—2444no—
334C2×Q8—611no—
335C2×Q8—1221no—
336C2×Q8—1222no—
337C2×Q8—1222no—
338C2×Q8—1222no—
339C2×Q8D4×D6111yescharacteristic
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—2442no—
344C22×C4—611no—
345C22×C4—611no—
346C22×C4—611no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—2442no—
350Q16—2442no—
351Q16—2442no—
352SD16—44no—
353SD16—2444no—
354SD16—2444no—
355SD16—2444no—
356D8—42no—
357D8—42no—
358C2×C8—21no—
359C2×C8—22no—
360C2×C8—611no—
361C2×C8—614no—
362C2×C8—611no—
363C2×C8—1222no—
364C2×C8—1221no—
365C2×C8—1222no—
366C2×C8—2442no—
367C2×C8D4×D6111yescharacteristic
368C2×C8D4×D6111yescharacteristic
369C4⋊C4—21no—
370C4⋊C4—611no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C4⋊C4D4×D6111yescharacteristic
378C22⋊C4—2442no—
379C22⋊C4—2442no—
380C42—21no—
381C42—611no—
382C42—611no—
383C42—611no—
384C42—1221no—
385C42—1221no—
386C42—1221no—
387C42C23×D6111yescharacteristic
388C23—42no—
389C23—611no—
390Q8—21no—
391Q8—612no—
392Q8—1224no—
393Q8—1221no—
394Q8—2442no—
395Q8—2442no—
396Q8#1451979612yescomplemented
397D4—42no—
398D4—42no—
399D4—42no—
400D4—42no—
401D4—82no—
402D4—4884no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—611no—
406C2×C4—612no—
407C2×C4—612no—
408C2×C4—611no—
409C2×C4—611no—
410C2×C4—1224no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—4884no—
416C2×C4—2441no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4C2×D4×D6111yescharacteristic
420C2×C4C2×D4×D6111yescharacteristic
421C2×C4#41×D6111yescharacteristic
422C8—21no—
423C8—42no—
424C8—41no—
425C8—612no—
426C8—1221no—
427C8—2442no—
428C8—2441no—
429C8SD16×D6112yes—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—612no—
434C22—1221no—
435C22—1221no—
436C22#121×D6111yescharacteristic
437C4—22no—
438C4—21no—
439C4—41no—
440C4—612no—
441C4—1221no—
442C4—1222no—
443C4—1221no—
444C4—4882no—
445C4—2441no—
446C4C2×SD16×D6111yescharacteristic
447C4#135×D6111yescharacteristic
448C4C2×#1451979611yescharacteristic
449C4AGL(1,ℤ/8)×D6111yescharacteristic
450C2—162no—
451C2—1222no—
452C2#460×D6111yescharacteristic
453C2#2145539811yescharacteristic
454C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1668C2
socleC22×C61C2×#121
prime radical#21456246C2
hypercenter#1668D61
hypocenterC61#19114
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1668

Complements

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

NH ≅ G/Ntype
2. #21455849C2semidirect
3. C8×Q8×D61C2semidirect
4. #21445190C2semidirect
5. #21445189C2semidirect
6. #21445176C2semidirect
7. #21445166C2semidirect
8. #21445165C2semidirect
9. #21444566C2semidirect
10. #21444465C2semidirect
11. C61×#1668C2semidirect
159. C8×D61SD16semidirect
163. Q8×C61#135semidirect
193. Dic122#135semidirect
229. #1668D61semidirect
274. C61#19114semidirect
396. Q8#14519796semidirect

Maximal subgroups

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

subgroupindex
#214562462
#214558492
C8×Q8×D612
#214451902
#214451892
#214451762
#214451662
#214451652
#214445662
#214444652
C61×#16682
#1911461