Finite Group Databasebeta

← #29662356

Subgroups — #29662356

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#21456248C211yesmaximal, characteristic
2#21455852C212yesmaximal, complemented
3C8×Q8×D61C211yesmaximal, characteristic, complemented
4#21445186C211yesmaximal, characteristic, complemented
5#21445180C211yesmaximal, characteristic, complemented
6#21445179C211yesmaximal, characteristic, complemented
7#21445170C212yesmaximal, complemented
8#21445169C212yesmaximal, complemented
9#21444566C211yesmaximal, characteristic, complemented
10#21444467C212yesmaximal, complemented
11C61×#1670C211yesmaximal, characteristic, complemented
12C2×#14519800—21no—
13C2×#14519800—21no—
14#16840313C2212yes—
15C4×Q8×D61C2211yescharacteristic
16#16840147—21no—
17#16840147—21no—
18#16840139C2211yescharacteristic
19#16840136C2212yes—
20#16840082—22no—
21#16840044C2212yes—
22#16839963C2212yes—
23#16839962C2212yes—
24#16839902—22no—
25Q8×#92418C2211yescharacteristic
26C8×Dic122—21no—
27C8×Dic122C2211yescharacteristic
28#16839886—21no—
29#16839886C2211yescharacteristic
30#16839849—22no—
31#16839847—22no—
32#16839814C2212yes—
33C4⋊C8×D61—21no—
34C4⋊C8×D61C2211yescharacteristic
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839512C2212yes—
39#16839508C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C61×#494C2211yescharacteristic
43C61×#475C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758226—22no—
48C2×Q8×D61D411yescharacteristic
49#14519800—44no—
50#14519800—44no—
51C2×#11758190—22no—
52C2×#11758201—22no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×D488—21no—
56C2×D488—21no—
57C2×C8×D61—21no—
58C2×C8×D61D411yescharacteristic
59C2×C8×D61D411yescharacteristic
60#14519774C2312yes—
61#14519766—22no—
62#14519752—22no—
63#14519752—22no—
64Q8×Dic61D411yescharacteristic
65C4×Dic122—21no—
66C4×Dic122C2311yescharacteristic
67#14519742—21no—
68#14519742D411yescharacteristic
69#14519739—22no—
70#14519739—22no—
71C4⋊C4×D61—21no—
72C4⋊C4×D61D411yescharacteristic
73C42×D61—21no—
74C42×D61C2311yescharacteristic
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78C4⋊#92418C2311yescharacteristic
79#14519706—21no—
80#14519706—21no—
81#14519706—21no—
82#14519700—22no—
83#14519699—22no—
84#14519697—22no—
85#14519689—22no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61—22no—
90C8×Dic61—21no—
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C2×Q16×C61—21no—
94C2×Q16×C61—21no—
95C4⋊Q8×C61C2312yes—
96C4×Q8×C61C2311yescharacteristic
97#106×C61—22no—
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2312yes—
100C4⋊C8×C61C2311yescharacteristic
101#103×C61—22no—
102C4×C8×C61—21no—
103C4×C8×C61C2311yescharacteristic
104C2×Q8×C61—22no—
105C2×Q8×C61C2×D411yescharacteristic
106#11758226—44no—
107#11758226—44no—
108Q8×D61—22no—
109Q8×D61—22no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—42no—
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—42no—
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61C2×D411yescharacteristic
119C2×C4×D61#4111yescharacteristic
120C2×Dic122—21no—
121C2×Dic122C2×D411yescharacteristic
122Q16×C61—42no—
123Q16×C61—42no—
124C2×C8×C61—21no—
125C2×C8×C61—22no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61C2×D411yescharacteristic
132C42×C61—21no—
133C42×C61C2411yescharacteristic
134#11758201—44no—
135#11758201—44no—
136#11758201—44no—
137#11758198—42no—
138#11758198—42no—
139C4⋊Dic61—21no—
140C4⋊Dic61C2×D411yescharacteristic
141#11758196—21no—
142#11758196—21no—
143#11758196—21no—
144C4×Dic61—21no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C4×Dic61C2×D411yescharacteristic
148C4×Dic61C2×D411yescharacteristic
149C4×Dic61#4111yescharacteristic
150C2×#92418—21no—
151C2×#92418—22no—
152C2×#92418C2×D411yescharacteristic
153C2×#92418C2×D411yescharacteristic
154D488—42no—
155D488—42no—
156#11758190—44no—
157C8×D61—22no—
158C8×D61—42no—
159C8×D61SD1614yescomplemented
160C22×D61—42no—
161C22×D61#12111yescharacteristic
162Q8×C61—21no—
163Q8×C61—42no—
164Q8×C61—42no—
165Q8×C61#13512yescomplemented
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—42no—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61#12111yescharacteristic
175C2×Dic61#12111yescharacteristic
176C2×Dic61#12111yescharacteristic
177D244—42no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—82no—
182D244—84no—
183C4×D61—24no—
184C4×D61—21no—
185C4×D61—21no—
186C4×D61—41no—
187C4×D61—42no—
188C4×D61—84no—
189C4×D61C2×SD1612yes—
190C4×D61AGL(1,ℤ/8)12yes—
191Dic122—24no—
192Dic122—21no—
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×C61C2×#12111yescharacteristic
204C2×D61—21no—
205C2×D61—21no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—82no—
209C2×D61#46012yes—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—41no—
213C4×C61—82no—
214C4×C61C22×SD1611yescharacteristic
215C4×C61C2×#13511yescharacteristic
216C4×C61C2×#13511yescharacteristic
217C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
218Dic61—21no—
219Dic61—22no—
220Dic61—21no—
221Dic61—41no—
222Dic61#46012yes—
223#3147—611no—
224#3103—612no—
225C2×C8×Q8—611no—
226#1670—611no—
227#1670D6111yescharacteristic, complemented
228#1668—611no—
229#1668—611no—
230#1666—612no—
231#1666—612no—
232#1564—611no—
233#1540—612no—
234C2×C61C2×#46011yescharacteristic
235C2×C61#305011yescharacteristic
236C2×C61#306711yescharacteristic
237D61—22no—
238D61—162no—
239C2×#135—1221no—
240C2×#135—1221no—
241#532—612no—
242C2×C4×Q8—611no—
243#495—1221no—
244#495—1221no—
245#494C2×D6111yescharacteristic
246#493—611no—
247#492—612no—
248#481—1222no—
249#475—612no—
250#475C2×D6112yes—
251#460—612no—
252#460—612no—
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#332C2×D6112yes—
272#331—612no—
273#331—612no—
274C61#1911411yescharacteristic, complemented, Sylow
275C2×#41—1222no—
276C22×Q8—611no—
277#135—2444no—
278#135—2444no—
279C2×Q16—21no—
280C2×Q16—21no—
281C2×SD16—1222no—
282C2×SD16—1222no—
283C2×SD16—1222no—
284C2×SD16—1222no—
285C2×D8—1221no—
286C2×D8—1221no—
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—1221no—
290C4⋊Q8C22×D6112yes—
291#127—612no—
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—22no—
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—22no—
319#103—1222no—
320#102—1222no—
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—22no—
335C2×Q8—611no—
336C2×Q8—1221no—
337C2×Q8—1222no—
338C2×Q8—1222no—
339C2×Q8D4×D6111yescharacteristic
340C2×D4—1222no—
341C2×D4—1222no—
342C2×D4—1222no—
343C2×D4—2442no—
344C22×C4—611no—
345C22×C4—611no—
346C22×C4—611no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—2442no—
350Q16—42no—
351Q16—42no—
352SD16—2444no—
353SD16—2444no—
354SD16—2444no—
355SD16—2444no—
356D8—2442no—
357D8—2442no—
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—22no—
371C4⋊C4—22no—
372C4⋊C4—611no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
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—611no—
389C23—2442no—
390Q8—21no—
391Q8—42no—
392Q8—42no—
393Q8—612no—
394Q8—1224no—
395Q8—1221no—
396Q8#1451980012yescomplemented
397D4—4882no—
398D4—4884no—
399D4—2442no—
400D4—2442no—
401D4—2442no—
402D4—2442no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—611no—
407C2×C4—612no—
408C2×C4—612no—
409C2×C4—611no—
410C2×C4—611no—
411C2×C4—1224no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—4884no—
417C2×C4—2441no—
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—612no—
431C22—1221no—
432C22—1221no—
433C22—4882no—
434C22—4882no—
435C22—4882no—
436C22#121×D6111yescharacteristic
437C4—22no—
438C4—21no—
439C4—41no—
440C4—82no—
441C4—612no—
442C4—1221no—
443C4—1222no—
444C4—1221no—
445C4—2441no—
446C4C2×SD16×D6111yescharacteristic
447C4#135×D6111yescharacteristic
448C4C2×#1451980011yescharacteristic
449C4AGL(1,ℤ/8)×D6111yescharacteristic
450C2—1222no—
451C2—9762no—
452C2#460×D6111yescharacteristic
453C2#2145540211yescharacteristic
454C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1670C2
socleC22×C61C2×#121
prime radical#21456248C2
hypercenter#1670D61
hypocenterC61#19114
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1670

Complements

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

NH ≅ G/Ntype
2. #21455852C2semidirect
3. C8×Q8×D61C2semidirect
4. #21445186C2semidirect
5. #21445180C2semidirect
6. #21445179C2semidirect
7. #21445170C2semidirect
8. #21445169C2semidirect
9. #21444566C2semidirect
10. #21444467C2semidirect
11. C61×#1670C2semidirect
159. C8×D61SD16semidirect
165. Q8×C61#135semidirect
193. Dic122#135semidirect
227. #1670D61semidirect
274. C61#19114semidirect
396. Q8#14519800semidirect

Maximal subgroups

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

subgroupindex
#214562482
#214558522
C8×Q8×D612
#214451862
#214451802
#214451792
#214451702
#214451692
#214445662
#214444672
C61×#16702
#1911461