Finite Group Databasebeta

← #30608334

Subgroups — #30608334

Order 15616.

420 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840671C211yesmaximal, characteristic, complemented
2C2×#16840669C211yesmaximal, characteristic, complemented
3#21460948C212yesmaximal, complemented
4#21460946C212yesmaximal, complemented
5#21456287C211yesmaximal, characteristic
6C2×#14519838—21no—
7C2×#14519836—21no—
8#16840671—21no—
9#16840671C2212yes—
10#16840669—21no—
11#16840669C2212yes—
12C2×#14519807—21no—
13C2×#14519802—21no—
14C2×#14519765C2211yescharacteristic
15#16840154C411yescharacteristic, complemented
16#16840154C2211yescharacteristic
17#16840149C411yescharacteristic, complemented
18#16840149C2211yescharacteristic
19#16840144—21no—
20#16840144—21no—
21C4×#11758194C411yescharacteristic, complemented
22C61×#532C411yescharacteristic, complemented
23C2×#11758217—41no—
24C2×C22⋊#6923—41no—
25#14519838—22no—
26#14519838—41no—
27#14519836—22no—
28#14519836—41no—
29C22×Dic122D411yescharacteristic
30C22×D244D411yescharacteristic
31#14519807—42no—
32#14519807—42no—
33#14519802—42no—
34#14519802—42no—
35C2×#11758202—21no—
36C2×#11758202—21no—
37C2×#11758200—21no—
38C2×#11758200—21no—
39C2×#11758201—21no—
40C2×#11758201—21no—
41C2×#11758199—21no—
42C2×#11758199—21no—
43C2×#11758194—21no—
44#14519765—41no—
45#14519765C2×C411yescharacteristic
46#14519765D411yescharacteristic
47#14519765D411yescharacteristic
48#14519765C2311yescharacteristic
49C2×#11758198—41no—
50#14519688C2×C411yescharacteristic, complemented
51#14519688C2×C411yescharacteristic, complemented
52C4×#92418—21no—
53C2×#41×C61—21no—
54C4⋊Q8×C61C2×C411yescharacteristic, complemented
55#127×C61C2×C411yescharacteristic, complemented
56#124×C61—21no—
57#121×C61—41no—
58C4×D4×C61—41no—
59C2×C42×C61C2×C411yescharacteristic
60C23×D61—21no—
61C22×#6923—81no—
62#41×C61—41no—
63#41×C61—41no—
64#41×C61—41no—
65#41×C61—41no—
66C2×Q8×C61—21no—
67C2×D4×C61—21no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C22×C4×C61—21no—
71C22×C4×C61—41no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×Dic61—21no—
74C2×D244—21no—
75C2×D244—41no—
76C2×D244—41no—
77C2×D244C22⋊C411yescharacteristic
78C2×D244C2×D411yescharacteristic
79C2×Dic122—21no—
80C2×Dic122—41no—
81C2×Dic122—41no—
82C2×Dic122C22⋊C411yescharacteristic
83C2×Dic122C2×D411yescharacteristic
84C22⋊#6923—42no—
85C22⋊#6923—81no—
86#11758217—42no—
87#11758217—81no—
88C2×#92420—81no—
89C4⋊C4×C61—41no—
90C22⋊C4×C61—42no—
91C42×C61—21no—
92C42×C61C22⋊C411yescharacteristic
93C42×C61C22×C411yescharacteristic
94#11758202—42no—
95#11758202—42no—
96#11758201—42no—
97#11758201—42no—
98#11758200—42no—
99#11758200—42no—
100#11758199—42no—
101#11758199—42no—
102#11758198—41no—
103#11758198—41no—
104#11758198—41no—
105#11758198—41no—
106#11758194—22no—
107#11758194—22no—
108C2×#92418—21no—
109C2×#92418—21no—
110C23×C61—41no—
111C23×C61#9911yescharacteristic
112C22×D61—21no—
113C22×D61—21no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—82no—
117C2×#6923—82no—
118C2×#6923—162no—
119Q8×C61—41no—
120Q8×C61—41no—
121D4×C61—41no—
122D4×C61—41no—
123D4×C61—82no—
124D4×C61—81no—
125D4×C61—81no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—41no—
131C2×C4×C61—81no—
132C2×C4×C61—81no—
133C2×C4×C61—81no—
134C2×C4×C61#9911yescharacteristic
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×Dic61—21no—
137C2×Dic61—21no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140D244—41no—
141D244—41no—
142D244—81no—
143Dic122—41no—
144Dic122—41no—
145Dic122—81no—
146#92420—82no—
147#92418—22no—
148#92418—22no—
149#13578—611nomaximal, Sylow
150C22×C61—41no—
151C22×C61—82no—
152C22×C61—81no—
153C22×C61#35311yescharacteristic
154C22×C61#35311yescharacteristic
155C22×C61C2×#9911yescharacteristic
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—82no—
159C2×D61—81no—
160C2×D61—81no—
161C2×D61—161no—
162#6923—162no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—41no—
166C4×C61—41no—
167C4×C61—81no—
168Dic61—41no—
169Dic61—41no—
170#3153—611no—
171C2×#355—611no—
172C2×#354—611no—
173#1407—612no—
174#1406—612no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61—161no—
179C2×C61C2×#35311yescharacteristic
180D61—81no—
181D61—81no—
182C2×#137—1221no—
183C2×AGL(1,ℤ/8)—1221no—
184#532#692311yescharacteristic, complemented
185C2×#124—611no—
186#497—611no—
187#497—611no—
188#496—611no—
189#496—611no—
190#495—1221no—
191#495—1221no—
192C2×#101—1221no—
193C2×#99—1221no—
194C4×M16—611no—
195#355—612no—
196#355—1221no—
197#354—612no—
198#354—1221no—
199C61#1357811yescharacteristic, complemented, Sylow
200C2×#41—21no—
201C22×Q8—611no—
202C22×D4—611no—
203#137—2442no—
204#137—2442no—
205AGL(1,ℤ/8)—2442no—
206AGL(1,ℤ/8)—2442no—
207C2×Q16—1221no—
208C2×Q16—1221no—
209C2×SD16—1221no—
210C2×SD16—1221no—
211C2×SD16—1221no—
212C2×SD16—1221no—
213C2×D8—1221no—
214C2×D8—1221no—
215C2×M16—1221no—
216C2×M16—2441no—
217C4⋊Q8C2×#692311yescharacteristic, complemented
218#127C2×#692311yescharacteristic, complemented
219#124—21no—
220#124—611no—
221#124—611no—
222#124—611no—
223#124—611no—
224#124—2441no—
225#121—41no—
226C4×D4—41no—
227C2×C22⋊C4—2441no—
228C2×C22⋊C4—2441no—
229C2×C42C2×#692311yescharacteristic
230#101—1222no—
231#101—2441no—
232#99—1222no—
233#99—2441no—
234#97—611no—
235#97—611no—
236C4×C8—1221no—
237C24—1221no—
238#41—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—611no—
244C2×Q8—611no—
245C2×Q8—1221no—
246C2×Q8—2441no—
247C2×Q8—2441no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—611no—
252C2×D4—611no—
253C2×D4—1221no—
254C2×D4—2441no—
255C2×D4—2441no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—1221no—
259C22×C4—4881no—
260C22×C4C22⋊#692311yescharacteristic
261Q16—2442no—
262Q16—2442no—
263SD16—2442no—
264SD16—2442no—
265SD16—2442no—
266SD16—2442no—
267D8—2442no—
268D8—2442no—
269M16—1222no—
270M16—1222no—
271M16—4881no—
272M16—2442no—
273C2×C8—1221no—
274C2×C8—1221no—
275C2×C8—4881no—
276C4⋊C4—41no—
277C22⋊C4—42no—
278C22⋊C4—4881no—
279C22⋊C4—2441no—
280C22⋊C4—2441no—
281C22⋊C4—2441no—
282C22⋊C4—2441no—
283C22⋊C4—2442no—
284C42—21no—
285C42C22⋊#692311yescharacteristic
286C42C22×#692311yescharacteristic
287C23—41no—
288C23—1221no—
289C23—1221no—
290C23—4882no—
291C23—2441no—
292C23—2441no—
293C23#1451983511yescharacteristic
294Q8—41no—
295Q8—41no—
296Q8—4881no—
297Q8—2441no—
298Q8—2441no—
299D4—41no—
300D4—41no—
301D4—82no—
302D4—81no—
303D4—81no—
304D4—4881no—
305D4—2441no—
306D4—2441no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—9762no—
318C2×C4—4882no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4#1451983511yescharacteristic
322C2×C4C2×C22⋊#692311yescharacteristic
323C8—1222no—
324C8—1222no—
325C8—4882no—
326C22—41no—
327C22—82no—
328C22—81no—
329C22—9761no—
330C22—4881no—
331C22—4881no—
332C22—4882no—
333C22—2441no—
334C22—2441no—
335C22#1684066611yescharacteristic
336C22#1684066611yescharacteristic
337C22C2×#1451983511yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—9762no—
344C4—2441no—
345C4—2441no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2—4881no—
351C2—4881no—
352C2C2×#1684066611yescharacteristic
subgroupG/H
centerC2C2×#16840666
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#532C4
socleC2×C61C2×#353
prime radical#21456287C2
cosocle#14519765C23
hypercenter#532#6923
hypocenterC61#13578
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#16840671C2semidirect
2. C2×#16840669C2semidirect
3. #21460948C2semidirect
4. #21460946C2semidirect
15. #16840154C4semidirect
17. #16840149C4semidirect
21. C4×#11758194C4semidirect
22. C61×#532C4semidirect
50. #14519688C2×C4semidirect
51. #14519688C2×C4semidirect
54. C4⋊Q8×C61C2×C4semidirect
55. #127×C61C2×C4semidirect
184. #532#6923semidirect
199. C61#13578semidirect
217. C4⋊Q8C2×#6923semidirect
218. #127C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168406712
C2×#168406692
#214609482
#214609462
#214562872
#1357861