Finite Group Databasebeta

← #30608330

Subgroups — #30608330

Order 15616.

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

normal subgroups only

This group has 408 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
1#21462112C211yesmaximal, characteristic, complemented
2#21462092C211yesmaximal, characteristic, complemented
3#21460948C212yesmaximal, complemented
4#21460946C212yesmaximal, complemented
5#21456264C211yesmaximal, characteristic
6#16840748—21no—
7#16840741—21no—
8#16840673—21no—
9#16840671C2212yes—
10#16840669C2212yes—
11#16840667—21no—
12C2×#14519797—21no—
13C2×#14519797—21no—
14#16840312C2211yescharacteristic
15#16840157—21no—
16#16840154C411yescharacteristic, complemented
17#16840154C2211yescharacteristic
18#16840152—21no—
19#16840149C411yescharacteristic, complemented
20#16840149C2211yescharacteristic
21C2×#14519688C411yescharacteristic, complemented
22C61×#532C411yescharacteristic, complemented
23D61⋊M16—41no—
24#14519856—41no—
25#14519838—22no—
26#14519837—41no—
27#14519836—22no—
28#14519835—41no—
29C2×#11758222D411yescharacteristic
30C2×#11758222D411yescharacteristic
31#14519797—42no—
32#14519797—42no—
33#14519797—42no—
34#14519797—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—
43C22×#92418—21no—
44#14519777D411yescharacteristic
45#14519774D411yescharacteristic
46#14519765C2×C411yescharacteristic
47#14519765C2311yescharacteristic
48#14519751—41no—
49C4×D244—41no—
50#14519688—21no—
51#14519688C2×C411yescharacteristic
52#14519688C2×C411yescharacteristic
53C2×#41×C61—21no—
54C4⋊Q8×C61C2×C411yescharacteristic
55#127×C61C2×C411yescharacteristic
56#124×C61—21no—
57#121×C61—41no—
58C4×D4×C61—41no—
59C2×C42×C61C2×C411yescharacteristic
60#41×C61—41no—
61#41×C61—41no—
62#41×C61—41no—
63#41×C61—41no—
64C2×Q8×C61—21no—
65C2×D4×C61—21no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C22×C4×C61—21no—
69C22×C4×C61—41no—
70C22×C4×C61C22⋊C411yescharacteristic
71C2×C22⋊D61—21no—
72C2×C22⋊D61—21no—
73#11758222—41no—
74#11758222—41no—
75#11758222—41no—
76#11758222—41no—
77C2×D244—41no—
78C2×D244C22⋊C411yescharacteristic
79C2×D244C2×D411yescharacteristic
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×Dic122C22⋊C411yescharacteristic
83C2×Dic122C2×D411yescharacteristic
84C22⋊#6923—42no—
85#11758217—42no—
86C4⋊#6923—81no—
87C4×#6923—81no—
88#11758213—81no—
89D61⋊C8—81no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—42no—
92C42×C61—21no—
93C42×C61C22⋊C411yescharacteristic
94C42×C61C22×C411yescharacteristic
95#11758202—42no—
96#11758202—42no—
97#11758201—42no—
98#11758201—42no—
99#11758200—42no—
100#11758200—42no—
101#11758199—42no—
102#11758199—42no—
103#11758198—41no—
104#11758198—41no—
105C4⋊Dic61—41no—
106C2×#92418—22no—
107C2×#92418—22no—
108C2×#92418—21no—
109C2×#92418—21no—
110C23×C61—41no—
111C23×C61#9911yescharacteristic
112C22×D61—21no—
113C22×D61—21no—
114C2×#6923—82no—
115Q8×C61—41no—
116Q8×C61—41no—
117D4×C61—41no—
118D4×C61—41no—
119D4×C61—82no—
120D4×C61—81no—
121D4×C61—81no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—81no—
128C2×C4×C61—81no—
129C2×C4×C61—81no—
130C2×C4×C61#9911yescharacteristic
131C2×C4×C61C2×C22⋊C411yescharacteristic
132C22⋊D61—81no—
133C22⋊D61—81no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136D244—41no—
137D244—41no—
138D244—82no—
139C4×D61—41no—
140C4×D61—41no—
141C4×D61—41no—
142C4×D61—41no—
143Dic122—41no—
144Dic122—41no—
145#92420—82no—
146#92418—22no—
147#92418—22no—
148#13577—611nomaximal, Sylow
149C22×C61—21no—
150C22×C61—41no—
151C22×C61—82no—
152C22×C61—81no—
153C22×C61C2×#9911yescharacteristic
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—81no—
157C2×D61—81no—
158#6923—162no—
159C4×C61—21no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—81no—
163C4×C61#35311yescharacteristic
164C4×C61#35311yescharacteristic
165Dic61—41no—
166Dic61—41no—
167#3150—611no—
168#2132—611no—
169#2126—611no—
170#1407—612no—
171#1406—612no—
172C2×C61—21no—
173C2×C61—41no—
174C2×C61—161no—
175C2×C61C2×#35311yescharacteristic
176D61—81no—
177D61—81no—
178C2×#135—1221no—
179C2×#135—1221no—
180#532—611no—
181#532#692311yescharacteristic, complemented
182#497—611no—
183#497—611no—
184#497—1221no—
185#496—611no—
186#496—611no—
187#496—1221no—
188#413—1221no—
189#410—1221no—
190C2×#97—611no—
191#356—1221no—
192#355—612no—
193#354—612no—
194#353—1221no—
195C61#1357711yescharacteristic, complemented, Sylow
196C2×#41—21no—
197C2×#41—611no—
198C2×#41—611no—
199#135—2442no—
200#135—2442no—
201#135—2442no—
202#135—2442no—
203C2×Q16—1221no—
204C2×Q16—1221no—
205C2×SD16—1221no—
206C2×SD16—1221no—
207C2×SD16—1221no—
208C2×SD16—1221no—
209C2×D8—1221no—
210C2×D8—1221no—
211C2×M16—2441no—
212C22×C8—1221no—
213C4⋊Q8—611no—
214C4⋊Q8C2×#692311yescharacteristic
215#127—611no—
216#127C2×#692311yescharacteristic
217#124—21no—
218#124—611no—
219#124—611no—
220#121—41no—
221#121—2441no—
222C4×D4—41no—
223C4×D4—2441no—
224#117—2441no—
225C2×C42C2×#692311yescharacteristic
226#101—1222no—
227#100—2441no—
228#99—1222no—
229#99—2441no—
230#97—611no—
231#97—611no—
232#97—1221no—
233#41—41no—
234#41—41no—
235#41—41no—
236#41—41no—
237#41—2441no—
238#41—2441no—
239#41—2441no—
240#41—2441no—
241C2×Q8—21no—
242C2×Q8—611no—
243C2×Q8—611no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—611no—
248C2×D4—611no—
249C2×D4—1221no—
250C2×D4—1221no—
251C2×D4—2441no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4C22⋊#692311yescharacteristic
257Q16—2442no—
258Q16—2442no—
259SD16—2442no—
260SD16—2442no—
261SD16—2442no—
262SD16—2442no—
263D8—2442no—
264D8—2442no—
265M16—4881no—
266M16—2442no—
267C2×C8—1222no—
268C2×C8—1222no—
269C2×C8—1221no—
270C2×C8—1221no—
271C2×C8—4881no—
272C4⋊C4—41no—
273C4⋊C4—4881no—
274C4⋊C4—2441no—
275C22⋊C4—42no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C22⋊C4—2442no—
279C42—21no—
280C42—4881no—
281C42C22⋊#692311yescharacteristic
282C42C22×#692311yescharacteristic
283C23—41no—
284C23—1221no—
285C23—1221no—
286C23#1451983511yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—2441no—
290Q8—2441no—
291D4—41no—
292D4—41no—
293D4—82no—
294D4—81no—
295D4—81no—
296D4—4882no—
297D4—4881no—
298D4—4881no—
299D4—2441no—
300D4—2441no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—1221no—
310C2×C4—1221no—
311C2×C4—4882no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4#1451983511yescharacteristic
317C2×C4C2×C22⋊#692311yescharacteristic
318C8—1222no—
319C8—1222no—
320C8—4882no—
321C22—21no—
322C22—41no—
323C22—82no—
324C22—81no—
325C22—4881no—
326C22—4881no—
327C22—2441no—
328C22—2441no—
329C22C2×#1451983511yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—81no—
334C4—9762no—
335C4—2441no—
336C4—2441no—
337C4#1684066611yescharacteristic
338C4#1684066611yescharacteristic
339C2—21no—
340C2—41no—
341C2—161no—
342C2—4881no—
343C2—4881no—
344C2C2×#1684066611yescharacteristic
subgroupG/H
centerC4#16840666
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#532C4
socleC2×C61C2×#353
prime radical#21456264C2
cosocle#14519765C23
hypercenter#532#6923
hypocenterC61#13577
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #21462112C2semidirect
2. #21462092C2semidirect
3. #21460948C2semidirect
4. #21460946C2semidirect
16. #16840154C4semidirect
19. #16840149C4semidirect
21. C2×#14519688C4semidirect
22. C61×#532C4semidirect
181. #532#6923semidirect
195. C61#13577semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214620922
#214609482
#214609462
#214562642
#1357761