Finite Group Databasebeta

← #30604798

Subgroups — #30604798

Order 15616.

444 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 88 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#21462048C211yesmaximal, characteristic, complemented
2C2×#16840638C211yesmaximal, characteristic, complemented
3#21460812C212yesmaximal, complemented
4#21460810C212yesmaximal, complemented
5#21455401C211yesmaximal, characteristic
6C22×#11758213—21no—
7C2×#14519851—21no—
8#16840657—21no—
9#16840657C2212yes—
10#16840638—21no—
11#16840638C2212yes—
12C2×#14519799C411yescharacteristic
13C22×C8×D61C2211yescharacteristic
14#16840274—21no—
15#16840004C411yescharacteristic
16#16840004C2211yescharacteristic
17#16839999C411yescharacteristic
18#16839999C2211yescharacteristic
19#16839992—21no—
20#16839992—21no—
21#16839756—21no—
22C61×#427C411yescharacteristic
23C2×#11758213—24no—
24C2×#11758213—22no—
25C2×#11758213—42no—
26C2×#11758213—41no—
27C22×#92420—41no—
28#14519851—42no—
29C2×#11758187—22no—
30C2×#11758222—21no—
31C22×C4×D61D411yescharacteristic
32#14519799—42no—
33C2×Dic244C2×C411yescharacteristic
34C2×#11758190—21no—
35C2×D488C2×C411yescharacteristic
36C22×#92418D411yescharacteristic
37C2×C8×D61—42no—
38C2×C8×D61C2×C411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41C2×C8×D61C2311yescharacteristic
42#14519761—41no—
43#14519757—21no—
44#14519757—21no—
45#14519752—21no—
46#14519752—21no—
47#14519747—41no—
48C4×D244—41no—
49C4×D244—41no—
50#14519700—21no—
51#14519700—21no—
52#14519697—21no—
53#14519697—21no—
54C22×C8×C61C2×C411yescharacteristic
55#117×C61—21no—
56#107×C61C2×C411yescharacteristic
57#107×C61C2×C411yescharacteristic
58#106×C61—21no—
59C23×D61SD1611yescharacteristic
60C22×C4×C61C22⋊C411yescharacteristic
61C2×C22⋊D61—41no—
62C22×Dic61D811yescharacteristic
63#11758222—42no—
64#11758222—42no—
65C2×D244—21no—
66C2×D244—41no—
67C2×D244—41no—
68C2×C4×D61—22no—
69C2×C4×D61—22no—
70C2×C4×D61—41no—
71C2×C4×D61C22⋊C411yescharacteristic
72C2×C4×D61D811yescharacteristic
73C2×C4×D61SD1611yescharacteristic
74C2×C4×D61C2×D411yescharacteristic
75C2×Dic122—21no—
76C2×#92420—44no—
77C2×#92420—42no—
78#11758213—24no—
79#11758213—44no—
80#11758213—42no—
81C2×C8×C61—22no—
82C2×C8×C61C22⋊C411yescharacteristic
83C2×C8×C61C22×C411yescharacteristic
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C22⋊C4×C61—41no—
87C42×C61—41no—
88#11758198—41no—
89#11758198—41no—
90#11758198—41no—
91#11758198—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—41no—
94C2×#92418—22no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C2×D411yescharacteristic
97Dic244—41no—
98D488—41no—
99#11758190—42no—
100C8×D61—41no—
101C8×D61—41no—
102C8×D61—41no—
103C8×D61—41no—
104#11758187—42no—
105C23×C61#10211yescharacteristic
106C22×D61—22no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61#10211yescharacteristic
111C22×D61C2×SD1611yescharacteristic
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61#9912yes—
115C2×C4×C61#10211yescharacteristic
116C2×C4×C61#10412yes—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C22⋊D61—82no—
119C2×Dic61—22no—
120C2×Dic61—41no—
121C2×Dic61#10211yescharacteristic
122C2×Dic61C2×D811yescharacteristic
123D244—41no—
124D244—41no—
125D244—82no—
126D244—82no—
127C4×D61—22no—
128C4×D61—22no—
129C4×D61—21no—
130C4×D61—21no—
131C4×D61—21no—
132C4×D61—21no—
133C4×D61—82no—
134Dic122—41no—
135Dic122—41no—
136#92420—44no—
137C8×C61—21no—
138C8×C61—21no—
139#92418—21no—
140#92418—21no—
141#11710—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#32712yes—
145C22×C61C2×#10211yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—44no—
153C2×D61—81no—
154C2×D61—81no—
155C2×D61—81no—
156C4×C61—81no—
157C4×C61—81no—
158C4×C61#32712yes—
159C4×C61C2×#9911yescharacteristic
160C4×C61C2×#10411yescharacteristic
161Dic61—21no—
162Dic61—21no—
163Dic61—81no—
164#3050—611no—
165#2112—611no—
166C2×#341—611no—
167#1350—612no—
168#1349—612no—
169C2×C61—22no—
170C2×C61C2×#32711yescharacteristic
171C2×C61#214311yescharacteristic
172C2×C61#214811yescharacteristic
173D61—41no—
174D61—41no—
175D61—161no—
176C2×#135—611no—
177C22×M16—1221no—
178C23×C8—611no—
179#525—1221no—
180C2×M32—1221no—
181#467—611no—
182#467—611no—
183#466—611no—
184#466—611no—
185#465—1221no—
186#465—1221no—
187#427#692311yescharacteristic
188#416—1221no—
189C22⋊C16—612no—
190C22⋊C16—1221no—
191#341—612no—
192#341—1221no—
193C61#1171011yescharacteristic, complemented, Sylow
194C2×#41—1221no—
195C23×C4—611no—
196#135—2442no—
197C2×Q16—611no—
198C2×SD16—1221no—
199C2×D8—611no—
200C2×M16—1222no—
201C2×M16—1224no—
202C2×M16—2441no—
203C2×M16—2442no—
204C22×C8—611no—
205C22×C8—611no—
206C22×C8—611no—
207C22×C8—611no—
208C22×C8—611no—
209C22×C8—2442no—
210C22×C8—2441no—
211C22×C8C2×#692311yescharacteristic
212#123—2441no—
213C22⋊Q8—1221no—
214C22⋊Q8—1221no—
215#121—1221no—
216#121—1221no—
217#120—2441no—
218C4×D4—2441no—
219C4×D4—2441no—
220#117—21no—
221M32—2442no—
222C2×C16—1222no—
223#107C2×#692311yescharacteristic
224#107C2×#692311yescharacteristic
225#106—21no—
226#103—1221no—
227#103—1221no—
228#102—1221no—
229#102—1221no—
230C24—611no—
231#41—2442no—
232#41—2442no—
233C2×Q8—1221no—
234C2×D4—1221no—
235C2×D4—2441no—
236C2×D4—2441no—
237C2×D4—2441no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—2441no—
246C22×C4C22⋊#692311yescharacteristic
247Q16—2441no—
248SD16—2442no—
249D8—2441no—
250M16—1224no—
251M16—2442no—
252M16—2444no—
253C2×C8—22no—
254C2×C8—611no—
255C2×C8—611no—
256C2×C8—1222no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8—2441no—
260C2×C8—2441no—
261C2×C8—2442no—
262C2×C8—2444no—
263C2×C8C22⋊#692311yescharacteristic
264C2×C8C22×#692311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—2441no—
268C4⋊C4—2441no—
269C22⋊C4—41no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C22⋊C4—2441no—
273C22⋊C4—2441no—
274C42—41no—
275C16—2442no—
276C23—611no—
277C23—611no—
278C23—1222no—
279C23—2441no—
280C23—2441no—
281C23—2441no—
282C23#1451983911yescharacteristic
283Q8—2441no—
284Q8—2441no—
285D4—4882no—
286D4—4882no—
287D4—4882no—
288D4—2441no—
289D4—2441no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—1222no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1222no—
300C2×C4—1222no—
301C2×C4—4882no—
302C2×C4—2441no—
303C2×C4#1451983612yes—
304C2×C4#1451983911yescharacteristic
305C2×C4#1451984212yes—
306C2×C4C2×C22⋊#692311yescharacteristic
307C8—21no—
308C8—21no—
309C8—1221no—
310C8—1221no—
311C8—2444no—
312C22—21no—
313C22—21no—
314C22—1221no—
315C22—1221no—
316C22—1221no—
317C22—1221no—
318C22—1221no—
319C22—1221no—
320C22—4881no—
321C22—4881no—
322C22—4881no—
323C22—2444no—
324C22#1684061412yes—
325C22C2×#1451983911yescharacteristic
326C4—81no—
327C4—81no—
328C4—1221no—
329C4—1221no—
330C4—4881no—
331C4#1684061412yes—
332C4C2×#1451983611yescharacteristic
333C4C2×#1451984211yescharacteristic
334C2—22no—
335C2—9761no—
336C2—2441no—
337C2—2441no—
338C2C2×#1684061411yescharacteristic
339C2#2146213311yescharacteristic
340C2#2146214711yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#427C4
socleC22×C61C2×#102
prime radical#21455401C2
cosocleC2×C8×D61C23
hypercenter#427#6923
hypocenterC61#11710
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #21462048C2semidirect
2. C2×#16840638C2semidirect
3. #21460812C2semidirect
4. #21460810C2semidirect
193. C61#11710semidirect

Maximal subgroups

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

subgroupindex
#214620482
C2×#168406382
#214608122
#214608102
#214554012
#1171061