Finite Group Databasebeta

← #30605424

Subgroups — #30605424

Order 15616.

422 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 80 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×#16840657C211yesmaximal, characteristic, complemented
2#21461695C211yesmaximal, characteristic, complemented
3#21460807C212yesmaximal, complemented
4#21460806C212yesmaximal, complemented
5#21455402C211yesmaximal, characteristic
6C22×#11758187—21no—
7C2×#14519845—21no—
8#16840704—21no—
9#16840657—21no—
10#16840657C2212yes—
11#16840636C2212yes—
12C2×#14519800C411yescharacteristic, complemented
13C22×C8×D61C2211yescharacteristic
14#16840275—21no—
15#16840000C2211yescharacteristic
16#16839999C2211yescharacteristic
17#16839998C411yescharacteristic, complemented
18#16839994—21no—
19#16839993—21no—
20#16839799C411yescharacteristic, complemented
21#16839757—21no—
22C61×#467C411yescharacteristic, complemented
23C2×C4⋊#6923—22no—
24C2×C22⋊#6923—41no—
25C2×#11758187—22no—
26C2×#11758187—42no—
27#14519845—42no—
28(C2×#6923)⋊C4—41no—
29C2×#11758226—21no—
30C22×C4×D61D411yescharacteristic
31#14519800—42no—
32C2×#11758201—21no—
33C2×D488C2×C411yescharacteristic
34C22×#92418D411yescharacteristic
35C2×C8×D61—42no—
36C2×C8×D61C2×C411yescharacteristic
37C2×C8×D61D411yescharacteristic
38C2×C8×D61D411yescharacteristic
39C2×C8×D61C2311yescharacteristic
40#14519762—41no—
41#14519753—21no—
42#14519752—21no—
43#14519751—21no—
44#14519747—41no—
45#14519739—41no—
46#14519737—41no—
47#14519735—21no—
48#14519711C2×C411yescharacteristic
49#14519708—21no—
50#14519701—21no—
51#14519699—21no—
52#14519697—21no—
53C2×Q16×C61C2×C411yescharacteristic
54C22×C8×C61C2×C411yescharacteristic
55C22⋊Q8×C61—21no—
56#107×C61C2×C411yescharacteristic
57#103×C61—21no—
58C23×D61D811yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—21no—
61C22×C4×C61C22⋊C411yescharacteristic
62C2×C22⋊D61—41no—
63C22×Dic61SD1611yescharacteristic
64#11758226—42no—
65#11758226—42no—
66C2×D244—21no—
67C2×D244—41no—
68C2×D244—41no—
69C2×C4×D61—41no—
70C2×C4×D61—41no—
71C2×C4×D61—41no—
72C2×C4×D61C22⋊C411yescharacteristic
73C2×C4×D61D811yescharacteristic
74C2×C4×D61SD1611yescharacteristic
75C2×C4×D61C2×D411yescharacteristic
76C22⋊#6923—82no—
77C4⋊#6923—42no—
78C4⋊#6923—42no—
79Q16×C61—41no—
80C2×C8×C61—41no—
81C2×C8×C61C22⋊C411yescharacteristic
82C2×C8×C61C22×C411yescharacteristic
83C4⋊C4×C61—21no—
84C4⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86#11758201—42no—
87#11758198—41no—
88#11758198—41no—
89#11758198—41no—
90#11758198—41no—
91C4⋊Dic61—21no—
92#11758196—41no—
93C4×Dic61—41no—
94C2×#92418—22no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C2×D411yescharacteristic
97D488—41no—
98C8×D61—22no—
99C8×D61—22no—
100C8×D61—41no—
101C8×D61—41no—
102#11758187—42no—
103C23×C61#10211yescharacteristic
104C22×D61—22no—
105C22×D61—22no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61#10211yescharacteristic
109C22×D61C2×D811yescharacteristic
110C2×#6923—82no—
111C2×#6923—82no—
112C2×#6923—82no—
113Q8×C61—41no—
114Q8×C61—41no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61#10211yescharacteristic
120C2×C4×C61C2×C22⋊C411yescharacteristic
121C22⋊D61—82no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61#10211yescharacteristic
125C2×Dic61C2×SD1611yescharacteristic
126D244—41no—
127D244—41no—
128D244—82no—
129D244—82no—
130C4×D61—21no—
131C4×D61—21no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—41no—
135C4×D61—41no—
136C4×D61—82no—
137C8×C61—21no—
138C8×C61—21no—
139#92418—21no—
140#92418#9912yes—
141#11885—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#36012yes—
145C22×C61C2×#10211yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—44no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—81no—
155C2×D61#50812yes—
156#6923—162no—
157C4×C61—21no—
158C4×C61—81no—
159C4×C61—81no—
160C4×C61C2×#9911yescharacteristic
161C4×C61#42111yescharacteristic
162Dic61—21no—
163Dic61—21no—
164Dic61—81no—
165#3050—611no—
166C2×C22⋊C16—611no—
167#1807—611no—
168#1348—612no—
169#1347—612no—
170C2×C61—22no—
171C2×C61#151911yescharacteristic
172C2×C61#213811yescharacteristic
173C2×C61C2×#36011yescharacteristic
174D61—22no—
175D61—41no—
176D61—161no—
177C2×#135—611no—
178C23×C8—611no—
179#525—1221no—
180C22×C16—1221no—
181#467—611no—
182#467#692311yescharacteristic, complemented
183#466—611no—
184#466—611no—
185#465—1221no—
186#465—1221no—
187#427—611no—
188C2×#106—1221no—
189#416—1221no—
190#380—1221no—
191C22⋊C16—612no—
192C22⋊C16—1221no—
193#340—612no—
194C61#1188511yescharacteristic, complemented, Sylow
195C2×#41—1221no—
196C23×C4—611no—
197#135—2442no—
198C2×Q16C2×#692311yescharacteristic
199C2×SD16—1221no—
200C2×D8—611no—
201C22×C8—611no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—611no—
205C22×C8—611no—
206C22×C8—2442no—
207C22×C8C2×#692311yescharacteristic
208#123—2441no—
209C22⋊Q8—21no—
210C22⋊Q8—1221no—
211#121—1221no—
212#121—1221no—
213#120—2441no—
214C4×D4—2441no—
215C4×D4—2441no—
216#117—1221no—
217C2×C4⋊C4—1222no—
218C2×C22⋊C4—2441no—
219C2×C16—1222no—
220C2×C16—2442no—
221#107—611no—
222#107C2×#692311yescharacteristic
223#106—1221no—
224#106—2442no—
225#103—21no—
226#103—1221no—
227#102—1221no—
228#102—1221no—
229#95—2441no—
230C24—611no—
231#41—2442no—
232#41—2442no—
233C2×Q8—21no—
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—2441no—
244C22×C4—2441no—
245C22×C4—2441no—
246C22×C4—2442no—
247C22×C4C22⋊#692311yescharacteristic
248Q16—41no—
249SD16—2442no—
250D8—2441no—
251C2×C8—41no—
252C2×C8—611no—
253C2×C8—611no—
254C2×C8—1222no—
255C2×C8—1222no—
256C2×C8—1222no—
257C2×C8—2441no—
258C2×C8—2441no—
259C2×C8C22⋊#692311yescharacteristic
260C2×C8C22×#692311yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—1221no—
264C4⋊C4—2441no—
265C4⋊C4—2442no—
266C4⋊C4—2442no—
267C22⋊C4—41no—
268C22⋊C4—4882no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C22⋊C4—2441no—
273C42—2441no—
274C16—2442no—
275C23—611no—
276C23—611no—
277C23—1222no—
278C23—1222no—
279C23—2441no—
280C23—2441no—
281C23#1451983911yescharacteristic
282Q8—41no—
283Q8—41no—
284D4—4882no—
285D4—4882no—
286D4—4882no—
287D4—2441no—
288D4—2441no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—4882no—
300C2×C4—4882no—
301C2×C4—4882no—
302C2×C4—4882no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4#1451983911yescharacteristic
308C2×C4C2×C22⋊#692311yescharacteristic
309C8—21no—
310C8—21no—
311C8—612no—
312C8—1221no—
313C22—21no—
314C22—21no—
315C22—612no—
316C22—1221no—
317C22—1221no—
318C22—1221no—
319C22—1221no—
320C22—1221no—
321C22—4881no—
322C22—4881no—
323C22—4881no—
324C22—2444no—
325C22#1684067912yes—
326C22C2×#1451983911yescharacteristic
327C4—21no—
328C4—81no—
329C4—81no—
330C4—1221no—
331C4—1221no—
332C4—9762no—
333C4—4881no—
334C4D61⋊#9911yescharacteristic
335C4#1684075911yescharacteristic
336C2—22no—
337C2—1222no—
338C2—9761no—
339C2—2441no—
340C2#2146125411yescharacteristic
341C2#2146212611yescharacteristic
342C2C2×#1684067911yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#467C4
socleC22×C61C2×#102
prime radical#21455402C2
cosocleC2×C8×D61C23
hypercenter#467#6923
hypocenterC61#11885
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#16840657C2semidirect
2. #21461695C2semidirect
3. #21460807C2semidirect
4. #21460806C2semidirect
12. C2×#14519800C4semidirect
17. #16839998C4semidirect
20. #16839799C4semidirect
22. C61×#467C4semidirect
182. #467#6923semidirect
194. C61#11885semidirect

Maximal subgroups

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

subgroupindex
C2×#168406572
#214616952
#214608072
#214608062
#214554022
#1188561