Finite Group Databasebeta

← #30605440

Subgroups — #30605440

Order 15616.

416 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 76 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
2#21461696C211yesmaximal, characteristic, complemented
3#21460807C212yesmaximal, complemented
4#21460806C212yesmaximal, complemented
5#21455402C211yesmaximal, characteristic
6C2×#14519851—21no—
7C2×#14519846—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—
25#14519851—42no—
26C2×#11758187—22no—
27#14519846—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—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61#10211yescharacteristic
110C22×D61C2×D811yescharacteristic
111C2×#6923—82no—
112C2×#6923—82no—
113C2×#6923—82no—
114Q8×C61—41no—
115Q8×C61—41no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#10211yescharacteristic
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C22⋊D61—82no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61#10211yescharacteristic
126C2×Dic61C2×SD1611yescharacteristic
127D244—41no—
128D244—41no—
129D244—82no—
130D244—81no—
131D244—81no—
132C4×D61—21no—
133C4×D61—21no—
134C4×D61—21no—
135C4×D61—21no—
136C4×D61—41no—
137C4×D61—41no—
138C4×D61—82no—
139C8×C61—21no—
140C8×C61—21no—
141#92418—21no—
142#92418#9912yes—
143#11890—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—21no—
147C22×C61C2×#10211yescharacteristic
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—21no—
152C2×D61—21no—
153C2×D61—42no—
154C2×D61—42no—
155C2×D61—81no—
156C2×D61—81no—
157C2×D61—81no—
158C2×D61#50911yescharacteristic
159C2×D61#51011yescharacteristic
160#6923—162no—
161C4×C61—21no—
162C4×C61—81no—
163C4×C61—81no—
164C4×C61C2×#9911yescharacteristic
165C4×C61#42111yescharacteristic
166Dic61—21no—
167Dic61—21no—
168Dic61—81no—
169#3050—611no—
170#2112—611no—
171#1808—611no—
172#1348—612no—
173#1347—612no—
174C2×C61—41no—
175C2×C61#151911yescharacteristic
176C2×C61#213911yescharacteristic
177C2×C61#214511yescharacteristic
178D61—21no—
179D61—21no—
180D61—41no—
181D61—161no—
182C2×#135—611no—
183C23×C8—611no—
184#525—1221no—
185C2×M32—1221no—
186#467—611no—
187#467#692311yescharacteristic, complemented
188#466—611no—
189#466—611no—
190#465—1221no—
191#465—1221no—
192#427—611no—
193C2×#107—1221no—
194#416—1221no—
195#380—1221no—
196C22⋊C16—612no—
197C22⋊C16—1221no—
198#340—612no—
199C61#1189011yescharacteristic, complemented, Sylow
200C2×#41—1221no—
201C23×C4—611no—
202#135—2442no—
203C2×Q16C2×#692311yescharacteristic
204C2×SD16—1221no—
205C2×D8—611no—
206C22×C8—611no—
207C22×C8—611no—
208C22×C8—611no—
209C22×C8—611no—
210C22×C8—611no—
211C22×C8—2442no—
212C22×C8C2×#692311yescharacteristic
213#123—2441no—
214C22⋊Q8—21no—
215C22⋊Q8—1221no—
216#121—1221no—
217#121—1221no—
218#120—2441no—
219C4×D4—2441no—
220C4×D4—2441no—
221#117—1221no—
222C2×C4⋊C4—1222no—
223C2×C22⋊C4—2441no—
224M32—2442no—
225C2×C16—1222no—
226#107—611no—
227#107—2442no—
228#107C2×#692311yescharacteristic
229#106—1221no—
230#103—21no—
231#103—1221no—
232#102—1221no—
233#102—1221no—
234#95—2441no—
235C24—611no—
236#41—2442no—
237#41—2442no—
238C2×Q8—21no—
239C2×D4—1221no—
240C2×D4—2441no—
241C2×D4—2441no—
242C2×D4—2441no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—2441no—
249C22×C4—2441no—
250C22×C4—2441no—
251C22×C4—2442no—
252C22×C4C22⋊#692311yescharacteristic
253Q16—41no—
254SD16—2442no—
255D8—2441no—
256C2×C8—41no—
257C2×C8—611no—
258C2×C8—611no—
259C2×C8—1222no—
260C2×C8—1222no—
261C2×C8—1222no—
262C2×C8—2441no—
263C2×C8—2441no—
264C2×C8C22⋊#692311yescharacteristic
265C2×C8C22×#692311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—1221no—
269C4⋊C4—2441no—
270C4⋊C4—2442no—
271C4⋊C4—2442no—
272C22⋊C4—41no—
273C22⋊C4—4882no—
274C22⋊C4—2441no—
275C22⋊C4—2441no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C42—2441no—
279C16—2442no—
280C23—611no—
281C23—611no—
282C23—1221no—
283C23—1221no—
284C23—2441no—
285C23—2441no—
286C23—2441no—
287C23#1451983911yescharacteristic
288Q8—41no—
289Q8—41no—
290D4—4881no—
291D4—4881no—
292D4—4882no—
293D4—4882no—
294D4—2441no—
295D4—2441no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—611no—
301C2×C4—611no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—4882no—
307C2×C4—4882no—
308C2×C4—4882no—
309C2×C4—4882no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4#1451983911yescharacteristic
315C2×C4C2×C22⋊#692311yescharacteristic
316C8—21no—
317C8—21no—
318C8—612no—
319C8—1221no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—611no—
324C22—611no—
325C22—1221no—
326C22—1221no—
327C22—1221no—
328C22—1221no—
329C22—1221no—
330C22—4881no—
331C22—4881no—
332C22—4881no—
333C22—2442no—
334C22—2442no—
335C22C2×#1451983911yescharacteristic
336C4—21no—
337C4—81no—
338C4—81no—
339C4—1221no—
340C4—1221no—
341C4—9762no—
342C4—4881no—
343C4D61⋊#9911yescharacteristic
344C4#1684075911yescharacteristic
345C2—41no—
346C2—1221no—
347C2—1221no—
348C2—9761no—
349C2—2441no—
350C2#2146125411yescharacteristic
351C2#2146212711yescharacteristic
352C2#2146214111yescharacteristic
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#11890
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. #21462048C2semidirect
2. #21461696C2semidirect
3. #21460807C2semidirect
4. #21460806C2semidirect
12. C2×#14519800C4semidirect
17. #16839998C4semidirect
20. #16839799C4semidirect
22. C61×#467C4semidirect
187. #467#6923semidirect
199. C61#11890semidirect

Maximal subgroups

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

subgroupindex
#214620482
#214616962
#214608072
#214608062
#214554022
#1189061