Finite Group Databasebeta

← #30605428

Subgroups — #30605428

Order 15616.

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

normal subgroups only

This group has 470 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
1D61⋊(C22⋊C16)C211yesmaximal, characteristic, complemented
2#21461696C211yesmaximal, characteristic, complemented
3#21460806C214yesmaximal, complemented
4C2×#16839999C211yesmaximal, characteristic
5C2×D61⋊C16—21no—
6C2×#14519846—21no—
7#16840704—21no—
8#16840657C2212yescomplemented
9C22⋊#11758187—21no—
10#16840636C2212yescomplemented
11C22×D488C411yescharacteristic, complemented
12C22×C8×D61C2211yescharacteristic
13C2×#14519752—21no—
14#16839999—21no—
15#16839999—21no—
16#16839999C412yescomplemented
17#16839999C2212yes—
18C2×#14519697—21no—
19C2×#107×C61C411yescharacteristic, complemented
20C2×C4⋊#6923—22no—
21C2×C22⋊#6923—41no—
22D61⋊C16—42no—
23C2×#11758187—22no—
24#14519846—42no—
25(C2×#6923)⋊C4—41no—
26C22×D244—21no—
27C22×D244—41no—
28C22×C4×D61D411yescharacteristic
29C2×D488—21no—
30C2×D488—42no—
31C2×D488C2×C412yes—
32C22×#92418D411yescharacteristic
33C2×C8×D61—42no—
34C2×C8×D61C2×C411yescharacteristic
35C2×C8×D61D411yescharacteristic
36C2×C8×D61D411yescharacteristic
37C2×C8×D61C2311yescharacteristic
38#14519752—22no—
39#14519752—22no—
40#14519752—41no—
41#14519752—41no—
42C2×#11758198—41no—
43#14519697—22no—
44#14519697—22no—
45C22×C8×C61C2×C411yescharacteristic
46C2×C4⋊C4×C61—21no—
47#107×C61—21no—
48#107×C61C2×C412yes—
49C23×D61—41no—
50C23×D61D811yescharacteristic
51C22×#6923—42no—
52C22×C4×C61—41no—
53C22×C4×C61C22⋊C411yescharacteristic
54C22×Dic61SD1611yescharacteristic
55C2×D244—22no—
56C2×D244—42no—
57C2×D244—42no—
58C2×D244—42no—
59C2×D244—42no—
60C2×D244—41no—
61C2×D244—82no—
62C2×D244—82no—
63C2×C4×D61—41no—
64C2×C4×D61—41no—
65C2×C4×D61C22⋊C411yescharacteristic
66C2×C4×D61D811yescharacteristic
67C2×C4×D61SD1611yescharacteristic
68C2×C4×D61C2×D411yescharacteristic
69C22⋊#6923—82no—
70C4⋊#6923—42no—
71C4⋊#6923—42no—
72C2×C8×C61—22no—
73C2×C8×C61C22⋊C411yescharacteristic
74C2×C8×C61C22×C411yescharacteristic
75C4⋊C4×C61—22no—
76C4⋊C4×C61—41no—
77#11758198—42no—
78#11758198—42no—
79C2×#92418—41no—
80C2×#92418C22⋊C411yescharacteristic
81C2×#92418C2×D411yescharacteristic
82D488—42no—
83D488—42no—
84C8×D61—22no—
85C8×D61—22no—
86C8×D61—41no—
87C8×D61—41no—
88#11758187—42no—
89C23×C61#10211yescharacteristic
90C22×D61—41no—
91C22×D61—41no—
92C22×D61—42no—
93C22×D61—82no—
94C22×D61—82no—
95C22×D61—82no—
96C22×D61#10211yescharacteristic
97C22×D61D1612yes—
98C22×D61SD3212yes—
99C22×D61C2×D811yescharacteristic
100C2×#6923—82no—
101C2×#6923—82no—
102C2×#6923—82no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—42no—
106C2×C4×C61—82no—
107C2×C4×C61#10211yescharacteristic
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×Dic61—41no—
110C2×Dic61#10211yescharacteristic
111C2×Dic61C2×SD1611yescharacteristic
112D244—42no—
113D244—42no—
114D244—82no—
115D244—84no—
116D244—82no—
117D244—82no—
118C4×D61—21no—
119C4×D61—21no—
120C4×D61—21no—
121C4×D61—21no—
122C4×D61—41no—
123C4×D61—41no—
124C8×C61—21no—
125C8×C61#9912yes—
126#92418—21no—
127#92418—21no—
128#11886—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61#35712yes—
132C22×C61C2×#10211yescharacteristic
133C2×D61—21no—
134C2×D61—21no—
135C2×D61—21no—
136C2×D61—21no—
137C2×D61—42no—
138C2×D61—42no—
139C2×D61—82no—
140C2×D61—82no—
141C2×D61—82no—
142C2×D61—164no—
143C2×D61#35712yes—
144C2×D61C2×D1611yescharacteristic
145C2×D61C2×SD3211yescharacteristic
146#6923—162no—
147C4×C61—21no—
148C4×C61—82no—
149C4×C61C2×#9911yescharacteristic
150C4×C61#42111yescharacteristic
151Dic61—21no—
152Dic61—21no—
153C2×#466—611no—
154C2×C22⋊C16—611no—
155#1808—611no—
156#1347—614no—
157C2×C61—22no—
158C2×C61#151911yescharacteristic
159C2×C61C2×#35711yescharacteristic
160C2×C61#214511yescharacteristic
161D61—21no—
162D61—22no—
163D61—21no—
164D61—162no—
165C22×D8—611no—
166C23×C8—611no—
167C2×#121—1221no—
168C22×C16—1221no—
169#466—612no—
170#466—612no—
171#466—1221no—
172#466—1221no—
173C2×#107—1221no—
174C2×#107#692311yescharacteristic, complemented
175C2×#102—1221no—
176#380—1221no—
177C22⋊C16—612no—
178C22⋊C16—1221no—
179#340—612no—
180C61#1188611yescharacteristic, complemented, Sylow
181C22×D4—1221no—
182C22×D4—2441no—
183C23×C4—611no—
184C2×D8—612no—
185C2×D8—1221no—
186C2×D8—2442no—
187C22×C8—611no—
188C22×C8—611no—
189C22×C8—611no—
190C22×C8—611no—
191C22×C8—611no—
192C22×C8—2442no—
193C22×C8C2×#692311yescharacteristic
194#121—1222no—
195#121—1222no—
196#121—2441no—
197#121—2441no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—1222no—
200C2×C22⋊C4—2441no—
201C2×C22⋊C4—2441no—
202C2×C16—1222no—
203C2×C16—2442no—
204#107—21no—
205#107—2442no—
206#107C2×#692312yes—
207#102—1222no—
208#102—1222no—
209#95—2441no—
210C24—611no—
211C24—2441no—
212C2×D4—1222no—
213C2×D4—4882no—
214C2×D4—4882no—
215C2×D4—2442no—
216C2×D4—2442no—
217C2×D4—2442no—
218C2×D4—2442no—
219C2×D4—2441no—
220C22×C4—41no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—2441no—
227C22×C4—2441no—
228C22×C4—2442no—
229C22×C4C22⋊#692311yescharacteristic
230D8—2442no—
231D8—2442no—
232C2×C8—22no—
233C2×C8—611no—
234C2×C8—611no—
235C2×C8—1222no—
236C2×C8—1222no—
237C2×C8—2441no—
238C2×C8—2441no—
239C2×C8—2441no—
240C2×C8C22⋊#692311yescharacteristic
241C2×C8C22×#692311yescharacteristic
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—2442no—
245C4⋊C4—2442no—
246C22⋊C4—4882no—
247C22⋊C4—2442no—
248C22⋊C4—2442no—
249C16—2442no—
250C23—611no—
251C23—611no—
252C23—612no—
253C23—612no—
254C23—4882no—
255C23—4882no—
256C23—4882no—
257C23—2441no—
258C23—2441no—
259C23—2442no—
260C23#1451983911yescharacteristic
261D4—4882no—
262D4—4882no—
263D4—4884no—
264D4—4882no—
265D4—2442no—
266D4—2442no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—82no—
271C2×C4—611no—
272C2×C4—611no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—4882no—
278C2×C4—4882no—
279C2×C4—4882no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4#1451983911yescharacteristic
284C2×C4C2×C22⋊#692311yescharacteristic
285C8—21no—
286C8—1221no—
287C8—1221no—
288C8#1451983612yes—
289C22—21no—
290C22—21no—
291C22—611no—
292C22—612no—
293C22—611no—
294C22—1221no—
295C22—1221no—
296C22—1221no—
297C22—1221no—
298C22—9764no—
299C22—4882no—
300C22—4882no—
301C22—4882no—
302C22—2442no—
303C22—2442no—
304C22#1684067412yes—
305C22C2×#1451983911yescharacteristic
306C4—21no—
307C4—82no—
308C4—1221no—
309C4—1221no—
310C4—9762no—
311C4C2×#1451983611yescharacteristic
312C4#1684076111yescharacteristic
313C2—22no—
314C2—1221no—
315C2—1222no—
316C2—1221no—
317C2—9762no—
318C2#2146125611yescharacteristic
319C2C2×#1684067411yescharacteristic
320C2#2146214111yescharacteristic
subgroupG/H
centerC22C2×#14519839
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC2×#107×C61C4
socleC22×C61C2×#102
prime radicalC2×#16839999C2
cosocleC2×C8×D61C23
hypercenterC2×#107#6923
hypocenterC61#11886
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. D61⋊(C22⋊C16)C2semidirect
2. #21461696C2semidirect
3. #21460806C2semidirect
8. #16840657C22semidirect
10. #16840636C22semidirect
11. C22×D488C4semidirect
16. #16839999C4semidirect
19. C2×#107×C61C4semidirect
174. C2×#107#6923semidirect
180. C61#11886semidirect

Maximal subgroups

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

subgroupindex
D61⋊(C22⋊C16)2
#214616962
#214608062
C2×#168399992
#1188661