Finite Group Databasebeta

← #30599064

Subgroups — #30599064

Order 15616.

388 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461381C211yesmaximal, characteristic, complemented
2#21461379C211yesmaximal, characteristic, complemented
3#21461307C212yesmaximal
4#21460725C212yesmaximal, complemented
5#21454527C211yesmaximal, characteristic
6#16840783C2212yes—
7#16840783C2212yes—
8#16840626—21no—
9#16840625—21no—
10#16840624—21no—
11#16840623—21no—
12#16840618—22no—
13#16840617—22no—
14C2×#14519785C411yescharacteristic
15C2×#14519784C411yescharacteristic
16#16840308C2211yescharacteristic
17#16839837—21no—
18#16839837—21no—
19#16839825—21no—
20#16839825—21no—
21#16839785C2211yescharacteristic
22#16839728C411yescharacteristic
23C2×C8×Dic61C2211yescharacteristic
24C61×#408C411yescharacteristic
25C4⋊#92420—22no—
26C4⋊#92420—22no—
27C22⋊#92420—22no—
28C22⋊#92420—22no—
29#14519832—22no—
30#14519832—22no—
31C2×#11758222C2×C411yescharacteristic
32#14519785—42no—
33#14519784—42no—
34C2×#11758194—21no—
35C2×#11758194C2×C411yescharacteristic
36C2×#11758189—21no—
37C22×#92418—21no—
38C22×#92418C2×C411yescharacteristic
39C2×C8×D61—21no—
40#14519775Q811yescharacteristic
41#14519773Q811yescharacteristic
42#14519766D411yescharacteristic
43#14519764D411yescharacteristic
44#14519749—21no—
45#14519749—21no—
46C4×C22⋊D61—21no—
47C4×C22⋊D61—21no—
48C2×C4×Dic61C2311yescharacteristic
49#14519706—21no—
50#14519706—21no—
51#14519689—21no—
52#14519689—21no—
53C8×Dic61D411yescharacteristic
54C8×Dic61D411yescharacteristic
55C8×Dic61D411yescharacteristic
56C8×Dic61D411yescharacteristic
57C2×#41×C61C2×C411yescharacteristic
58C2×M16×C61C2×C411yescharacteristic
59C22×C8×C61C2×C411yescharacteristic
60C22⋊C8×C61—21no—
61C22⋊C8×C61—21no—
62#41×C61—41no—
63#41×C61—41no—
64C2×Q8×C61C4⋊C411yescharacteristic
65C2×D4×C61—21no—
66C2×D4×C61C4⋊C411yescharacteristic
67C22×C4×C61—21no—
68C22×C4×C61C22×C411yescharacteristic
69C2×C22⋊D61—21no—
70C2×C22⋊D61—21no—
71C2×C22⋊D61—21no—
72C22×Dic61#4111yescharacteristic
73C22×Dic61#4111yescharacteristic
74#11758222—41no—
75#11758222—41no—
76C2×D244C4⋊C411yescharacteristic
77C2×C4×D61—21no—
78C2×Dic122C4⋊C411yescharacteristic
79C2×#92420—42no—
80C2×#92420—42no—
81M16×C61—41no—
82C2×C8×C61—21no—
83C2×C8×C61—41no—
84C2×C8×C61C22⋊C411yescharacteristic
85C2×C8×C61C22⋊C411yescharacteristic
86C22⋊Dic61—21no—
87C22⋊Dic61—21no—
88#11758198—21no—
89#11758198—21no—
90#11758196—21no—
91#11758196—21no—
92C4×Dic61C2×D411yescharacteristic
93C4×Dic61C2×D411yescharacteristic
94C4×Dic61C2×D411yescharacteristic
95C4×Dic61C2×Q811yescharacteristic
96#11758194—42no—
97#11758194—41no—
98C2×#92418—22no—
99C2×#92418—21no—
100C2×#92418—42no—
101C2×#92418C22⋊C411yescharacteristic
102C2×#92418C22⋊C411yescharacteristic
103#11758189—42no—
104C8×D61—42no—
105C23×C61—21no—
106C23×C61#11711yescharacteristic
107C22×D61—21no—
108Q8×C61—41no—
109D4×C61—41no—
110D4×C61—42no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61C2×C22⋊C411yescharacteristic
117C2×C4×C61C2×C4⋊C411yescharacteristic
118C22⋊D61—42no—
119C22⋊D61—42no—
120C22⋊D61—42no—
121C2×Dic61—21no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61#12111yescharacteristic
125C2×Dic61#12111yescharacteristic
126C2×Dic61C22⋊Q811yescharacteristic
127C2×Dic61C22⋊Q811yescharacteristic
128D244—41no—
129C4×D61—41no—
130C4×D61—41no—
131Dic122—41no—
132#92420—82no—
133#92420—82no—
134C8×C61—21no—
135C8×C61—21no—
136C8×C61—41no—
137#92418—21no—
138#92418—41no—
139#92418#10012yes—
140#8634—611nomaximal, Sylow
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—41no—
145C22×C61—41no—
146C22×C61—41no—
147C22×C61#38011yescharacteristic
148C2×D61—41no—
149C2×D61—41no—
150C2×D61—41no—
151C4×C61—21no—
152C4×C61—41no—
153C4×C61C2×#10011yescharacteristic
154C4×C61#41311yescharacteristic
155Dic61—21no—
156Dic61—21no—
157Dic61—21no—
158Dic61—21no—
159Dic61—41no—
160#2931—611no—
161#1568—611no—
162#1568—611no—
163#1536—612no—
164#1307—612no—
165C2×C61—41no—
166C2×C61—81no—
167C2×C61#180011yescharacteristic
168C2×C61#181411yescharacteristic
169C2×C61#181411yescharacteristic
170D61—81no—
171C2×#131—611no—
172C2×#131—611no—
173#532—611no—
174#436—1221no—
175#436—1221no—
176C8×D4—1221no—
177C8×D4—1221no—
178#432—612no—
179#432—612no—
180#423—611no—
181#408—611no—
182#408#692311yescharacteristic
183C2×C4×C8—611no—
184#335—1221no—
185#335—1221no—
186#334—1221no—
187#334—1221no—
188#329—1222no—
189#329—1222no—
190C61#863411yescharacteristic, complemented, Sylow
191C2×#41—611no—
192C2×#41C2×#692311yescharacteristic
193#131—2442no—
194#131—2442no—
195C2×M16—611no—
196C2×M16—1221no—
197C2×M16—1221no—
198C2×M16C2×#692311yescharacteristic
199C22×C8—611no—
200C22×C8—1221no—
201C22×C8—1221no—
202C22×C8C2×#692311yescharacteristic
203C4⋊Q8—611no—
204#127—611no—
205#124—611no—
206#124—611no—
207#121—1221no—
208#121—1221no—
209C4×D4—1221no—
210C4×D4—1221no—
211C2×C42—611no—
212C4⋊C8—1221no—
213C4⋊C8—1221no—
214C4⋊C8—1222no—
215C4⋊C8—1222no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—1221no—
219C22⋊C8—1221no—
220C22⋊C8—1222no—
221C22⋊C8—1222no—
222#97—1222no—
223#97—1222no—
224C4×C8—611no—
225C4×C8—611no—
226C4×C8—611no—
227C4×C8—611no—
228#41—41no—
229#41—41no—
230#41—2441no—
231#41—2441no—
232C2×Q8—611no—
233C2×Q8C4⋊#692311yescharacteristic
234C2×D4—21no—
235C2×D4—611no—
236C2×D4—1221no—
237C2×D4—1221no—
238C2×D4—1221no—
239C2×D4C4⋊#692311yescharacteristic
240C22×C4—21no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—1221no—
244C22×C4C22×#692311yescharacteristic
245M16—41no—
246M16—2442no—
247M16—2442no—
248M16—2441no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—611no—
252C2×C8—611no—
253C2×C8—1222no—
254C2×C8—1221no—
255C2×C8—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8—2442no—
259C2×C8C22⋊#692311yescharacteristic
260C2×C8C22⋊#692311yescharacteristic
261C4⋊C4—1221no—
262C4⋊C4—1221no—
263C22⋊C4—1221no—
264C22⋊C4—1221no—
265C22⋊C4—1221no—
266C22⋊C4—1221no—
267C42—611no—
268C42—611no—
269C42—611no—
270C42—611no—
271C23—21no—
272C23—1221no—
273C23#1451985611yescharacteristic
274Q8—41no—
275Q8—2441no—
276D4—41no—
277D4—42no—
278D4—2441no—
279D4—2442no—
280D4—2442no—
281D4—2442no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—611no—
288C2×C4—611no—
289C2×C4—611no—
290C2×C4—611no—
291C2×C4—1221no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4C2×C22⋊#692311yescharacteristic
297C2×C4C2×C4⋊#692311yescharacteristic
298C8—21no—
299C8—21no—
300C8—41no—
301C8—612no—
302C8—1221no—
303C8—4882no—
304C8—4882no—
305C8—2441no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—2441no—
313C22—2441no—
314C22—2441no—
315C22#1684070411yescharacteristic
316C4—21no—
317C4—41no—
318C4—1221no—
319C4—1221no—
320C4—1221no—
321C4—1221no—
322C4—2441no—
323C4D61⋊#10011yescharacteristic
324C4#1684074911yescharacteristic
325C2—41no—
326C2—81no—
327C2—4881no—
328C2#2146168411yescharacteristic
329C2#2146170511yescharacteristic
330C2#2146170611yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#408C4
socleC22×C61#380
prime radical#16840308C22
cosocleC2×C4×Dic61C23
hypercenter#408#6923
hypocenterC61#8634
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #21461381C2semidirect
2. #21461379C2semidirect
4. #21460725C2semidirect
190. C61#8634semidirect

Maximal subgroups

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

subgroupindex
#214613812
#214613792
#214613072
#214607252
#214545272
#863461