Finite Group Databasebeta

← #29597113

Subgroups — #29597113

Order 15616.

480 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#14519722C211yesmaximal, characteristic
2#21454686C211yesmaximal, characteristic, complemented
3C2×#16839577C211yesmaximal, characteristic, complemented
4#21447534C211yesmaximal, characteristic
5#21446727C211yesmaximal, characteristic
6#21443907C211yesmaximal, characteristic, complemented
7C61×#1382C211yesmaximal, characteristic, complemented
8C22×Dic244C2211yescharacteristic
9C22×D488C2211yescharacteristic
10#16840239—21no—
11C2×#14519722—22no—
12C2×#14519722—22no—
13C2×#14519722C414yescomplemented
14C2×#14519717C2211yescharacteristic
15#16839868—22no—
16#16839856—22no—
17#16839770—21no—
18#16839768—21no—
19C2×#14519693—21no—
20#16839717C2211yescharacteristic
21#16839577—22no—
22#16839562—22no—
23C22×C16×C61C2211yescharacteristic
24C2×#108×C61C2211yescharacteristic
25C61×#427C2211yescharacteristic
26C22×Dic122—21no—
27C22×D244—21no—
28C2×Dic244—21no—
29C2×Dic244—21no—
30C2×Dic244—21no—
31C2×Dic244C2×C412yes—
32C2×Dic244D412yes—
33C2×Dic244D412yes—
34C2×D488—21no—
35C2×D488—21no—
36C2×D488—21no—
37C2×D488—21no—
38C2×D488—21no—
39C2×D488C2×C412yes—
40C2×#11758194D411yescharacteristic
41C4×Dic122—42no—
42#14519742—24no—
43#14519736D411yescharacteristic
44#14519733—42no—
45#14519722—28no—
46#14519717—21no—
47#14519717—21no—
48C2×#11758185—21no—
49#14519700—24no—
50#14519699—42no—
51#14519693—42no—
52#14519687—21no—
53C8×Dic61—21no—
54C2×M16×C61—21no—
55C22×C8×C61C2311yescharacteristic
56#117×C61—21no—
57C2×C16×C61—21no—
58C2×C16×C61—21no—
59C2×C16×C61C2×C412yes—
60#108×C61—21no—
61#108×C61—21no—
62#107×C61D412yes—
63#106×C61—21no—
64C23×D61—41no—
65C22×C4×C61C2×D411yescharacteristic
66C22×Dic61—41no—
67C2×D244—22no—
68C2×D244—41no—
69C2×D244—41no—
70C2×D244—41no—
71C2×D244—41no—
72C2×D244—41no—
73C2×Dic122—22no—
74C2×Dic122—22no—
75C2×Dic122—22no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×Dic122—41no—
79M16×C61—41no—
80M16×C61—41no—
81C2×C8×C61—41no—
82C2×C8×C61C22×C411yescharacteristic
83C2×C8×C61C2×D411yescharacteristic
84C2×C8×C61C2×D411yescharacteristic
85C2×C8×C61C2×D411yescharacteristic
86C2×C8×C61#4111yescharacteristic
87C2×C8×C61#4111yescharacteristic
88C4⋊C4×C61—22no—
89C22⋊C4×C61—41no—
90C42×C61—41no—
91C22⋊Dic61—21no—
92C4⋊Dic61D812yes—
93#11758196—44no—
94C4×Dic61—21no—
95C4×Dic61—44no—
96#11758194—21no—
97#11758194—21no—
98C2×#92418—21no—
99Dic244—22no—
100Dic244—22no—
101Dic244—22no—
102Dic244C22⋊C414yes—
103D488—22no—
104D488—22no—
105D488—41no—
106D488—41no—
107C16×C61—22no—
108#11758185—41no—
109C23×C61AGL(1,ℤ/8)11yescharacteristic
110C22×D61—42no—
111C22×D61—81no—
112C22×D61—81no—
113C22×D61—162no—
114C2×C4×C61—42no—
115C2×C4×C61C4×D411yescharacteristic
116C2×C4×C61#12011yescharacteristic
117C2×C4×C61#12111yescharacteristic
118C2×C4×C61C2×D811yescharacteristic
119C2×C4×C61#13511yescharacteristic
120C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
121C2×Dic61—22no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125D244—42no—
126D244—42no—
127D244—81no—
128D244—81no—
129Dic122—24no—
130Dic122—42no—
131Dic122—42no—
132Dic122—42no—
133C8×C61—81no—
134C8×C61C2×C22⋊C411yescharacteristic
135C8×C61C4×D411yescharacteristic
136C8×C61#12111yescharacteristic
137C8×C61#12311yescharacteristic
138#92418—41no—
139#13018—611nomaximal, Sylow
140C22×C61—22no—
141C22×C61#44211yescharacteristic
142C22×C61#44711yescharacteristic
143C22×C61#46311yescharacteristic
144C2×D61—82no—
145C2×D61—162no—
146C2×D61—162no—
147C2×D61—162no—
148C4×C61—82no—
149C4×C61#38611yescharacteristic
150C4×C61C4×D811yescharacteristic
151C4×C61#44911yescharacteristic
152C4×C61#45911yescharacteristic
153Dic61—44no—
154Dic61—42no—
155C22×SD32—611no—
156#2950—611no—
157C2×#362—611no—
158#2140—611no—
159#1978—611no—
160#1382D6111yescharacteristic, complemented
161#1379—611no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61#187911yescharacteristic
165C2×C61#219312yes—
166D61—162no—
167C22×Q16—611no—
168C22×D8—611no—
169#519—1221no—
170C2×SD32—614no—
171C2×SD32—1222no—
172C2×SD32—1222no—
173C2×M32—611no—
174C22×C16C2×D6111yescharacteristic
175#441—1222no—
176C4×Q16—1222no—
177C2×#108C2×D6111yescharacteristic
178#427C2×D6111yescharacteristic
179#419—1221no—
180#418—1221no—
181C2×#100—1221no—
182#405—611no—
183#362—1222no—
184#357—1222no—
185C61#1301811yescharacteristic, complemented, Sylow
186C22×Q8—1221no—
187C22×D4—1221no—
188C2×Q16—612no—
189C2×Q16—612no—
190C2×Q16—612no—
191C2×Q16—1221no—
192C2×Q16—1221no—
193C2×Q16—1221no—
194C2×D8—612no—
195C2×D8—1221no—
196C2×D8—1221no—
197C2×D8—1221no—
198C2×D8—1221no—
199C2×D8—1221no—
200C2×M16—21no—
201C2×M16—611no—
202C22×C8C22×D6111yescharacteristic
203C4×Q8—1224no—
204C4×Q8—2442no—
205#117—21no—
206#117—611no—
207#117—2442no—
208SD32—1228no—
209M32—1221no—
210M32—1221no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—1221no—
214C2×C16C4×D6112yes—
215#108—21no—
216#108—21no—
217#107D24412yes—
218#106—21no—
219#103—1224no—
220#102—2442no—
221#100—2442no—
222#97—1221no—
223C4×C8—1221no—
224C24—2441no—
225C2×Q8—1222no—
226C2×Q8—1222no—
227C2×Q8—1222no—
228C2×Q8—2441no—
229C2×Q8—2441no—
230C2×Q8—2441no—
231C2×D4—1222no—
232C2×D4—2441no—
233C2×D4—2441no—
234C2×D4—2441no—
235C2×D4—2441no—
236C2×D4—2441no—
237C22×C4—2441no—
238C22×C4D4×D6111yescharacteristic
239Q16—614no—
240Q16—1222no—
241Q16—1222no—
242Q16—1222no—
243D8—1222no—
244D8—1222no—
245D8—2441no—
246D8—2441no—
247M16—41no—
248M16—41no—
249M16—1221no—
250M16—1221no—
251C2×C8—41no—
252C2×C8—1221no—
253C2×C8C2×C4×D6111yescharacteristic
254C2×C8C2×D24411yescharacteristic
255C2×C8#1175822211yescharacteristic
256C2×C8D4×D6111yescharacteristic
257C2×C8D4×D6111yescharacteristic
258C2×C8#1175822611yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—612no—
261C4⋊C4—2444no—
262C22⋊C4—41no—
263C22⋊C4—1221no—
264C42—41no—
265C42—1221no—
266C42—2444no—
267C16—22no—
268C16—2441no—
269C23—9762no—
270C23—4881no—
271C23—4881no—
272C23—2442no—
273C23#1451980411yescharacteristic
274Q8—1224no—
275Q8—2442no—
276Q8—2442no—
277Q8—2442no—
278D4—4881no—
279D4—4881no—
280D4—2442no—
281D4—2442no—
282C2×C4—42no—
283C2×C4—1222no—
284C2×C4—2442no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4#1451973911yescharacteristic
288C2×C4#1451974711yescharacteristic
289C2×C4#1451975011yescharacteristic
290C2×C4D8×D6111yescharacteristic
291C2×C4#1451980011yescharacteristic
292C2×C4#1451980411yescharacteristic
293C8—81no—
294C8—2441no—
295C8C22⋊C4×D6111yescharacteristic
296C8C4×D24411yescharacteristic
297C8#1451975211yescharacteristic
298C8#1451976211yescharacteristic
299C22—22no—
300C22—9762no—
301C22—9762no—
302C22—9762no—
303C22—4882no—
304C22#1683987311yescharacteristic
305C22#1683989711yescharacteristic
306C22#1683998011yescharacteristic
307C4—82no—
308C4—2444no—
309C4—2442no—
310C4#1683964011yescharacteristic
311C4#1683984411yescharacteristic
312C4#1683991011yescharacteristic
313C4#1683995611yescharacteristic
314C2—21no—
315C2—21no—
316C2—9762no—
317C2#2144616811yescharacteristic
318C2#2144778412yes—
subgroupG/H
centerC22#16839897
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC61×#1382C2
socleC22×C61#447
prime radicalC22×D488C22
cosocleC22×C8×C61C23
hypercenter#1382D61
hypocenterC61#13018
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1382

Complements

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

NH ≅ G/Ntype
2. #21454686C2semidirect
3. C2×#16839577C2semidirect
6. #21443907C2semidirect
7. C61×#1382C2semidirect
13. C2×#14519722C4semidirect
160. #1382D61semidirect
185. C61#13018semidirect

Maximal subgroups

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

subgroupindex
C22×#145197222
#214546862
C2×#168395772
#214475342
#214467272
#214439072
C61×#13822
#1301861