Finite Group Databasebeta

← #29833483

Subgroups — #29833483

Order 15616.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840203C211yesmaximal, characteristic
2#21454735C211yesmaximal, characteristic
3#21454687C211yesmaximal, characteristic, complemented
4#21447713C218yesmaximal, complemented
5#21447593C211yesmaximal, characteristic, complemented
6#21447556C211yesmaximal, characteristic
7#21447423C211yesmaximal, characteristic, complemented
8C2×C61×#362C211yesmaximal, characteristic, complemented
9C2×#14519799C2211yescharacteristic
10#16840528—21no—
11#16840240—21no—
12#16840203C414yescomplemented
13#16840203C2214yes—
14C2×#14519723C412yescomplemented
15C2×#14519720C412yescomplemented
16C2×#14519717C2211yescharacteristic
17#16839883C2214yes—
18#16839878—22no—
19#16839869C2214yescomplemented
20Q16×Dic61—22no—
21#16839800C2211yescharacteristic
22#16839778—21no—
23#16839771—21no—
24#16839717C2211yescharacteristic
25#16839586C2214yescomplemented
26#16839573C2214yes—
27#16839539C2214yescomplemented
28C22×Q16×C61C2211yescharacteristic
29C2×M32×C61C2211yescharacteristic
30C2×#108×C61C2211yescharacteristic
31C61×#362C2214yescomplemented
32C2×#11758222—21no—
33#14519799—21no—
34#14519799—21no—
35#14519799C2×C412yes—
36#14519799C2312yes—
37C2×Dic244C2×C411yescharacteristic
38C2×#11758190—21no—
39C2×D488C2×C411yescharacteristic
40#14519785—24no—
41#14519783—42no—
42C2×#11758194D411yescharacteristic
43M16×D61—42no—
44Q8×Dic61—24no—
45Q8×Dic61—42no—
46#14519736—21no—
47#14519736—42no—
48#14519736D411yescharacteristic
49#14519723C2×C418yes—
50#14519720C2×C418yes—
51#14519717C2×C412yes—
52#14519717C2312yes—
53C2×#11758185C2×C412yes—
54#14519711—21no—
55#14519709C2312yes—
56#14519703—24no—
57#14519701—24no—
58#14519687C2312yes—
59C8×Dic61D412yes—
60C22×Q8×C61—21no—
61C2×Q16×C61—21no—
62C2×Q16×C61—21no—
63C2×Q16×C61—21no—
64C2×Q16×C61C2×C412yes—
65C2×Q16×C61C2×C412yes—
66C2×Q16×C61C2312yes—
67C2×M16×C61—21no—
68C22×C8×C61C2311yescharacteristic
69M32×C61—21no—
70M32×C61C2312yes—
71C2×C16×C61—21no—
72#108×C61—21no—
73#108×C61C2312yes—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61—22no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—41no—
80C22×C4×C61—41no—
81C22×C4×C61C2×D411yescharacteristic
82C2×C22⋊D61—41no—
83#11758222—22no—
84#11758222—22no—
85#11758222—41no—
86#11758222—41no—
87C2×D244—21no—
88C2×C4×D61—41no—
89C2×Dic122—21no—
90Q16×C61—22no—
91Q16×C61—22no—
92Q16×C61—22no—
93Q16×C61C22×C414yes—
94M16×C61—22no—
95M16×C61—41no—
96C2×C8×C61—41no—
97C2×C8×C61C22×C411yescharacteristic
98C2×C8×C61C22×C411yescharacteristic
99C2×C8×C61C2×D411yescharacteristic
100C2×C8×C61#4111yescharacteristic
101C2×C8×C61#4111yescharacteristic
102C2×C8×C61C2411yescharacteristic
103C22⋊Dic61—21no—
104C22⋊Dic61—41no—
105C4⋊Dic61—21no—
106C4⋊Dic61—22no—
107C4⋊Dic61—44no—
108C4×Dic61—44no—
109C4×Dic61—41no—
110C4×Dic61C2×D412yes—
111#11758194—21no—
112#11758194—21no—
113C2×#92418C2×D412yes—
114Dic244—21no—
115Dic244C22×C412yes—
116D488—21no—
117D488C22×C412yes—
118#11758190—22no—
119#11758190—22no—
120#11758189—44no—
121C8×D61—44no—
122C16×C61—22no—
123#11758185C22×C414yes—
124C23×C61AGL(1,ℤ/8)11yescharacteristic
125C22×D61—41no—
126Q8×C61—24no—
127Q8×C61—42no—
128Q8×C61—42no—
129Q8×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61C4×D411yescharacteristic
134C2×C4×C61C4×D411yescharacteristic
135C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C61C22×D411yescharacteristic
139C22⋊D61—42no—
140C22⋊D61—42no—
141C2×Dic61—22no—
142C2×Dic61—41no—
143C2×Dic61—42no—
144D244—22no—
145D244—41no—
146C4×D61—42no—
147C4×D61—42no—
148Dic122—22no—
149Dic122—41no—
150C8×C61—42no—
151C8×C61C4×D411yescharacteristic
152C8×C61C4×D411yescharacteristic
153C8×C61C23×C411yescharacteristic
154C8×C61C2×#4111yescharacteristic
155#92418—22no—
156#33470—611nomaximal, Sylow
157C22×C61—41no—
158C22×C61#44211yescharacteristic
159C22×C61#44211yescharacteristic
160C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
161C2×D61—42no—
162C2×D61—161no—
163C4×C61—44no—
164C4×C61#44211yescharacteristic
165C4×C61#44211yescharacteristic
166C4×C61C2×C4×D411yescharacteristic
167C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
168Dic61—42no—
169Dic61—42no—
170Dic61—82no—
171C2×#510—611no—
172#2957—611no—
173#2950—611no—
174#2180—618no—
175#2155—611no—
176C2×#362D6111yescharacteristic, complemented
177#2145—611no—
178#2110—611no—
179C2×C61—21no—
180C2×C61—21no—
181C2×C61—21no—
182C2×C61C2×#44211yescharacteristic
183D61—82no—
184C2×#135—611no—
185C22×Q16C2×D6111yescharacteristic
186#568—1221no—
187#519—1221no—
188#510—614no—
189#510—614no—
190C2×Q32—612no—
191C2×SD32—612no—
192C2×M32—611no—
193C2×M32C2×D6111yescharacteristic
194#444—614no—
195#443—1222no—
196#441—614no—
197C4×Q16—1222no—
198C2×#108C2×D6111yescharacteristic
199#427—611no—
200#421—1221no—
201#419—1221no—
202#405—611no—
203#365—614no—
204#362C2×D6114yescomplemented
205#360—614no—
206#347—614no—
207C61#3347011yescharacteristic, complemented, Sylow
208C2×#41—1221no—
209C22×Q8—21no—
210#135—612no—
211#135—612no—
212#135—1221no—
213#135—1221no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—611no—
218C2×Q16C4×D6112yes—
219C2×Q16C4×D6112yes—
220C2×Q16C22×D6112yes—
221C2×SD16—1221no—
222C2×D8—611no—
223#131—1224no—
224#131—2442no—
225C2×M16—21no—
226C2×M16—611no—
227C2×M16—2442no—
228C22×C8C22×D6111yescharacteristic
229C4×Q8—1224no—
230C4×Q8—2442no—
231#117—611no—
232#117—1221no—
233#117—2442no—
234Q32—618no—
235SD32—618no—
236M32—21no—
237M32—612no—
238M32—612no—
239M32C22×D6112yes—
240C2×C16—21no—
241C2×C16—612no—
242#108—21no—
243#108C22×D6112yes—
244#107—1221no—
245#106—612no—
246#104—1224no—
247#103—1224no—
248#97—612no—
249C4×C8—612no—
250#41—1222no—
251#41—1222no—
252#41—2441no—
253#41—2441no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—1221no—
261C2×D4—1221no—
262C2×D4—2441no—
263C22×C4—41no—
264C22×C4—2441no—
265C22×C4D4×D6111yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—22no—
269Q16—612no—
270Q16—1221no—
271Q16C2×C4×D6114yes—
272SD16—1222no—
273SD16—1222no—
274D8—612no—
275D8—1221no—
276M16—22no—
277M16—41no—
278M16—1221no—
279M16—1221no—
280M16—2444no—
281C2×C8—41no—
282C2×C8—612no—
283C2×C8—2444no—
284C2×C8C2×C4×D6111yescharacteristic
285C2×C8C2×C4×D6111yescharacteristic
286C2×C8D4×D6111yescharacteristic
287C2×C8#1175822411yescharacteristic
288C2×C8#1175822411yescharacteristic
289C2×C8C23×D6111yescharacteristic
290C4⋊C4—1221no—
291C4⋊C4—1222no—
292C4⋊C4—2444no—
293C22⋊C4—1221no—
294C22⋊C4—2441no—
295C42—612no—
296C42—2444no—
297C42—2441no—
298C16—22no—
299C16—614no—
300C23—2441no—
301C23#1451980411yescharacteristic
302Q8—24no—
303Q8—42no—
304Q8—42no—
305Q8—42no—
306Q8—1222no—
307Q8—2441no—
308D4—1222no—
309D4—2442no—
310D4—2442no—
311D4—2441no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—1222no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2442no—
319C2×C4—2442no—
320C2×C4#1451973711yescharacteristic
321C2×C4#1451973711yescharacteristic
322C2×C4#1451980111yescharacteristic
323C2×C4#1451980111yescharacteristic
324C2×C4#1451980411yescharacteristic
325C2×C4C2×D4×D6111yescharacteristic
326C8—42no—
327C8—1222no—
328C8#1451973711yescharacteristic
329C8#1451973711yescharacteristic
330C8C22×C4×D6111yescharacteristic
331C8C2×#1175822411yescharacteristic
332C22—41no—
333C22—9761no—
334C22—2442no—
335C22#1683987111yescharacteristic
336C22#1683987111yescharacteristic
337C22C2×#1451980411yescharacteristic
338C4—44no—
339C4—4882no—
340C4—2442no—
341C4—2442no—
342C4#1683987111yescharacteristic
343C4#1683987111yescharacteristic
344C4C2×#1451973711yescharacteristic
345C4C2×#1451980111yescharacteristic
346C2—21no—
347C2—21no—
348C2—21no—
349C2—4882no—
350C2C2×#1683987111yescharacteristic
subgroupG/H
centerC2C2×#16839871
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC2×C61×#362C2
socleC2×C61C2×#442
prime radicalC2×#14519799C22
cosocleC2×C8×C61C24
hypercenterC2×#362D61
hypocenterC61#33470
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
3. #21454687C2semidirect
4. #21447713C2semidirect
5. #21447593C2semidirect
7. #21447423C2semidirect
8. C2×C61×#362C2semidirect
12. #16840203C4semidirect
14. C2×#14519723C4semidirect
15. C2×#14519720C4semidirect
19. #16839869C22semidirect
25. #16839586C22semidirect
27. #16839539C22semidirect
31. C61×#362C22semidirect
176. C2×#362D61semidirect
204. #362C2×D61semidirect
207. C61#33470semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
#214547352
#214546872
#214477132
#214475932
#214475562
#214474232
C2×C61×#3622
#3347061