Finite Group Databasebeta

← #29834915

Subgroups — #29834915

Order 15616.

566 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455463C211yesmaximal, characteristic
2#21454427C211yesmaximal, characteristic, complemented
3#21448032C218yesmaximal, complemented
4#21447593C211yesmaximal, characteristic, complemented
5#21447592C211yesmaximal, characteristic, complemented
6C2×#16839575C211yesmaximal, characteristic, complemented
7#21447439C211yesmaximal, characteristic, complemented
8C2×C61×#361C211yesmaximal, characteristic, complemented
9#16840528—21no—
10C2×#14519783C2211yescharacteristic
11#16840329—21no—
12C2×#14519717C2211yescharacteristic
13#16840012C2214yes—
14#16839995—22no—
15#16839810—21no—
16#16839809C2214yescomplemented
17C2×#14519713C2211yescharacteristic
18#108×D61—22no—
19#16839800C2211yescharacteristic
20#16839767—21no—
21#16839586C2214yescomplemented
22#16839585C2214yescomplemented
23#16839575C2214yes—
24#16839543C2214yescomplemented
25C22×D8×C61C2211yescharacteristic
26C2×M32×C61C2211yescharacteristic
27C2×#108×C61C2211yescharacteristic
28C61×#361C2214yes—
29C2×#11758222D411yescharacteristic
30#14519785—42no—
31#14519783—21no—
32#14519783—21no—
33#14519783—21no—
34#14519783—21no—
35#14519783—42no—
36C2×#11758194—21no—
37C2×#11758194D411yescharacteristic
38C2×#11758189C2312yes—
39M16×D61—24no—
40C2×C8×D61D412yes—
41#14519773—41no—
42#14519764—41no—
43#14519748—24no—
44#14519746—42no—
45#14519736—21no—
46#14519717—21no—
47#14519717C2312yes—
48C2×#11758185—21no—
49#14519713—21no—
50#14519713C2312yes—
51#14519712—24no—
52#14519711—21no—
53#14519709C2312yes—
54#14519698—24no—
55C22×D4×C61—21no—
56C2×D8×C61—21no—
57C2×D8×C61—21no—
58C2×D8×C61—21no—
59C2×D8×C61—21no—
60C2×D8×C61—21no—
61C2×D8×C61C2312yes—
62C2×M16×C61—21no—
63C22×C8×C61C2311yescharacteristic
64M32×C61—21no—
65M32×C61C2312yes—
66C2×C16×C61—21no—
67#108×C61—21no—
68#108×C61C2312yes—
69C24×C61—41no—
70C2×D4×C61—22no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—44no—
78C2×C22⋊D61SD1612yes—
79#11758222—21no—
80#11758222—21no—
81#11758222—21no—
82#11758222—21no—
83C2×D244SD1611yescharacteristic
84C2×C4×D61C2×D412yes—
85C2×Dic122SD1611yescharacteristic
86D8×C61—22no—
87D8×C61—22no—
88D8×C61—41no—
89D8×C61—41no—
90M16×C61—22no—
91M16×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61C2×D411yescharacteristic
94C2×C8×C61#4111yescharacteristic
95C2×C8×C61#4111yescharacteristic
96C2×C8×C61#4111yescharacteristic
97C2×C8×C61#4111yescharacteristic
98C2×C8×C61C2411yescharacteristic
99C22⋊Dic61—41no—
100C22⋊Dic61—44no—
101C4⋊Dic61—22no—
102C4×Dic61—41no—
103#11758194—21no—
104#11758194—21no—
105#11758194—22no—
106#11758194—41no—
107C2×#92418—41no—
108C2×#92418C2×D412yes—
109#11758189—22no—
110#11758189—22no—
111#11758189—44no—
112C8×D61—22no—
113C8×D61—22no—
114C8×D61—44no—
115C16×C61—22no—
116#11758185—22no—
117C23×C61—42no—
118C23×C61—81no—
119C23×C61—81no—
120C23×C61—162no—
121C23×C61C2×SD1611yescharacteristic
122C22×D61C2×SD1612yes—
123D4×C61—42no—
124D4×C61—42no—
125D4×C61—81no—
126D4×C61—81no—
127C2×C4×C61#12311yescharacteristic
128C2×C4×C61#12311yescharacteristic
129C2×C4×C61C2×SD1611yescharacteristic
130C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C61C22×D411yescharacteristic
133C22⋊D61—22no—
134C22⋊D61—22no—
135C22⋊D61—164no—
136C2×Dic61—42no—
137C2×Dic61C2×SD1612yes—
138D244—21no—
139D244—21no—
140C4×D61—22no—
141C4×D61—22no—
142Dic122—21no—
143Dic122—21no—
144C8×C61—42no—
145C8×C61#12311yescharacteristic
146C8×C61#12311yescharacteristic
147C8×C61C2×#4111yescharacteristic
148C8×C61C2×#4111yescharacteristic
149#92418—22no—
150#92418—42no—
151#33657—611nomaximal, Sylow
152C22×C61—41no—
153C22×C61—84no—
154C22×C61—82no—
155C22×C61—162no—
156C22×C61—162no—
157C22×C61#48111yescharacteristic
158C22×C61#48111yescharacteristic
159C22×C61C22×SD1611yescharacteristic
160C2×D61—22no—
161C2×D61—42no—
162C4×C61#48111yescharacteristic
163C4×C61#48111yescharacteristic
164C4×C61C2×#12311yescharacteristic
165C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
166Dic61—22no—
167Dic61—82no—
168#3058—611no—
169#2916—611no—
170#2238—618no—
171#2155—611no—
172#2155—611no—
173C2×#361—611no—
174C2×#361D6111yescharacteristic, complemented
175#2115—611no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61—162no—
180C2×C61C2×#48111yescharacteristic
181D61—42no—
182C22×D8C2×D6111yescharacteristic
183#568—1221no—
184C2×#131—611no—
185#535—1221no—
186C2×M32—611no—
187C2×M32C2×D6111yescharacteristic
188#469—614no—
189#466—1222no—
190#430—614no—
191#430—1221no—
192C2×#108—611no—
193C2×#108—1222no—
194C2×#108C2×D6111yescharacteristic
195#427—611no—
196#418—1221no—
197#365—614no—
198#365—614no—
199#361—614no—
200#361C2×D6114yes—
201#349—614no—
202C61#3365711yescharacteristic, complemented, Sylow
203C2×#41—611no—
204C22×D4—21no—
205C2×D8—21no—
206C2×D8—21no—
207C2×D8—21no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8C22×D6112yes—
211#131—1221no—
212#131—1221no—
213#131—1221no—
214#131—1221no—
215#131—2442no—
216#131—2442no—
217C2×M16—21no—
218C2×M16—611no—
219C2×M16—612no—
220C2×M16—1224no—
221C2×M16—1221no—
222C22×C8—612no—
223C22×C8C22×D6111yescharacteristic
224#127—2441no—
225#124—2441no—
226#121—1224no—
227#120—2442no—
228#117—1221no—
229M32—21no—
230M32—612no—
231M32—1221no—
232M32C22×D6112yes—
233C2×C16—21no—
234C2×C16—1221no—
235#108—21no—
236#108—612no—
237#108—1224no—
238#108—1221no—
239#108C22×D6112yes—
240#107—1221no—
241#106—612no—
242#102—1224no—
243C24—41no—
244#41—1221no—
245#41—1221no—
246#41—1221no—
247#41—1221no—
248C2×Q8—611no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—612no—
256C2×D4—611no—
257C2×D4—2444no—
258C22×C4—612no—
259C22×C4D4×D6111yescharacteristic
260D8—22no—
261D8—22no—
262D8—41no—
263D8—41no—
264M16—22no—
265M16—41no—
266M16—1221no—
267M16—1221no—
268M16—1222no—
269M16—1222no—
270M16—1222no—
271M16—2444no—
272M16—2441no—
273C2×C8—41no—
274C2×C8—612no—
275C2×C8—1222no—
276C2×C8—1222no—
277C2×C8—2444no—
278C2×C8—2441no—
279C2×C8#1175822211yescharacteristic
280C2×C8#1175822211yescharacteristic
281C2×C8D4×D6111yescharacteristic
282C2×C8#1175822411yescharacteristic
283C2×C8#1175822411yescharacteristic
284C2×C8C23×D6111yescharacteristic
285C4⋊C4—1222no—
286C22⋊C4—2444no—
287C22⋊C4—2441no—
288C42—2441no—
289C16—22no—
290C16—1222no—
291C23—42no—
292C23—81no—
293C23—81no—
294C23—162no—
295C23—612no—
296C23SD16×D6111yescharacteristic
297Q8—1221no—
298Q8—1221no—
299D4—42no—
300D4—42no—
301D4—81no—
302D4—81no—
303D4—1222no—
304D4—1222no—
305D4—1221no—
306D4—1221no—
307D4—9764no—
308C2×C4—612no—
309C2×C4—1222no—
310C2×C4—1222no—
311C2×C4—2442no—
312C2×C4#1451975911yescharacteristic
313C2×C4#1451975911yescharacteristic
314C2×C4SD16×D6111yescharacteristic
315C2×C4#1451980111yescharacteristic
316C2×C4#1451980111yescharacteristic
317C2×C4C2×D4×D6111yescharacteristic
318C8—42no—
319C8—1222no—
320C8—2442no—
321C8#1451975911yescharacteristic
322C8#1451975911yescharacteristic
323C8C2×#1175822411yescharacteristic
324C8C2×#1175822211yescharacteristic
325C22—41no—
326C22—84no—
327C22—82no—
328C22—162no—
329C22—162no—
330C22—1222no—
331C22—2442no—
332C22#1684007911yescharacteristic
333C22#1684007911yescharacteristic
334C22C2×SD16×D6111yescharacteristic
335C4—1222no—
336C4—4882no—
337C4#1684007911yescharacteristic
338C4#1684007911yescharacteristic
339C4C2×#1451975911yescharacteristic
340C4C2×#1451980111yescharacteristic
341C2—21no—
342C2—21no—
343C2—21no—
344C2—162no—
345C2—2442no—
346C2C2×#1684007911yescharacteristic
subgroupG/H
centerC2C2×#16840079
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#361C2
socleC2×C61C2×#481
prime radical#21455463C2
hypercenterC2×#361D61
hypocenterC61#33657
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#361

Complements

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

NH ≅ G/Ntype
2. #21454427C2semidirect
3. #21448032C2semidirect
4. #21447593C2semidirect
5. #21447592C2semidirect
6. C2×#16839575C2semidirect
7. #21447439C2semidirect
8. C2×C61×#361C2semidirect
16. #16839809C22semidirect
21. #16839586C22semidirect
22. #16839585C22semidirect
24. #16839543C22semidirect
174. C2×#361D61semidirect
202. C61#33657semidirect

Maximal subgroups

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

subgroupindex
#214554632
#214544272
#214480322
#214475932
#214475922
C2×#168395752
#214474392
C2×C61×#3612
#3365761