Finite Group Databasebeta

← #29834918

Subgroups — #29834918

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#21455468C211yesmaximal, characteristic
2#21454427C211yesmaximal, characteristic, complemented
3#21448035C218yesmaximal, complemented
4#21447591C211yesmaximal, characteristic, complemented
5C2×#16839576C211yesmaximal, characteristic, complemented
6C2×#16839574C211yesmaximal, characteristic, complemented
7#21447439C211yesmaximal, characteristic, complemented
8C61×#2155C211yesmaximal, characteristic, complemented
9C22×D488C2211yescharacteristic
10#16840528—21no—
11C2×#14519783C2211yescharacteristic
12#16840331—21no—
13C2×#14519717C2211yescharacteristic
14#16840016C2214yes—
15#16839999—22no—
16#16839810—21no—
17#16839809C2214yescomplemented
18C2×#14519713C2211yescharacteristic
19#108×D61—22no—
20#16839766—21no—
21#16839584C2214yescomplemented
22#16839576C2214yes—
23#16839574C2214yes—
24#16839543C2214yescomplemented
25C2×M32×C61C2211yescharacteristic
26C2×#108×C61C2211yescharacteristic
27C61×#427C2211yescharacteristic
28C61×#365C2214yescomplemented
29C2×#11758222D411yescharacteristic
30C22×D244—21no—
31C2×D488—21no—
32C2×D488—21no—
33C2×D488—21no—
34C2×D488—21no—
35C2×D488—21no—
36C2×D488C2312yes—
37#14519785—42no—
38#14519783—21no—
39#14519783—21no—
40#14519783—21no—
41#14519783—21no—
42#14519783—42no—
43C2×#11758194—21no—
44C2×#11758194D411yescharacteristic
45C2×#11758189C2312yes—
46M16×D61—24no—
47C2×C8×D61D412yes—
48#14519774—41no—
49#14519765—41no—
50#14519752—24no—
51#14519747—42no—
52#14519717—21no—
53#14519717C2312yes—
54C2×#11758185—21no—
55#14519713—21no—
56#14519713C2312yes—
57#14519712—24no—
58#14519697—24no—
59C2×M16×C61—21no—
60C22×C8×C61C2311yescharacteristic
61#117×C61—21no—
62M32×C61—21no—
63M32×C61C2312yes—
64C2×C16×C61—21no—
65#108×C61—21no—
66#108×C61C2312yes—
67#107×C61—21no—
68#106×C61C2312yes—
69C23×D61—41no—
70C22×C4×C61C2×D411yescharacteristic
71C2×C22⋊D61SD1612yes—
72#11758222—21no—
73#11758222—21no—
74#11758222—21no—
75#11758222—21no—
76C2×D244—22no—
77C2×D244—41no—
78C2×D244—41no—
79C2×D244—41no—
80C2×D244—41no—
81C2×D244—41no—
82C2×D244—44no—
83C2×D244SD1611yescharacteristic
84C2×C4×D61C2×D412yes—
85C2×Dic122SD1611yescharacteristic
86M16×C61—22no—
87M16×C61—41no—
88C2×C8×C61—41no—
89C2×C8×C61C2×D411yescharacteristic
90C2×C8×C61#4111yescharacteristic
91C2×C8×C61#4111yescharacteristic
92C2×C8×C61#4111yescharacteristic
93C2×C8×C61#4111yescharacteristic
94C2×C8×C61C2411yescharacteristic
95C4⋊C4×C61—22no—
96C22⋊C4×C61—41no—
97C42×C61—41no—
98#11758198—44no—
99#11758194—21no—
100#11758194—21no—
101#11758194—22no—
102#11758194—41no—
103C2×#92418—41no—
104C2×#92418C2×D412yes—
105D488—22no—
106D488—22no—
107D488—41no—
108D488—41no—
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×C61C2×SD1611yescharacteristic
118C22×D61—42no—
119C22×D61—81no—
120C22×D61—81no—
121C22×D61—162no—
122C22×D61C2×SD1612yes—
123C2×C4×C61—42no—
124C2×C4×C61#12311yescharacteristic
125C2×C4×C61#12311yescharacteristic
126C2×C4×C61C2×SD1611yescharacteristic
127C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C61C22×D411yescharacteristic
130C22⋊D61—22no—
131C22⋊D61—22no—
132C2×Dic61C2×SD1612yes—
133D244—21no—
134D244—21no—
135D244—42no—
136D244—42no—
137D244—81no—
138D244—81no—
139D244—164no—
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#48111yescharacteristic
154C22×C61#48111yescharacteristic
155C22×C61C22×SD1611yescharacteristic
156C2×D61—22no—
157C2×D61—42no—
158C2×D61—84no—
159C2×D61—82no—
160C2×D61—162no—
161C2×D61—162no—
162C4×C61—82no—
163C4×C61#48111yescharacteristic
164C4×C61#48111yescharacteristic
165C4×C61C2×#12311yescharacteristic
166C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
167Dic61—22no—
168#3058—611no—
169#2916—611no—
170#2238—618no—
171#2155—611no—
172#2155D6111yescharacteristic, complemented
173C2×#361—611no—
174C2×#361—611no—
175#2115—611no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61C2×#48111yescharacteristic
180D61—42no—
181D61—162no—
182C22×D8—611no—
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#427C2×D6111yescharacteristic
196#418—1221no—
197#365—614no—
198#365C2×D6114yescomplemented
199#361—614no—
200#361—614no—
201#349—614no—
202C61#3365711yescharacteristic, complemented, Sylow
203C2×#41—611no—
204C22×D4—1221no—
205C2×D8—612no—
206C2×D8—1221no—
207C2×D8—1221no—
208C2×D8—1221no—
209C2×D8—1221no—
210C2×D8—1221no—
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—21no—
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—21no—
241#106C22×D6112yes—
242#102—1224no—
243C24—2441no—
244#41—1221no—
245#41—1221no—
246#41—1221no—
247#41—1221no—
248C2×Q8—611no—
249C2×D4—612no—
250C2×D4—611no—
251C2×D4—1222no—
252C2×D4—2444no—
253C2×D4—2441no—
254C2×D4—2441no—
255C2×D4—2441no—
256C2×D4—2441no—
257C2×D4—2441no—
258C22×C4—612no—
259C22×C4D4×D6111yescharacteristic
260D8—1222no—
261D8—1222no—
262D8—2441no—
263D8—2441no—
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#1175822611yescharacteristic
283C2×C8#1175822611yescharacteristic
284C2×C8C23×D6111yescharacteristic
285C4⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—2444no—
288C42—41no—
289C16—22no—
290C16—1222no—
291C23—612no—
292C23—9762no—
293C23—4881no—
294C23—4881no—
295C23—2442no—
296C23SD16×D6111yescharacteristic
297Q8—1221no—
298Q8—1221no—
299D4—1222no—
300D4—1222no—
301D4—1221no—
302D4—1221no—
303D4—9764no—
304D4—4881no—
305D4—4881no—
306D4—2442no—
307D4—2442no—
308C2×C4—42no—
309C2×C4—612no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4#1451976211yescharacteristic
313C2×C4#1451976211yescharacteristic
314C2×C4SD16×D6111yescharacteristic
315C2×C4#1451980411yescharacteristic
316C2×C4#1451980411yescharacteristic
317C2×C4C2×D4×D6111yescharacteristic
318C8—42no—
319C8—1222no—
320C8—2442no—
321C8#1451976211yescharacteristic
322C8#1451976211yescharacteristic
323C8C2×#1175822211yescharacteristic
324C8C2×#1175822611yescharacteristic
325C22—41no—
326C22—1222no—
327C22—9762no—
328C22—9762no—
329C22—4884no—
330C22—4882no—
331C22—2442no—
332C22#1684008211yescharacteristic
333C22#1684008211yescharacteristic
334C22C2×SD16×D6111yescharacteristic
335C4—82no—
336C4—1222no—
337C4#1684008211yescharacteristic
338C4#1684008211yescharacteristic
339C4C2×#1451976211yescharacteristic
340C4C2×#1451980411yescharacteristic
341C2—21no—
342C2—21no—
343C2—21no—
344C2—9762no—
345C2—2442no—
346C2C2×#1684008211yescharacteristic
subgroupG/H
centerC2C2×#16840082
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2155C2
socleC2×C61C2×#481
prime radical#21455468C2
hypercenter#2155D61
hypocenterC61#33657
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
2. #21454427C2semidirect
3. #21448035C2semidirect
4. #21447591C2semidirect
5. C2×#16839576C2semidirect
6. C2×#16839574C2semidirect
7. #21447439C2semidirect
8. C61×#2155C2semidirect
17. #16839809C22semidirect
21. #16839584C22semidirect
24. #16839543C22semidirect
28. C61×#365C22semidirect
172. #2155D61semidirect
198. #365C2×D61semidirect
202. C61#33657semidirect

Maximal subgroups

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

subgroupindex
#214554682
#214544272
#214480352
#214475912
C2×#168395762
C2×#168395742
#214474392
C61×#21552
#3365761