Finite Group Databasebeta

← C22×#17986038 #32861104

Subgroups — C22×#17986038 #32861104

Order 18176.

3182 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11765848C211yesmaximal, characteristic
2C22×#14556086C211yesmaximal, characteristic
3C22×C4⋊C4×D71C211yesmaximal, characteristic
4C2×#17986038C2124yesmaximal, direct product factor
5C22×#14556062C211yesmaximal, characteristic, complemented
6C22×#14556061C211yesmaximal, characteristic, complemented
7C22×#14556059C211yesmaximal, characteristic, complemented
8C22×#106×C71C211yesmaximal, characteristic, complemented
9C23×C4×D71—21no—
10C23×C4×D71C2211yescharacteristic
11C22×#11765848C4116yescomplemented
12C22×#11765848C22112yes—
13C23×C71⋊C8C2211yescharacteristic
14C2×#14556086—26no—
15C2×#14556086C22112yes—
16C22×C4⋊Dic71C2211yescharacteristic
17C22×C4⋊Dic71C2211yescharacteristic
18C22×Dic71⋊C4—21no—
19C2×C4⋊C4×D71—26no—
20C2×C4⋊C4×D71C22112yes—
21C22×(C2×D71)⋊C4—21no—
22C22×C4×Dic71—21no—
23#17986038C22164yesdirect product factor
24C2×#14556062C22112yescomplemented
25C2×#14556061C22112yescomplemented
26C2×#14556059C22112yescomplemented
27C23×C8×C71C2211yescharacteristic
28C22×C4⋊C4×C71C2211yescharacteristic
29C22×C4⋊C4×C71C2211yescharacteristic
30C2×#106×C71C22112yescomplemented
31C24×D71D411yescharacteristic
32C23×Dic71—21no—
33C23×Dic71—21no—
34C23×Dic71D411yescharacteristic
35C22×C4×D71—212no—
36C22×C4×D71—48no—
37C22×C4×D71C2×C418yes—
38C22×C4×D71D416yes—
39C22×C4×D71Q818yes—
40C22×C4×D71C2316yes—
41C2×#11765848C2×C4196yescomplemented
42C2×#11765848C23116yes—
43C22×C71⋊C8C2×C418yescomplemented
44C22×C71⋊C8C2316yes—
45#14556086—224no—
46#14556086C23116yes—
47C2×C4⋊Dic71—23no—
48C2×C4⋊Dic71—23no—
49C2×C4⋊Dic71C2316yes—
50C2×C4⋊Dic71C2316yes—
51C2×Dic71⋊C4—212no—
52C4⋊C4×D71—224no—
53C4⋊C4×D71C23116yes—
54C2×(C2×D71)⋊C4—212no—
55C2×C4×Dic71—212no—
56#14556062C23116yescomplemented
57#14556061C23116yescomplemented
58#14556059C23116yescomplemented
59C23×C4×C71—21no—
60C23×C4×C71—21no—
61C23×C4×C71C2311yescharacteristic
62C22×C8×C71C2×C418yescomplemented
63C22×C8×C71C2316yes—
64C2×C4⋊C4×C71—23no—
65C2×C4⋊C4×C71—23no—
66C2×C4⋊C4×C71C2316yes—
67C2×C4⋊C4×C71C2316yes—
68#106×C71C23116yescomplemented
69C23×D71—24no—
70C23×D71—24no—
71C23×D71C4⋊C418yes—
72C23×D71C2×D416yes—
73C22×Dic71—26no—
74C22×Dic71—26no—
75C22×Dic71—44no—
76C22×Dic71—44no—
77C22×Dic71C4⋊C418yes—
78C22×Dic71C2×D416yes—
79C2×C4×D71—216no—
80C2×C4×D71—448no—
81C2×C4×D71C4⋊C4148yes—
82C2×C4×D71C22×C4124yes—
83C2×C4×D71C2×D4112yes—
84C2×C4×D71C2×Q8124yes—
85C2×C4×D71C2414yes—
86#11765848C22×C4164yescomplemented
87C2×C71⋊C8C22×C4124yescomplemented
88C2×C71⋊C8C2414yes—
89C4⋊Dic71—26no—
90C4⋊Dic71—26no—
91C4⋊Dic71C2414yes—
92C4⋊Dic71C2414yes—
93Dic71⋊C4—216no—
94(C2×D71)⋊C4—216no—
95C4×Dic71—216no—
96C24×C71C2×D411yescharacteristic
97C22×C4×C71—26no—
98C22×C4×C71—26no—
99C22×C4×C71—44no—
100C22×C4×C71—44no—
101C22×C4×C71C22×C414yes—
102C22×C4×C71C2×D413yes—
103C22×C4×C71C2×Q814yes—
104C22×C4×C71C2413yes—
105C2×C8×C71C22×C4124yescomplemented
106C2×C8×C71C2414yes—
107C4⋊C4×C71—26no—
108C4⋊C4×C71—26no—
109C4⋊C4×C71C2414yes—
110C4⋊C4×C71C2414yes—
111C23×C71C2×C4⋊C414yes—
112C23×C71AGL(1,ℤ/8)14yes—
113C23×C71#13714yes—
114C23×C71C22×D413yes—
115C22×D71—212no—
116C22×D71—212no—
117C22×D71—232no—
118C22×D71C2×C4⋊C4124yes—
119C22×D71C22×D414yes—
120C2×C4×C71—24no—
121C2×C4×C71—24no—
122C2×C4×C71—412no—
123C2×C4×C71—412no—
124C2×C4×C71C2×C4⋊C4112yes—
125C2×C4×C71C23×C416yes—
126C2×C4×C71C22×D413yes—
127C2×C4×C71C22×Q816yes—
128C2×C4×C71C2511yescharacteristic
129C2×Dic71—24no—
130C2×Dic71—24no—
131C2×Dic71—412no—
132C2×Dic71—412no—
133C2×Dic71C2×C4⋊C4124yes—
134C2×Dic71C22×D414yes—
135C4×D71—432no—
136C4×D71C2×C4⋊C4148yes—
137C4×D71C23×C418yes—
138C4×D71C22×Q818yes—
139C8×C71C23×C418yescomplemented
140C71⋊C8C23×C418yescomplemented
141C22×C71#428116yescomplemented
142C22×C71C22×C4⋊C416yes—
143C22×C71C2×AGL(1,ℤ/8)16yes—
144C22×C71C2×#13716yes—
145C22×C71C23×D411yescharacteristic
146C2×D71—248no—
147C2×D71—24no—
148C2×D71—24no—
149C2×D71C22×C4⋊C418yes—
150C4×C71—44no—
151C4×C71—44no—
152C4×C71C22×C4⋊C416yes—
153C4×C71C24×C411yescharacteristic
154C4×C71C23×Q811yescharacteristic
155Dic71—44no—
156Dic71—44no—
157Dic71C22×C4⋊C418yes—
158C22×#428—711nomaximal, Sylow
159C2×C71C2×#428112yescomplemented
160C2×C71C23×C4⋊C411yescharacteristic
161C2×C71C22×AGL(1,ℤ/8)11yescharacteristic
162C2×C71C22×#13711yescharacteristic
163D71—28no—
164C23×M16—711no—
165C22×#117—711no—
166C23×C4⋊C4—711no—
167C2×#428—7124no—
168C22×#107—711no—
169C22×#107—711no—
170C22×#106—711no—
171C22×#106D7111yescharacteristic, complemented
172C71C22×#42811yescharacteristic, complemented, Sylow
173C24×C4—711no—
174C24×C4—1421no—
175C22×M16—7116no—
176C22×M16—7112no—
177C23×C8—711no—
178C23×C8C2×D7111yescharacteristic
179C2×#117—7112no—
180C2×#117—1426no—
181C22×C4⋊C4—711no—
182C22×C4⋊C4—711no—
183C22×C4⋊C4—7112no—
184C22×C4⋊C4—1426no—
185C22×C4⋊C4—1421no—
186C22×C4⋊C4C2×D7111yescharacteristic
187C22×C4⋊C4C2×D7111yescharacteristic
188C22×C22⋊C4—1421no—
189C22×C42—1421no—
190#428—7164no—
191C2×#107—7112no—
192C2×#107—7112no—
193C2×#106—7112no—
194C2×#106C2×D71112yescomplemented
195C25—711no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4—2848no—
199C23×C4—711no—
200C23×C4—718no—
201C23×C4—716no—
202C23×C4—718no—
203C23×C4—716no—
204C23×C4—14212no—
205C23×C4—1421no—
206C23×C4—1421no—
207C23×C4C22×D7111yescharacteristic
208C2×M16—7196no—
209C2×M16—7116no—
210C22×C8—718no—
211C22×C8—716no—
212C22×C8C4×D7118yescomplemented
213C22×C8C22×D7116yes—
214#117—7116no—
215#117—14224no—
216C2×C4⋊C4—23no—
217C2×C4⋊C4—23no—
218C2×C4⋊C4—716no—
219C2×C4⋊C4—716no—
220C2×C4⋊C4—7116no—
221C2×C4⋊C4—14224no—
222C2×C4⋊C4—14212no—
223C2×C4⋊C4—1423no—
224C2×C4⋊C4—1423no—
225C2×C4⋊C4C22×D7116yes—
226C2×C4⋊C4C22×D7116yes—
227C2×C22⋊C4—14212no—
228C2×C42—14212no—
229#107—7116no—
230#107—7116no—
231#106—7116no—
232#106C22×D71116yescomplemented
233C24—716no—
234C24—718no—
235C24—1424no—
236C24—1424no—
237C24D4×D7111yescharacteristic
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—28448no—
243C22×C4—2844no—
244C22×C4—2844no—
245C22×C4—716no—
246C22×C4—718no—
247C22×C4—7124no—
248C22×C4—7112no—
249C22×C4—7124no—
250C22×C4—714no—
251C22×C4—7148no—
252C22×C4—14216no—
253C22×C4—1426no—
254C22×C4—1426no—
255C22×C4C2×C4×D7114yes—
256C22×C4D4×D7113yes—
257C22×C4Q8×D7114yes—
258C22×C4C23×D7113yes—
259M16—7164no—
260C2×C8—7124no—
261C2×C8—714no—
262C2×C8C2×C4×D71124yescomplemented
263C2×C8C23×D7114yes—
264C4⋊C4—26no—
265C4⋊C4—26no—
266C4⋊C4—714no—
267C4⋊C4—714no—
268C4⋊C4—14216no—
269C4⋊C4—1426no—
270C4⋊C4—1426no—
271C4⋊C4C23×D7114yes—
272C4⋊C4C23×D7114yes—
273C22⋊C4—14216no—
274C42—14216no—
275C23—714no—
276C23—7124no—
277C23—14212no—
278C23—14212no—
279C23—14232no—
280C23C4⋊C4×D7114yes—
281C23#1455615514yes—
282C23#1455615714yes—
283C23C2×D4×D7113yes—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—412no—
287C2×C4—412no—
288C2×C4—28432no—
289C2×C4—28412no—
290C2×C4—28412no—
291C2×C4—714no—
292C2×C4—7124no—
293C2×C4—718no—
294C2×C4—718no—
295C2×C4—7148no—
296C2×C4—1424no—
297C2×C4—1424no—
298C2×C4C4⋊C4×D71112yes—
299C2×C4C22×C4×D7116yes—
300C2×C4C2×D4×D7113yes—
301C2×C4C2×Q8×D7116yes—
302C2×C4C24×D7111yescharacteristic
303C8—718no—
304C8C22×C4×D7118yescomplemented
305C22—718no—
306C22—14248no—
307C22—1424no—
308C22—1424no—
309C22#17986038116yesdirect product factor
310C22C2×C4⋊C4×D7116yes—
311C22C2×#1455615516yes—
312C22C2×#1455615716yes—
313C22C22×D4×D7111yescharacteristic
314C4—44no—
315C4—44no—
316C4—2844no—
317C4—2844no—
318C4—718no—
319C4C2×C4⋊C4×D7116yes—
320C4C23×C4×D7111yescharacteristic
321C4C22×Q8×D7111yescharacteristic
322C2—1428no—
323C2C2×#17986038112yesdirect product factor
324C2C22×C4⋊C4×D7111yescharacteristic
325C2C22×#1455615511yescharacteristic
326C2C22×#1455615711yescharacteristic
subgroupG/H
centerC24D4×D71
commutatorC4×C71C24×C4
FrattiniC2×C4C24×D71
FittingC22×#106×C71C2
socleC24×C71C2×D4
prime radicalC24×D71D4
cosocleC2×C4×C71C25
hypercenterC22×#106D71
hypocenterC71C22×#428
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#106

Complements

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

NH ≅ G/Ntype
4. C2×#17986038C2direct (N × H)
5. C22×#14556062C2semidirect
6. C22×#14556061C2semidirect
7. C22×#14556059C2semidirect
8. C22×#106×C71C2semidirect
11. C22×#11765848C4semidirect
23. #17986038C22direct (N × H)
24. C2×#14556062C22semidirect
25. C2×#14556061C22semidirect
26. C2×#14556059C22semidirect
30. C2×#106×C71C22semidirect
41. C2×#11765848C2×C4semidirect
43. C22×C71⋊C8C2×C4semidirect
56. #14556062C23semidirect
57. #14556061C23semidirect
58. #14556059C23semidirect
62. C22×C8×C71C2×C4semidirect
68. #106×C71C23semidirect
86. #11765848C22×C4semidirect
87. C2×C71⋊C8C22×C4semidirect
105. C2×C8×C71C22×C4semidirect
139. C8×C71C23×C4semidirect
140. C71⋊C8C23×C4semidirect
141. C22×C71#428semidirect
159. C2×C71C2×#428semidirect
171. C22×#106D71semidirect
172. C71C22×#428semidirect
194. C2×#106C2×D71semidirect
212. C22×C8C4×D71semidirect
232. #106C22×D71semidirect
262. C2×C8C2×C4×D71semidirect
304. C8C22×C4×D71semidirect
309. C22#17986038direct (N × H)
323. C2C2×#17986038direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658482
C22×#145560862
C22×C4⋊C4×D712
C2×#179860382
C22×#145560622
C22×#145560612
C22×#145560592
C22×#106×C712
C22×#42871