Finite Group Databasebeta

← C22×#10623086 #15660718

Subgroups — C22×#10623086 #15660718

Order 2816.

12230 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#4341C212yesmaximal, complemented
2C2×#10623086C2148yesmaximal, direct product factor
3C22×#66922C211yesmaximal, characteristic, complemented
4C22×#66939C214yesmaximal, complemented
5C22×#66950C212yesmaximal, complemented
6C22×#66983C212yesmaximal, complemented
7C22×C4×D44C211yesmaximal, characteristic, complemented
8C23×C4⋊Dic11C212yesmaximal, complemented
9C22×C4×D4×C11C211yesmaximal, characteristic, complemented
10C23×D4×C11C2211yescharacteristic
11C24×C4×C11C2212yes—
12C23×C22⋊D11C2214yes—
13C24×Dic11C2214yes—
14C22×#4341—232no—
15C22×#4341C22148yescomplemented
16C23×D44C2211yescharacteristic
17C23×C4×D11C2212yes—
18C23×Dic22C2212yes—
19C2×C4×D4×C11C22124yescomplemented
20C22×C11×C4⋊C4C2211yescharacteristic
21C22×C11×C22⋊C4C2212yes—
22C22×C42×C11C2211yescharacteristic
23C2×#66983C22148yescomplemented
24C22×(C2×D11)⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2214yes—
26C22×C4⋊Dic11C2212yes—
27C22×C4⋊Dic11C2212yes—
28C22×C4⋊Dic11C2211yescharacteristic
29C22×C4⋊Dic11C2214yes—
30C22×C4⋊Dic11C22148yescomplemented
31#10623086C221256yesdirect product factor
32C2×#66950C22148yescomplemented
33C2×#66939C22196yescomplemented
34C2×C4×D44C22124yescomplemented
35C2×#66922C22124yescomplemented
36C25×C11C2312yes—
37C24×D11C2312yes—
38C22×D4×C11D4116yescomplemented
39C22×D4×C11C23112yes—
40C23×C4×C11—216no—
41C23×C4×C11C2311yescharacteristic
42C23×C4×C11C2312yes—
43C23×C4×C11C23124yes—
44C23×C4×C11C2311yescharacteristic
45C23×C4×C11C2311yescharacteristic
46C22×C22⋊D11—232no—
47C22×C22⋊D11C23148yes—
48C23×Dic11—232no—
49C23×Dic11C23148yes—
50C23×Dic11C2314yes—
51C23×Dic11C2312yes—
52C2×#4341—2384no—
53C2×#4341C231128yescomplemented
54C22×D44—28no—
55C22×D44C23112yes—
56C22×C4×D11—216no—
57C22×C4×D11C23124yes—
58C22×Dic22—216no—
59C22×Dic22C23124yes—
60C4×D4×C11C23164yescomplemented
61C2×C11×C4⋊C4C23112yes—
62C2×C11×C22⋊C4C23124yes—
63C2×C42×C11C23112yes—
64#66983C231128yescomplemented
65C2×(C2×D11)⋊C4C23124yes—
66C2×(C2×D11)⋊C4C23148yes—
67C2×C4⋊Dic11C23124yes—
68C2×C4⋊Dic11C23124yes—
69C2×C4⋊Dic11C23112yes—
70C2×C4⋊Dic11C23148yes—
71C2×C4⋊Dic11C231128yescomplemented
72#66950C231128yescomplemented
73#66939C231256yescomplemented
74C4×D44C23164yescomplemented
75#66922C23164yescomplemented
76C22×#547—111nomaximal, Sylow
77C24×C11—28no—
78C24×C11—28no—
79C24×C11C2×D4116yes—
80C24×C11C24112yes—
81C24×C11C2411yescharacteristic
82C23×D11—28no—
83C23×D11—28no—
84C23×D11—28no—
85C23×D11C24112yes—
86C2×D4×C11C2×D4196yescomplemented
87C2×D4×C11C24116yes—
88C22×C4×C11—28no—
89C22×C4×C11—24no—
90C22×C4×C11—296no—
91C22×C4×C11C2×D418yes—
92C22×C4×C11#4118yescomplemented
93C22×C4×C11C2416yes—
94C22×C4×C11C24112yes—
95C22×C4×C11C24132yes—
96C22×C4×C11C2416yes—
97C22×C4×C11C2416yes—
98C2×C22⋊D11—2192no—
99C2×C22⋊D11C24164yes—
100C22×Dic11—216no—
101C22×Dic11—28no—
102C22×Dic11—2192no—
103C22×Dic11C24164yes—
104C22×Dic11C24124yes—
105C22×Dic11C24112yes—
106#4341—2512no—
107C2×D44—248no—
108C2×D44C24116yes—
109C2×C4×D11—296no—
110C2×C4×D11C24132yes—
111C2×Dic22—296no—
112C2×Dic22C24132yes—
113C11×C4⋊C4C24116yes—
114C11×C22⋊C4C24132yes—
115C42×C11C24116yes—
116(C2×D11)⋊C4C24132yes—
117(C2×D11)⋊C4C24164yes—
118C4⋊Dic11C24132yes—
119C4⋊Dic11C24132yes—
120C4⋊Dic11C24116yes—
121C4⋊Dic11C24164yes—
122C23×#41—112no—
123C2×#547—1148no—
124C22×C4⋊Q8—111no—
125C22×#123—114no—
126C22×C22⋊Q8—112no—
127C22×#121—112no—
128C22×C4×D4—111no—
129C22×C4×D4D1111yescharacteristic, complemented
130C23×C4⋊C4—112no—
131C23×C11—264no—
132C23×C11—224no—
133C23×C11—224no—
134C23×C11C22×D4148yes—
135C23×C11C22×D414yes—
136C23×C11C2×#4114yes—
137C23×C112−1+414yes—
138C23×C11C2518yes—
139C23×C11C2513yes—
140C22×D11—224no—
141C22×D11—224no—
142C22×D11—224no—
143C22×D11—432no—
144C22×D11C2518yes—
145D4×C11C22×D4164yescomplemented
146C2×C4×C11—224no—
147C2×C4×C11—212no—
148C2×C4×C11—264no—
149C2×C4×C11C22×D4124yes—
150C2×C4×C11C2×#41124yescomplemented
151C2×C4×C11C2514yes—
152C2×C4×C11C2518yes—
153C2×C4×C11C2514yes—
154C2×C4×C11C2514yes—
155C22⋊D11—2128no—
156C2×Dic11—248no—
157C2×Dic11—224no—
158C2×Dic11—2128no—
159C2×Dic11C25116yes—
160C2×Dic11C2518yes—
161D44—232no—
162C4×D11—264no—
163Dic22—264no—
164C22×#41—1148no—
165C22×#41—2232no—
166C23×Q8—112no—
167C23×D4—114no—
168C23×D4—111no—
169C23×D4C2×D1111yescharacteristic
170C24×C4—114no—
171C24×C4—112no—
172C24×C4C2×D1112yes—
173#547—11256no—
174C2×C4⋊Q8—1124no—
175C2×#123—1196no—
176C2×C22⋊Q8—1148no—
177C2×#121—1148no—
178C2×C4×D4—1124no—
179C2×C4×D4C2×D11124yescomplemented
180C22×C4⋊C4—111no—
181C22×C4⋊C4—112no—
182C22×C4⋊C4—114no—
183C22×C4⋊C4—112no—
184C22×C4⋊C4—1148no—
185C22×C4⋊C4C2×D1111yescharacteristic
186C22×C22⋊C4—112no—
187C22×C22⋊C4—114no—
188C22×C22⋊C4C2×D1112yes—
189C22×C42C2×D1111yescharacteristic
190C22×C11—28no—
191C22×C11—28no—
192C22×C11—296no—
193C22×C11#547116yescomplemented
194C22×C11C23×D4116yes—
195C22×C11C23×D416yes—
196C22×C11C22×#4116yes—
197C22×C11C2×2−1+416yes—
198C22×C11C2611yescharacteristic
199C2×D11—28no—
200C2×D11—28no—
201C2×D11—28no—
202C2×D11—448no—
203C4×C11—28no—
204C4×C11—24no—
205C4×C11C23×D418yes—
206C4×C11C22×#4118yescomplemented
207Dic11—216no—
208Dic11—28no—
209C25—112no—
210C25C22×D1112yes—
211C2×#41—11128no—
212C2×#41—22384no—
213C22×Q8—1124no—
214C22×Q8—2216no—
215C22×D4—1148no—
216C22×D4—1112no—
217C22×D4—2232no—
218C22×D4—228no—
219C22×D4D44116yescomplemented
220C22×D4C22×D11112yes—
221C23×C4—216no—
222C23×C4—1148no—
223C23×C4—112no—
224C23×C4—114no—
225C23×C4—1124no—
226C23×C4—2232no—
227C23×C4—2216no—
228C23×C4C22×D11124yes—
229C23×C4C22×D1111yescharacteristic
230C23×C4C22×D1112yes—
231C23×C4C22×D1111yescharacteristic
232C23×C4C22×D1111yescharacteristic
233C4⋊Q8—1164no—
234#123—11256no—
235C22⋊Q8—11128no—
236#121—11128no—
237C4×D4—1164no—
238C4×D4C22×D11164yescomplemented
239C2×C4⋊C4—1112no—
240C2×C4⋊C4—1124no—
241C2×C4⋊C4—1148no—
242C2×C4⋊C4—11128no—
243C2×C4⋊C4—1124no—
244C2×C4⋊C4C22×D11112yes—
245C2×C22⋊C4—1124no—
246C2×C22⋊C4—1148no—
247C2×C22⋊C4C22×D11124yes—
248C2×C42C22×D11112yes—
249C2×C11—216no—
250C2×C11C2×#547112yescomplemented
251C2×C11C24×D411yescharacteristic
252C2×C11C23×#4111yescharacteristic
253C2×C11C22×2−1+411yescharacteristic
254D11—48no—
255C24—28no—
256C24—28no—
257C24—1112no—
258C24—228no—
259C24—228no—
260C24—228no—
261C24C2×D44116yes—
262C24C23×D11112yes—
263C24C23×D1111yescharacteristic
264#41—22512no—
265C2×Q8—1132no—
266C2×Q8—2296no—
267C2×D4—1164no—
268C2×D4—1116no—
269C2×D4—22192no—
270C2×D4—2248no—
271C2×D4C2×D44196yescomplemented
272C2×D4C23×D11116yes—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—296no—
276C22×C4—1112no—
277C22×C4—1124no—
278C22×C4—1164no—
279C22×C4—1132no—
280C22×C4—228no—
281C22×C4—2216no—
282C22×C4—22192no—
283C22×C4—2296no—
284C22×C4C2×D4418yes—
285C22×C4#434118yescomplemented
286C22×C4C23×D1116yes—
287C22×C4C23×D11112yes—
288C22×C4C23×D11132yes—
289C22×C4C23×D1116yes—
290C22×C4C23×D1116yes—
291C4⋊C4—1116no—
292C4⋊C4—1132no—
293C4⋊C4—1164no—
294C4⋊C4—1132no—
295C4⋊C4C23×D11116yes—
296C22⋊C4—1132no—
297C22⋊C4—1164no—
298C22⋊C4C23×D11132yes—
299C42C23×D11116yes—
300C11C22×#54711yescharacteristic, complemented, Sylow
301C23—264no—
302C23—224no—
303C23—224no—
304C23—118no—
305C23—2224no—
306C23—2224no—
307C23—2224no—
308C23—4432no—
309C23C22×D44148yes—
310C23C22×D4414yes—
311C23C2×#434114yes—
312C23#6704314yes—
313C23C24×D1118yes—
314C23C24×D1113yes—
315Q8—2264no—
316D4—22128no—
317D4—2232no—
318D4C22×D44164yescomplemented
319C2×C4—224no—
320C2×C4—212no—
321C2×C4—264no—
322C2×C4—118no—
323C2×C4—1116no—
324C2×C4—2224no—
325C2×C4—2248no—
326C2×C4—22128no—
327C2×C4—2264no—
328C2×C4C22×D44124yes—
329C2×C4C2×#4341124yescomplemented
330C2×C4C24×D1114yes—
331C2×C4C24×D1118yes—
332C2×C4C24×D1114yes—
333C2×C4C24×D1114yes—
334C22—28no—
335C22—28no—
336C22—296no—
337C22—228no—
338C22—228no—
339C22—228no—
340C22—4448no—
341C22#10623086116yesdirect product factor
342C22C23×D44116yes—
343C22C23×D4416yes—
344C22C22×#434116yes—
345C22C2×#6704316yes—
346C22C25×D1111yescharacteristic
347C4—28no—
348C4—24no—
349C4—228no—
350C4—2216no—
351C4C23×D4418yes—
352C4C22×#434118yescomplemented
353C2—216no—
354C2—448no—
355C2C2×#10623086112yesdirect product factor
356C2C24×D4411yescharacteristic
357C2C23×#434111yescharacteristic
358C2C22×#6704311yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC22×C4×D4×C11C2
socleC24×C11C24
hypercenterC22×C4×D4D11
hypocenterC11C22×#547
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C23×#4341C2semidirect
2. C2×#10623086C2direct (N × H)
3. C22×#66922C2semidirect
4. C22×#66939C2semidirect
5. C22×#66950C2semidirect
6. C22×#66983C2semidirect
7. C22×C4×D44C2semidirect
8. C23×C4⋊Dic11C2semidirect
9. C22×C4×D4×C11C2semidirect
15. C22×#4341C22semidirect
19. C2×C4×D4×C11C22semidirect
23. C2×#66983C22semidirect
30. C22×C4⋊Dic11C22semidirect
31. #10623086C22direct (N × H)
32. C2×#66950C22semidirect
33. C2×#66939C22semidirect
34. C2×C4×D44C22semidirect
35. C2×#66922C22semidirect
38. C22×D4×C11D4semidirect
53. C2×#4341C23semidirect
60. C4×D4×C11C23semidirect
64. #66983C23semidirect
71. C2×C4⋊Dic11C23semidirect
72. #66950C23semidirect
73. #66939C23semidirect
74. C4×D44C23semidirect
75. #66922C23semidirect
86. C2×D4×C11C2×D4semidirect
92. C22×C4×C11#41semidirect
129. C22×C4×D4D11semidirect
145. D4×C11C22×D4semidirect
150. C2×C4×C11C2×#41semidirect
179. C2×C4×D4C2×D11semidirect
193. C22×C11#547semidirect
206. C4×C11C22×#41semidirect
219. C22×D4D44semidirect
238. C4×D4C22×D11semidirect
250. C2×C11C2×#547semidirect
271. C2×D4C2×D44semidirect
285. C22×C4#4341semidirect
300. C11C22×#547semidirect
318. D4C22×D44semidirect
329. C2×C4C2×#4341semidirect
341. C22#10623086direct (N × H)
352. C4C22×#4341semidirect
355. C2C2×#10623086direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43412
C2×#106230862
C22×#669222
C22×#669392
C22×#669502
C22×#669832
C22×C4×D442
C23×C4⋊Dic112
C22×C4×D4×C112
C22×#54711