Finite Group Databasebeta

← C22×#10623257 #15660719

Subgroups — C22×#10623257 #15660719

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×#4343C212yesmaximal, complemented
2C2×#10623257C2148yesmaximal, direct product factor
3C22×#66947C214yesmaximal, complemented
4C22×#66999C212yesmaximal, complemented
5C22×#66948C212yesmaximal, complemented
6C22×C4×D44C211yesmaximal, characteristic, complemented
7C22×C4×D44C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×D11C212yesmaximal, complemented
9C22×C11×C4⋊Q8C211yesmaximal, characteristic, complemented
10C23×Q8×C11C2212yes—
11C22×#4343—232no—
12C22×#4343C22148yescomplemented
13C23×D44C2214yes—
14C23×D44C2211yescharacteristic
15C23×D44C2211yescharacteristic
16C23×C4×D11C2214yes—
17C23×C4×D11C2212yes—
18C23×C4×D11C2212yes—
19C2×C11×C4⋊Q8C22124yescomplemented
20C22×C11×C4⋊C4C2212yes—
21C22×C11×C4⋊C4C2212yes—
22C22×C42×C11C2211yescharacteristic
23C2×#66999C22148yescomplemented
24C22×(C2×D11)⋊C4C2212yes—
25C22×(C2×D11)⋊C4C2212yes—
26C22×(C2×D11)⋊C4C2214yes—
27C22×C4⋊Dic11C2211yescharacteristic
28C22×C4⋊Dic11C2211yescharacteristic
29C22×Dic11⋊C4C2214yes—
30#10623257C221256yesdirect product factor
31C2×#66948C22148yescomplemented
32C2×#66947C22196yescomplemented
33C2×C4⋊C4×D11C22148yescomplemented
34C2×C4×D44C22124yescomplemented
35C2×C4×D44C22124yescomplemented
36C24×D11C2312yes—
37C24×D11C2312yes—
38C22×Q8×C11—216no—
39C22×Q8×C11C23124yes—
40C23×C4×C11C2314yes—
41C23×C4×C11C2311yescharacteristic
42C23×C4×C11C2311yescharacteristic
43C23×C4×C11C2311yescharacteristic
44C23×Dic11C2312yes—
45C23×Dic11C2312yes—
46C2×#4343—2384no—
47C2×#4343C231128yescomplemented
48C22×D44—232no—
49C22×D44—28no—
50C22×D44D4116yescomplemented
51C22×D44C23112yes—
52C22×D44C23148yes—
53C22×D44C23112yes—
54C22×C4×D11—232no—
55C22×C4×D11—216no—
56C22×C4×D11—216no—
57C22×C4×D11C23124yes—
58C22×C4×D11C23124yes—
59C22×C4×D11C23148yes—
60C11×C4⋊Q8C23164yescomplemented
61C2×C11×C4⋊C4C23124yes—
62C2×C11×C4⋊C4C23124yes—
63C2×C42×C11C23112yes—
64#66999C231128yescomplemented
65C2×(C2×D11)⋊C4C23124yes—
66C2×(C2×D11)⋊C4C23124yes—
67C2×(C2×D11)⋊C4C23148yes—
68C2×C4⋊Dic11C23112yes—
69C2×C4⋊Dic11C23112yes—
70C2×Dic11⋊C4C23148yes—
71#66948C231128yescomplemented
72#66947C231256yescomplemented
73C4⋊C4×D11C231128yescomplemented
74C4×D44C23164yescomplemented
75C4×D44C23164yescomplemented
76C22×#547—111nomaximal, Sylow
77C24×C11C2411yescharacteristic
78C23×D11—28no—
79C23×D11—28no—
80C23×D11—28no—
81C23×D11—28no—
82C23×D11—28no—
83C23×D11C2×D4116yes—
84C23×D11C24112yes—
85C23×D11C24112yes—
86C2×Q8×C11—296no—
87C2×Q8×C11C24132yes—
88C22×C4×C11—216no—
89C22×C4×C11—24no—
90C22×C4×C11C2×D418yes—
91C22×C4×C11#4118yescomplemented
92C22×C4×C11C24124yes—
93C22×C4×C11C2416yes—
94C22×C4×C11C2416yes—
95C22×C4×C11C2416yes—
96C22×Dic11—28no—
97C22×Dic11—28no—
98C22×Dic11C24112yes—
99C22×Dic11C24112yes—
100#4343—2512no—
101C2×D44—2192no—
102C2×D44—248no—
103C2×D44C2×D4196yescomplemented
104C2×D44C24116yes—
105C2×D44C24164yes—
106C2×D44C24116yes—
107C2×C4×D11—2192no—
108C2×C4×D11—296no—
109C2×C4×D11—296no—
110C2×C4×D11C24132yes—
111C2×C4×D11C24132yes—
112C2×C4×D11C24164yes—
113C11×C4⋊C4C24132yes—
114C11×C4⋊C4C24132yes—
115C42×C11C24116yes—
116(C2×D11)⋊C4C24132yes—
117(C2×D11)⋊C4C24132yes—
118(C2×D11)⋊C4C24164yes—
119C4⋊Dic11C24116yes—
120C4⋊Dic11C24116yes—
121Dic11⋊C4C24164yes—
122C23×#41—112no—
123C2×#547—1148no—
124C22×C4⋊Q8D1111yescharacteristic, complemented
125C22×#123—114no—
126C22×C22⋊Q8—112no—
127C22×#121—112no—
128C22×C4×D4—111no—
129C22×C4×D4—111no—
130C23×C4⋊C4—112no—
131C23×C11C22×D414yes—
132C23×C11C2×#4114yes—
133C23×C112−1+414yes—
134C23×C11C2513yes—
135C22×D11—264no—
136C22×D11—224no—
137C22×D11—224no—
138C22×D11—224no—
139C22×D11—224no—
140C22×D11—224no—
141C22×D11—432no—
142C22×D11C22×D4148yes—
143C22×D11C2518yes—
144C22×D11C2518yes—
145Q8×C11—264no—
146C2×C4×C11—248no—
147C2×C4×C11—212no—
148C2×C4×C11C22×D4124yes—
149C2×C4×C11C2×#41124yescomplemented
150C2×C4×C11C25116yes—
151C2×C4×C11C2514yes—
152C2×C4×C11C2514yes—
153C2×C4×C11C2514yes—
154C2×Dic11—224no—
155C2×Dic11—224no—
156C2×Dic11C2518yes—
157C2×Dic11C2518yes—
158D44—2128no—
159D44—232no—
160D44C22×D4164yescomplemented
161C4×D11—2128no—
162C4×D11—264no—
163C4×D11—264no—
164C22×#41—1148no—
165C22×#41—2232no—
166C23×Q8C2×D1112yes—
167C23×D4—111no—
168C23×D4—114no—
169C23×D4—111no—
170C24×C4—114no—
171C24×C4—112no—
172C24×C4—112no—
173#547—11256no—
174C2×C4⋊Q8C2×D11124yescomplemented
175C2×#123—1196no—
176C2×C22⋊Q8—1148no—
177C2×#121—1148no—
178C2×C4×D4—1124no—
179C2×C4×D4—1124no—
180C22×C4⋊C4—114no—
181C22×C4⋊C4—111no—
182C22×C4⋊C4—111no—
183C22×C4⋊C4—1148no—
184C22×C4⋊C4C2×D1112yes—
185C22×C4⋊C4C2×D1112yes—
186C22×C22⋊C4—112no—
187C22×C22⋊C4—112no—
188C22×C22⋊C4—114no—
189C22×C42C2×D1111yescharacteristic
190C22×C11#547116yescomplemented
191C22×C11C23×D416yes—
192C22×C11C22×#4116yes—
193C22×C11C2×2−1+416yes—
194C22×C11C2611yescharacteristic
195C2×D11—28no—
196C2×D11—28no—
197C2×D11—296no—
198C2×D11—28no—
199C2×D11—28no—
200C2×D11—28no—
201C2×D11—448no—
202C2×D11C23×D4116yes—
203C4×C11—216no—
204C4×C11—24no—
205C4×C11C23×D418yes—
206C4×C11C22×#4118yescomplemented
207Dic11—28no—
208Dic11—28no—
209C25—112no—
210C25—112no—
211C2×#41—11128no—
212C2×#41—22384no—
213C22×Q8—216no—
214C22×Q8C22×D11124yes—
215C22×D4—1116no—
216C22×D4—1112no—
217C22×D4—1112no—
218C22×D4—1148no—
219C22×D4—2232no—
220C22×D4—228no—
221C23×C4—112no—
222C23×C4—112no—
223C23×C4—1124no—
224C23×C4—1148no—
225C23×C4—1124no—
226C23×C4—2232no—
227C23×C4—2216no—
228C23×C4—2216no—
229C23×C4C22×D1111yescharacteristic
230C23×C4C22×D1114yes—
231C23×C4C22×D1111yescharacteristic
232C23×C4C22×D1111yescharacteristic
233C4⋊Q8C22×D11164yescomplemented
234#123—11256no—
235C22⋊Q8—11128no—
236#121—11128no—
237C4×D4—1164no—
238C4×D4—1164no—
239C2×C4⋊C4—1112no—
240C2×C4⋊C4—1148no—
241C2×C4⋊C4—1112no—
242C2×C4⋊C4—11128no—
243C2×C4⋊C4C22×D11124yes—
244C2×C4⋊C4C22×D11124yes—
245C2×C22⋊C4—1124no—
246C2×C22⋊C4—1124no—
247C2×C22⋊C4—1148no—
248C2×C42C22×D11112yes—
249C2×C11C2×#547112yescomplemented
250C2×C11C24×D411yescharacteristic
251C2×C11C23×#4111yescharacteristic
252C2×C11C22×2−1+411yescharacteristic
253D11—216no—
254D11—48no—
255C24—1116no—
256C24—1112no—
257C24—1112no—
258C24—228no—
259C24—228no—
260C24—228no—
261C24—228no—
262C24—228no—
263C24C23×D1111yescharacteristic
264#41—22512no—
265C2×Q8—296no—
266C2×Q8C23×D11132yes—
267C2×D4—1196no—
268C2×D4—1116no—
269C2×D4—1116no—
270C2×D4—1164no—
271C2×D4—22192no—
272C2×D4—2248no—
273C22×C4—216no—
274C22×C4—24no—
275C22×C4—1112no—
276C22×C4—1112no—
277C22×C4—1132no—
278C22×C4—1164no—
279C22×C4—1132no—
280C22×C4—22192no—
281C22×C4—228no—
282C22×C4—228no—
283C22×C4—2296no—
284C22×C4—2296no—
285C22×C4D4×D1118yes—
286C22×C4#434318yescomplemented
287C22×C4C23×D1116yes—
288C22×C4C23×D11124yes—
289C22×C4C23×D1116yes—
290C22×C4C23×D1116yes—
291C4⋊C4—1116no—
292C4⋊C4—1164no—
293C4⋊C4—1116no—
294C4⋊C4C23×D11132yes—
295C4⋊C4C23×D11132yes—
296C22⋊C4—1132no—
297C22⋊C4—1132no—
298C22⋊C4—1164no—
299C42C23×D11116yes—
300C11C22×#54711yescharacteristic, complemented, Sylow
301C23—1148no—
302C23—118no—
303C23—118no—
304C23—2264no—
305C23—2224no—
306C23—2224no—
307C23—2224no—
308C23—2224no—
309C23—2224no—
310C23—4432no—
311C23C2×D4×D1114yes—
312C23C2×#434314yes—
313C23#6704014yes—
314C23C24×D1113yes—
315Q8—264no—
316D4—1164no—
317D4—22128no—
318D4—2232no—
319C2×C4—248no—
320C2×C4—212no—
321C2×C4—118no—
322C2×C4—118no—
323C2×C4—22128no—
324C2×C4—2224no—
325C2×C4—2224no—
326C2×C4—2264no—
327C2×C4—2264no—
328C2×C4C2×D4×D11124yes—
329C2×C4C2×#4343124yescomplemented
330C2×C4C24×D1114yes—
331C2×C4C24×D11116yes—
332C2×C4C24×D1114yes—
333C2×C4C24×D1114yes—
334C22—1116no—
335C22—228no—
336C22—228no—
337C22—2296no—
338C22—228no—
339C22—228no—
340C22—228no—
341C22—4448no—
342C22#10623257116yesdirect product factor
343C22C22×D4×D1116yes—
344C22C22×#434316yes—
345C22C2×#6704016yes—
346C22C25×D1111yescharacteristic
347C4—216no—
348C4—24no—
349C4—228no—
350C4—228no—
351C4C22×D4×D1118yes—
352C4C22×#434318yescomplemented
353C2—2216no—
354C2—448no—
355C2C2×#10623257112yesdirect product factor
356C2C23×D4×D1111yescharacteristic
357C2C23×#434311yescharacteristic
358C2C22×#6704011yescharacteristic
subgroupG/H
centerC24C23×D11
commutator, cosocleC22×C11C26
FrattiniC22C25×D11
FittingC22×C11×C4⋊Q8C2
socleC24×C11C24
hypercenterC22×C4⋊Q8D11
hypocenterC11C22×#547
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C23×#4343C2semidirect
2. C2×#10623257C2direct (N × H)
3. C22×#66947C2semidirect
4. C22×#66999C2semidirect
5. C22×#66948C2semidirect
6. C22×C4×D44C2semidirect
7. C22×C4×D44C2semidirect
8. C22×C4⋊C4×D11C2semidirect
9. C22×C11×C4⋊Q8C2semidirect
12. C22×#4343C22semidirect
19. C2×C11×C4⋊Q8C22semidirect
23. C2×#66999C22semidirect
30. #10623257C22direct (N × H)
31. C2×#66948C22semidirect
32. C2×#66947C22semidirect
33. C2×C4⋊C4×D11C22semidirect
34. C2×C4×D44C22semidirect
35. C2×C4×D44C22semidirect
47. C2×#4343C23semidirect
50. C22×D44D4semidirect
60. C11×C4⋊Q8C23semidirect
64. #66999C23semidirect
71. #66948C23semidirect
72. #66947C23semidirect
73. C4⋊C4×D11C23semidirect
74. C4×D44C23semidirect
75. C4×D44C23semidirect
91. C22×C4×C11#41semidirect
103. C2×D44C2×D4semidirect
124. C22×C4⋊Q8D11semidirect
149. C2×C4×C11C2×#41semidirect
160. D44C22×D4semidirect
174. C2×C4⋊Q8C2×D11semidirect
190. C22×C11#547semidirect
206. C4×C11C22×#41semidirect
233. C4⋊Q8C22×D11semidirect
249. C2×C11C2×#547semidirect
286. C22×C4#4343semidirect
300. C11C22×#547semidirect
329. C2×C4C2×#4343semidirect
342. C22#10623257direct (N × H)
352. C4C22×#4343semidirect
355. C2C2×#10623257direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#43432
C2×#106232572
C22×#669472
C22×#669992
C22×#669482
C22×C4×D442
C22×C4⋊C4×D112
C22×C11×C4⋊Q82
C22×#54711