Finite Group Databasebeta

← C2×#13077065 #15617219

Subgroups — C2×#13077065 #15617219

Order 2816.

5206 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#67001C212yesmaximal, complemented
2#13077065C2116yesmaximal, direct product factor
3C2×#10622847C216yesmaximal, complemented
4C2×#10622820C213yesmaximal, complemented
5C22×C22⋊(C11⋊C8)C213yesmaximal, complemented
6C23×C11×#41C211yesmaximal, characteristic
7C22×C11×#41—216no—
8C22×C11×#41C418yes—
9C22×C11×#41C411yescharacteristic
10C22×C11×#41C413yes—
11C22×C11×#41C2213yes—
12C22×C11×#41C2218yes—
13C22×C11×#41C2211yescharacteristic
14C23×Q8×C11C411yescharacteristic
15C23×D4×C11C413yes—
16C24×C4×C11C2213yes—
17C2×#67001—216no—
18C2×#67001C22116yescomplemented
19C22×#4318C2216yescomplemented
20C23×C11⋊C8C2216yescomplemented
21#10622847C22148yescomplemented
22#10622820C22124yescomplemented
23C2×C22⋊(C11⋊C8)C2211yescharacteristic
24C2×C22⋊(C11⋊C8)C2213yes—
25C2×C22⋊(C11⋊C8)C22124yescomplemented
26C2×C22⋊(C11⋊C8)C2216yescomplemented
27C2×C22⋊(C11⋊C8)C2216yescomplemented
28C25×C11C2×C413yes—
29C2×C11×#41—264no—
30C2×C11×#41—264no—
31C2×C11×#41—28no—
32C2×C11×#41—224no—
33C2×C11×#41—224no—
34C2×C11×#41C2×C418yes—
35C2×C11×#41C2×C4124yes—
36C2×C11×#41C2×C4112yes—
37C2×C11×#41C2×C414yes—
38C2×C11×#41D4116yes—
39C2×C11×#41C23112yes—
40C2×C11×#41C2314yes—
41C22×Q8×C11—28no—
42C22×Q8×C11C2×C418yes—
43C22×Q8×C11C2×C413yes—
44C22×Q8×C11C2×C411yescharacteristic
45C22×D4×C11—224no—
46C22×D4×C11C2×C416yes—
47C22×D4×C11C2×C4124yes—
48C22×D4×C11C2×C413yes—
49C22×D4×C11C2×C413yes—
50C23×C4×C11—224no—
51C23×C4×C11C2×C413yes—
52C23×C4×C11C2×C413yes—
53C23×C4×C11C2×C4112yes—
54C23×C4×C11C2×C413yes—
55C23×C4×C11C2311yescharacteristic
56C23×C4×C11C2313yes—
57C23×C4×C11C23112yes—
58C23×C4×C11C2313yes—
59#67001—264no—
60C2×#4318—224no—
61C2×#4318C23124yescomplemented
62C22×C11⋊C8—224no—
63C22×C11⋊C8C2312yes—
64C22×C11⋊C8C2316yes—
65C22×C11⋊C8C23124yescomplemented
66C22⋊(C11⋊C8)C2314yes—
67C22⋊(C11⋊C8)C23112yes—
68C22⋊(C11⋊C8)C23124yescomplemented
69C22⋊(C11⋊C8)C23124yescomplemented
70C2×#2880—111nomaximal, Sylow
71C24×C11—212no—
72C24×C11—212no—
73C24×C11—46no—
74C24×C11C22×C413yes—
75C24×C11C22×C4112yes—
76C24×C11C22×C413yes—
77C24×C11C22×C411yescharacteristic
78C11×#41—232no—
79C11×#41—296no—
80C11×#41—248no—
81C11×#41—248no—
82C11×#41C22⋊C4132yes—
83C11×#41C2×D4132yes—
84C2×Q8×C11—232no—
85C2×Q8×C11—212no—
86C2×Q8×C11C22⋊C418yes—
87C2×Q8×C11C22×C4112yes—
88C2×Q8×C11C22×C414yes—
89C2×D4×C11—224no—
90C2×D4×C11—296no—
91C2×D4×C11—212no—
92C2×D4×C11C22⋊C4124yes—
93C2×D4×C11C22×C4124yes—
94C2×D4×C11C22×C4112yes—
95C2×D4×C11C22×C4112yes—
96C22×C4×C11—212no—
97C22×C4×C11—212no—
98C22×C4×C11—212no—
99C22×C4×C11—248no—
100C22×C4×C11—24no—
101C22×C4×C11—212no—
102C22×C4×C11—248no—
103C22×C4×C11C22×C411yescharacteristic
104C22×C4×C11C22×C412yes—
105C22×C4×C11C22×C413yes—
106C22×C4×C11C22×C4112yes—
107C22×C4×C11C22×C4112yes—
108C22×C4×C11C22×C4112yes—
109C22×C4×C11C22×C416yes—
110C22×C4×C11C22×C413yes—
111C22×C4×C11C22×C416yes—
112C22×C4×C11C2×D4124yes—
113C22×C4×C11C2412yes—
114C22×C4×C11C2416yes—
115C22×C4×C11C2416yes—
116C22×C4×C11C2411yescharacteristic
117#4318—248no—
118C2×C11⋊C8—24no—
119C2×C11⋊C8—212no—
120C2×C11⋊C8—248no—
121C2×C11⋊C8C2414yes—
122C2×C11⋊C8C24112yes—
123C23×#41D1111yescharacteristic
124C22×#131—112no—
125#2880—1116no—
126C2×#408—116no—
127C2×#407—113no—
128C22×C22⋊C8—113no—
129C23×C11—22no—
130C23×C11—26no—
131C23×C11—22no—
132C23×C11—26no—
133C23×C11—224no—
134C23×C11—26no—
135C23×C11—26no—
136C23×C11—224no—
137C23×C11—26no—
138C23×C11—412no—
139C23×C11—448no—
140C23×C11C2×C22⋊C4112yes—
141C23×C11#13114yescomplemented
142C23×C11C23×C416yes—
143C23×C11C23×C416yes—
144C23×C11C23×C412yes—
145C23×C11C23×C411yescharacteristic
146Q8×C11—224no—
147Q8×C11C2×C22⋊C4116yes—
148D4×C11—248no—
149D4×C11—224no—
150D4×C11C2×C22⋊C4148yes—
151C2×C4×C11—22no—
152C2×C4×C11—26no—
153C2×C4×C11—224no—
154C2×C4×C11—24no—
155C2×C4×C11—224no—
156C2×C4×C11—224no—
157C2×C4×C11—212no—
158C2×C4×C11—24no—
159C2×C4×C11—212no—
160C2×C4×C11C2×C22⋊C4112yes—
161C2×C4×C11C2×C22⋊C4124yes—
162C2×C4×C11C23×C412yes—
163C2×C4×C11C23×C416yes—
164C2×C4×C11C23×C411yescharacteristic
165C2×C4×C11C23×C416yes—
166C2×C4×C11C22×D414yes—
167C2×C4×C11C22×D4124yes—
168C2×C4×C11C2511yescharacteristic
169C11⋊C8—24no—
170C11⋊C8—212no—
171C22×#41—216no—
172C22×#41Dic1118yes—
173C22×#41Dic1111yescharacteristic
174C22×#41Dic1113yes—
175C22×#41C2×D1113yes—
176C22×#41C2×D1118yes—
177C22×#41C2×D1111yescharacteristic
178C23×Q8Dic1111yescharacteristic
179C23×D4Dic1113yes—
180C24×C4C2×D1113yes—
181C2×#131—1116no—
182C2×#131—2216no—
183C22×M16—116no—
184C23×C8—116no—
185#408—1148no—
186#407—1124no—
187C2×C22⋊C8—111no—
188C2×C22⋊C8—113no—
189C2×C22⋊C8—116no—
190C2×C22⋊C8—1124no—
191C2×C22⋊C8—116no—
192C22×C11—28no—
193C22×C11—224no—
194C22×C11—22no—
195C22×C11—26no—
196C22×C11—22no—
197C22×C11—26no—
198C22×C11—26no—
199C22×C11—26no—
200C22×C11—26no—
201C22×C11—448no—
202C22×C11—412no—
203C22×C11C22×C22⋊C4112yes—
204C22×C11C22×C22⋊C412yes—
205C22×C11C2×#13112yes—
206C22×C11C2×#13114yescomplemented
207C22×C11#56812yes—
208C22×C11C24×C411yescharacteristic
209C4×C11—22no—
210C4×C11—26no—
211C4×C11C22×C22⋊C412yes—
212C4×C11C22×C22⋊C4112yes—
213C4×C11C23×D412yes—
214C25C2×Dic1113yes—
215C2×#41—264no—
216C2×#41—264no—
217C2×#41—28no—
218C2×#41—224no—
219C2×#41—224no—
220C2×#41C2×Dic1118yes—
221C2×#41C2×Dic11124yes—
222C2×#41C2×Dic11112yes—
223C2×#41C2×Dic1114yes—
224C2×#41C22⋊D11116yes—
225C2×#41C22×D11112yes—
226C2×#41C22×D1114yes—
227C22×Q8—28no—
228C22×Q8C2×Dic1118yes—
229C22×Q8C2×Dic1113yes—
230C22×Q8C2×Dic1111yescharacteristic
231C22×D4—224no—
232C22×D4C2×Dic1116yes—
233C22×D4C2×Dic11124yes—
234C22×D4C2×Dic1113yes—
235C22×D4C2×Dic1113yes—
236C23×C4—224no—
237C23×C4C2×Dic1113yes—
238C23×C4C2×Dic1113yes—
239C23×C4C2×Dic11112yes—
240C23×C4C2×Dic1113yes—
241C23×C4C22×D11112yes—
242C23×C4C22×D1111yescharacteristic
243C23×C4C22×D1113yes—
244C23×C4C22×D1113yes—
245#131—2264no—
246C2×M16—1124no—
247C2×M16—2224no—
248C22×C8—1124no—
249C22×C8—112no—
250C22×C8—116no—
251C22×C8—2224no—
252C22⋊C8—114no—
253C22⋊C8—1112no—
254C22⋊C8—1124no—
255C22⋊C8—1124no—
256C2×C11—24no—
257C2×C11—212no—
258C2×C11—46no—
259C2×C11#288014yescomplemented
260C2×C11C23×C22⋊C411yescharacteristic
261C2×C11C22×#13111yescharacteristic
262C2×C11C2×#56811yescharacteristic
263C24—212no—
264C24—212no—
265C24—46no—
266C24C22×Dic1113yes—
267C24C22×Dic11112yes—
268C24C22×Dic1111yescharacteristic
269C24C22×Dic1113yes—
270#41—232no—
271#41—296no—
272#41—248no—
273#41—248no—
274#41C22⋊Dic11132yes—
275#41C2×C22⋊D11132yes—
276C2×Q8—232no—
277C2×Q8—212no—
278C2×Q8C22⋊Dic1118yes—
279C2×Q8C22×Dic11112yes—
280C2×Q8C22×Dic1114yes—
281C2×D4—224no—
282C2×D4—296no—
283C2×D4—212no—
284C2×D4C22⋊Dic11124yes—
285C2×D4C22×Dic11124yes—
286C2×D4C22×Dic11112yes—
287C2×D4C22×Dic11112yes—
288C22×C4—212no—
289C22×C4—212no—
290C22×C4—212no—
291C22×C4—248no—
292C22×C4—24no—
293C22×C4—212no—
294C22×C4—248no—
295C22×C4C22×Dic1111yescharacteristic
296C22×C4C22×Dic1112yes—
297C22×C4C22×Dic1113yes—
298C22×C4C22×Dic11112yes—
299C22×C4C22×Dic11112yes—
300C22×C4C22×Dic11112yes—
301C22×C4C22×Dic1116yes—
302C22×C4C22×Dic1113yes—
303C22×C4C22×Dic1116yes—
304C22×C4C2×C22⋊D11124yes—
305C22×C4C23×D1112yes—
306C22×C4C23×D1116yes—
307C22×C4C23×D1116yes—
308C22×C4C23×D1111yescharacteristic
309M16—2248no—
310C2×C8—114no—
311C2×C8—1112no—
312C2×C8—224no—
313C2×C8—2212no—
314C2×C8—2248no—
315C11C2×#288011yescharacteristic, complemented, Sylow
316C23—26no—
317C23—22no—
318C23—26no—
319C23—22no—
320C23—26no—
321C23—224no—
322C23—26no—
323C23—224no—
324C23—26no—
325C23—412no—
326C23—448no—
327C23#6700114yescomplemented
328C23C2×C22⋊Dic11112yes—
329C23C23×Dic1116yes—
330C23C23×Dic1112yes—
331C23C23×Dic1116yes—
332C23C23×Dic1111yescharacteristic
333Q8—224no—
334Q8C2×C22⋊Dic11116yes—
335D4—248no—
336D4—224no—
337D4C2×C22⋊Dic11148yes—
338C2×C4—22no—
339C2×C4—26no—
340C2×C4—224no—
341C2×C4—24no—
342C2×C4—224no—
343C2×C4—224no—
344C2×C4—212no—
345C2×C4—24no—
346C2×C4—212no—
347C2×C4C2×C22⋊Dic11112yes—
348C2×C4C2×C22⋊Dic11124yes—
349C2×C4C23×Dic1112yes—
350C2×C4C23×Dic1116yes—
351C2×C4C23×Dic1111yescharacteristic
352C2×C4C23×Dic1116yes—
353C2×C4C22×C22⋊D1114yes—
354C2×C4C22×C22⋊D11124yes—
355C2×C4C24×D1111yescharacteristic
356C8—224no—
357C8—2212no—
358C22—28no—
359C22—224no—
360C22—22no—
361C22—26no—
362C22—22no—
363C22—26no—
364C22—26no—
365C22—26no—
366C22—26no—
367C22—448no—
368C22—412no—
369C22C2×#6700112yes—
370C22C2×#6700114yescomplemented
371C22#1062335012yes—
372C22C22×C22⋊Dic11112yes—
373C22C22×C22⋊Dic1112yes—
374C22C24×Dic1111yescharacteristic
375C4—22no—
376C4—26no—
377C4C22×C22⋊Dic1112yes—
378C4C22×C22⋊Dic11112yes—
379C4C23×C22⋊D1112yes—
380C2—24no—
381C2—212no—
382C2—46no—
383C2#1307706514yesdirect product factor
384C2C23×C22⋊Dic1111yescharacteristic
385C2C22×#6700111yescharacteristic
386C2C2×#1062335011yescharacteristic
subgroupG/H
centerC22×C4C23×D11
commutatorC22×C11C24×C4
FrattiniC2×C4C24×D11
Fitting, prime radicalC23×C11×#41C2
socleC23×C11C23×C4
cosocleC2×C4×C11C25
hypercenterC23×#41D11
hypocenterC11C2×#2880
LCSG ▷ C22×C11 ▷ C11
UCSC1 ◁ C22×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C22×#67001C2semidirect
2. #13077065C2direct (N × H)
3. C2×#10622847C2semidirect
4. C2×#10622820C2semidirect
5. C22×C22⋊(C11⋊C8)C2semidirect
18. C2×#67001C22semidirect
19. C22×#4318C22semidirect
20. C23×C11⋊C8C22semidirect
21. #10622847C22semidirect
22. #10622820C22semidirect
25. C2×C22⋊(C11⋊C8)C22semidirect
26. C2×C22⋊(C11⋊C8)C22semidirect
27. C2×C22⋊(C11⋊C8)C22semidirect
61. C2×#4318C23semidirect
65. C22×C11⋊C8C23semidirect
68. C22⋊(C11⋊C8)C23semidirect
69. C22⋊(C11⋊C8)C23semidirect
141. C23×C11#131semidirect
206. C22×C11C2×#131semidirect
259. C2×C11#2880semidirect
315. C11C2×#2880semidirect
327. C23#67001semidirect
370. C22C2×#67001semidirect
383. C2#13077065direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#670012
#130770652
C2×#106228472
C2×#106228202
C22×C22⋊(C11⋊C8)2
C23×C11×#412
C2×#288011