Finite Group Databasebeta

← C2×Q8×#1200 #14478856

Subgroups — C2×Q8×#1200 #14478856

Order 1920.

3328 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×D15C211yesmaximal, characteristic
2C2×C5×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
4Q8×#1200C2116yesmaximal, direct product factor
5C2×#91437C213yesmaximal
6C2×C4×#1200C213yesmaximal
7C2×#91340C213yesmaximal, complemented
8C2×#91333C213yesmaximal, complemented
9C2×C4×Q8×D5—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C2×Q8×D15C4116yescomplemented
12C2×Q8×D15C2218yes—
13C22×C4×D15C2213yes—
14C22×Dic30C2213yes—
15C22×#1200C2213yes—
16C22×#1200C2211yescharacteristic
17C5×Q8×Dic3C2218yescomplemented
18C2×C5×C4⋊Dic3C2213yes—
19C2×C4×C5×Dic3C2213yes—
20C3×Q8×Dic5C2218yescomplemented
21C2×C3×C4⋊Dic5C2213yes—
22C2×C3×C4×Dic5C2213yes—
23C2×#6669C2216yes—
24C2×Dic3×Dic5C2213yes—
25C2×Dic3×Dic5C2213yes—
26#91437C22124yes—
27C4×#1200C22124yes—
28#91340C22124yescomplemented
29#91333C22124yescomplemented
30C2×C4×D3×Q8—51nomaximal, Hall
31C22×Q8×D5—31no—
32C2×C4×C5×Q8—31no—
33C2×Q8×Dic5D311yescharacteristic, complemented
34C4×Q8×D5—316no—
35C2×D5×C4⋊C4—33no—
36C2×#4033—33no—
37C2×C42×D5—33no—
38C2×C4×Dic10—33no—
39C23×D15C2311yescharacteristic
40C2×C3×C5×Q8C2×C418yescomplemented
41C2×C3×C5×Q8C2314yes—
42C22×C3×C4×C5C2313yes—
43C22×Dic15C2313yes—
44Q8×D15C2×C4164yescomplemented
45C2×C4×D15C2×C4124yes—
46C2×C4×D15C23112yes—
47C2×Dic30C2×C4124yescomplemented
48C2×Dic30C23112yes—
49C22×C5×Dic3C2313yes—
50C22×C5×Dic3C2311yescharacteristic
51C22×C3×Dic5C2313yes—
52C22×C3×Dic5C2311yescharacteristic
53C2×#1200—212no—
54C2×#1200Q818yesdirect product factor
55C2×#1200C23112yes—
56C2×#1200C2314yes—
57C5×C4⋊Dic3C23112yes—
58C4×C5×Dic3C23112yes—
59C3×C4⋊Dic5C23112yes—
60C3×C4×Dic5C23112yes—
61#6669C23124yes—
62Dic3×Dic5C23112yes—
63Dic3×Dic5C23112yes—
64C22×D3×Q8—51no—
65C2×C3×C4×Q8—51no—
66C2×Q8×Dic3D511yescharacteristic, complemented
67C4×D3×Q8—516no—
68C2×D3×C4⋊C4—53no—
69C2×#887—53no—
70C2×C42×D3—53no—
71C2×C4×Dic6—53no—
72C22×C5×Q8C2×D311yescharacteristic
73C2×Q8×D5—316no—
74C2×Q8×D5—38no—
75C22×C4×D5—33no—
76C22×C4×D5—31no—
77C22×C4×D5—33no—
78C22×Dic10—33no—
79C4×C5×Q8—38no—
80C2×C5×C4⋊C4—33no—
81C2×C42×C5—33no—
82Q8×Dic5C2×D318yescomplemented
83C2×C4⋊Dic5C2×D313yes—
84C2×#683—36no—
85C2×C4×Dic5—33no—
86C2×C4×Dic5—33no—
87C2×C4×Dic5C2×D313yes—
88D5×C4⋊C4—324no—
89#4033—324no—
90C42×D5—324no—
91C4×Dic10—324no—
92C22×C4×Q8—151noSylow
93C23×C3×C5C2411yescharacteristic
94C22×D15C22×C414yes—
95C22×D15C2×Q814yes—
96C22×D15#4114yes—
97C22×D15C2412yes—
98C3×C5×Q8C22×C4116yescomplemented
99C2×C3×C4×C5C22×C4112yes—
100C2×C3×C4×C5C2416yes—
101C2×Dic15C22×C4112yes—
102C2×Dic15C2416yes—
103C4×D15C22×C4148yes—
104Dic30C22×C4148yescomplemented
105C2×C5×Dic3—26no—
106C2×C5×Dic3C2×Q814yescomplemented
107C2×C5×Dic3C2416yes—
108C2×C5×Dic3C2412yes—
109C2×C3×Dic5—26no—
110C2×C3×Dic5C2×Q814yescomplemented
111C2×C3×Dic5C2416yes—
112C2×C3×Dic5C2412yes—
113#1200—224no—
114#1200C2×Q8116yesdirect product factor
115C22×C3×Q8C2×D511yescharacteristic
116C2×D3×Q8—516no—
117C2×D3×Q8—58no—
118C22×C4×D3—53no—
119C22×C4×D3—51no—
120C22×C4×D3—53no—
121C22×Dic6—53no—
122C3×C4×Q8—58no—
123C2×C3×C4⋊C4—53no—
124C2×C3×C42—53no—
125Q8×Dic3C2×D518yescomplemented
126C2×C4⋊Dic3C2×D513yes—
127C2×Dic3⋊C4—56no—
128C2×C4×Dic3—53no—
129C2×C4×Dic3—53no—
130C2×C4×Dic3C2×D513yes—
131D3×C4⋊C4—524no—
132#887—524no—
133C42×D3—524no—
134C4×Dic6—524no—
135C23×D5—31no—
136C2×C5×Q8C4×D318yescomplemented
137C2×C5×Q8C22×D314yes—
138C22×C4×C5—31no—
139C22×C4×C5—33no—
140C22×C4×C5C22×D313yes—
141C22×Dic5—33no—
142C22×Dic5C22×D313yes—
143C22×Dic5C22×D311yescharacteristic
144Q8×D5—364no—
145C2×C4×D5—324no—
146C2×C4×D5—312no—
147C2×C4×D5—38no—
148C2×C4×D5—34no—
149C2×C4×D5—312no—
150C2×C4×D5—612no—
151C2×Dic10—324no—
152C2×Dic10—312no—
153C5×C4⋊C4—312no—
154C42×C5—312no—
155C4⋊Dic5C22×D3112yes—
156#683—324no—
157C4×Dic5—312no—
158C4×Dic5—312no—
159C4×Dic5C22×D3112yes—
160C23×Q8—151no—
161C2×C4×Q8—31no—
162C2×C4×Q8—51no—
163C2×C4×Q8—153no—
164C2×C4×Q8—153no—
165C2×C4×Q8—1516no—
166C22×C4⋊C4—153no—
167C22×C42—153no—
168C22×C3×C5C23×C412yes—
169C22×C3×C5C22×Q812yes—
170C22×C3×C5C2×#4112yes—
171C22×C3×C5C2511yescharacteristic
172C2×D15C4×Q8116yescomplemented
173C2×D15C23×C414yes—
174C2×D15C22×Q814yes—
175C2×D15C2×#4114yes—
176C3×C4×C5C23×C4112yes—
177Dic15C23×C4112yes—
178C3×Dic5—26no—
179C3×Dic5C22×Q814yescomplemented
180C5×Dic3—26no—
181C5×Dic3C22×Q814yescomplemented
182C23×D3—51no—
183C2×C3×Q8C4×D518yescomplemented
184C2×C3×Q8C22×D514yes—
185C22×C3×C4—51no—
186C22×C3×C4—53no—
187C22×C3×C4C22×D513yes—
188C22×Dic3—53no—
189C22×Dic3C22×D513yes—
190C22×Dic3C22×D511yescharacteristic
191D3×Q8—564no—
192C2×C4×D3—524no—
193C2×C4×D3—512no—
194C2×C4×D3—58no—
195C2×C4×D3—54no—
196C2×C4×D3—512no—
197C2×C4×D3—1012no—
198C2×Dic6—524no—
199C2×Dic6—512no—
200C3×C4⋊C4—512no—
201C3×C42—512no—
202C4⋊Dic3C22×D5112yes—
203Dic3⋊C4—524no—
204C4×Dic3—512no—
205C4×Dic3—512no—
206C4×Dic3C22×D5112yes—
207C23×C5C23×D311yescharacteristic
208C22×D5—34no—
209C22×D5—34no—
210C22×D5—34no—
211C22×D5—32no—
212C5×Q8C2×C4×D3116yescomplemented
213C2×C4×C5—34no—
214C2×C4×C5—32no—
215C2×C4×C5—36no—
216C2×C4×C5—66no—
217C2×C4×C5C2×C4×D3112yes—
218C2×C4×C5C23×D316yes—
219C2×Dic5—26no—
220C2×Dic5—312no—
221C2×Dic5—36no—
222C2×Dic5D3×Q814yescomplemented
223C2×Dic5C23×D316yes—
224C2×Dic5C23×D312yes—
225C4×D5—348no—
226C4×D5—316no—
227C4×D5—624no—
228Dic10—348no—
229C22×Q8—1516no—
230C22×Q8—158no—
231C22×Q8—153no—
232C22×Q8D3×D511yescharacteristic
233C23×C4—151no—
234C23×C4—153no—
235C23×C4—153no—
236C4×Q8—38no—
237C4×Q8—58no—
238C4×Q8—1524no—
239C4×Q8—1524no—
240C2×C4⋊C4—33no—
241C2×C4⋊C4—53no—
242C2×C4⋊C4—1524no—
243C2×C4⋊C4—156no—
244C2×C42—33no—
245C2×C42—53no—
246C2×C42—1524no—
247C2×C42—153no—
248C2×C42—153no—
249C2×C3×C5C2×C4×Q814yescomplemented
250C2×C3×C5C24×C411yescharacteristic
251C2×C3×C5C23×Q811yescharacteristic
252C2×C3×C5C22×#4111yescharacteristic
253D15C2×C4×Q818yescomplemented
254C23×C3C23×D511yescharacteristic
255C22×D3—54no—
256C22×D3—54no—
257C22×D3—54no—
258C22×D3—52no—
259C3×Q8C2×C4×D5116yescomplemented
260C2×C3×C4—54no—
261C2×C3×C4—52no—
262C2×C3×C4—56no—
263C2×C3×C4—106no—
264C2×C3×C4C2×C4×D5112yes—
265C2×C3×C4C23×D516yes—
266C2×Dic3—26no—
267C2×Dic3—512no—
268C2×Dic3—56no—
269C2×Dic3Q8×D514yescomplemented
270C2×Dic3C23×D516yes—
271C2×Dic3C23×D512yes—
272C4×D3—548no—
273C4×D3—516no—
274C4×D3—1024no—
275Dic6—548no—
276C22×C5C22×C4×D312yes—
277C22×C5C2×D3×Q812yes—
278C22×C5D3×#4112yes—
279C22×C5C24×D311yescharacteristic
280C2×D5—34no—
281C2×D5—34no—
282C2×D5—34no—
283C2×D5—316no—
284C4×C5—34no—
285C4×C5—66no—
286C4×C5C22×C4×D3112yes—
287Dic5—26no—
288Dic5—312no—
289Dic5C2×D3×Q814yescomplemented
290C24—151no—
291C2×Q8—1564no—
292C2×Q8—1524no—
293C2×Q8—1512no—
294C2×Q8#120018yesdirect product factor
295C2×Q8C2×D3×D514yes—
296C22×C4—31no—
297C22×C4—33no—
298C22×C4—51no—
299C22×C4—53no—
300C22×C4—154no—
301C22×C4—158no—
302C22×C4—1512no—
303C22×C4—153no—
304C22×C4—1524no—
305C22×C4—1512no—
306C22×C4—3012no—
307C22×C4C2×D3×D513yes—
308C4⋊C4—312no—
309C4⋊C4—512no—
310C4⋊C4—1524no—
311C42—312no—
312C42—512no—
313C42—1512no—
314C42—1512no—
315C3×C5C22×C4×Q811yescharacteristic, complemented, Hall
316C22×C3C22×C4×D512yes—
317C22×C3C2×Q8×D512yes—
318C22×C3D5×#4112yes—
319C22×C3C24×D511yescharacteristic
320C2×D3—516no—
321C2×D3—54no—
322C2×D3—54no—
323C2×D3—54no—
324C3×C4—54no—
325C3×C4—106no—
326C3×C4C22×C4×D5112yes—
327Dic3—26no—
328Dic3—512no—
329Dic3C2×Q8×D514yescomplemented
330C2×C5C4×D3×Q814yescomplemented
331C2×C5C23×C4×D311yescharacteristic
332C2×C5C22×D3×Q811yescharacteristic
333C2×C5C2×D3×#4111yescharacteristic
334D5—38no—
335C23—154no—
336C23—154no—
337C23—154no—
338C23—152no—
339C23C22×D3×D511yescharacteristic
340Q8—1548no—
341Q8C2×#1200116yesdirect product factor
342C2×C4—34no—
343C2×C4—32no—
344C2×C4—36no—
345C2×C4—54no—
346C2×C4—52no—
347C2×C4—56no—
348C2×C4—66no—
349C2×C4—106no—
350C2×C4—1516no—
351C2×C4—1512no—
352C2×C4—156no—
353C2×C4—1548no—
354C2×C4—3024no—
355C2×C4C2×#1200112yes—
356C2×C4C22×D3×D516yes—
357C2×C3C4×Q8×D514yescomplemented
358C2×C3C23×C4×D511yescharacteristic
359C2×C3C22×Q8×D511yescharacteristic
360C2×C3C2×D5×#4111yescharacteristic
361D3—58no—
362C5C2×C4×D3×Q811yescharacteristic, complemented, Sylow
363C22—154no—
364C22—1516no—
365C22—154no—
366C22—154no—
367C22D3×Q8×D512yes—
368C22#9203412yes—
369C22C22×#120012yes—
370C22C23×D3×D511yescharacteristic
371C4—34no—
372C4—54no—
373C4—66no—
374C4—106no—
375C4—1512no—
376C4C22×#1200112yes—
377C3C2×C4×Q8×D511yescharacteristic, complemented, Sylow
378C2—158no—
379C2Q8×#120014yesdirect product factor
380C2C2×D3×Q8×D511yescharacteristic
381C2C2×#9203411yescharacteristic
382C2C23×#120011yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutatorC2×C3×C5C24×C4
FrattiniC22C23×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C24
prime radicalC23×D15C23
cosocleC22×C3×C5C25
hypercenterC22×Q8D3×D5
hypocenterC3×C5C22×C4×Q8
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C5×Q8×Dic3C2semidirect
3. C2×C3×Q8×Dic5C2semidirect
4. Q8×#1200C2direct (N × H)
7. C2×#91340C2semidirect
8. C2×#91333C2semidirect
11. C2×Q8×D15C4semidirect
17. C5×Q8×Dic3C22semidirect
20. C3×Q8×Dic5C22semidirect
28. #91340C22semidirect
29. #91333C22semidirect
33. C2×Q8×Dic5D3semidirect
40. C2×C3×C5×Q8C2×C4semidirect
44. Q8×D15C2×C4semidirect
47. C2×Dic30C2×C4semidirect
54. C2×#1200Q8direct (N × H)
66. C2×Q8×Dic3D5semidirect
82. Q8×Dic5C2×D3semidirect
98. C3×C5×Q8C22×C4semidirect
104. Dic30C22×C4semidirect
106. C2×C5×Dic3C2×Q8semidirect
110. C2×C3×Dic5C2×Q8semidirect
114. #1200C2×Q8direct (N × H)
125. Q8×Dic3C2×D5semidirect
136. C2×C5×Q8C4×D3semidirect
172. C2×D15C4×Q8semidirect
179. C3×Dic5C22×Q8semidirect
181. C5×Dic3C22×Q8semidirect
183. C2×C3×Q8C4×D5semidirect
212. C5×Q8C2×C4×D3semidirect
222. C2×Dic5D3×Q8semidirect
249. C2×C3×C5C2×C4×Q8semidirect
253. D15C2×C4×Q8semidirect
259. C3×Q8C2×C4×D5semidirect
269. C2×Dic3Q8×D5semidirect
289. Dic5C2×D3×Q8semidirect
294. C2×Q8#1200direct (N × H)
315. C3×C5C22×C4×Q8semidirect
329. Dic3C2×Q8×D5semidirect
330. C2×C5C4×D3×Q8semidirect
341. Q8C2×#1200direct (N × H)
357. C2×C3C4×Q8×D5semidirect
362. C5C2×C4×D3×Q8semidirect
377. C3C2×C4×Q8×D5semidirect
379. C2Q8×#1200direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D152
C2×C5×Q8×Dic32
C2×C3×Q8×Dic52
Q8×#12002
C2×#914372
C2×C4×#12002
C2×#913402
C2×#913332
C2×C4×Q8×D53
C2×C4×D3×Q85