Finite Group Databasebeta

← C2×Q8×#1201 #14482954

Subgroups — C2×Q8×#1201 #14482954

Order 1920.

3136 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×D3×Q8C211yesmaximal, characteristic, complemented
2C22×C3×Q8×D5C211yesmaximal, characteristic, complemented
3C2×Q8×Dic15C211yesmaximal, characteristic, complemented
4Q8×#1201C2116yesmaximal, direct product factor
5C2×C4×#1201C213yesmaximal
6C2×#91385C213yesmaximal, complemented
7C2×#91369C213yesmaximal, complemented
8C2×#91359C213yesmaximal, complemented
9C2×Q8×C22⋊D5—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C2×C5×D3×Q8—28no—
12C2×C5×D3×Q8C2218yescomplemented
13C22×C4×C5×D3C2213yes—
14C22×C5×Dic6C2213yes—
15C2×C3×Q8×D5—28no—
16C2×C3×Q8×D5C2218yescomplemented
17C22×C3×C4×D5C2213yes—
18C22×C3×Dic10C2213yes—
19C22×#1201C2211yescharacteristic
20Q8×Dic15C2218yescomplemented
21C2×C4⋊Dic15C2213yes—
22C2×C4×Dic15C2213yes—
23C2×#6669C2216yes—
24C2×#6669C2213yes—
25C2×(C2×D3)⋊Dic5C2213yes—
26C2×#6665C2213yes—
27C4×#1201C22124yes—
28#91385C22124yescomplemented
29#91369C22124yescomplemented
30#91359C22124yescomplemented
31C2×Q8×#67—51nomaximal, Hall
32C23×C5×Q8—31no—
33C22×Q8×D5D311yescharacteristic, complemented
34Q8×C22⋊D5—316no—
35C2×#4092—33no—
36C2×Q8×Dic5—31no—
37C2×#4090—33no—
38C2×C4×C22⋊D5—33no—
39C2×#4070—33no—
40C23×C5×D3C2311yescharacteristic
41C23×C3×D5C2311yescharacteristic
42C2×C3×C5×Q8D418yescomplemented
43C2×C3×C5×Q8C2314yes—
44C22×C3×C4×C5C2313yes—
45C22×Dic15C2313yes—
46C22×Dic15C2311yescharacteristic
47C22×C5×Dic3C2313yes—
48C5×D3×Q8—232no—
49C2×C4×C5×D3—212no—
50C2×C4×C5×D3C23112yes—
51C2×C5×Dic6—212no—
52C2×C5×Dic6C23112yes—
53C22×C3×Dic5C2313yes—
54C3×Q8×D5—232no—
55C2×C3×C4×D5—212no—
56C2×C3×C4×D5C23112yes—
57C2×C3×Dic10—212no—
58C2×C3×Dic10C23112yes—
59C2×#1201Q818yesdirect product factor
60C2×#1201C2314yes—
61C4⋊Dic15C23112yes—
62C4×Dic15C23112yes—
63#6669C23112yes—
64#6669C23124yes—
65(C2×D3)⋊Dic5C23112yes—
66#6665C23112yes—
67C23×C3×Q8—51no—
68C22×D3×Q8D511yescharacteristic, complemented
69Q8×#67—516no—
70C2×#946—53no—
71C2×Q8×Dic3—51no—
72C2×#944—53no—
73C2×C4×#67—53no—
74C2×#924—53no—
75C22×C5×Q8—38no—
76C22×C5×Q8—33no—
77C22×C5×Q8—68no—
78C22×C5×Q8C2×D311yescharacteristic
79C23×C4×C5—33no—
80C22×C22⋊D5—31no—
81C2×Q8×D5—28no—
82C2×Q8×D5C2×D318yescomplemented
83C22×C4×D5C2×D313yes—
84C22×Dic10C2×D313yes—
85C2×C22⋊Dic5—33no—
86#4092—324no—
87Q8×Dic5—38no—
88#4090—324no—
89C4×C22⋊D5—324no—
90C2×#685—33no—
91C2×C4⋊Dic5—33no—
92#4070—324no—
93C2×#683—36no—
94C2×#683—33no—
95C2×C4×Dic5—33no—
96C2×D4×Q8—151noSylow
97C23×C3×C5C2411yescharacteristic
98C22×C5×D3—22no—
99C22×C5×D3—22no—
100C22×C5×D3C2×Q814yescomplemented
101C22×C5×D3C2412yes—
102C22×C3×D5—22no—
103C22×C3×D5—22no—
104C22×C3×D5C2×Q814yescomplemented
105C22×C3×D5C2412yes—
106C3×C5×Q8C2×D4116yescomplemented
107C2×C3×C4×C5C2×D4112yes—
108C2×C3×C4×C5C2416yes—
109C2×Dic15—26no—
110C2×Dic15C2×Q814yescomplemented
111C2×Dic15C2416yes—
112C2×Dic15C2412yes—
113C2×C5×Dic3—26no—
114C2×C5×Dic3C2416yes—
115C4×C5×D3—224no—
116C5×Dic6—224no—
117C2×C3×Dic5—26no—
118C2×C3×Dic5C2416yes—
119C3×C4×D5—224no—
120C3×Dic10—224no—
121#1201C2×Q8116yesdirect product factor
122C22×C3×Q8—58no—
123C22×C3×Q8—53no—
124C22×C3×Q8—108no—
125C22×C3×Q8C2×D511yescharacteristic
126C23×C3×C4—53no—
127C22×#67—51no—
128C2×D3×Q8—28no—
129C2×D3×Q8C2×D518yescomplemented
130C22×C4×D3C2×D513yes—
131C22×Dic6C2×D513yes—
132C2×#217—53no—
133#946—524no—
134Q8×Dic3—58no—
135#944—524no—
136C4×#67—524no—
137C2×(C2×D3)⋊C4—53no—
138C2×C4⋊Dic3—53no—
139#924—524no—
140C2×Dic3⋊C4—56no—
141C2×Dic3⋊C4—53no—
142C2×C4×Dic3—53no—
143C24×C5—31no—
144C23×D5C22×D311yescharacteristic
145C2×C5×Q8—312no—
146C2×C5×Q8—632no—
147C2×C5×Q8—612no—
148C2×C5×Q8#6718yescomplemented
149C2×C5×Q8C22×D314yes—
150C22×C4×C5—33no—
151C22×C4×C5—312no—
152C22×C4×C5—612no—
153C22×C4×C5C22×D313yes—
154C2×C22⋊D5—38no—
155C2×C22⋊D5—34no—
156C22×Dic5—31no—
157C22×Dic5—33no—
158C22×Dic5C22×D313yes—
159Q8×D5—232no—
160C2×C4×D5—212no—
161C2×C4×D5C22×D3112yes—
162C2×Dic10—212no—
163C2×Dic10C22×D3112yes—
164C22⋊Dic5—312no—
165#685—312no—
166C4⋊Dic5—312no—
167#683—324no—
168#683—312no—
169C4×Dic5—312no—
170C23×Q8—31no—
171C23×Q8—51no—
172D4×Q8—1516no—
173C2×C4⋊Q8—153no—
174C2×C22⋊Q8—153no—
175C2×C22⋊Q8—153no—
176C2×C4×Q8—151no—
177C2×C4×D4—153no—
178C22×C3×C5C22×D412yes—
179C22×C3×C5C22×Q812yes—
180C22×C3×C52−1+412yes—
181C22×C3×C5C2511yescharacteristic
182C2×C5×D3—28no—
183C2×C5×D3—22no—
184C2×C5×D3—22no—
185C2×C5×D3C22×Q814yescomplemented
186C2×C3×D5—28no—
187C2×C3×D5—22no—
188C2×C3×D5—22no—
189C2×C3×D5C22×Q814yescomplemented
190C3×C4×C5C22×D4112yes—
191Dic15—26no—
192Dic15C22×Q814yescomplemented
193C3×Dic5—26no—
194C5×Dic3—26no—
195C24×C3—51no—
196C23×D3C22×D511yescharacteristic
197C2×C3×Q8—512no—
198C2×C3×Q8—1032no—
199C2×C3×Q8—1012no—
200C2×C3×Q8C22⋊D518yescomplemented
201C2×C3×Q8C22×D514yes—
202C22×C3×C4—53no—
203C22×C3×C4—512no—
204C22×C3×C4—1012no—
205C22×C3×C4C22×D513yes—
206C2×#67—58no—
207C2×#67—54no—
208C22×Dic3—51no—
209C22×Dic3—53no—
210C22×Dic3C22×D513yes—
211D3×Q8—232no—
212C2×C4×D3—212no—
213C2×C4×D3C22×D5112yes—
214C2×Dic6—212no—
215C2×Dic6C22×D5112yes—
216#217—512no—
217(C2×D3)⋊C4—512no—
218C4⋊Dic3—512no—
219Dic3⋊C4—524no—
220Dic3⋊C4—512no—
221C4×Dic3—512no—
222C23×C5—34no—
223C23×C5—32no—
224C23×C5—62no—
225C23×C5—62no—
226C23×C5C23×D311yescharacteristic
227C22×D5—22no—
228C22×D5—22no—
229C22×D5D3×Q814yescomplemented
230C22×D5C23×D312yes—
231C5×Q8—624no—
232C5×Q8C2×#67116yescomplemented
233C2×C4×C5—36no—
234C2×C4×C5—66no—
235C2×C4×C5—624no—
236C2×C4×C5C2×#67112yes—
237C2×C4×C5C23×D316yes—
238C22⋊D5—316no—
239C2×Dic5—26no—
240C2×Dic5—34no—
241C2×Dic5—32no—
242C2×Dic5—36no—
243C2×Dic5—66no—
244C2×Dic5C23×D316yes—
245C4×D5—224no—
246Dic10—224no—
247C22×Q8—38no—
248C22×Q8—33no—
249C22×Q8—58no—
250C22×Q8—53no—
251C22×Q8—68no—
252C22×Q8—108no—
253C22×Q8D3×D511yescharacteristic
254C22×D4—151no—
255C23×C4—33no—
256C23×C4—53no—
257C4⋊Q8—1524no—
258C22⋊Q8—1524no—
259C22⋊Q8—1524no—
260C4×Q8—158no—
261C4×D4—1524no—
262C2×C4⋊C4—153no—
263C2×C4⋊C4—156no—
264C2×C4⋊C4—153no—
265C2×C22⋊C4—153no—
266C2×C22⋊C4—153no—
267C2×C42—153no—
268C2×C3×C5D4×Q814yescomplemented
269C2×C3×C5C23×D411yescharacteristic
270C2×C3×C5C23×Q811yescharacteristic
271C2×C3×C5C2×2−1+411yescharacteristic
272C3×D5—24no—
273C5×D3—24no—
274C23×C3—54no—
275C23×C3—52no—
276C23×C3—102no—
277C23×C3—102no—
278C23×C3C23×D511yescharacteristic
279C22×D3—22no—
280C22×D3—22no—
281C22×D3Q8×D514yescomplemented
282C22×D3C23×D512yes—
283C3×Q8—1024no—
284C3×Q8C2×C22⋊D5116yescomplemented
285C2×C3×C4—56no—
286C2×C3×C4—106no—
287C2×C3×C4—1024no—
288C2×C3×C4C2×C22⋊D5112yes—
289C2×C3×C4C23×D516yes—
290#67—516no—
291C2×Dic3—26no—
292C2×Dic3—54no—
293C2×Dic3—52no—
294C2×Dic3—56no—
295C2×Dic3—106no—
296C2×Dic3C23×D516yes—
297C4×D3—224no—
298Dic6—224no—
299C22×C5—34no—
300C22×C5—62no—
301C22×C5—62no—
302C22×C5—68no—
303C22×C5C2×D3×Q812yes—
304C22×C5#100712yes—
305C22×C5C22×#6712yes—
306C22×C5C24×D311yescharacteristic
307C2×D5—22no—
308C2×D5—22no—
309C2×D5—28no—
310C2×D5C2×D3×Q814yescomplemented
311C4×C5—66no—
312C4×C5C22×#67112yes—
313Dic5—26no—
314Dic5—34no—
315Dic5—66no—
316C24—31no—
317C24—51no—
318C2×Q8—312no—
319C2×Q8—512no—
320C2×Q8—632no—
321C2×Q8—612no—
322C2×Q8—1032no—
323C2×Q8—1012no—
324C2×Q8#120118yesdirect product factor
325C2×Q8C2×D3×D514yes—
326C2×D4—154no—
327C2×D4—158no—
328C22×C4—33no—
329C22×C4—312no—
330C22×C4—53no—
331C22×C4—512no—
332C22×C4—612no—
333C22×C4—1012no—
334C22×C4—151no—
335C22×C4—153no—
336C22×C4C2×D3×D513yes—
337C4⋊C4—1512no—
338C4⋊C4—1524no—
339C4⋊C4—1512no—
340C22⋊C4—1512no—
341C22⋊C4—1512no—
342C42—1512no—
343C3×C5C2×D4×Q811yescharacteristic, complemented, Hall
344C22×C3—54no—
345C22×C3—108no—
346C22×C3—102no—
347C22×C3—102no—
348C22×C3C2×Q8×D512yes—
349C22×C3#414712yes—
350C22×C3C22×C22⋊D512yes—
351C22×C3C24×D511yescharacteristic
352C2×D3—28no—
353C2×D3—22no—
354C2×D3—22no—
355C2×D3C2×Q8×D514yescomplemented
356C3×C4—106no—
357C3×C4C22×C22⋊D5112yes—
358Dic3—26no—
359Dic3—54no—
360Dic3—106no—
361C2×C5—64no—
362C2×C5Q8×#6714yescomplemented
363C2×C5C22×D3×Q811yescharacteristic
364C2×C5C2×#100711yescharacteristic
365C2×C5C23×#6711yescharacteristic
366D5—24no—
367C23—34no—
368C23—32no—
369C23—54no—
370C23—52no—
371C23—62no—
372C23—62no—
373C23—102no—
374C23—102no—
375C23C22×D3×D511yescharacteristic
376Q8—624no—
377Q8—1024no—
378Q8C2×#1201116yesdirect product factor
379D4—1516no—
380C2×C4—36no—
381C2×C4—56no—
382C2×C4—66no—
383C2×C4—624no—
384C2×C4—106no—
385C2×C4—1024no—
386C2×C4—154no—
387C2×C4—152no—
388C2×C4—156no—
389C2×C4—306no—
390C2×C4C2×#1201112yes—
391C2×C4C22×D3×D516yes—
392C2×C3—104no—
393C2×C3Q8×C22⋊D514yescomplemented
394C2×C3C22×Q8×D511yescharacteristic
395C2×C3C2×#414711yescharacteristic
396C2×C3C23×C22⋊D511yescharacteristic
397D3—24no—
398C5C2×Q8×#6711yescharacteristic, complemented, Sylow
399C22—34no—
400C22—54no—
401C22—62no—
402C22—68no—
403C22—62no—
404C22—102no—
405C22—108no—
406C22—102no—
407C22#9202912yes—
408C22D3×Q8×D512yes—
409C22C22×#120112yes—
410C22C23×D3×D511yescharacteristic
411C4—66no—
412C4—106no—
413C4—154no—
414C4—306no—
415C4C22×#1201112yes—
416C3C2×Q8×C22⋊D511yescharacteristic, complemented, Sylow
417C2—64no—
418C2—104no—
419C2Q8×#120114yesdirect product factor
420C2C2×#9202911yescharacteristic
421C2C2×D3×Q8×D511yescharacteristic
422C2C23×#120111yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C24
prime radicalC22×#1201C22
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×D4×Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C5×D3×Q8C2semidirect
2. C22×C3×Q8×D5C2semidirect
3. C2×Q8×Dic15C2semidirect
4. Q8×#1201C2direct (N × H)
6. C2×#91385C2semidirect
7. C2×#91369C2semidirect
8. C2×#91359C2semidirect
12. C2×C5×D3×Q8C22semidirect
16. C2×C3×Q8×D5C22semidirect
20. Q8×Dic15C22semidirect
28. #91385C22semidirect
29. #91369C22semidirect
30. #91359C22semidirect
33. C22×Q8×D5D3semidirect
42. C2×C3×C5×Q8D4semidirect
59. C2×#1201Q8direct (N × H)
68. C22×D3×Q8D5semidirect
82. C2×Q8×D5C2×D3semidirect
100. C22×C5×D3C2×Q8semidirect
104. C22×C3×D5C2×Q8semidirect
106. C3×C5×Q8C2×D4semidirect
110. C2×Dic15C2×Q8semidirect
121. #1201C2×Q8direct (N × H)
129. C2×D3×Q8C2×D5semidirect
148. C2×C5×Q8#67semidirect
185. C2×C5×D3C22×Q8semidirect
189. C2×C3×D5C22×Q8semidirect
192. Dic15C22×Q8semidirect
200. C2×C3×Q8C22⋊D5semidirect
229. C22×D5D3×Q8semidirect
232. C5×Q8C2×#67semidirect
268. C2×C3×C5D4×Q8semidirect
281. C22×D3Q8×D5semidirect
284. C3×Q8C2×C22⋊D5semidirect
310. C2×D5C2×D3×Q8semidirect
324. C2×Q8#1201direct (N × H)
343. C3×C5C2×D4×Q8semidirect
355. C2×D3C2×Q8×D5semidirect
362. C2×C5Q8×#67semidirect
378. Q8C2×#1201direct (N × H)
393. C2×C3Q8×C22⋊D5semidirect
398. C5C2×Q8×#67semidirect
416. C3C2×Q8×C22⋊D5semidirect
419. C2Q8×#1201direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×D3×Q82
C22×C3×Q8×D52
C2×Q8×Dic152
Q8×#12012
C2×C4×#12012
C2×#913852
C2×#913692
C2×#913592
C2×Q8×C22⋊D53
C2×Q8×#675