Finite Group Databasebeta

← C2×C3×Q8×#67 #11902646

Subgroups — C2×C3×Q8×#67 #11902646

Order 1152.

3561 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×Q8C211yesmaximal, characteristic, complemented
2C22×C3×D3×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×#67C2116yesmaximal, direct product factor
4C2×C3×#946C213yesmaximal, complemented
5C2×C3×Q8×Dic3C211yesmaximal, characteristic, complemented
6C2×C3×#944C213yesmaximal, complemented
7C2×C3×C4×#67C213yesmaximal
8C2×C3×#924C213yesmaximal, complemented
9C2×C3×D4×Q8—31nomaximal
10C2×Q8×#67C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8—28no—
12C22×C32×Q8C2213yes—
13C22×C32×Q8C2211yescharacteristic
14C22×C32×Q8C2218yescomplemented
15C23×C32×C4C2213yes—
16C22×C3×#67C2211yescharacteristic
17C2×C3×D3×Q8—28no—
18C2×C3×D3×Q8C2218yescomplemented
19C22×C3×C4×D3C2213yes—
20C22×C3×Dic6C2213yes—
21C2×C3×#217C2213yes—
22C3×#946C22124yescomplemented
23C3×Q8×Dic3C2218yescomplemented
24C3×#944C22124yescomplemented
25C3×C4×#67C22124yes—
26C2×C3×(C2×D3)⋊C4C2213yes—
27C2×C3×C4⋊Dic3C2213yes—
28C3×#924C22124yescomplemented
29C2×C3×Dic3⋊C4C2216yes—
30C2×C3×Dic3⋊C4C2213yes—
31C2×C3×C4×Dic3C2213yes—
32C23×C3×Q8—21no—
33C23×C3×Q8—31no—
34C23×C3×Q8D311yescharacteristic, complemented
35C23×C3×Q8C2×C311yescharacteristic, complemented
36C22×D3×Q8C2×C311yescharacteristic, complemented
37C3×D4×Q8—316no—
38C2×C3×C4⋊Q8—33no—
39C2×C3×C22⋊Q8—33no—
40C2×C3×C22⋊Q8—33no—
41C2×C3×C4×Q8—31no—
42C2×C3×C4×D4—33no—
43Q8×#67C2×C3116yesdirect product factor
44C2×#946C2×C313yescomplemented
45C2×Q8×Dic3C2×C311yescharacteristic, complemented
46C2×#944C2×C313yescomplemented
47C2×C4×#67C2×C313yes—
48C2×#924C2×C313yescomplemented
49C24×C32C2311yescharacteristic
50C23×C3×D3C2311yescharacteristic
51C2×C32×Q8—232no—
52C2×C32×Q8—212no—
53C2×C32×Q8D418yescomplemented
54C2×C32×Q8C23112yes—
55C2×C32×Q8C2314yes—
56C22×C32×C4—212no—
57C22×C32×C4C2313yes—
58C22×C32×C4C23112yes—
59C22×C32×C4C2313yes—
60C2×C3×#67Q818yesdirect product factor
61C2×C3×#67C2314yes—
62C22×C3×Dic3C2313yes—
63C22×C3×Dic3C2313yes—
64C22×C3×Dic3C2311yescharacteristic
65C3×D3×Q8—232no—
66C2×C3×C4×D3—212no—
67C2×C3×C4×D3C23112yes—
68C2×C3×Dic6—212no—
69C2×C3×Dic6C23112yes—
70C3×#217C23112yes—
71C3×(C2×D3)⋊C4C23112yes—
72C3×C4⋊Dic3C23112yes—
73C3×Dic3⋊C4C23124yes—
74C3×Dic3⋊C4C23112yes—
75C3×C4×Dic3C23112yes—
76C2×D4×Q8—31noSylow
77C22×C3×Q8—216no—
78C22×C3×Q8—28no—
79C22×C3×Q8—28no—
80C22×C3×Q8—23no—
81C22×C3×Q8—21no—
82C22×C3×Q8—28no—
83C22×C3×Q8—38no—
84C22×C3×Q8—33no—
85C22×C3×Q8—68no—
86C22×C3×Q8C2×D313yes—
87C22×C3×Q8C2×D311yescharacteristic
88C22×C3×Q8C2×D318yescomplemented
89C22×C3×Q8C22×C313yes—
90C22×C3×Q8C22×C311yescharacteristic
91C22×C3×Q8C22×C318yescomplemented
92C22×C3×D4—31no—
93C23×C3×C4—23no—
94C23×C3×C4—33no—
95C23×C3×C4C2×D313yes—
96C23×C3×C4C22×C313yes—
97C22×#67C22×C311yescharacteristic
98C2×D3×Q8—28no—
99C2×D3×Q8C22×C318yescomplemented
100C22×C4×D3C22×C313yes—
101C22×Dic6C22×C313yes—
102C3×C4⋊Q8—324no—
103C3×C22⋊Q8—324no—
104C3×C22⋊Q8—324no—
105C3×C4×Q8—38no—
106C3×C4×D4—324no—
107C2×C3×C4⋊C4—33no—
108C2×C3×C4⋊C4—36no—
109C2×C3×C4⋊C4—33no—
110C2×C3×C22⋊C4—33no—
111C2×C3×C22⋊C4—33no—
112C2×C3×C42—33no—
113C2×#217C22×C313yes—
114#946C22×C3124yescomplemented
115Q8×Dic3C22×C318yescomplemented
116#944C22×C3124yescomplemented
117C4×#67C22×C3124yes—
118C2×(C2×D3)⋊C4C22×C313yes—
119C2×C4⋊Dic3C22×C313yes—
120#924C22×C3124yescomplemented
121C2×Dic3⋊C4C22×C316yes—
122C2×Dic3⋊C4C22×C313yes—
123C2×C4×Dic3C22×C313yes—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32C2×Q814yescomplemented
127C23×C32C2412yes—
128C23×C32C2411yescharacteristic
129C22×C3×D3—22no—
130C22×C3×D3—22no—
131C22×C3×D3C2×Q814yescomplemented
132C22×C3×D3C2412yes—
133C32×Q8—224no—
134C32×Q8C2×D4116yescomplemented
135C2×C32×C4—26no—
136C2×C32×C4—224no—
137C2×C32×C4C2×D4112yes—
138C2×C32×C4C2416yes—
139C2×C32×C4C2416yes—
140C3×#67C2×Q8116yesdirect product factor
141C2×C3×Dic3—26no—
142C2×C3×Dic3—26no—
143C2×C3×Dic3C2×Q814yescomplemented
144C2×C3×Dic3C2416yes—
145C2×C3×Dic3C2416yes—
146C2×C3×Dic3C2412yes—
147C3×C4×D3—224no—
148C3×Dic6—224no—
149C23×Q8—31no—
150C23×Q8C3×D311yescharacteristic, complemented
151D4×Q8—316no—
152C2×C4⋊Q8—33no—
153C2×C22⋊Q8—33no—
154C2×C22⋊Q8—33no—
155C2×C4×Q8—31no—
156C2×C4×D4—33no—
157C24×C3—21no—
158C24×C3—31no—
159C24×C3C22×D311yescharacteristic
160C24×C3C23×C311yescharacteristic
161C23×D3C23×C311yescharacteristic
162C2×C3×Q8—264no—
163C2×C3×Q8—224no—
164C2×C3×Q8—232no—
165C2×C3×Q8—232no—
166C2×C3×Q8—212no—
167C2×C3×Q8—212no—
168C2×C3×Q8—28no—
169C2×C3×Q8—212no—
170C2×C3×Q8—24no—
171C2×C3×Q8—312no—
172C2×C3×Q8—632no—
173C2×C3×Q8—612no—
174C2×C3×Q8#6718yesdirect product factor
175C2×C3×Q8C3×D418yescomplemented
176C2×C3×Q8C22×D3112yes—
177C2×C3×Q8C22×D314yes—
178C2×C3×Q8C23×C3112yes—
179C2×C3×Q8C23×C314yes—
180C2×C3×D4—38no—
181C2×C3×D4—34no—
182C22×C3×C4—224no—
183C22×C3×C4—212no—
184C22×C3×C4—212no—
185C22×C3×C4—23no—
186C22×C3×C4—212no—
187C22×C3×C4—23no—
188C22×C3×C4—31no—
189C22×C3×C4—33no—
190C22×C3×C4—33no—
191C22×C3×C4—312no—
192C22×C3×C4—612no—
193C22×C3×C4C22×D3112yes—
194C22×C3×C4C22×D313yes—
195C22×C3×C4C22×D313yes—
196C22×C3×C4C23×C313yes—
197C22×C3×C4C23×C3112yes—
198C22×C3×C4C23×C313yes—
199C2×#67C3×Q818yesdirect product factor
200C2×#67C23×C314yes—
201C22×Dic3C23×C313yes—
202C22×Dic3C23×C313yes—
203C22×Dic3C23×C311yescharacteristic
204D3×Q8—232no—
205C2×C4×D3—212no—
206C2×C4×D3C23×C3112yes—
207C2×Dic6—212no—
208C2×Dic6C23×C3112yes—
209C3×C4⋊C4—312no—
210C3×C4⋊C4—324no—
211C3×C4⋊C4—312no—
212C3×C22⋊C4—312no—
213C3×C22⋊C4—312no—
214C3×C42—312no—
215#217C23×C3112yes—
216(C2×D3)⋊C4C23×C3112yes—
217C4⋊Dic3C23×C3112yes—
218Dic3⋊C4C23×C3124yes—
219Dic3⋊C4C23×C3112yes—
220C4×Dic3C23×C3112yes—
221C22×C32—28no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32C22×D412yes—
225C22×C32C22×Q812yes—
226C22×C32C22×Q814yescomplemented
227C22×C322−1+412yes—
228C22×C32C2511yescharacteristic
229C2×C3×D3—28no—
230C2×C3×D3—22no—
231C2×C3×D3—22no—
232C2×C3×D3C22×Q814yescomplemented
233C32×C4—26no—
234C32×C4C22×D4112yes—
235C3×Dic3—26no—
236C3×Dic3—26no—
237C3×Dic3C22×Q814yescomplemented
238C22×Q8—28no—
239C22×Q8—38no—
240C22×Q8—33no—
241C22×Q8—68no—
242C22×Q8C2×C3×D313yes—
243C22×Q8C2×C3×D311yescharacteristic
244C22×Q8C2×C3×D318yescomplemented
245C22×D4—31no—
246C23×C4—33no—
247C23×C4C2×C3×D313yes—
248C4⋊Q8—324no—
249C22⋊Q8—324no—
250C22⋊Q8—324no—
251C4×Q8—38no—
252C4×D4—324no—
253C2×C4⋊C4—33no—
254C2×C4⋊C4—33no—
255C2×C4⋊C4—36no—
256C2×C22⋊C4—33no—
257C2×C22⋊C4—33no—
258C2×C42—33no—
259C23×C3—24no—
260C23×C3—24no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—24no—
268C23×C3—34no—
269C23×C3—32no—
270C23×C3—62no—
271C23×C3—62no—
272C23×C3D3×Q814yescomplemented
273C23×C3C2×C3×Q814yescomplemented
274C23×C3C23×D312yes—
275C23×C3C23×D311yescharacteristic
276C23×C3C24×C312yes—
277C23×C3C24×C311yescharacteristic
278C22×D3—22no—
279C22×D3—22no—
280C22×D3C2×C3×Q814yescomplemented
281C22×D3C24×C312yes—
282C3×Q8—248no—
283C3×Q8—224no—
284C3×Q8—224no—
285C3×Q8—216no—
286C3×Q8—624no—
287C3×Q8C2×#67116yesdirect product factor
288C3×Q8C2×C3×D4116yescomplemented
289C3×D4—316no—
290C2×C3×C4—212no—
291C2×C3×C4—248no—
292C2×C3×C4—26no—
293C2×C3×C4—26no—
294C2×C3×C4—224no—
295C2×C3×C4—224no—
296C2×C3×C4—26no—
297C2×C3×C4—26no—
298C2×C3×C4—212no—
299C2×C3×C4—34no—
300C2×C3×C4—32no—
301C2×C3×C4—36no—
302C2×C3×C4—36no—
303C2×C3×C4—66no—
304C2×C3×C4—66no—
305C2×C3×C4—624no—
306C2×C3×C4C2×#67112yes—
307C2×C3×C4C2×C3×D4112yes—
308C2×C3×C4C23×D316yes—
309C2×C3×C4C23×D316yes—
310C2×C3×C4C24×C316yes—
311C2×C3×C4C24×C316yes—
312#67C2×C3×Q8116yesdirect product factor
313C2×Dic3—26no—
314C2×Dic3—26no—
315C2×Dic3C2×C3×Q814yescomplemented
316C2×Dic3C24×C316yes—
317C2×Dic3C24×C316yes—
318C2×Dic3C24×C312yes—
319C4×D3—224no—
320Dic6—224no—
321C2×C32—24no—
322C2×C32D4×Q814yescomplemented
323C2×C32C23×D411yescharacteristic
324C2×C32C23×Q811yescharacteristic
325C2×C32C2×2−1+411yescharacteristic
326C3×D3—24no—
327C24—31no—
328C24C22×C3×D311yescharacteristic
329C2×Q8—232no—
330C2×Q8—212no—
331C2×Q8—312no—
332C2×Q8—632no—
333C2×Q8—612no—
334C2×Q8C3×#6718yesdirect product factor
335C2×Q8C22×C3×D3112yes—
336C2×Q8C22×C3×D314yes—
337C2×D4—34no—
338C2×D4—38no—
339C22×C4—212no—
340C22×C4—31no—
341C22×C4—33no—
342C22×C4—33no—
343C22×C4—312no—
344C22×C4—612no—
345C22×C4C22×C3×D313yes—
346C22×C4C22×C3×D3112yes—
347C22×C4C22×C3×D313yes—
348C4⋊C4—312no—
349C4⋊C4—312no—
350C4⋊C4—324no—
351C22⋊C4—312no—
352C22⋊C4—312no—
353C42—312no—
354C22×C3—24no—
355C22×C3—216no—
356C22×C3—24no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—28no—
362C22×C3—28no—
363C22×C3—22no—
364C22×C3—24no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—34no—
369C22×C3—62no—
370C22×C3—62no—
371C22×C3—68no—
372C22×C3C2×D3×Q812yes—
373C22×C3C2×D3×Q814yescomplemented
374C22×C3#100712yes—
375C22×C3C22×#6712yes—
376C22×C3C22×C3×D412yes—
377C22×C3C22×C3×Q812yes—
378C22×C3C22×C3×Q814yescomplemented
379C22×C3C3×2−1+412yes—
380C22×C3C24×D311yescharacteristic
381C22×C3C25×C311yescharacteristic
382C2×D3—22no—
383C2×D3—22no—
384C2×D3—28no—
385C2×D3C22×C3×Q814yescomplemented
386C3×C4—212no—
387C3×C4—26no—
388C3×C4—26no—
389C3×C4—212no—
390C3×C4—34no—
391C3×C4—66no—
392C3×C4—66no—
393C3×C4C22×#67112yes—
394C3×C4C22×C3×D4112yes—
395Dic3—26no—
396Dic3—26no—
397Dic3C22×C3×Q814yescomplemented
398C32C2×D4×Q811yescharacteristic, complemented, Sylow
399C23—22no—
400C23—22no—
401C23—34no—
402C23—32no—
403C23—62no—
404C23—62no—
405C23C3×D3×Q814yescomplemented
406C23C23×C3×D312yes—
407C23C23×C3×D311yescharacteristic
408Q8—224no—
409Q8—624no—
410Q8C2×C3×#67116yesdirect product factor
411D4—316no—
412C2×C4—26no—
413C2×C4—224no—
414C2×C4—34no—
415C2×C4—32no—
416C2×C4—36no—
417C2×C4—36no—
418C2×C4—66no—
419C2×C4—66no—
420C2×C4—624no—
421C2×C4C2×C3×#67112yes—
422C2×C4C23×C3×D316yes—
423C2×C4C23×C3×D316yes—
424C2×C3—28no—
425C2×C3—24no—
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—64no—
432C2×C3Q8×#6714yesdirect product factor
433C2×C3C3×D4×Q814yescomplemented
434C2×C3C22×D3×Q811yescharacteristic
435C2×C3C2×#100711yescharacteristic
436C2×C3C23×#6711yescharacteristic
437C2×C3C23×C3×D411yescharacteristic
438C2×C3C23×C3×Q811yescharacteristic
439C2×C3C2×C3×2−1+411yescharacteristic
440D3—24no—
441C22—28no—
442C22—22no—
443C22—22no—
444C22—34no—
445C22—68no—
446C22—62no—
447C22—62no—
448C22C2×C3×D3×Q812yes—
449C22C2×C3×D3×Q814yescomplemented
450C22C3×#100712yes—
451C22C22×C3×#6712yes—
452C22C24×C3×D311yescharacteristic
453C4—26no—
454C4—34no—
455C4—66no—
456C4—66no—
457C4C22×C3×#67112yes—
458C3—21no—
459C3C2×Q8×#6711yescharacteristic, direct product factor
460C3C2×C3×D4×Q811yescharacteristic, complemented
461C2—24no—
462C2—64no—
463C2C3×Q8×#6714yesdirect product factor
464C2C22×C3×D3×Q811yescharacteristic
465C2C2×C3×#100711yescharacteristic
466C2C23×C3×#6711yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC23×C32×Q8C2
socleC23×C32C24
prime radicalC22×C3×#67C22
hypercenterC23×C3×Q8D3
hypocenterC3C2×C3×D4×Q8
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C23×C32×Q8C2semidirect
2. C22×C3×D3×Q8C2semidirect
3. C3×Q8×#67C2direct (N × H)
4. C2×C3×#946C2semidirect
5. C2×C3×Q8×Dic3C2semidirect
6. C2×C3×#944C2semidirect
8. C2×C3×#924C2semidirect
10. C2×Q8×#67C3direct (N × H)
14. C22×C32×Q8C22semidirect
18. C2×C3×D3×Q8C22semidirect
22. C3×#946C22semidirect
23. C3×Q8×Dic3C22semidirect
24. C3×#944C22semidirect
28. C3×#924C22semidirect
34. C23×C3×Q8D3semidirect
35. C23×C3×Q8C2×C3semidirect
36. C22×D3×Q8C2×C3semidirect
43. Q8×#67C2×C3direct (N × H)
44. C2×#946C2×C3semidirect
45. C2×Q8×Dic3C2×C3semidirect
46. C2×#944C2×C3semidirect
48. C2×#924C2×C3semidirect
53. C2×C32×Q8D4semidirect
60. C2×C3×#67Q8direct (N × H)
88. C22×C3×Q8C2×D3semidirect
91. C22×C3×Q8C22×C3semidirect
99. C2×D3×Q8C22×C3semidirect
114. #946C22×C3semidirect
115. Q8×Dic3C22×C3semidirect
116. #944C22×C3semidirect
120. #924C22×C3semidirect
126. C23×C32C2×Q8semidirect
131. C22×C3×D3C2×Q8semidirect
134. C32×Q8C2×D4semidirect
140. C3×#67C2×Q8direct (N × H)
143. C2×C3×Dic3C2×Q8semidirect
150. C23×Q8C3×D3semidirect
174. C2×C3×Q8#67direct (N × H)
175. C2×C3×Q8C3×D4semidirect
199. C2×#67C3×Q8direct (N × H)
226. C22×C32C22×Q8semidirect
232. C2×C3×D3C22×Q8semidirect
237. C3×Dic3C22×Q8semidirect
244. C22×Q8C2×C3×D3semidirect
272. C23×C3D3×Q8semidirect
273. C23×C3C2×C3×Q8semidirect
280. C22×D3C2×C3×Q8semidirect
287. C3×Q8C2×#67direct (N × H)
288. C3×Q8C2×C3×D4semidirect
312. #67C2×C3×Q8direct (N × H)
315. C2×Dic3C2×C3×Q8semidirect
322. C2×C32D4×Q8semidirect
334. C2×Q8C3×#67direct (N × H)
373. C22×C3C2×D3×Q8semidirect
378. C22×C3C22×C3×Q8semidirect
385. C2×D3C22×C3×Q8semidirect
397. Dic3C22×C3×Q8semidirect
398. C32C2×D4×Q8semidirect
405. C23C3×D3×Q8semidirect
410. Q8C2×C3×#67direct (N × H)
432. C2×C3Q8×#67direct (N × H)
433. C2×C3C3×D4×Q8semidirect
449. C22C2×C3×D3×Q8semidirect
459. C3C2×Q8×#67direct (N × H)
460. C3C2×C3×D4×Q8semidirect
463. C2C3×Q8×#67direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×Q82
C22×C3×D3×Q82
C3×Q8×#672
C2×C3×#9462
C2×C3×Q8×Dic32
C2×C3×#9442
C2×C3×C4×#672
C2×C3×#9242
C2×C3×D4×Q83
C2×Q8×#673