Finite Group Databasebeta

← C32×Q8×#67 #13179192

Subgroups — C32×Q8×#67 #13179192

Order 1728.

2304 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×Q8C211yesmaximal, characteristic, complemented
2C2×C32×D3×Q8C211yesmaximal, characteristic, complemented
3C32×#946C213yesmaximal, complemented
4C32×Q8×Dic3C211yesmaximal, characteristic, complemented
5C32×#944C213yesmaximal, complemented
6C32×C4×#67C213yesmaximal
7C32×#924C213yesmaximal, complemented
8C32×D4×Q8—31nomaximal
9C3×Q8×#67C314yesmaximal, direct product factor
10C2×C33×Q8—24no—
11C2×C33×Q8C2213yes—
12C2×C33×Q8C2211yescharacteristic
13C22×C33×C4C2213yes—
14C2×C32×#67C2211yescharacteristic
15C32×D3×Q8—24no—
16C2×C32×C4×D3C2213yes—
17C2×C32×Dic6C2213yes—
18C32×#217C2213yes—
19C32×(C2×D3)⋊C4C2213yes—
20C32×C4⋊Dic3C2213yes—
21C32×Dic3⋊C4C2216yes—
22C32×Dic3⋊C4C2213yes—
23C32×C4×Dic3C2213yes—
24C22×C32×Q8—24no—
25C22×C32×Q8—31no—
26C22×C32×Q8D311yescharacteristic, complemented
27C22×C32×Q8C2×C314yescomplemented
28C2×C3×D3×Q8C2×C314yescomplemented
29C32×C4⋊Q8—33no—
30C32×C22⋊Q8—33no—
31C32×C22⋊Q8—33no—
32C32×C4×Q8—31no—
33C32×C4×D4—33no—
34C3×#946C2×C3112yescomplemented
35C3×Q8×Dic3C2×C314yescomplemented
36C3×#944C2×C3112yescomplemented
37C3×C4×#67C2×C3112yes—
38C3×#924C2×C3112yescomplemented
39C23×C33C2311yescharacteristic
40C22×C32×D3C2311yescharacteristic
41C33×Q8—26no—
42C33×Q8D414yescomplemented
43C2×C33×C4—26no—
44C2×C33×C4C2313yes—
45C2×C33×C4C2313yes—
46C32×#67Q814yesdirect product factor
47C2×C32×Dic3C2313yes—
48C2×C32×Dic3C2313yes—
49C2×C32×Dic3C2311yescharacteristic
50C32×C4×D3—26no—
51C32×Dic6—26no—
52C3×D4×Q8—34no—
53Q8×#67C3211yescharacteristic, direct product factor
54C2×C32×Q8—232no—
55C2×C32×Q8—212no—
56C2×C32×Q8—24no—
57C2×C32×Q8—24no—
58C2×C32×Q8—216no—
59C2×C32×Q8—33no—
60C2×C32×Q8—64no—
61C2×C32×Q8C2×D313yes—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×Q8C22×C3112yes—
64C2×C32×Q8C22×C314yes—
65C2×C32×D4—31no—
66C22×C32×C4—212no—
67C22×C32×C4—33no—
68C22×C32×C4C2×D313yes—
69C22×C32×C4C22×C3112yes—
70C2×C3×#67C22×C314yes—
71C3×D3×Q8—216no—
72C2×C3×C4×D3C22×C3112yes—
73C2×C3×Dic6C22×C3112yes—
74C32×C4⋊C4—33no—
75C32×C4⋊C4—33no—
76C32×C4⋊C4—36no—
77C32×C22⋊C4—33no—
78C32×C22⋊C4—33no—
79C32×C42—33no—
80C3×#217C22×C3112yes—
81C3×(C2×D3)⋊C4C22×C3112yes—
82C3×C4⋊Dic3C22×C3112yes—
83C3×Dic3⋊C4C22×C3124yes—
84C3×Dic3⋊C4C22×C3112yes—
85C3×C4×Dic3C22×C3112yes—
86C22×C33—21no—
87C22×C33—21no—
88C22×C33C2×Q812yescomplemented
89C22×C33C2411yescharacteristic
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3C2×Q812yescomplemented
93C33×C4—23no—
94C33×C4C2×D416yes—
95C32×Dic3—23no—
96C32×Dic3—23no—
97C32×Dic3C2×Q812yescomplemented
98C22×C3×Q8—24no—
99C22×C3×Q8—34no—
100C22×C3×Q8C3×D314yescomplemented
101C22×C3×Q8C2×C3211yescharacteristic, complemented
102C2×D3×Q8C2×C3211yescharacteristic, complemented
103C3×C4⋊Q8—312no—
104C3×C22⋊Q8—312no—
105C3×C22⋊Q8—312no—
106C3×C4×Q8—34no—
107C3×C4×D4—312no—
108#946C2×C3213yescomplemented
109Q8×Dic3C2×C3211yescharacteristic, complemented
110#944C2×C3213yescomplemented
111C4×#67C2×C3213yes—
112#924C2×C3213yescomplemented
113C23×C32—24no—
114C23×C32—31no—
115C23×C32C22×D311yescharacteristic
116C23×C32C23×C314yes—
117C22×C3×D3C23×C314yes—
118C32×Q8—248no—
119C32×Q8—26no—
120C32×Q8—224no—
121C32×Q8—216no—
122C32×Q8—66no—
123C32×Q8#6714yesdirect product factor
124C32×Q8C3×D4116yescomplemented
125C32×D4—34no—
126C2×C32×C4—248no—
127C2×C32×C4—26no—
128C2×C32×C4—224no—
129C2×C32×C4—212no—
130C2×C32×C4—212no—
131C2×C32×C4—31no—
132C2×C32×C4—33no—
133C2×C32×C4—33no—
134C2×C32×C4—66no—
135C2×C32×C4C22×D313yes—
136C2×C32×C4C22×D313yes—
137C2×C32×C4C23×C3112yes—
138C2×C32×C4C23×C3112yes—
139C3×#67C3×Q8116yesdirect product factor
140C2×C3×Dic3C23×C3112yes—
141C2×C3×Dic3C23×C3112yes—
142C2×C3×Dic3C23×C314yes—
143C3×C4×D3—224no—
144C3×Dic6—224no—
145D4×Q8—31noSylow
146C2×C33—22no—
147C2×C33C22×D411yescharacteristic
148C2×C33C22×Q811yescharacteristic
149C2×C332−1+411yescharacteristic
150C32×D3—22no—
151C2×C3×Q8—232no—
152C2×C3×Q8—216no—
153C2×C3×Q8—24no—
154C2×C3×Q8—212no—
155C2×C3×Q8—24no—
156C2×C3×Q8—312no—
157C2×C3×Q8—616no—
158C2×C3×Q8C2×C3×D3112yes—
159C2×C3×Q8C2×C3×D314yes—
160C2×C3×Q8C22×C3213yes—
161C2×C3×Q8C22×C3211yescharacteristic
162C2×C3×D4—34no—
163C22×C3×C4—212no—
164C22×C3×C4—312no—
165C22×C3×C4C2×C3×D3112yes—
166C22×C3×C4C22×C3213yes—
167C2×#67C22×C3211yescharacteristic
168D3×Q8—24no—
169C2×C4×D3C22×C3213yes—
170C2×Dic6C22×C3213yes—
171C3×C4⋊C4—312no—
172C3×C4⋊C4—324no—
173C3×C4⋊C4—312no—
174C3×C22⋊C4—312no—
175C3×C22⋊C4—312no—
176C3×C42—312no—
177#217C22×C3213yes—
178(C2×D3)⋊C4C22×C3213yes—
179C4⋊Dic3C22×C3213yes—
180Dic3⋊C4C22×C3216yes—
181Dic3⋊C4C22×C3213yes—
182C4×Dic3C22×C3213yes—
183C22×C32—28no—
184C22×C32—28no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—24no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—28no—
191C22×C32—32no—
192C22×C32—61no—
193C22×C32—61no—
194C22×C32D3×Q812yescomplemented
195C22×C32C2×C3×Q818yescomplemented
196C22×C32C23×D311yescharacteristic
197C22×C32C24×C314yes—
198C2×C3×D3—24no—
199C2×C3×D3—24no—
200C2×C3×D3C2×C3×Q818yescomplemented
201C32×C4—224no—
202C32×C4—23no—
203C32×C4—212no—
204C32×C4—224no—
205C32×C4—32no—
206C32×C4—63no—
207C32×C4—63no—
208C32×C4C2×#6716yes—
209C32×C4C2×C3×D4124yes—
210C3×Dic3—212no—
211C3×Dic3—212no—
212C3×Dic3C2×C3×Q818yescomplemented
213C22×Q8—31no—
214C22×Q8C32×D311yescharacteristic, complemented
215C4⋊Q8—33no—
216C22⋊Q8—33no—
217C22⋊Q8—33no—
218C4×Q8—31no—
219C4×D4—33no—
220C33D4×Q811yescharacteristic, complemented, Sylow
221C23×C3—24no—
222C23×C3—34no—
223C23×C3C22×C3×D314yes—
224C23×C3C23×C3211yescharacteristic
225C22×D3C23×C3211yescharacteristic
226C3×Q8—248no—
227C3×Q8—224no—
228C3×Q8—26no—
229C3×Q8—216no—
230C3×Q8—624no—
231C3×Q8C3×#67116yesdirect product factor
232C3×Q8C32×D414yescomplemented
233C3×D4—316no—
234C2×C3×C4—248no—
235C2×C3×C4—224no—
236C2×C3×C4—26no—
237C2×C3×C4—212no—
238C2×C3×C4—212no—
239C2×C3×C4—34no—
240C2×C3×C4—312no—
241C2×C3×C4—312no—
242C2×C3×C4—624no—
243C2×C3×C4C22×C3×D3112yes—
244C2×C3×C4C22×C3×D3112yes—
245C2×C3×C4C23×C3213yes—
246C2×C3×C4C23×C3213yes—
247#67C32×Q814yesdirect product factor
248C2×Dic3C23×C3213yes—
249C2×Dic3C23×C3213yes—
250C2×Dic3C23×C3211yescharacteristic
251C4×D3—26no—
252Dic6—26no—
253C2×C32—216no—
254C2×C32—22no—
255C2×C32—28no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—62no—
260C2×C32C2×D3×Q811yescharacteristic
261C2×C32#100711yescharacteristic
262C2×C32C22×#6711yescharacteristic
263C2×C32C22×C3×D414yes—
264C2×C32C22×C3×Q814yes—
265C2×C32C3×2−1+414yes—
266C3×D3—28no—
267C2×Q8—24no—
268C2×Q8—33no—
269C2×Q8—64no—
270C2×Q8C2×C32×D313yes—
271C2×Q8C2×C32×D311yescharacteristic
272C2×D4—31no—
273C22×C4—33no—
274C22×C4C2×C32×D313yes—
275C4⋊C4—33no—
276C4⋊C4—33no—
277C4⋊C4—36no—
278C22⋊C4—33no—
279C22⋊C4—33no—
280C42—33no—
281C22×C3—28no—
282C22×C3—28no—
283C22×C3—24no—
284C22×C3—24no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—24no—
288C22×C3—28no—
289C22×C3—38no—
290C22×C3—64no—
291C22×C3—64no—
292C22×C3C3×D3×Q818yescomplemented
293C22×C3C2×C32×Q812yescomplemented
294C22×C3C23×C3×D314yes—
295C22×C3C24×C3211yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3C2×C32×Q812yescomplemented
299C3×C4—224no—
300C3×C4—212no—
301C3×C4—23no—
302C3×C4—224no—
303C3×C4—38no—
304C3×C4—612no—
305C3×C4—612no—
306C3×C4C2×C3×#67124yes—
307C3×C4C2×C32×D416yes—
308Dic3—23no—
309Dic3—23no—
310Dic3C2×C32×Q812yescomplemented
311C32—24no—
312C32Q8×#6711yescharacteristic, direct product factor
313C32C3×D4×Q814yescomplemented
314C23—31no—
315C23C22×C32×D311yescharacteristic
316Q8—26no—
317Q8—66no—
318Q8C32×#6714yesdirect product factor
319D4—34no—
320C2×C4—26no—
321C2×C4—31no—
322C2×C4—33no—
323C2×C4—33no—
324C2×C4—66no—
325C2×C4C22×C32×D313yes—
326C2×C4C22×C32×D313yes—
327C2×C3—216no—
328C2×C3—28no—
329C2×C3—22no—
330C2×C3—24no—
331C2×C3—24no—
332C2×C3—24no—
333C2×C3—68no—
334C2×C3C2×C3×D3×Q814yes—
335C2×C3C3×#100714yes—
336C2×C3C22×C3×#6714yes—
337C2×C3C22×C32×D411yescharacteristic
338C2×C3C22×C32×Q811yescharacteristic
339C2×C3C32×2−1+411yescharacteristic
340D3—22no—
341C22—21no—
342C22—21no—
343C22—32no—
344C22—61no—
345C22—61no—
346C22C32×D3×Q812yescomplemented
347C22C23×C32×D311yescharacteristic
348C4—23no—
349C4—32no—
350C4—63no—
351C4—63no—
352C4C2×C32×#6716yes—
353C3—24no—
354C3C3×Q8×#6714yesdirect product factor
355C3C32×D4×Q811yescharacteristic, complemented
356C2—22no—
357C2—62no—
358C2C2×C32×D3×Q811yescharacteristic
359C2C32×#100711yescharacteristic
360C2C22×C32×#6711yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC22×C33×Q8C2
socleC22×C33C24
prime radicalC2×C32×#67C22
hypercenterC22×C32×Q8D3
hypocenterC3C32×D4×Q8
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C22×C32×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C33×Q8C2semidirect
2. C2×C32×D3×Q8C2semidirect
3. C32×#946C2semidirect
4. C32×Q8×Dic3C2semidirect
5. C32×#944C2semidirect
7. C32×#924C2semidirect
9. C3×Q8×#67C3direct (N × H)
26. C22×C32×Q8D3semidirect
27. C22×C32×Q8C2×C3semidirect
28. C2×C3×D3×Q8C2×C3semidirect
34. C3×#946C2×C3semidirect
35. C3×Q8×Dic3C2×C3semidirect
36. C3×#944C2×C3semidirect
38. C3×#924C2×C3semidirect
42. C33×Q8D4semidirect
46. C32×#67Q8direct (N × H)
53. Q8×#67C32direct (N × H)
88. C22×C33C2×Q8semidirect
92. C2×C32×D3C2×Q8semidirect
97. C32×Dic3C2×Q8semidirect
100. C22×C3×Q8C3×D3semidirect
101. C22×C3×Q8C2×C32semidirect
102. C2×D3×Q8C2×C32semidirect
108. #946C2×C32semidirect
109. Q8×Dic3C2×C32semidirect
110. #944C2×C32semidirect
112. #924C2×C32semidirect
123. C32×Q8#67direct (N × H)
124. C32×Q8C3×D4semidirect
139. C3×#67C3×Q8direct (N × H)
194. C22×C32D3×Q8semidirect
195. C22×C32C2×C3×Q8semidirect
200. C2×C3×D3C2×C3×Q8semidirect
212. C3×Dic3C2×C3×Q8semidirect
214. C22×Q8C32×D3semidirect
220. C33D4×Q8semidirect
231. C3×Q8C3×#67direct (N × H)
232. C3×Q8C32×D4semidirect
247. #67C32×Q8direct (N × H)
292. C22×C3C3×D3×Q8semidirect
293. C22×C3C2×C32×Q8semidirect
298. C2×D3C2×C32×Q8semidirect
310. Dic3C2×C32×Q8semidirect
312. C32Q8×#67direct (N × H)
313. C32C3×D4×Q8semidirect
318. Q8C32×#67direct (N × H)
346. C22C32×D3×Q8semidirect
354. C3C3×Q8×#67direct (N × H)
355. C3C32×D4×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C33×Q82
C2×C32×D3×Q82
C32×#9462
C32×Q8×Dic32
C32×#9442
C32×C4×#672
C32×#9242
C32×D4×Q83
C3×Q8×#673