Finite Group Databasebeta

← Q8×C9×#67 #13153650

Subgroups — Q8×C9×#67 #13153650

Order 1728.

996 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×C3×Q8×C9C211yesmaximal, characteristic, complemented
2C2×D3×Q8×C9C211yesmaximal, characteristic, complemented
3C9×#946C213yesmaximal, complemented
4Q8×C9×Dic3C211yesmaximal, characteristic, complemented
5C9×#944C213yesmaximal, complemented
6C4×C9×#67C213yesmaximal
7C9×#924C213yesmaximal, complemented
8C3×Q8×#67C311yesmaximal, characteristic
9D4×Q8×C9—31nomaximal
10C2×C3×Q8×C9—24no—
11C2×C3×Q8×C9C2213yes—
12C2×C3×Q8×C9C2211yescharacteristic
13C22×C3×C4×C9C2213yes—
14C2×C9×#67C2211yescharacteristic
15D3×Q8×C9—24no—
16C2×C4×D3×C9C2213yes—
17C2×C9×Dic6C2213yes—
18C9×#217C2213yes—
19C9×(C2×D3)⋊C4C2213yes—
20C9×C4⋊Dic3C2213yes—
21C9×Dic3⋊C4C2216yes—
22C9×Dic3⋊C4C2213yes—
23C4×C9×Dic3C2213yes—
24C22×C32×Q8C2×C311yescharacteristic
25C2×C3×D3×Q8C2×C311yescharacteristic
26C3×#946C2×C313yes—
27C3×Q8×Dic3C2×C311yescharacteristic
28C3×#944C2×C313yes—
29C3×C4×#67C2×C313yes—
30C3×#924C2×C313yes—
31C22×Q8×C9—21no—
32C22×Q8×C9—31no—
33C22×Q8×C9D311yescharacteristic, complemented
34C9×C4⋊Q8—33no—
35C9×C22⋊Q8—33no—
36C9×C22⋊Q8—33no—
37C4×Q8×C9—31no—
38C4×D4×C9—33no—
39C23×C3×C9C2311yescharacteristic
40C22×D3×C9C2311yescharacteristic
41C3×Q8×C9—26no—
42C3×Q8×C9D414yescomplemented
43C2×C3×C4×C9—26no—
44C2×C3×C4×C9C2313yes—
45C2×C3×C4×C9C2313yes—
46C9×#67Q814yesdirect product factor
47C2×C9×Dic3C2313yes—
48C2×C9×Dic3C2313yes—
49C2×C9×Dic3C2311yescharacteristic
50C4×D3×C9—26no—
51C9×Dic6—26no—
52C3×D4×Q8—31no—
53Q8×#67C911yescharacteristic, direct product factor
54C2×C32×Q8—24no—
55C2×C32×Q8C22×C313yes—
56C2×C32×Q8C22×C311yescharacteristic
57C22×C32×C4C22×C313yes—
58C2×C3×#67C22×C311yescharacteristic
59C3×D3×Q8—24no—
60C2×C3×C4×D3C22×C313yes—
61C2×C3×Dic6C22×C313yes—
62C3×#217C22×C313yes—
63C3×(C2×D3)⋊C4C22×C313yes—
64C3×C4⋊Dic3C22×C313yes—
65C3×Dic3⋊C4C22×C316yes—
66C3×Dic3⋊C4C22×C313yes—
67C3×C4×Dic3C22×C313yes—
68C2×Q8×C9—28no—
69C2×Q8×C9—23no—
70C2×Q8×C9—21no—
71C2×Q8×C9—24no—
72C2×Q8×C9—33no—
73C2×Q8×C9—64no—
74C2×Q8×C9C2×D313yes—
75C2×Q8×C9C2×D311yescharacteristic
76C2×D4×C9—31no—
77C22×C4×C9—23no—
78C22×C4×C9—33no—
79C22×C4×C9C2×D313yes—
80C9×C4⋊C4—33no—
81C9×C4⋊C4—33no—
82C9×C4⋊C4—36no—
83C9×C22⋊C4—33no—
84C9×C22⋊C4—33no—
85C42×C9—33no—
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9C2×Q812yescomplemented
89C22×C3×C9C2411yescharacteristic
90C2×D3×C9—21no—
91C2×D3×C9—21no—
92C2×D3×C9C2×Q812yescomplemented
93C3×C4×C9—23no—
94C3×C4×C9C2×D416yes—
95C9×Dic3—23no—
96C9×Dic3—23no—
97C9×Dic3C2×Q812yescomplemented
98C22×C3×Q8—21no—
99C22×C3×Q8—31no—
100C22×C3×Q8C2×C911yescharacteristic, complemented
101C22×C3×Q8C3×D311yescharacteristic
102C2×D3×Q8C2×C911yescharacteristic, complemented
103C3×C4⋊Q8—33no—
104C3×C22⋊Q8—33no—
105C3×C22⋊Q8—33no—
106C3×C4×Q8—31no—
107C3×C4×D4—33no—
108#946C2×C913yescomplemented
109Q8×Dic3C2×C911yescharacteristic, complemented
110#944C2×C913yescomplemented
111C4×#67C2×C913yes—
112#924C2×C913yescomplemented
113C23×C32C23×C311yescharacteristic
114C22×C3×D3C23×C311yescharacteristic
115C32×Q8—26no—
116C32×Q8C3×D414yes—
117C2×C32×C4—26no—
118C2×C32×C4C23×C313yes—
119C2×C32×C4C23×C313yes—
120C3×#67C3×Q814yes—
121C2×C3×Dic3C23×C313yes—
122C2×C3×Dic3C23×C313yes—
123C2×C3×Dic3C23×C311yescharacteristic
124C3×C4×D3—26no—
125C3×Dic6—26no—
126C23×C9—21no—
127C23×C9—31no—
128C23×C9C22×D311yescharacteristic
129Q8×C9—212no—
130Q8×C9—26no—
131Q8×C9—24no—
132Q8×C9—66no—
133Q8×C9#6714yesdirect product factor
134D4×C9—34no—
135C2×C4×C9—212no—
136C2×C4×C9—26no—
137C2×C4×C9—23no—
138C2×C4×C9—23no—
139C2×C4×C9—31no—
140C2×C4×C9—33no—
141C2×C4×C9—33no—
142C2×C4×C9—66no—
143C2×C4×C9C22×D313yes—
144C2×C4×C9C22×D313yes—
145D4×Q8—31noSylow
146C2×C3×C9—22no—
147C2×C3×C9C22×D411yescharacteristic
148C2×C3×C9C22×Q811yescharacteristic
149C2×C3×C92−1+411yescharacteristic
150D3×C9—22no—
151C2×C3×Q8—28no—
152C2×C3×Q8—24no—
153C2×C3×Q8—24no—
154C2×C3×Q8—23no—
155C2×C3×Q8—21no—
156C2×C3×Q8—33no—
157C2×C3×Q8—64no—
158C2×C3×Q8C22×C913yes—
159C2×C3×Q8C22×C911yescharacteristic
160C2×C3×Q8C2×C3×D313yes—
161C2×C3×Q8C2×C3×D311yescharacteristic
162C2×C3×D4—31no—
163C22×C3×C4—23no—
164C22×C3×C4—33no—
165C22×C3×C4C22×C913yes—
166C22×C3×C4C2×C3×D313yes—
167C2×#67C22×C911yescharacteristic
168D3×Q8—24no—
169C2×C4×D3C22×C913yes—
170C2×Dic6C22×C913yes—
171C3×C4⋊C4—33no—
172C3×C4⋊C4—36no—
173C3×C4⋊C4—33no—
174C3×C22⋊C4—33no—
175C3×C22⋊C4—33no—
176C3×C42—33no—
177#217C22×C913yes—
178(C2×D3)⋊C4C22×C913yes—
179C4⋊Dic3C22×C913yes—
180Dic3⋊C4C22×C916yes—
181Dic3⋊C4C22×C913yes—
182C4×Dic3C22×C913yes—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32C2×C3×Q812yes—
186C22×C32C24×C311yescharacteristic
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3C2×C3×Q812yes—
190C32×C4—23no—
191C32×C4C2×C3×D416yes—
192C3×Dic3—23no—
193C3×Dic3—23no—
194C3×Dic3C2×C3×Q812yes—
195C22×C9—22no—
196C22×C9—22no—
197C22×C9—21no—
198C22×C9—21no—
199C22×C9—21no—
200C22×C9—22no—
201C22×C9—32no—
202C22×C9—61no—
203C22×C9—61no—
204C22×C9D3×Q812yescomplemented
205C22×C9C23×D311yescharacteristic
206C4×C9—26no—
207C4×C9—23no—
208C4×C9—26no—
209C4×C9—32no—
210C4×C9—63no—
211C4×C9—63no—
212C4×C9C2×#6716yes—
213C22×Q8—31no—
214C22×Q8D3×C911yescharacteristic, complemented
215C4⋊Q8—33no—
216C22⋊Q8—33no—
217C22⋊Q8—33no—
218C4×Q8—31no—
219C4×D4—33no—
220C3×C9D4×Q811yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—31no—
223C23×C3C23×C911yescharacteristic
224C23×C3C22×C3×D311yescharacteristic
225C22×D3C23×C911yescharacteristic
226C3×Q8—212no—
227C3×Q8—26no—
228C3×Q8—26no—
229C3×Q8—24no—
230C3×Q8—66no—
231C3×Q8D4×C914yescomplemented
232C3×Q8C3×#6714yes—
233C3×D4—34no—
234C2×C3×C4—212no—
235C2×C3×C4—26no—
236C2×C3×C4—26no—
237C2×C3×C4—23no—
238C2×C3×C4—23no—
239C2×C3×C4—31no—
240C2×C3×C4—33no—
241C2×C3×C4—33no—
242C2×C3×C4—66no—
243C2×C3×C4C23×C913yes—
244C2×C3×C4C23×C913yes—
245C2×C3×C4C22×C3×D313yes—
246C2×C3×C4C22×C3×D313yes—
247#67Q8×C914yesdirect product factor
248C2×Dic3C23×C913yes—
249C2×Dic3C23×C913yes—
250C2×Dic3C23×C911yescharacteristic
251C4×D3—26no—
252Dic6—26no—
253C2×C32—22no—
254C2×C32C22×C3×D411yescharacteristic
255C2×C32C22×C3×Q811yescharacteristic
256C2×C32C3×2−1+411yescharacteristic
257C3×D3—22no—
258C2×C9—24no—
259C2×C9—22no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—62no—
264C2×C9C2×D3×Q811yescharacteristic
265C2×C9#100711yescharacteristic
266C2×C9C22×#6711yescharacteristic
267C2×Q8—24no—
268C2×Q8—33no—
269C2×Q8—64no—
270C2×Q8C2×D3×C913yes—
271C2×Q8C2×D3×C911yescharacteristic
272C2×D4—31no—
273C22×C4—33no—
274C22×C4C2×D3×C913yes—
275C4⋊C4—33no—
276C4⋊C4—33no—
277C4⋊C4—36no—
278C22⋊C4—33no—
279C22⋊C4—33no—
280C42—33no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—32no—
290C22×C3—61no—
291C22×C3—61no—
292C22×C3C2×Q8×C912yescomplemented
293C22×C3C24×C911yescharacteristic
294C22×C3C3×D3×Q812yes—
295C22×C3C23×C3×D311yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3C2×Q8×C912yescomplemented
299C3×C4—26no—
300C3×C4—23no—
301C3×C4—23no—
302C3×C4—26no—
303C3×C4—32no—
304C3×C4—63no—
305C3×C4—63no—
306C3×C4C2×D4×C916yes—
307C3×C4C2×C3×#6716yes—
308Dic3—23no—
309Dic3—23no—
310Dic3C2×Q8×C912yescomplemented
311C32C3×D4×Q811yescharacteristic
312C9—21no—
313C9Q8×#6711yescharacteristic, direct product factor
314C23—31no—
315C23C22×D3×C911yescharacteristic
316Q8—26no—
317Q8—66no—
318Q8C9×#6714yesdirect product factor
319D4—34no—
320C2×C4—26no—
321C2×C4—31no—
322C2×C4—33no—
323C2×C4—33no—
324C2×C4—66no—
325C2×C4C22×D3×C913yes—
326C2×C4C22×D3×C913yes—
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—62no—
334C2×C3C22×D4×C911yescharacteristic
335C2×C3C22×Q8×C911yescharacteristic
336C2×C3C9×2−1+411yescharacteristic
337C2×C3C2×C3×D3×Q811yescharacteristic
338C2×C3C3×#100711yescharacteristic
339C2×C3C22×C3×#6711yescharacteristic
340D3—22no—
341C22—21no—
342C22—21no—
343C22—32no—
344C22—61no—
345C22—61no—
346C22D3×Q8×C912yescomplemented
347C22C23×D3×C911yescharacteristic
348C4—23no—
349C4—32no—
350C4—63no—
351C4—63no—
352C4C2×C9×#6716yes—
353C3—21no—
354C3D4×Q8×C911yescharacteristic, complemented
355C3C3×Q8×#6711yescharacteristic
356C2—22no—
357C2—62no—
358C2C2×D3×Q8×C911yescharacteristic
359C2C9×#100711yescharacteristic
360C2C22×C9×#6711yescharacteristic
subgroupG/H
centerC22×C9C23×D3
commutatorC22×C3C24×C9
FrattiniC22×C3C23×C3×D3
FittingC22×C3×Q8×C9C2
socle, cosocleC22×C32C24×C3
prime radicalC2×C3×#67C22×C3
hypercenterC22×Q8×C9D3
hypocenterC3D4×Q8×C9
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C22×Q8×C9

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8×C9C2semidirect
2. C2×D3×Q8×C9C2semidirect
3. C9×#946C2semidirect
4. Q8×C9×Dic3C2semidirect
5. C9×#944C2semidirect
7. C9×#924C2semidirect
33. C22×Q8×C9D3semidirect
42. C3×Q8×C9D4semidirect
46. C9×#67Q8direct (N × H)
53. Q8×#67C9direct (N × H)
88. C22×C3×C9C2×Q8semidirect
92. C2×D3×C9C2×Q8semidirect
97. C9×Dic3C2×Q8semidirect
100. C22×C3×Q8C2×C9semidirect
102. C2×D3×Q8C2×C9semidirect
108. #946C2×C9semidirect
109. Q8×Dic3C2×C9semidirect
110. #944C2×C9semidirect
112. #924C2×C9semidirect
133. Q8×C9#67direct (N × H)
204. C22×C9D3×Q8semidirect
214. C22×Q8D3×C9semidirect
220. C3×C9D4×Q8semidirect
231. C3×Q8D4×C9semidirect
247. #67Q8×C9direct (N × H)
292. C22×C3C2×Q8×C9semidirect
298. C2×D3C2×Q8×C9semidirect
310. Dic3C2×Q8×C9semidirect
313. C9Q8×#67direct (N × H)
318. Q8C9×#67direct (N × H)
346. C22D3×Q8×C9semidirect
354. C3D4×Q8×C9semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Q8×C92
C2×D3×Q8×C92
C9×#9462
Q8×C9×Dic32
C9×#9442
C4×C9×#672
C9×#9242
C3×Q8×#673
D4×Q8×C93