Finite Group Databasebeta

← C3×Q8×#611 #13153645

Subgroups — C3×Q8×#611 #13153645

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×C3×Q8×D9C211yesmaximal, characteristic, complemented
3C3×#63508C213yesmaximal, complemented
4C3×Q8×Dic9C211yesmaximal, characteristic, complemented
5C3×#63506C213yesmaximal, complemented
6C3×C4×#611C213yesmaximal
7C3×#63486C213yesmaximal, complemented
8C3×Q8×#67—31nomaximal
9Q8×#611C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9—24no—
11C2×C3×Q8×C9C2213yes—
12C2×C3×Q8×C9C2211yescharacteristic
13C22×C3×C4×C9C2213yes—
14C2×C3×#611C2211yescharacteristic
15C3×Q8×D9—24no—
16C2×C3×C4×D9C2213yes—
17C2×C3×Dic18C2213yes—
18C3×#3676C2213yes—
19C3×(C2×D9)⋊C4C2213yes—
20C3×C4⋊Dic9C2213yes—
21C3×Dic9⋊C4C2216yes—
22C3×Dic9⋊C4C2213yes—
23C3×C4×Dic9C2213yes—
24C22×C32×Q8D311yescharacteristic
25C2×C3×D3×Q8—31no—
26C3×#946—33no—
27C3×Q8×Dic3—31no—
28C3×#944—33no—
29C3×C4×#67—33no—
30C3×#924—33no—
31C22×Q8×C9—21no—
32C22×Q8×C9C2×C311yescharacteristic, complemented
33C2×Q8×D9C2×C311yescharacteristic, complemented
34#63508C2×C313yescomplemented
35Q8×Dic9C2×C311yescharacteristic, complemented
36#63506C2×C313yescomplemented
37C4×#611C2×C313yes—
38#63486C2×C313yescomplemented
39C23×C3×C9C2311yescharacteristic
40C22×C3×D9C2311yescharacteristic
41C3×Q8×C9—26no—
42C3×Q8×C9D414yescomplemented
43C2×C3×C4×C9—26no—
44C2×C3×C4×C9C2313yes—
45C2×C3×C4×C9C2313yes—
46C3×#611Q814yesdirect product factor
47C2×C3×Dic9C2313yes—
48C2×C3×Dic9C2313yes—
49C2×C3×Dic9C2311yescharacteristic
50C3×C4×D9—26no—
51C3×Dic18—26no—
52C3×D4×Q8—91no—
53Q8×#67—31no—
54C2×C32×Q8—24no—
55C2×C32×Q8C2×D313yes—
56C2×C32×Q8C2×D311yescharacteristic
57C22×C32×C4C2×D313yes—
58C2×C3×#67—31no—
59C3×D3×Q8—64no—
60C2×C3×C4×D3—33no—
61C2×C3×Dic6—33no—
62C3×#217—33no—
63C3×(C2×D3)⋊C4—33no—
64C3×C4⋊Dic3—33no—
65C3×Dic3⋊C4—36no—
66C3×Dic3⋊C4—33no—
67C3×C4×Dic3—33no—
68C2×Q8×C9—28no—
69C2×Q8×C9—23no—
70C2×Q8×C9—21no—
71C2×Q8×C9—24no—
72C2×Q8×C9C22×C313yes—
73C2×Q8×C9C22×C311yescharacteristic
74C22×C4×C9—23no—
75C22×C4×C9C22×C313yes—
76C2×#611C22×C311yescharacteristic
77Q8×D9—24no—
78C2×C4×D9C22×C313yes—
79C2×Dic18C22×C313yes—
80#3676C22×C313yes—
81(C2×D9)⋊C4C22×C313yes—
82C4⋊Dic9C22×C313yes—
83Dic9⋊C4C22×C313yes—
84Dic9⋊C4C22×C316yes—
85C4×Dic9C22×C313yes—
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9C2×Q812yescomplemented
89C22×C3×C9C2411yescharacteristic
90C2×C3×D9—21no—
91C2×C3×D9—21no—
92C2×C3×D9C2×Q812yescomplemented
93C3×C4×C9—23no—
94C3×C4×C9C2×D416yes—
95C3×Dic9—23no—
96C3×Dic9—23no—
97C3×Dic9C2×Q812yescomplemented
98C22×C3×Q8—21no—
99C22×C3×Q8—91no—
100C22×C3×Q8D911yescharacteristic, complemented
101C22×C3×Q8C3×D311yescharacteristic
102C2×D3×Q8—31no—
103C3×C4⋊Q8—93no—
104C3×C22⋊Q8—93no—
105C3×C22⋊Q8—93no—
106C3×C4×Q8—91no—
107C3×C4×D4—93no—
108#946—33no—
109Q8×Dic3—31no—
110#944—33no—
111C4×#67—33no—
112#924—33no—
113C23×C32C22×D311yescharacteristic
114C22×C3×D3—31no—
115C32×Q8—26no—
116C32×Q8#6714yes—
117C2×C32×C4—26no—
118C2×C32×C4C22×D313yes—
119C2×C32×C4C22×D313yes—
120C3×#67—34no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—33no—
123C2×C3×Dic3—33no—
124C3×C4×D3—66no—
125C3×Dic6—66no—
126C23×C9—21no—
127C23×C9C23×C311yescharacteristic
128C22×D9C23×C311yescharacteristic
129Q8×C9—212no—
130Q8×C9—26no—
131Q8×C9—24no—
132Q8×C9C3×D414yescomplemented
133C2×C4×C9—212no—
134C2×C4×C9—26no—
135C2×C4×C9—23no—
136C2×C4×C9—23no—
137C2×C4×C9C23×C313yes—
138C2×C4×C9C23×C313yes—
139#611C3×Q814yesdirect product factor
140C2×Dic9C23×C313yes—
141C2×Dic9C23×C313yes—
142C2×Dic9C23×C311yescharacteristic
143C4×D9—26no—
144Dic18—26no—
145D4×Q8—91noSylow
146C2×C3×C9—22no—
147C2×C3×C9C22×D411yescharacteristic
148C2×C3×C9C22×Q811yescharacteristic
149C2×C3×C92−1+411yescharacteristic
150C3×D9—22no—
151C2×C3×Q8—28no—
152C2×C3×Q8—24no—
153C2×C3×Q8—24no—
154C2×C3×Q8—23no—
155C2×C3×Q8—21no—
156C2×C3×Q8—93no—
157C2×C3×Q8—184no—
158C2×C3×Q8C2×D913yes—
159C2×C3×Q8C2×D911yescharacteristic
160C2×C3×Q8C2×C3×D313yes—
161C2×C3×Q8C2×C3×D311yescharacteristic
162C2×C3×D4—91no—
163C22×C3×C4—23no—
164C22×C3×C4—93no—
165C22×C3×C4C2×D913yes—
166C22×C3×C4C2×C3×D313yes—
167C2×#67—31no—
168D3×Q8—64no—
169C2×C4×D3—33no—
170C2×Dic6—33no—
171C3×C4⋊C4—93no—
172C3×C4⋊C4—96no—
173C3×C4⋊C4—93no—
174C3×C22⋊C4—93no—
175C3×C22⋊C4—93no—
176C3×C42—93no—
177#217—33no—
178(C2×D3)⋊C4—33no—
179C4⋊Dic3—33no—
180Dic3⋊C4—36no—
181Dic3⋊C4—33no—
182C4×Dic3—33no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32D3×Q812yes—
186C22×C32C23×D311yescharacteristic
187C2×C3×D3—32no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C32×C4—23no—
191C32×C4C2×#6716yes—
192C3×Dic3—32no—
193C3×Dic3—63no—
194C3×Dic3—63no—
195C22×C9—22no—
196C22×C9—22no—
197C22×C9—21no—
198C22×C9—21no—
199C22×C9—21no—
200C22×C9—22no—
201C22×C9C2×C3×Q812yescomplemented
202C22×C9C24×C311yescharacteristic
203C2×D9—21no—
204C2×D9—21no—
205C2×D9C2×C3×Q812yescomplemented
206C4×C9—26no—
207C4×C9—23no—
208C4×C9—26no—
209C4×C9C2×C3×D416yes—
210Dic9—23no—
211Dic9—23no—
212Dic9C2×C3×Q812yescomplemented
213C22×Q8—91no—
214C22×Q8C3×D911yescharacteristic, complemented
215C4⋊Q8—93no—
216C22⋊Q8—93no—
217C22⋊Q8—93no—
218C4×Q8—91no—
219C4×D4—93no—
220C3×C9D4×Q811yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—91no—
223C23×C3C22×D911yescharacteristic
224C23×C3C22×C3×D311yescharacteristic
225C22×D3—31no—
226C3×Q8—212no—
227C3×Q8—26no—
228C3×Q8—26no—
229C3×Q8—24no—
230C3×Q8—186no—
231C3×Q8#61114yesdirect product factor
232C3×Q8C3×#6714yes—
233C3×D4—94no—
234C2×C3×C4—212no—
235C2×C3×C4—26no—
236C2×C3×C4—26no—
237C2×C3×C4—23no—
238C2×C3×C4—23no—
239C2×C3×C4—91no—
240C2×C3×C4—93no—
241C2×C3×C4—93no—
242C2×C3×C4—186no—
243C2×C3×C4C22×D913yes—
244C2×C3×C4C22×D913yes—
245C2×C3×C4C22×C3×D313yes—
246C2×C3×C4C22×C3×D313yes—
247#67—34no—
248C2×Dic3—31no—
249C2×Dic3—33no—
250C2×Dic3—33no—
251C4×D3—66no—
252Dic6—66no—
253C2×C32—22no—
254C2×C32C2×D3×Q811yescharacteristic
255C2×C32#100711yescharacteristic
256C2×C32C22×#6711yescharacteristic
257C3×D3—62no—
258C2×C9—24no—
259C2×C9—22no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9C22×C3×D411yescharacteristic
264C2×C9C22×C3×Q811yescharacteristic
265C2×C9C3×2−1+411yescharacteristic
266D9—22no—
267C2×Q8—24no—
268C2×Q8—93no—
269C2×Q8—184no—
270C2×Q8C2×C3×D913yes—
271C2×Q8C2×C3×D911yescharacteristic
272C2×D4—91no—
273C22×C4—93no—
274C22×C4C2×C3×D913yes—
275C4⋊C4—93no—
276C4⋊C4—93no—
277C4⋊C4—96no—
278C22⋊C4—93no—
279C22⋊C4—93no—
280C42—93no—
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—92no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3Q8×D912yescomplemented
293C22×C3C23×D911yescharacteristic
294C22×C3C3×D3×Q812yes—
295C22×C3C23×C3×D311yescharacteristic
296C2×D3—32no—
297C2×D3—61no—
298C2×D3—61no—
299C3×C4—26no—
300C3×C4—23no—
301C3×C4—23no—
302C3×C4—26no—
303C3×C4—92no—
304C3×C4—183no—
305C3×C4—183no—
306C3×C4C2×#61116yes—
307C3×C4C2×C3×#6716yes—
308Dic3—32no—
309Dic3—63no—
310Dic3—63no—
311C32Q8×#6711yescharacteristic
312C9—21no—
313C9C3×D4×Q811yescharacteristic, complemented
314C23—91no—
315C23C22×C3×D911yescharacteristic
316Q8—26no—
317Q8—186no—
318Q8C3×#61114yesdirect product factor
319D4—94no—
320C2×C4—26no—
321C2×C4—91no—
322C2×C4—93no—
323C2×C4—93no—
324C2×C4—186no—
325C2×C4C22×C3×D913yes—
326C2×C4C22×C3×D913yes—
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—182no—
334C2×C3C2×Q8×D911yescharacteristic
335C2×C3#6371311yescharacteristic
336C2×C3C22×#61111yescharacteristic
337C2×C3C2×C3×D3×Q811yescharacteristic
338C2×C3C3×#100711yescharacteristic
339C2×C3C22×C3×#6711yescharacteristic
340D3—62no—
341C22—21no—
342C22—21no—
343C22—92no—
344C22—181no—
345C22—181no—
346C22C3×Q8×D912yescomplemented
347C22C23×C3×D911yescharacteristic
348C4—23no—
349C4—92no—
350C4—183no—
351C4—183no—
352C4C2×C3×#61116yes—
353C3—21no—
354C3Q8×#61111yescharacteristic, direct product factor
355C3C3×Q8×#6711yescharacteristic
356C2—22no—
357C2—182no—
358C2C2×C3×Q8×D911yescharacteristic
359C2C3×#6371311yescharacteristic
360C2C22×C3×#61111yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC22×C3×Q8×C9C2
socleC22×C32C23×D3
prime radicalC2×C3×#611C22
hypercenterC22×C3×Q8D9
hypocenterC9C3×D4×Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8×C9C2semidirect
2. C2×C3×Q8×D9C2semidirect
3. C3×#63508C2semidirect
4. C3×Q8×Dic9C2semidirect
5. C3×#63506C2semidirect
7. C3×#63486C2semidirect
9. Q8×#611C3direct (N × H)
32. C22×Q8×C9C2×C3semidirect
33. C2×Q8×D9C2×C3semidirect
34. #63508C2×C3semidirect
35. Q8×Dic9C2×C3semidirect
36. #63506C2×C3semidirect
38. #63486C2×C3semidirect
42. C3×Q8×C9D4semidirect
46. C3×#611Q8direct (N × H)
88. C22×C3×C9C2×Q8semidirect
92. C2×C3×D9C2×Q8semidirect
97. C3×Dic9C2×Q8semidirect
100. C22×C3×Q8D9semidirect
132. Q8×C9C3×D4semidirect
139. #611C3×Q8direct (N × H)
201. C22×C9C2×C3×Q8semidirect
205. C2×D9C2×C3×Q8semidirect
212. Dic9C2×C3×Q8semidirect
214. C22×Q8C3×D9semidirect
220. C3×C9D4×Q8semidirect
231. C3×Q8#611direct (N × H)
292. C22×C3Q8×D9semidirect
313. C9C3×D4×Q8semidirect
318. Q8C3×#611direct (N × H)
346. C22C3×Q8×D9semidirect
354. C3Q8×#611direct (N × H)

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×C3×Q8×D92
C3×#635082
C3×Q8×Dic92
C3×#635062
C3×C4×#6112
C3×#634862
C3×Q8×#673
Q8×#6113