Finite Group Databasebeta

← C4⋊Q8×(C3⋊D3)⋊C3 #13153320

Subgroups — C4⋊Q8×(C3⋊D3)⋊C3 #13153320

Order 1728.

1083 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C212yesmaximal
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#64153⋊C3C212yesmaximal, complemented
4C4⋊C4×(C3⋊D3)⋊C3C214yesmaximal
5#64097⋊C3C214yesmaximal, complemented
6C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
7#64078⋊C3C211yesmaximal, characteristic, complemented
8C3⋊D3×C4⋊Q8C311yesmaximal, characteristic, complemented
9C3×D3×C4⋊Q8—31nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11Q8×(C3⋊D3)⋊C3—28no—
12C2×C4×(C3⋊D3)⋊C3C2214yes—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×#634⋊C3C2212yes—
16C2×#634⋊C3C2214yes—
17C4⋊C4×H(3,𝔽3)C2214yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2214yes—
20#3750⋊C3C2218yes—
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C4×(C3⋊Dic3)⋊C3C2212yes—
23C2×Q8×C3⋊D3C2×C312yes—
24C2×C3×D3×Q8—32no—
25C32×C4⋊Q8—21no—
26C32×C4⋊Q8D311yescharacteristic, complemented
27C32×C4⋊Q8C2×C311yescharacteristic, complemented
28#64153C2×C312yescomplemented
29C4⋊C4×C3⋊D3C2×C314yes—
30#64097C2×C314yescomplemented
31C42×C3⋊D3C2×C311yescharacteristic
32#64078C2×C311yescharacteristic, complemented
33C3×#944—32no—
34C3×D3×C4⋊C4—34no—
35C3×#888—34no—
36C3×C42×D3—31no—
37C3×#869—31no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39Q8×H(3,𝔽3)—24no—
40C2×C4×H(3,𝔽3)C2314yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2312yes—
43C2×(C3⋊Dic3)⋊C3C2314yes—
44C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C4×(C3⋊D3)⋊C3—28no—
47C4×(C3⋊D3)⋊C3D414yes—
48C4×(C3⋊D3)⋊C3Q818yescomplemented
49#634⋊C3—24no—
50#634⋊C3—28no—
51C2×C3×C4⋊Q8—91no—
52D3×C4⋊Q8—31no—
53D3×C4⋊Q8—91no—
54C2×C32×Q8—22no—
55C2×C32×Q8C2×D312yes—
56C2×C32×Q8C22×C312yes—
57Q8×C3⋊D3—28no—
58C2×C4×C3⋊D3C22×C314yes—
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×C4×C3⋊D3C22×C312yes—
61C2×#634C22×C312yes—
62C2×#634C22×C314yes—
63C3×D3×Q8—68no—
64C2×C3×C4×D3—32no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—34no—
67C2×C3×Dic6—34no—
68C2×C3×Dic6—32no—
69C32×C4⋊C4—24no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C22×C314yes—
72C32×C42—21no—
73C32×C42C2×D311yescharacteristic
74C32×C42C22×C311yescharacteristic
75C4⋊(C3⋊Dic3)C22×C314yes—
76#3750C22×C318yes—
77C4×C3⋊Dic3C22×C312yes—
78C4×C3⋊Dic3C22×C311yescharacteristic
79C3×C4⋊Dic3—34no—
80C3×Dic3⋊C4—38no—
81C3×C4×Dic3—32no—
82C3×C4×Dic3—31no—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×(C3⋊D3)⋊C3C2×D412yes—
85C2×(C3⋊D3)⋊C3C2×Q814yes—
86C4×H(3,𝔽3)—24no—
87C4×H(3,𝔽3)C2×D412yes—
88C4×H(3,𝔽3)C2×Q814yescomplemented
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3C2×D412yes—
91(C3⋊Dic3)⋊C3C2×Q814yescomplemented
92C22×C3×Q8—92no—
93C2×D3×Q8—32no—
94C2×D3×Q8—92no—
95C3×C4⋊Q8—31no—
96C3×C4⋊Q8—31no—
97C3×C4⋊Q8—61no—
98C3×C4⋊Q8—91no—
99C3×C4⋊Q8—94no—
100C3×C4⋊Q8—92no—
101C3×C4⋊Q8C3×D311yescharacteristic
102C2×C3×C4⋊C4—94no—
103C2×C3×C42—91no—
104#944—32no—
105#944—92no—
106D3×C4⋊C4—34no—
107D3×C4⋊C4—94no—
108#888—34no—
109#888—94no—
110C42×D3—31no—
111C42×D3—91no—
112#869—31no—
113#869—91no—
114C22×C3⋊D3C23×C311yescharacteristic
115C22×C3×D3—31no—
116C32×Q8—24no—
117C32×Q8—24no—
118C32×Q8—44no—
119C2×C32×C4—22no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D312yes—
125C2×C32×C4C23×C314yes—
126C2×C32×C4C23×C311yescharacteristic
127C2×C32×C4C23×C312yes—
128C2×C3⋊Dic3C23×C314yes—
129C2×C3⋊Dic3C23×C311yescharacteristic
130C2×C3⋊Dic3C23×C312yes—
131C4×C3⋊D3—28no—
132C4×C3⋊D3C3×D414yes—
133C4×C3⋊D3C3×Q818yescomplemented
134#634—24no—
135#634—28no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—34no—
139C3×C4×D3—34no—
140C3×C4×D3—38no—
141C3×C4×D3—68no—
142C3×Dic6—68no—
143C3×Dic6—64no—
144C2×C4⋊Q8—91noSylow
145C2×H(3,𝔽3)C22×D411yescharacteristic
146C2×H(3,𝔽3)C22×Q812yes—
147(C3⋊D3)⋊C3C4⋊Q814yesdirect product factor
148C2×C3×Q8—32no—
149C2×C3×Q8—32no—
150C2×C3×Q8—62no—
151C2×C3×Q8—92no—
152C2×C3×Q8—94no—
153C2×C3×Q8—188no—
154C2×C3×Q8C2×C3×D312yes—
155C22×C3×C4—92no—
156C22×C3×C4—91no—
157C22×C3×C4—94no—
158D3×Q8—68no—
159D3×Q8—188no—
160C2×C4×D3—32no—
161C2×C4×D3—31no—
162C2×C4×D3—34no—
163C2×C4×D3—92no—
164C2×C4×D3—91no—
165C2×C4×D3—94no—
166C2×Dic6—32no—
167C2×Dic6—34no—
168C2×Dic6—92no—
169C2×Dic6—94no—
170C3×C4⋊C4—34no—
171C3×C4⋊C4—34no—
172C3×C4⋊C4—64no—
173C3×C4⋊C4—98no—
174C3×C4⋊C4—94no—
175C3×C4⋊C4C2×C3×D314yes—
176C3×C42—31no—
177C3×C42—31no—
178C3×C42—61no—
179C3×C42—92no—
180C3×C42—91no—
181C3×C42C2×C3×D311yescharacteristic
182C4⋊Dic3—34no—
183C4⋊Dic3—94no—
184Dic3⋊C4—38no—
185Dic3⋊C4—98no—
186C4×Dic3—32no—
187C4×Dic3—31no—
188C4×Dic3—92no—
189C4×Dic3—91no—
190C22×C32—21no—
191C22×C32C23×D311yescharacteristic
192C22×C32C24×C311yescharacteristic
193C2×C3⋊D3C2×C3×D412yes—
194C2×C3⋊D3C2×C3×Q814yes—
195C2×C3×D3—32no—
196C2×C3×D3—34no—
197C32×C4—24no—
198C32×C4—24no—
199C32×C4—22no—
200C32×C4—24no—
201C32×C4—44no—
202C32×C4D3×D412yes—
203C32×C4D3×Q814yescomplemented
204C32×C4C2×C3×D412yes—
205C32×C4C2×C3×Q814yescomplemented
206C3⋊Dic3—24no—
207C3⋊Dic3C2×C3×D412yes—
208C3⋊Dic3C2×C3×Q814yescomplemented
209C3×Dic3—32no—
210C3×Dic3—34no—
211C3×Dic3—64no—
212C22×Q8—92no—
213C4⋊Q8—92no—
214C4⋊Q8—94no—
215C4⋊Q8—91no—
216C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, direct product factor
217C2×C4⋊C4—94no—
218C2×C42—91no—
219H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
220C23×C3—91no—
221C22×D3—31no—
222C22×D3—91no—
223C3×Q8—24no—
224C3×Q8—64no—
225C3×Q8—64no—
226C3×Q8—124no—
227C3×Q8—184no—
228C3×Q8—188no—
229C2×C3×C4—32no—
230C2×C3×C4—31no—
231C2×C3×C4—32no—
232C2×C3×C4—31no—
233C2×C3×C4—34no—
234C2×C3×C4—34no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—64no—
238C2×C3×C4—98no—
239C2×C3×C4—94no—
240C2×C3×C4—92no—
241C2×C3×C4—91no—
242C2×C3×C4—94no—
243C2×C3×C4—188no—
244C2×C3×C4C22×C3×D314yes—
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×C3×C4C22×C3×D312yes—
247C2×Dic3—32no—
248C2×Dic3—31no—
249C2×Dic3—34no—
250C2×Dic3—92no—
251C2×Dic3—91no—
252C2×Dic3—94no—
253C4×D3—34no—
254C4×D3—38no—
255C4×D3—68no—
256C4×D3—94no—
257C4×D3—98no—
258C4×D3—188no—
259Dic6—68no—
260Dic6—64no—
261Dic6—188no—
262Dic6—184no—
263C2×C32—21no—
264C2×C32—22no—
265C2×C32C2×D3×D411yescharacteristic
266C2×C32C2×D3×Q812yes—
267C2×C32C22×C3×D411yescharacteristic
268C2×C32C22×C3×Q812yes—
269C3⋊D3C3×C4⋊Q814yescomplemented
270C3×D3—34no—
271C2×Q8—94no—
272C2×Q8—92no—
273C2×Q8—188no—
274C2×Q8C2×(C3⋊D3)⋊C312yes—
275C22×C4—92no—
276C22×C4—91no—
277C22×C4—94no—
278C4⋊C4—98no—
279C4⋊C4—94no—
280C4⋊C4C2×(C3⋊D3)⋊C314yes—
281C42—91no—
282C42—92no—
283C42C2×(C3⋊D3)⋊C311yescharacteristic
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—61no—
287C22×C3—92no—
288C22×C3—94no—
289C22×C3C23×C3×D311yescharacteristic
290C2×D3—32no—
291C2×D3—34no—
292C2×D3—92no—
293C2×D3—94no—
294C3×C4—24no—
295C3×C4—32no—
296C3×C4—32no—
297C3×C4—34no—
298C3×C4—34no—
299C3×C4—64no—
300C3×C4—64no—
301C3×C4—62no—
302C3×C4—64no—
303C3×C4—92no—
304C3×C4—94no—
305C3×C4—124no—
306C3×C4—184no—
307C3×C4C3×D3×D412yes—
308C3×C4C3×D3×Q814yes—
309Dic3—32no—
310Dic3—34no—
311Dic3—64no—
312Dic3—92no—
313Dic3—94no—
314Dic3—184no—
315C32—21no—
316C32D3×C4⋊Q811yescharacteristic, complemented
317C32C2×C3×C4⋊Q811yescharacteristic, complemented
318C23—91no—
319Q8—24no—
320Q8—188no—
321Q8—184no—
322C2×C4—92no—
323C2×C4—91no—
324C2×C4—94no—
325C2×C4—98no—
326C2×C4—94no—
327C2×C4—188no—
328C2×C4C22×(C3⋊D3)⋊C314yes—
329C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
330C2×C4C22×(C3⋊D3)⋊C312yes—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—32no—
334C2×C3—32no—
335C2×C3—61no—
336C2×C3—62no—
337C2×C3—94no—
338C2×C3C2×C3×D3×D411yescharacteristic
339C2×C3C2×C3×D3×Q812yes—
340D3—34no—
341D3—94no—
342C22—92no—
343C22—94no—
344C22C23×(C3⋊D3)⋊C311yescharacteristic
345C4—24no—
346C4—92no—
347C4—94no—
348C4—184no—
349C4D4×(C3⋊D3)⋊C312yes—
350C4Q8×(C3⋊D3)⋊C314yescomplemented
351C3—31no—
352C3—31no—
353C3—61no—
354C3C3×D3×C4⋊Q811yescharacteristic
355C2—94no—
356C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
357C2C2×Q8×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C2×C3×C4⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #64153⋊C3C2semidirect
5. #64097⋊C3C2semidirect
7. #64078⋊C3C2semidirect
8. C3⋊D3×C4⋊Q8C3semidirect
26. C32×C4⋊Q8D3semidirect
27. C32×C4⋊Q8C2×C3semidirect
28. #64153C2×C3semidirect
30. #64097C2×C3semidirect
32. #64078C2×C3semidirect
48. C4×(C3⋊D3)⋊C3Q8semidirect
88. C4×H(3,𝔽3)C2×Q8semidirect
91. (C3⋊Dic3)⋊C3C2×Q8semidirect
133. C4×C3⋊D3C3×Q8semidirect
147. (C3⋊D3)⋊C3C4⋊Q8direct (N × H)
203. C32×C4D3×Q8semidirect
205. C32×C4C2×C3×Q8semidirect
208. C3⋊Dic3C2×C3×Q8semidirect
216. C4⋊Q8(C3⋊D3)⋊C3direct (N × H)
219. H(3,𝔽3)C2×C4⋊Q8semidirect
269. C3⋊D3C3×C4⋊Q8semidirect
316. C32D3×C4⋊Q8semidirect
317. C32C2×C3×C4⋊Q8semidirect
350. C4Q8×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
H(3,𝔽3)×C4⋊Q82
#64153⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64097⋊C32
C42×(C3⋊D3)⋊C32
#64078⋊C32
C3⋊D3×C4⋊Q83
C3×D3×C4⋊Q83