Finite Group Databasebeta

← C2×Q16×(C3⋊D3)⋊C3 #13153463

Subgroups — C2×Q16×(C3⋊D3)⋊C3 #13153463

Order 1728.

963 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C212yesmaximal
2C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3756⋊C3C212yesmaximal, complemented
4Q16×(C3⋊D3)⋊C3C218yesmaximal, direct product factor
5C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
6C2×#3746⋊C3C211yesmaximal, characteristic, complemented
7C2×Q16×C3⋊D3C311yesmaximal, characteristic, complemented
8C2×C3×D3×Q16—31nomaximal
9C2×Q8×H(3,𝔽3)C2212yes—
10Q8×(C3⋊D3)⋊C3—24no—
11Q8×(C3⋊D3)⋊C3C2218yes—
12C2×C4×(C3⋊D3)⋊C3—22no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#634⋊C3—22no—
15C2×#634⋊C3C2212yes—
16Q16×H(3,𝔽3)C2214yescomplemented
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18#3756⋊C3C2218yescomplemented
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C8×(C3⋊D3)⋊C3C2214yes—
21#3746⋊C3C2214yescomplemented
22C2×Q8×C3⋊D3C2×C312yes—
23C2×C3×D3×Q8—32no—
24C2×C32×Q16—21no—
25C2×C32×Q16D311yescharacteristic, complemented
26C2×C32×Q16C2×C311yescharacteristic, complemented
27C2×#3756C2×C312yescomplemented
28Q16×C3⋊D3C2×C318yescomplemented
29C2×#3746C2×C311yescharacteristic, complemented
30C2×C8×C3⋊D3C2×C311yescharacteristic
31C2×C3×#216—32no—
32C3×D3×Q16—38no—
33C2×C3×Dic12—31no—
34C2×C3×D3×C8—31no—
35C22×(C3⋊D3)⋊C3D411yescharacteristic
36Q8×H(3,𝔽3)—22no—
37Q8×H(3,𝔽3)C2314yes—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C4×(C3⋊D3)⋊C3—28no—
43C4×(C3⋊D3)⋊C3D412yes—
44C4×(C3⋊D3)⋊C3C2312yes—
45#634⋊C3—22no—
46#634⋊C3—28no—
47#634⋊C3C2314yes—
48C8×H(3,𝔽3)C2312yes—
49(C3⋊(C3⋊C8))⋊C3C2312yes—
50C22×C3×Q16—91no—
51C2×D3×Q16—31no—
52C2×D3×Q16—91no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2×D312yes—
55C2×C32×Q8C22×C312yes—
56Q8×C3⋊D3—24no—
57Q8×C3⋊D3C22×C318yes—
58C2×C4×C3⋊D3—22no—
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×#634—22no—
61C2×#634C22×C312yes—
62C3×D3×Q8—38no—
63C3×D3×Q8—64no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—62no—
66C2×C3×Dic6—32no—
67C2×C3×Dic6—62no—
68C32×Q16—24no—
69C32×Q16C2×D314yescomplemented
70C32×Q16C22×C314yescomplemented
71C2×C32×C8—21no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C22×C311yescharacteristic
74#3756C22×C318yescomplemented
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3746C22×C314yescomplemented
77C8×C3⋊D3C22×C314yes—
78C3×#216—38no—
79C2×C3×C3⋊C8—31no—
80C3×Dic12—34no—
81C3×D3×C8—34no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3Q1614yesdirect product factor
84C2×(C3⋊D3)⋊C3C2×D412yes—
85C4×H(3,𝔽3)—24no—
86C4×H(3,𝔽3)C2×D411yescharacteristic
87C4×H(3,𝔽3)C2411yescharacteristic
88(C3⋊Dic3)⋊C3—24no—
89(C3⋊Dic3)⋊C3C2×D412yes—
90C22×C3×Q8—92no—
91C2×D3×Q8—32no—
92C2×D3×Q8—92no—
93C2×C3×Q16—31no—
94C2×C3×Q16—31no—
95C2×C3×Q16—61no—
96C2×C3×Q16—91no—
97C2×C3×Q16—98no—
98C2×C3×Q16—92no—
99C2×C3×Q16C3×D311yescharacteristic
100C22×C3×C8—91no—
101C2×#216—32no—
102C2×#216—92no—
103D3×Q16—38no—
104D3×Q16—98no—
105C2×Dic12—31no—
106C2×Dic12—91no—
107C2×D3×C8—31no—
108C2×D3×C8—91no—
109C22×C3⋊D3C3×D411yescharacteristic
110C22×C3×D3—31no—
111C32×Q8—22no—
112C32×Q8—22no—
113C32×Q8—24no—
114C32×Q8—42no—
115C32×Q8C22×D314yes—
116C32×Q8C23×C314yes—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—42no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3C3×D411yescharacteristic
125C4×C3⋊D3—28no—
126C4×C3⋊D3C3×D412yes—
127C4×C3⋊D3C23×C312yes—
128#634—22no—
129#634—28no—
130#634C23×C314yes—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—62no—
133C3×C4×D3—32no—
134C3×C4×D3—32no—
135C3×C4×D3—68no—
136C3×Dic6—34no—
137C3×Dic6—68no—
138C3×Dic6—62no—
139C32×C8—22no—
140C32×C8C22×D312yes—
141C32×C8C23×C312yes—
142C3⋊(C3⋊C8)C23×C312yes—
143C3×C3⋊C8—32no—
144C22×Q16—91noSylow
145C2×H(3,𝔽3)C2×Q1612yescomplemented
146C2×H(3,𝔽3)C22×D411yescharacteristic
147(C3⋊D3)⋊C3C2×Q1614yesdirect product factor
148C2×C3×Q8—32no—
149C2×C3×Q8—32no—
150C2×C3×Q8—62no—
151C2×C3×Q8—92no—
152C2×C3×Q8—98no—
153C2×C3×Q8—182no—
154C2×C3×Q8—184no—
155C2×C3×Q8C2×C3×D312yes—
156C22×C3×C4—91no—
157C22×C3×C4—182no—
158D3×Q8—38no—
159D3×Q8—64no—
160D3×Q8—98no—
161D3×Q8—184no—
162C2×C4×D3—31no—
163C2×C4×D3—62no—
164C2×C4×D3—91no—
165C2×C4×D3—182no—
166C2×Dic6—32no—
167C2×Dic6—62no—
168C2×Dic6—92no—
169C2×Dic6—182no—
170C3×Q16—34no—
171C3×Q16—34no—
172C3×Q16—64no—
173C3×Q16—94no—
174C3×Q16—98no—
175C3×Q16C2×C3×D314yes—
176C2×C3×C8—31no—
177C2×C3×C8—31no—
178C2×C3×C8—61no—
179C2×C3×C8—94no—
180C2×C3×C8—91no—
181C2×C3×C8C2×C3×D311yescharacteristic
182#216—38no—
183#216—98no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—91no—
186Dic12—34no—
187Dic12—94no—
188D3×C8—34no—
189D3×C8—94no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32C2×C3×D411yescharacteristic
193C2×C3⋊D3C3×Q1614yescomplemented
194C2×C3⋊D3C2×C3×D412yes—
195C2×C3×D3—34no—
196C2×C3×D3—32no—
197C32×C4—24no—
198C32×C4—24no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—44no—
202C32×C4D3×D411yescharacteristic
203C32×C4C2×C3×D411yescharacteristic
204C32×C4C23×D311yescharacteristic
205C32×C4C24×C311yescharacteristic
206C3⋊Dic3—24no—
207C3⋊Dic3C2×C3×D412yes—
208C3×Dic3—32no—
209C3×Dic3—64no—
210C22×Q8—92no—
211C2×Q16—92no—
212C2×Q16—98no—
213C2×Q16—91no—
214C2×Q16(C3⋊D3)⋊C311yescharacteristic, direct product factor
215C22×C8—91no—
216H(3,𝔽3)C22×Q1611yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—31no—
219C22×D3—91no—
220C3×Q8—22no—
221C3×Q8—34no—
222C3×Q8—34no—
223C3×Q8—62no—
224C3×Q8—62no—
225C3×Q8—64no—
226C3×Q8—94no—
227C3×Q8—122no—
228C3×Q8—182no—
229C3×Q8—188no—
230C3×Q8C22×C3×D314yes—
231C2×C3×C4—22no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—61no—
237C2×C3×C4—92no—
238C2×C3×C4—92no—
239C2×C3×C4—91no—
240C2×C3×C4—122no—
241C2×C3×C4—188no—
242C2×C3×C4—182no—
243C2×C3×C4C22×C3×D311yescharacteristic
244C2×Dic3—31no—
245C2×Dic3—62no—
246C2×Dic3—91no—
247C2×Dic3—182no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—68no—
251C4×D3—92no—
252C4×D3—92no—
253C4×D3—188no—
254Dic6—34no—
255Dic6—68no—
256Dic6—62no—
257Dic6—94no—
258Dic6—188no—
259Dic6—182no—
260C3×C8—32no—
261C3×C8—32no—
262C3×C8—62no—
263C3×C8—92no—
264C3×C8C22×C3×D312yes—
265C3⋊C8—32no—
266C3⋊C8—92no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32D3×Q1612yescomplemented
270C2×C32C2×C3×Q1612yescomplemented
271C2×C32C2×D3×D411yescharacteristic
272C2×C32C22×C3×D411yescharacteristic
273C3⋊D3C2×C3×Q1614yescomplemented
274C3×D3—34no—
275C2×Q8—98no—
276C2×Q8—92no—
277C2×Q8—182no—
278C2×Q8—184no—
279C2×Q8C2×(C3⋊D3)⋊C312yes—
280C22×C4—91no—
281C22×C4—182no—
282Q16—98no—
283Q16—94no—
284Q16C2×(C3⋊D3)⋊C314yesdirect product factor
285C2×C8—91no—
286C2×C8—94no—
287C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—61no—
291C22×C3—92no—
292C22×C3—94no—
293C22×C3C3×D3×D411yescharacteristic
294C2×D3—34no—
295C2×D3—32no—
296C2×D3—92no—
297C2×D3—94no—
298C3×C4—24no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—64no—
304C3×C4—64no—
305C3×C4—61no—
306C3×C4—61no—
307C3×C4—92no—
308C3×C4—124no—
309C3×C4—184no—
310C3×C4C3×D3×D411yescharacteristic
311C3×C4C23×C3×D311yescharacteristic
312Dic3—32no—
313Dic3—64no—
314Dic3—92no—
315Dic3—184no—
316C32—21no—
317C32C2×D3×Q1611yescharacteristic, complemented
318C32C22×C3×Q1611yescharacteristic, complemented
319C23—91no—
320Q8—22no—
321Q8—94no—
322Q8—188no—
323Q8—182no—
324Q8C22×(C3⋊D3)⋊C314yes—
325C2×C4—22no—
326C2×C4—91no—
327C2×C4—92no—
328C2×C4—92no—
329C2×C4—188no—
330C2×C4—182no—
331C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
332C8—92no—
333C8C22×(C3⋊D3)⋊C312yes—
334C2×C3—32no—
335C2×C3—32no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—62no—
339C2×C3—61no—
340C2×C3—94no—
341C2×C3C3×D3×Q1612yes—
342C2×C3C2×C3×D3×D411yescharacteristic
343D3—34no—
344D3—94no—
345C22—94no—
346C22—92no—
347C22D4×(C3⋊D3)⋊C311yescharacteristic
348C4—24no—
349C4—92no—
350C4—184no—
351C4D4×(C3⋊D3)⋊C311yescharacteristic
352C4C23×(C3⋊D3)⋊C311yescharacteristic
353C3—31no—
354C3—31no—
355C3—61no—
356C3C2×C3×D3×Q1611yescharacteristic
357C2—94no—
358C2Q16×(C3⋊D3)⋊C312yesdirect product factor
359C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×Q16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×Q16(C3⋊D3)⋊C3
hypocenterC32C22×C3×Q16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
2. C2×Q16×H(3,𝔽3)C2semidirect
3. C2×#3756⋊C3C2semidirect
4. Q16×(C3⋊D3)⋊C3C2direct (N × H)
6. C2×#3746⋊C3C2semidirect
7. C2×Q16×C3⋊D3C3semidirect
16. Q16×H(3,𝔽3)C22semidirect
18. #3756⋊C3C22semidirect
21. #3746⋊C3C22semidirect
25. C2×C32×Q16D3semidirect
26. C2×C32×Q16C2×C3semidirect
27. C2×#3756C2×C3semidirect
28. Q16×C3⋊D3C2×C3semidirect
29. C2×#3746C2×C3semidirect
69. C32×Q16C2×D3semidirect
70. C32×Q16C22×C3semidirect
74. #3756C22×C3semidirect
76. #3746C22×C3semidirect
83. C2×(C3⋊D3)⋊C3Q16direct (N × H)
145. C2×H(3,𝔽3)C2×Q16semidirect
147. (C3⋊D3)⋊C3C2×Q16direct (N × H)
193. C2×C3⋊D3C3×Q16semidirect
214. C2×Q16(C3⋊D3)⋊C3direct (N × H)
216. H(3,𝔽3)C22×Q16semidirect
269. C2×C32D3×Q16semidirect
270. C2×C32C2×C3×Q16semidirect
273. C3⋊D3C2×C3×Q16semidirect
284. Q16C2×(C3⋊D3)⋊C3direct (N × H)
317. C32C2×D3×Q16semidirect
318. C32C22×C3×Q16semidirect
358. C2Q16×(C3⋊D3)⋊C3direct (N × H)

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
C2×Q16×H(3,𝔽3)2
C2×#3756⋊C32
Q16×(C3⋊D3)⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#3746⋊C32
C2×Q16×C3⋊D33
C2×C3×D3×Q163