Finite Group Databasebeta

← C2×#64106⋊C3 #13152440

Subgroups — C2×#64106⋊C3 #13152440

Order 1728.

1219 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C212yesmaximal
5C2×C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal, complemented
6#64106⋊C3C218yesmaximal, direct product factor
7C2×#64106C311yesmaximal, characteristic, complemented
8C2×C3×#897—31nomaximal
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3—24no—
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C2×#634⋊C3—24no—
17C2×#634⋊C3C2214yescomplemented
18C4⋊C4×H(3,𝔽3)C2214yescomplemented
19#3752⋊C3C2218yes—
20C4⋊((C3⋊Dic3)⋊C3)C2218yescomplemented
21C22×C4×C3⋊D3C2×C311yescharacteristic
22C22×#634C2×C311yescharacteristic, complemented
23C22×C3×C4×D3—31no—
24C22×C3×Dic6—31no—
25C2×C32×C4⋊C4—21no—
26C2×C32×C4⋊C4D311yescharacteristic, complemented
27C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
28C2×#3752C2×C312yes—
29C2×C4⋊(C3⋊Dic3)C2×C312yescomplemented
30#64106C2×C318yescomplemented
31C2×C3×(C2×D3)⋊C4—32no—
32C2×C3×C4⋊Dic3—32no—
33C3×#897—38no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3Q814yescomplemented
38C22×(C3⋊D3)⋊C3C2312yes—
39C2×C4×H(3,𝔽3)—24no—
40C2×C4×H(3,𝔽3)D414yes—
41C2×C4×H(3,𝔽3)C2314yes—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×(C3⋊Dic3)⋊C3—24no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C4×(C3⋊D3)⋊C3—28no—
48#634⋊C3—28no—
49C2×C3×C22⋊Q8—91no—
50C2×#897—31no—
51C2×#897—91no—
52C23×C3⋊D3C22×C311yescharacteristic
53C23×C3×D3—31no—
54C22×C32×C4—21no—
55C22×C32×C4—22no—
56C22×C32×C4C2×D312yes—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C22×C312yes—
59C22×C32×C4C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C312yes—
61C22×C3⋊Dic3C22×C311yescharacteristic
62C2×C4×C3⋊D3—24no—
63C2×C4×C3⋊D3C22×C314yes—
64C2×#634—24no—
65C2×#634C22×C314yescomplemented
66C22×C3×Dic3—31no—
67C22×C3×Dic3—32no—
68C2×C3×C4×D3—34no—
69C2×C3×C4×D3—64no—
70C2×C3×Dic6—34no—
71C2×C3×Dic6—64no—
72C32×C4⋊C4—24no—
73C32×C4⋊C4C2×D314yescomplemented
74C32×C4⋊C4C22×C314yescomplemented
75#3752C22×C318yes—
76C4⋊(C3⋊Dic3)C22×C318yescomplemented
77C3×(C2×D3)⋊C4—38no—
78C3×C4⋊Dic3—38no—
79C22×H(3,𝔽3)C2×D412yes—
80C22×H(3,𝔽3)C2×Q812yes—
81C22×H(3,𝔽3)#4112yes—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3—28no—
84C2×(C3⋊D3)⋊C3—22no—
85C2×(C3⋊D3)⋊C3—22no—
86C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
87C4×H(3,𝔽3)—24no—
88C4×H(3,𝔽3)C2×D414yes—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—22no—
91C22×C3×Q8—91no—
92C23×C3×C4—91no—
93C22×C4×D3—31no—
94C22×C4×D3—91no—
95C22×Dic6—31no—
96C22×Dic6—91no—
97C3×C22⋊Q8—98no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—61no—
101C2×C3×C4⋊C4—92no—
102C2×C3×C4⋊C4C3×D311yescharacteristic
103C2×C3×C22⋊C4—92no—
104C2×(C2×D3)⋊C4—32no—
105C2×(C2×D3)⋊C4—92no—
106C2×C4⋊Dic3—32no—
107C2×C4⋊Dic3—92no—
108#897—38no—
109#897—98no—
110C23×C32—21no—
111C23×C32C22×D311yescharacteristic
112C23×C32C23×C311yescharacteristic
113C22×C3⋊D3—22no—
114C22×C3⋊D3—22no—
115C22×C3⋊D3C3×Q814yescomplemented
116C22×C3⋊D3C23×C312yes—
117C22×C3×D3—34no—
118C22×C3×D3—32no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—22no—
125C2×C32×C4—24no—
126C2×C32×C4—44no—
127C2×C32×C4D1214yes—
128C2×C32×C4C3×D414yes—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D312yes—
131C2×C32×C4C23×C314yes—
132C2×C32×C4C23×C312yes—
133C2×C3⋊Dic3—24no—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3C23×C312yes—
136C2×C3⋊Dic3C23×C314yes—
137C4×C3⋊D3—28no—
138#634—28no—
139C2×C3×Dic3—32no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—62no—
143C3×C4×D3—68no—
144C3×Dic6—68no—
145C2×C22⋊Q8—91noSylow
146C2×H(3,𝔽3)C22⋊Q814yescomplemented
147C2×H(3,𝔽3)C22×D411yescharacteristic
148C2×H(3,𝔽3)C22×Q811yescharacteristic
149C2×H(3,𝔽3)C2×#4111yescharacteristic
150(C3⋊D3)⋊C3—24no—
151C24×C3—91no—
152C23×D3—31no—
153C23×D3—91no—
154C2×C3×Q8—94no—
155C2×C3×Q8—184no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—32no—
159C22×C3×C4—32no—
160C22×C3×C4—61no—
161C22×C3×C4—62no—
162C22×C3×C4—94no—
163C22×C3×C4—92no—
164C22×C3×C4—91no—
165C22×C3×C4—184no—
166C22×C3×C4C2×C3×D312yes—
167C22×C3×C4C2×C3×D311yescharacteristic
168C22×Dic3—31no—
169C22×Dic3—32no—
170C22×Dic3—91no—
171C22×Dic3—92no—
172C2×C4×D3—34no—
173C2×C4×D3—64no—
174C2×C4×D3—94no—
175C2×C4×D3—184no—
176C2×Dic6—34no—
177C2×Dic6—64no—
178C2×Dic6—94no—
179C2×Dic6—184no—
180C3×C4⋊C4—34no—
181C3×C4⋊C4—34no—
182C3×C4⋊C4—64no—
183C3×C4⋊C4—98no—
184C3×C4⋊C4C2×C3×D314yes—
185C3×C22⋊C4—98no—
186(C2×D3)⋊C4—38no—
187(C2×D3)⋊C4—98no—
188C4⋊Dic3—38no—
189C4⋊Dic3—98no—
190C22×C32—21no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32C2×D1212yes—
195C22×C32#23712yes—
196C22×C32D3×Q812yes—
197C22×C32C2×C3×D412yes—
198C22×C32C2×C3×Q812yes—
199C22×C32C3×#4112yes—
200C22×C32C23×D311yescharacteristic
201C22×C32C24×C311yescharacteristic
202C2×C3⋊D3—28no—
203C2×C3⋊D3—22no—
204C2×C3⋊D3—22no—
205C2×C3⋊D3C2×C3×Q814yescomplemented
206C2×C3×D3—34no—
207C2×C3×D3—68no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—44no—
214C32×C4C2×D1214yes—
215C32×C4C2×C3×D414yes—
216C3⋊Dic3—24no—
217C3⋊Dic3—22no—
218C3×Dic3—64no—
219C3×Dic3—62no—
220C22×Q8—91no—
221C23×C4—91no—
222C22⋊Q8—98no—
223C2×C4⋊C4—92no—
224C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
225C2×C22⋊C4—92no—
226H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3—92no—
231C23×C3—94no—
232C23×C3—182no—
233C23×C3—182no—
234C23×C3C22×C3×D311yescharacteristic
235C22×D3—32no—
236C22×D3—34no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—94no—
240C22×D3—92no—
241C22×D3—182no—
242C22×D3—182no—
243C3×Q8—188no—
244C2×C3×C4—24no—
245C2×C3×C4—34no—
246C2×C3×C4—34no—
247C2×C3×C4—32no—
248C2×C3×C4—32no—
249C2×C3×C4—34no—
250C2×C3×C4—34no—
251C2×C3×C4—64no—
252C2×C3×C4—64no—
253C2×C3×C4—64no—
254C2×C3×C4—62no—
255C2×C3×C4—64no—
256C2×C3×C4—94no—
257C2×C3×C4—92no—
258C2×C3×C4—124no—
259C2×C3×C4—188no—
260C2×C3×C4—182no—
261C2×C3×C4—184no—
262C2×C3×C4C3×D1214yes—
263C2×C3×C4C22×C3×D314yes—
264C2×C3×C4C22×C3×D312yes—
265C2×Dic3—32no—
266C2×Dic3—34no—
267C2×Dic3—64no—
268C2×Dic3—62no—
269C2×Dic3—92no—
270C2×Dic3—94no—
271C2×Dic3—184no—
272C2×Dic3—182no—
273C4×D3—68no—
274C4×D3—188no—
275Dic6—68no—
276Dic6—188no—
277C2×C32—24no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32#89714yescomplemented
282C2×C32C3×C22⋊Q814yescomplemented
283C2×C32C22×D1211yescharacteristic
284C2×C32C2×#23711yescharacteristic
285C2×C32C2×D3×Q811yescharacteristic
286C2×C32C22×C3×D411yescharacteristic
287C2×C32C22×C3×Q811yescharacteristic
288C2×C32C2×C3×#4111yescharacteristic
289C3⋊D3—24no—
290C3×D3—64no—
291C24—91no—
292C2×Q8—94no—
293C2×Q8—184no—
294C22×C4—92no—
295C22×C4—91no—
296C22×C4—94no—
297C22×C4—184no—
298C22×C4C2×(C3⋊D3)⋊C312yes—
299C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
300C4⋊C4—98no—
301C4⋊C4C2×(C3⋊D3)⋊C314yescomplemented
302C22⋊C4—98no—
303C22×C3—31no—
304C22×C3—31no—
305C22×C3—32no—
306C22×C3—32no—
307C22×C3—32no—
308C22×C3—32no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—61no—
312C22×C3—62no—
313C22×C3—62no—
314C22×C3—62no—
315C22×C3—94no—
316C22×C3—182no—
317C22×C3—182no—
318C22×C3—188no—
319C22×C3C2×C3×D1212yes—
320C22×C3C3×#23712yes—
321C22×C3C3×D3×Q812yes—
322C22×C3C23×C3×D311yescharacteristic
323C2×D3—34no—
324C2×D3—68no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—94no—
328C2×D3—182no—
329C2×D3—182no—
330C2×D3—188no—
331C3×C4—24no—
332C3×C4—34no—
333C3×C4—34no—
334C3×C4—64no—
335C3×C4—64no—
336C3×C4—64no—
337C3×C4—124no—
338C3×C4—182no—
339C3×C4—184no—
340C3×C4C2×C3×D1214yes—
341Dic3—64no—
342Dic3—62no—
343Dic3—184no—
344Dic3—182no—
345C32—21no—
346C32C2×#89711yescharacteristic, complemented
347C32C2×C3×C22⋊Q811yescharacteristic, complemented
348C23—92no—
349C23—94no—
350C23—182no—
351C23—182no—
352C23C22×(C3⋊D3)⋊C311yescharacteristic
353Q8—188no—
354C2×C4—24no—
355C2×C4—94no—
356C2×C4—92no—
357C2×C4—182no—
358C2×C4—184no—
359C2×C4—188no—
360C2×C4C4⋊((C3⋊D3)⋊C3)14yes—
361C2×C4C22×(C3⋊D3)⋊C314yes—
362C2×C4C22×(C3⋊D3)⋊C312yes—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—34no—
370C2×C3—34no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—64no—
375C2×C3—184no—
376C2×C3C3×#89714yes—
377C2×C3C22×C3×D1211yescharacteristic
378C2×C3C2×C3×#23711yescharacteristic
379C2×C3C2×C3×D3×Q811yescharacteristic
380D3—64no—
381D3—184no—
382C22—94no—
383C22—182no—
384C22—182no—
385C22—188no—
386C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
387C22#3830⋊C312yes—
388C22Q8×(C3⋊D3)⋊C312yes—
389C22C23×(C3⋊D3)⋊C311yescharacteristic
390C4—24no—
391C4—182no—
392C4—184no—
393C4C2×C4⋊((C3⋊D3)⋊C3)14yes—
394C3—31no—
395C3—31no—
396C3—61no—
397C3C2×C3×#89711yescharacteristic
398C2—184no—
399C2#64106⋊C314yesdirect product factor
400C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
401C2C2×#3830⋊C311yescharacteristic
402C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. C2×C4⋊C4×H(3,𝔽3)C2semidirect
5. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. #64106⋊C3C2direct (N × H)
7. C2×#64106C3semidirect
17. C2×#634⋊C3C22semidirect
18. C4⋊C4×H(3,𝔽3)C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C22×#634C2×C3semidirect
26. C2×C32×C4⋊C4D3semidirect
27. C2×C32×C4⋊C4C2×C3semidirect
29. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
30. #64106C2×C3semidirect
37. C22×(C3⋊D3)⋊C3Q8semidirect
65. C2×#634C22×C3semidirect
73. C32×C4⋊C4C2×D3semidirect
74. C32×C4⋊C4C22×C3semidirect
76. C4⋊(C3⋊Dic3)C22×C3semidirect
86. C2×(C3⋊D3)⋊C3C2×Q8semidirect
115. C22×C3⋊D3C3×Q8semidirect
146. C2×H(3,𝔽3)C22⋊Q8semidirect
205. C2×C3⋊D3C2×C3×Q8semidirect
224. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
226. H(3,𝔽3)C2×C22⋊Q8semidirect
281. C2×C32#897semidirect
282. C2×C32C3×C22⋊Q8semidirect
301. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
346. C32C2×#897semidirect
347. C32C2×C3×C22⋊Q8semidirect
399. C2#64106⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C22×#634⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
#64106⋊C32
C2×#641063
C2×C3×#8973