Finite Group Databasebeta

← C2×#64101⋊C3 #13152434

Subgroups — C2×#64101⋊C3 #13152434

Order 1728.

1247 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C212yesmaximal
4C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
5C2×C4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
6#64101⋊C3C218yesmaximal, direct product factor
7C2×#64101C311yesmaximal, characteristic, complemented
8C2×C3×#892—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)⋊C3C418yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C4⋊C4×H(3,𝔽3)C2214yescomplemented
17#3752⋊C3C2218yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
19C4×(C3⋊Dic3)⋊C3C2218yescomplemented
20C22×C4×C3⋊D3C2×C311yescharacteristic
21C22×C3×C4×D3—31no—
22C2×C32×C4⋊C4—21no—
23C2×C32×C4⋊C4D311yescharacteristic, complemented
24C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
25C2×#3752C2×C312yes—
26C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
27C2×C4×C3⋊Dic3C2×C312yescomplemented
28#64101C2×C318yescomplemented
29C2×C3×(C2×D3)⋊C4—32no—
30C2×C3×C4⋊Dic3—31no—
31C2×C3×C4×Dic3—32no—
32C3×#892—38no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—22no—
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3C2×C414yes—
37C22×(C3⋊D3)⋊C3C2312yes—
38C2×C4×H(3,𝔽3)—24no—
39C2×C4×H(3,𝔽3)C2×C414yes—
40C2×C4×H(3,𝔽3)C2314yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×(C3⋊Dic3)⋊C3—24no—
43C2×(C3⋊Dic3)⋊C3C2×C414yes—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C4×(C3⋊D3)⋊C3C2×C4116yescomplemented
47C2×C3×#117—91no—
48C2×#892—31no—
49C2×#892—91no—
50C23×C3⋊D3C22×C311yescharacteristic
51C23×C3×D3—31no—
52C22×C32×C4—21no—
53C22×C32×C4—22no—
54C22×C32×C4C2×D312yes—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C312yes—
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C312yes—
59C22×C3⋊Dic3C22×C311yescharacteristic
60C2×C4×C3⋊D3C3×C418yescomplemented
61C2×C4×C3⋊D3C22×C314yes—
62C22×C3×Dic3—31no—
63C22×C3×Dic3—32no—
64C2×C3×C4×D3—38no—
65C2×C3×C4×D3—34no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4C2×D314yescomplemented
68C32×C4⋊C4C22×C314yescomplemented
69#3752C22×C318yes—
70C4⋊(C3⋊Dic3)C22×C314yescomplemented
71C4×C3⋊Dic3C22×C318yescomplemented
72C3×(C2×D3)⋊C4—38no—
73C3×C4⋊Dic3—34no—
74C3×C4×Dic3—38no—
75C22×H(3,𝔽3)C22×C412yes—
76C22×H(3,𝔽3)#4112yes—
77C22×H(3,𝔽3)#4112yes—
78C22×H(3,𝔽3)C2411yescharacteristic
79C2×(C3⋊D3)⋊C3—28no—
80C2×(C3⋊D3)⋊C3—22no—
81C2×(C3⋊D3)⋊C3—22no—
82C2×(C3⋊D3)⋊C3C22×C414yes—
83C4×H(3,𝔽3)—24no—
84C4×H(3,𝔽3)C22×C414yes—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3C22×C414yes—
87C23×C3×C4—91no—
88C22×C4×D3—31no—
89C22×C4×D3—91no—
90C3×#117—98no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4—61no—
94C2×C3×C4⋊C4—91no—
95C2×C3×C4⋊C4C3×D311yescharacteristic
96C2×C3×C22⋊C4—92no—
97C2×C3×C42—92no—
98C2×(C2×D3)⋊C4—32no—
99C2×(C2×D3)⋊C4—92no—
100C2×C4⋊Dic3—31no—
101C2×C4⋊Dic3—91no—
102C2×C4×Dic3—32no—
103C2×C4×Dic3—92no—
104#892—38no—
105#892—98no—
106C23×C32—21no—
107C23×C32C22×D311yescharacteristic
108C23×C32C23×C311yescharacteristic
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3C2×C3×C414yes—
112C22×C3⋊D3C23×C312yes—
113C22×C3×D3—34no—
114C22×C3×D3—32no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—22no—
121C2×C32×C4—24no—
122C2×C32×C4—44no—
123C2×C32×C4C4×D314yes—
124C2×C32×C4C2×C3×C414yes—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D312yes—
127C2×C32×C4C23×C314yes—
128C2×C32×C4C23×C312yes—
129C2×C3⋊Dic3—24no—
130C2×C3⋊Dic3C2×C3×C414yes—
131C2×C3⋊Dic3C23×C314yes—
132C2×C3⋊Dic3C23×C312yes—
133C4×C3⋊D3C2×C3×C4116yescomplemented
134C2×C3×Dic3—34no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—64no—
138C3×C4×D3—316no—
139C2×#117—91noSylow
140C2×H(3,𝔽3)#11714yescomplemented
141C2×H(3,𝔽3)C23×C411yescharacteristic
142C2×H(3,𝔽3)C2×#4111yescharacteristic
143C2×H(3,𝔽3)C2×#4111yescharacteristic
144(C3⋊D3)⋊C3—24no—
145C24×C3—91no—
146C23×D3—31no—
147C23×D3—91no—
148C22×C3×C4—31no—
149C22×C3×C4—31no—
150C22×C3×C4—32no—
151C22×C3×C4—32no—
152C22×C3×C4—61no—
153C22×C3×C4—62no—
154C22×C3×C4—94no—
155C22×C3×C4—98no—
156C22×C3×C4—91no—
157C22×C3×C4—92no—
158C22×C3×C4C2×C3×D312yes—
159C22×C3×C4C2×C3×D311yescharacteristic
160C22×Dic3—31no—
161C22×Dic3—32no—
162C22×Dic3—91no—
163C22×Dic3—92no—
164C2×C4×D3—38no—
165C2×C4×D3—34no—
166C2×C4×D3—94no—
167C2×C4×D3—98no—
168C3×C4⋊C4—34no—
169C3×C4⋊C4—34no—
170C3×C4⋊C4—64no—
171C3×C4⋊C4—94no—
172C3×C4⋊C4C2×C3×D314yes—
173C3×C22⋊C4—98no—
174C3×C42—98no—
175(C2×D3)⋊C4—38no—
176(C2×D3)⋊C4—98no—
177C4⋊Dic3—34no—
178C4⋊Dic3—94no—
179C4×Dic3—38no—
180C4×Dic3—98no—
181C22×C32—21no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32C2×C4×D312yes—
186C22×C32#23712yes—
187C22×C32#23912yes—
188C22×C32C22×C3×C412yes—
189C22×C32C3×#4112yes—
190C22×C32C3×#4112yes—
191C22×C32C23×D311yescharacteristic
192C22×C32C24×C311yescharacteristic
193C2×C3⋊D3—28no—
194C2×C3⋊D3—22no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3C22×C3×C414yes—
197C2×C3×D3—34no—
198C2×C3×D3—68no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—44no—
205C32×C4C2×C4×D314yes—
206C32×C4C22×C3×C414yes—
207C3⋊Dic3—24no—
208C3⋊Dic3C22×C3×C414yes—
209C3×Dic3—34no—
210C3×Dic3—64no—
211C23×C4—91no—
212#117—98no—
213C2×C4⋊C4—91no—
214C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
215C2×C22⋊C4—92no—
216C2×C42—92no—
217H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
218C23×C3—31no—
219C23×C3—31no—
220C23×C3—61no—
221C23×C3—92no—
222C23×C3—94no—
223C23×C3—182no—
224C23×C3—182no—
225C23×C3C22×C3×D311yescharacteristic
226C22×D3—32no—
227C22×D3—34no—
228C22×D3—62no—
229C22×D3—62no—
230C22×D3—94no—
231C22×D3—92no—
232C22×D3—182no—
233C22×D3—182no—
234C2×C3×C4—24no—
235C2×C3×C4—34no—
236C2×C3×C4—34no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—34no—
240C2×C3×C4—34no—
241C2×C3×C4—64no—
242C2×C3×C4—64no—
243C2×C3×C4—64no—
244C2×C3×C4—62no—
245C2×C3×C4—64no—
246C2×C3×C4—916no—
247C2×C3×C4—92no—
248C2×C3×C4—94no—
249C2×C3×C4—94no—
250C2×C3×C4—124no—
251C2×C3×C4—184no—
252C2×C3×C4C3×C4×D314yes—
253C2×C3×C4C22×C3×D314yes—
254C2×C3×C4C22×C3×D312yes—
255C2×Dic3—32no—
256C2×Dic3—34no—
257C2×Dic3—34no—
258C2×Dic3—64no—
259C2×Dic3—94no—
260C2×Dic3—92no—
261C2×Dic3—94no—
262C2×Dic3—184no—
263C4×D3—316no—
264C4×D3—916no—
265C2×C32—24no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32#89214yescomplemented
270C2×C32C3×#11714yescomplemented
271C2×C32C22×C4×D311yescharacteristic
272C2×C32C2×#23711yescharacteristic
273C2×C32C2×#23911yescharacteristic
274C2×C32C23×C3×C411yescharacteristic
275C2×C32C2×C3×#4111yescharacteristic
276C2×C32C2×C3×#4111yescharacteristic
277C3⋊D3—24no—
278C3×D3—64no—
279C24—91no—
280C22×C4—91no—
281C22×C4—98no—
282C22×C4—94no—
283C22×C4—92no—
284C22×C4C2×(C3⋊D3)⋊C312yes—
285C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
286C4⋊C4—94no—
287C4⋊C4C2×(C3⋊D3)⋊C314yescomplemented
288C22⋊C4—98no—
289C42—98no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—32no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—61no—
299C22×C3—62no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—94no—
303C22×C3—182no—
304C22×C3—182no—
305C22×C3—188no—
306C22×C3C2×C3×C4×D312yes—
307C22×C3C3×#23712yes—
308C22×C3C3×#23912yes—
309C22×C3C23×C3×D311yescharacteristic
310C2×D3—34no—
311C2×D3—68no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—94no—
315C2×D3—182no—
316C2×D3—182no—
317C2×D3—188no—
318C3×C4—24no—
319C3×C4—34no—
320C3×C4—34no—
321C3×C4—64no—
322C3×C4—64no—
323C3×C4—64no—
324C3×C4—94no—
325C3×C4—124no—
326C3×C4—184no—
327C3×C4C2×C3×C4×D314yes—
328Dic3—34no—
329Dic3—64no—
330Dic3—94no—
331Dic3—184no—
332C32—21no—
333C32C2×#89211yescharacteristic, complemented
334C32C2×C3×#11711yescharacteristic, complemented
335C23—92no—
336C23—94no—
337C23—182no—
338C23—182no—
339C23C22×(C3⋊D3)⋊C311yescharacteristic
340C2×C4—24no—
341C2×C4—92no—
342C2×C4—94no—
343C2×C4—916no—
344C2×C4—94no—
345C2×C4—184no—
346C2×C4C4×(C3⋊D3)⋊C314yes—
347C2×C4C22×(C3⋊D3)⋊C314yes—
348C2×C4C22×(C3⋊D3)⋊C312yes—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—34no—
356C2×C3—34no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—64no—
361C2×C3—184no—
362C2×C3C3×#89214yes—
363C2×C3C22×C3×C4×D311yescharacteristic
364C2×C3C2×C3×#23711yescharacteristic
365C2×C3C2×C3×#23911yescharacteristic
366D3—64no—
367D3—184no—
368C22—94no—
369C22—182no—
370C22—182no—
371C22—188no—
372C22C2×C4×(C3⋊D3)⋊C312yes—
373C22#3830⋊C312yes—
374C22#8872212yes—
375C22C23×(C3⋊D3)⋊C311yescharacteristic
376C4—24no—
377C4—94no—
378C4—184no—
379C4C2×C4×(C3⋊D3)⋊C314yes—
380C3—31no—
381C3—31no—
382C3—61no—
383C3C2×C3×#89211yescharacteristic
384C2—184no—
385C2#64101⋊C314yesdirect product factor
386C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
387C2C2×#3830⋊C311yescharacteristic
388C2C2×#8872211yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×C32C24×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#117
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
6. #64101⋊C3C2direct (N × H)
7. C2×#64101C3semidirect
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
16. C4⋊C4×H(3,𝔽3)C22semidirect
18. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
19. C4×(C3⋊Dic3)⋊C3C22semidirect
23. C2×C32×C4⋊C4D3semidirect
24. C2×C32×C4⋊C4C2×C3semidirect
26. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
27. C2×C4×C3⋊Dic3C2×C3semidirect
28. #64101C2×C3semidirect
46. C4×(C3⋊D3)⋊C3C2×C4semidirect
60. C2×C4×C3⋊D3C3×C4semidirect
67. C32×C4⋊C4C2×D3semidirect
68. C32×C4⋊C4C22×C3semidirect
70. C4⋊(C3⋊Dic3)C22×C3semidirect
71. C4×C3⋊Dic3C22×C3semidirect
133. C4×C3⋊D3C2×C3×C4semidirect
140. C2×H(3,𝔽3)#117semidirect
214. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
217. H(3,𝔽3)C2×#117semidirect
269. C2×C32#892semidirect
270. C2×C32C3×#117semidirect
287. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
333. C32C2×#892semidirect
334. C32C2×C3×#117semidirect
385. C2#64101⋊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
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×C4×(C3⋊Dic3)⋊C32
#64101⋊C32
C2×#641013
C2×C3×#8923