Finite Group Databasebeta

← C2×#64138⋊C3 #13153556

Subgroups — C2×#64138⋊C3 #13153556

Order 1728.

1433 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C211yesmaximal, characteristic
3C2×#3757⋊C3C211yesmaximal, characteristic, complemented
4#64138⋊C3C218yesmaximal, direct product factor
5C2×#3752⋊C3C212yesmaximal, complemented
6C2×#3750⋊C3C212yesmaximal, complemented
7C2×#64138C311yesmaximal, characteristic, complemented
8C2×C3×#929—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)—24no—
12C22×C4×H(3,𝔽3)C2212yes—
13C22×C4×H(3,𝔽3)C2214yescomplemented
14C2×#638⋊C3—24no—
15C2×#638⋊C3C2214yes—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2212yes—
18#3757⋊C3C2214yescomplemented
19#3752⋊C3C2218yescomplemented
20#3750⋊C3C2218yescomplemented
21C23×C32×C4—21no—
22C23×C32×C4D311yescharacteristic, complemented
23C23×C32×C4C2×C311yescharacteristic, complemented
24C22×#638C2×C311yescharacteristic
25C22×C3×#67—31no—
26C2×#3757C2×C311yescharacteristic, complemented
27#64138C2×C318yescomplemented
28C2×#3752C2×C312yescomplemented
29C2×#3750C2×C312yescomplemented
30C2×C3×#217—31no—
31C3×#929—38no—
32C2×C3×(C2×D3)⋊C4—32no—
33C2×C3×Dic3⋊C4—32no—
34C23×H(3,𝔽3)—24no—
35C23×H(3,𝔽3)D414yes—
36C23×H(3,𝔽3)C2312yes—
37C23×H(3,𝔽3)C2311yescharacteristic
38C22×(C3⋊D3)⋊C3—22no—
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3C2312yes—
41C2×C4×H(3,𝔽3)—24no—
42C2×C4×H(3,𝔽3)—28no—
43C2×C4×H(3,𝔽3)C2314yes—
44#638⋊C3—28no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2314yes—
49C2×C3×#123—91no—
50C2×#929—31no—
51C2×#929—91no—
52C24×C32—21no—
53C24×C32C2×D311yescharacteristic
54C24×C32C22×C311yescharacteristic
55C23×C3⋊D3C22×C311yescharacteristic
56C23×C3×D3—31no—
57C22×C32×C4—28no—
58C22×C32×C4—22no—
59C22×C32×C4—24no—
60C22×C32×C4—24no—
61C22×C32×C4—24no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C2×D314yescomplemented
64C22×C32×C4C22×C312yes—
65C22×C32×C4C22×C314yescomplemented
66C2×#638—24no—
67C2×#638C22×C314yes—
68C22×C3⋊Dic3C22×C311yescharacteristic
69C22×C3⋊Dic3C22×C312yes—
70C2×C3×#67—34no—
71C2×C3×#67—64no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—32no—
74#3757C22×C314yescomplemented
75#3752C22×C318yescomplemented
76#3750C22×C318yescomplemented
77C3×#217—34no—
78C3×(C2×D3)⋊C4—38no—
79C3×Dic3⋊C4—38no—
80C22×H(3,𝔽3)—28no—
81C22×H(3,𝔽3)—24no—
82C22×H(3,𝔽3)C2×D414yes—
83C22×H(3,𝔽3)C2×D412yes—
84C22×H(3,𝔽3)#4114yes—
85C22×H(3,𝔽3)C2411yescharacteristic
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—24no—
88C2×(C3⋊D3)⋊C3—44no—
89C4×H(3,𝔽3)—24no—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—24no—
92C22×C3×D4—91no—
93C23×C3×C4—31no—
94C23×C3×C4—31no—
95C23×C3×C4—61no—
96C23×C3×C4C3×D311yescharacteristic
97C22×#67—31no—
98C22×#67—91no—
99C3×#123—98no—
100C2×C3×C4⋊C4—92no—
101C2×C3×C22⋊C4—92no—
102C2×C3×C22⋊C4—91no—
103C2×#217—31no—
104C2×#217—91no—
105#929—38no—
106#929—98no—
107C2×(C2×D3)⋊C4—32no—
108C2×(C2×D3)⋊C4—92no—
109C2×Dic3⋊C4—32no—
110C2×Dic3⋊C4—92no—
111C23×C32—28no—
112C23×C32—24no—
113C23×C32—24no—
114C23×C32—24no—
115C23×C32—22no—
116C23×C32—21no—
117C23×C32#6714yes—
118C23×C32C3×D414yes—
119C23×C32C22×D312yes—
120C23×C32C22×D311yescharacteristic
121C23×C32C23×C312yes—
122C23×C32C23×C311yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—24no—
125C22×C3⋊D3C23×C312yes—
126C22×C3×D3—32no—
127C22×C3×D3—64no—
128C22×C3×D3—62no—
129C2×C32×C4—28no—
130C2×C32×C4—216no—
131C2×C32×C4—24no—
132C2×C32×C4—28no—
133C2×C32×C4—24no—
134C2×C32×C4—28no—
135C2×C32×C4—24no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C23×C314yes—
138#638—28no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—24no—
141C2×C3⋊Dic3C23×C312yes—
142C2×C3⋊Dic3C23×C314yes—
143C3×#67—68no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—62no—
148C2×#123—91noSylow
149C2×H(3,𝔽3)—24no—
150C2×H(3,𝔽3)#12314yescomplemented
151C2×H(3,𝔽3)C22×D411yescharacteristic
152C2×H(3,𝔽3)C2×#4112yes—
153(C3⋊D3)⋊C3—42no—
154C24×C3—31no—
155C24×C3—31no—
156C24×C3—61no—
157C24×C3—91no—
158C24×C3C2×C3×D311yescharacteristic
159C23×D3—31no—
160C23×D3—91no—
161C2×C3×D4—94no—
162C2×C3×D4—184no—
163C22×C3×C4—24no—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4—34no—
167C22×C3×C4—34no—
168C22×C3×C4—64no—
169C22×C3×C4—64no—
170C22×C3×C4—68no—
171C22×C3×C4—62no—
172C22×C3×C4—64no—
173C22×C3×C4—92no—
174C22×C3×C4—91no—
175C22×C3×C4C2×C3×D312yes—
176C22×C3×C4C2×C3×D314yes—
177C2×#67—34no—
178C2×#67—64no—
179C2×#67—94no—
180C2×#67—184no—
181C22×Dic3—31no—
182C22×Dic3—32no—
183C22×Dic3—91no—
184C22×Dic3—92no—
185C3×C4⋊C4—98no—
186C3×C22⋊C4—98no—
187C3×C22⋊C4—94no—
188#217—34no—
189#217—94no—
190(C2×D3)⋊C4—38no—
191(C2×D3)⋊C4—98no—
192Dic3⋊C4—38no—
193Dic3⋊C4—98no—
194C22×C32—216no—
195C22×C32—28no—
196C22×C32—28no—
197C22×C32—28no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—21no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—22no—
204C22×C32#23514yes—
205C22×C32C2×#6714yes—
206C22×C32C2×#6712yes—
207C22×C32C2×C3×D414yes—
208C22×C32C2×C3×D412yes—
209C22×C32C3×#4114yes—
210C22×C32C23×D311yescharacteristic
211C22×C32C24×C311yescharacteristic
212C2×C3⋊D3—22no—
213C2×C3⋊D3—24no—
214C2×C3⋊D3—44no—
215C2×C3×D3—64no—
216C2×C3×D3—62no—
217C2×C3×D3—124no—
218C32×C4—28no—
219C32×C4—24no—
220C32×C4—24no—
221C3⋊Dic3—22no—
222C3⋊Dic3—24no—
223C3×Dic3—64no—
224C3×Dic3—62no—
225C22×D4—91no—
226C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
227#123—98no—
228C2×C4⋊C4—92no—
229C2×C22⋊C4—91no—
230C2×C22⋊C4—92no—
231H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
232C23×C3—24no—
233C23×C3—34no—
234C23×C3—34no—
235C23×C3—32no—
236C23×C3—32no—
237C23×C3—31no—
238C23×C3—31no—
239C23×C3—64no—
240C23×C3—64no—
241C23×C3—68no—
242C23×C3—64no—
243C23×C3—62no—
244C23×C3—61no—
245C23×C3—92no—
246C23×C3—182no—
247C23×C3—184no—
248C23×C3C3×#6714yes—
249C23×C3C22×C3×D312yes—
250C23×C3C22×C3×D311yescharacteristic
251C22×D3—32no—
252C22×D3—64no—
253C22×D3—62no—
254C22×D3—92no—
255C22×D3—182no—
256C22×D3—184no—
257C3×D4—188no—
258C2×C3×C4—24no—
259C2×C3×C4—28no—
260C2×C3×C4—34no—
261C2×C3×C4—34no—
262C2×C3×C4—64no—
263C2×C3×C4—64no—
264C2×C3×C4—68no—
265C2×C3×C4—68no—
266C2×C3×C4—68no—
267C2×C3×C4—616no—
268C2×C3×C4—64no—
269C2×C3×C4—94no—
270C2×C3×C4—92no—
271C2×C3×C4—184no—
272C2×C3×C4—182no—
273C2×C3×C4C22×C3×D314yes—
274#67—68no—
275#67—188no—
276C2×Dic3—32no—
277C2×Dic3—34no—
278C2×Dic3—64no—
279C2×Dic3—62no—
280C2×Dic3—92no—
281C2×Dic3—94no—
282C2×Dic3—184no—
283C2×Dic3—182no—
284C2×C32—28no—
285C2×C32—24no—
286C2×C32—24no—
287C2×C32—24no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32#92914yescomplemented
291C2×C32C3×#12314yescomplemented
292C2×C32C2×#23512yes—
293C2×C32C22×#6711yescharacteristic
294C2×C32C22×C3×D411yescharacteristic
295C2×C32C2×C3×#4112yes—
296C3⋊D3—42no—
297C3×D3—122no—
298C24—91no—
299C24C2×(C3⋊D3)⋊C311yescharacteristic
300C2×D4—94no—
301C2×D4—184no—
302C22×C4—24no—
303C22×C4—92no—
304C22×C4—91no—
305C22×C4C2×(C3⋊D3)⋊C312yes—
306C22×C4C2×(C3⋊D3)⋊C314yescomplemented
307C4⋊C4—98no—
308C22⋊C4—94no—
309C22⋊C4—98no—
310C22×C3—28no—
311C22×C3—24no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—34no—
315C22×C3—34no—
316C22×C3—32no—
317C22×C3—32no—
318C22×C3—34no—
319C22×C3—34no—
320C22×C3—64no—
321C22×C3—64no—
322C22×C3—68no—
323C22×C3—68no—
324C22×C3—68no—
325C22×C3—616no—
326C22×C3—61no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3—62no—
330C22×C3—182no—
331C22×C3—184no—
332C22×C3—364no—
333C22×C3C3×#23514yes—
334C22×C3C2×C3×#6714yes—
335C22×C3C2×C3×#6712yes—
336C22×C3C23×C3×D311yescharacteristic
337C2×D3—64no—
338C2×D3—62no—
339C2×D3—124no—
340C2×D3—182no—
341C2×D3—184no—
342C2×D3—364no—
343C3×C4—24no—
344C3×C4—64no—
345C3×C4—64no—
346C3×C4—68no—
347C3×C4—182no—
348C3×C4—184no—
349Dic3—64no—
350Dic3—62no—
351Dic3—182no—
352Dic3—184no—
353C32—21no—
354C32C2×#92911yescharacteristic, complemented
355C32C2×C3×#12311yescharacteristic, complemented
356C23—24no—
357C23—92no—
358C23—184no—
359C23—182no—
360C23#638⋊C314yes—
361C23C22×(C3⋊D3)⋊C312yes—
362C23C22×(C3⋊D3)⋊C311yescharacteristic
363D4—188no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—94no—
367C2×C4—92no—
368C2×C4—184no—
369C2×C4—182no—
370C2×C4C22×(C3⋊D3)⋊C314yes—
371C2×C3—24no—
372C2×C3—32no—
373C2×C3—32no—
374C2×C3—34no—
375C2×C3—34no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—64no—
379C2×C3—64no—
380C2×C3—68no—
381C2×C3—62no—
382C2×C3—64no—
383C2×C3—61no—
384C2×C3—362no—
385C2×C3C3×#92914yes—
386C2×C3C2×C3×#23512yes—
387C2×C3C22×C3×#6711yescharacteristic
388D3—122no—
389D3—362no—
390C22—24no—
391C22—28no—
392C22—184no—
393C22—182no—
394C22—364no—
395C22#3828⋊C314yes—
396C22C2×#638⋊C314yes—
397C22C2×#638⋊C312yes—
398C22C23×(C3⋊D3)⋊C311yescharacteristic
399C4—24no—
400C4—184no—
401C4—182no—
402C3—31no—
403C3—31no—
404C3—61no—
405C3C2×C3×#92911yescharacteristic
406C2—24no—
407C2—362no—
408C2#64138⋊C314yesdirect product factor
409C2C2×#3828⋊C312yes—
410C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC23×C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC22×#638⋊C3C2
hypercenterC23×C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×H(3,𝔽3)C2semidirect
3. C2×#3757⋊C3C2semidirect
4. #64138⋊C3C2direct (N × H)
5. C2×#3752⋊C3C2semidirect
6. C2×#3750⋊C3C2semidirect
7. C2×#64138C3semidirect
13. C22×C4×H(3,𝔽3)C22semidirect
18. #3757⋊C3C22semidirect
19. #3752⋊C3C22semidirect
20. #3750⋊C3C22semidirect
22. C23×C32×C4D3semidirect
23. C23×C32×C4C2×C3semidirect
26. C2×#3757C2×C3semidirect
27. #64138C2×C3semidirect
28. C2×#3752C2×C3semidirect
29. C2×#3750C2×C3semidirect
63. C22×C32×C4C2×D3semidirect
65. C22×C32×C4C22×C3semidirect
74. #3757C22×C3semidirect
75. #3752C22×C3semidirect
76. #3750C22×C3semidirect
150. C2×H(3,𝔽3)#123semidirect
226. C23×C4(C3⋊D3)⋊C3semidirect
231. H(3,𝔽3)C2×#123semidirect
290. C2×C32#929semidirect
291. C2×C32C3×#123semidirect
306. C22×C4C2×(C3⋊D3)⋊C3semidirect
354. C32C2×#929semidirect
355. C32C2×C3×#123semidirect
408. C2#64138⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4×H(3,𝔽3)2
C22×#638⋊C32
C2×#3757⋊C32
#64138⋊C32
C2×#3752⋊C32
C2×#3750⋊C32
C2×#641383
C2×C3×#9293