Finite Group Databasebeta

← C2×#64155⋊C3 #13153659

Subgroups — C2×#64155⋊C3 #13153659

Order 1728.

1283 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3#64155⋊C3C218yesmaximal, direct product factor
4C2×#3752⋊C3C212yesmaximal
5C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
6C2×#3750⋊C3C212yesmaximal, complemented
7C2×#64155C311yesmaximal, characteristic, complemented
8C2×C3×#946—31nomaximal
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×Q8×H(3,𝔽3)—24no—
11C2×Q8×H(3,𝔽3)C2214yescomplemented
12C22×C4×H(3,𝔽3)C2212yes—
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3—24no—
17C2×C4×(C3⋊D3)⋊C3C2214yes—
18#3752⋊C3C2218yes—
19C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
20#3750⋊C3C2218yescomplemented
21C22×C32×Q8—21no—
22C22×C32×Q8D311yescharacteristic, complemented
23C22×C32×Q8C2×C311yescharacteristic, complemented
24C22×C4×C3⋊D3C2×C311yescharacteristic
25C22×C3×C4×D3—31no—
26#64155C2×C318yescomplemented
27C2×#3752C2×C312yes—
28C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
29C2×#3750C2×C312yescomplemented
30C3×#946—38no—
31C2×C3×(C2×D3)⋊C4—32no—
32C2×C3×C4⋊Dic3—31no—
33C2×C3×Dic3⋊C4—32no—
34C23×H(3,𝔽3)C2311yescharacteristic
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3Q814yescomplemented
38C22×(C3⋊D3)⋊C3C2312yes—
39Q8×H(3,𝔽3)—28no—
40C2×C4×H(3,𝔽3)—24no—
41C2×C4×H(3,𝔽3)D414yes—
42C2×C4×H(3,𝔽3)C2314yes—
43C2×C4×H(3,𝔽3)C2312yes—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48C4×(C3⋊D3)⋊C3—28no—
49C2×C3×C22⋊Q8—91no—
50C2×#946—31no—
51C2×#946—91no—
52C23×C3⋊D3C22×C311yescharacteristic
53C23×C3×D3—31no—
54C2×C32×Q8—28no—
55C2×C32×Q8—24no—
56C2×C32×Q8—24no—
57C2×C32×Q8—24no—
58C2×C32×Q8C2×D314yescomplemented
59C2×C32×Q8C22×C314yescomplemented
60C22×C32×C4—22no—
61C22×C32×C4—21no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C312yes—
65C22×C32×C4C22×C311yescharacteristic
66C22×C3⋊Dic3C22×C311yescharacteristic
67C22×C3⋊Dic3C22×C312yes—
68C2×C4×C3⋊D3—24no—
69C2×C4×C3⋊D3C22×C314yes—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—32no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—64no—
74#3752C22×C318yes—
75C4⋊(C3⋊Dic3)C22×C314yescomplemented
76#3750C22×C318yescomplemented
77C3×(C2×D3)⋊C4—38no—
78C3×C4⋊Dic3—34no—
79C3×Dic3⋊C4—38no—
80C22×H(3,𝔽3)C2×D412yes—
81C22×H(3,𝔽3)C2×Q812yes—
82C22×H(3,𝔽3)#4112yes—
83C22×H(3,𝔽3)C2411yescharacteristic
84C2×(C3⋊D3)⋊C3—28no—
85C2×(C3⋊D3)⋊C3—22no—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3C2×Q814yescomplemented
88C4×H(3,𝔽3)—24no—
89C4×H(3,𝔽3)C2×D414yes—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—24no—
92C22×C3×Q8—31no—
93C22×C3×Q8—31no—
94C22×C3×Q8—61no—
95C22×C3×Q8C3×D311yescharacteristic
96C23×C3×C4—91no—
97C22×C4×D3—31no—
98C22×C4×D3—91no—
99C3×C22⋊Q8—98no—
100C2×C3×C4⋊C4—92no—
101C2×C3×C4⋊C4—91no—
102C2×C3×C22⋊C4—92no—
103#946—38no—
104#946—98no—
105C2×(C2×D3)⋊C4—32no—
106C2×(C2×D3)⋊C4—92no—
107C2×C4⋊Dic3—31no—
108C2×C4⋊Dic3—91no—
109C2×Dic3⋊C4—32no—
110C2×Dic3⋊C4—92no—
111C23×C32—21no—
112C23×C32C22×D311yescharacteristic
113C23×C32C23×C311yescharacteristic
114C22×C3⋊D3—22no—
115C22×C3⋊D3—22no—
116C22×C3⋊D3C3×Q814yescomplemented
117C22×C3⋊D3C23×C312yes—
118C22×C3×D3—34no—
119C22×C3×D3—32no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C32×Q8—216no—
123C32×Q8—28no—
124C32×Q8—28no—
125C2×C32×C4—28no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—22no—
131C2×C32×C4#6714yes—
132C2×C32×C4C3×D414yes—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D312yes—
135C2×C32×C4C23×C314yes—
136C2×C32×C4C23×C312yes—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—24no—
139C2×C3⋊Dic3C23×C312yes—
140C2×C3⋊Dic3C23×C314yes—
141C4×C3⋊D3—28no—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—62no—
146C3×C4×D3—68no—
147C2×C22⋊Q8—91noSylow
148C2×H(3,𝔽3)C22⋊Q814yescomplemented
149C2×H(3,𝔽3)C22×D411yescharacteristic
150C2×H(3,𝔽3)C22×Q811yescharacteristic
151C2×H(3,𝔽3)C2×#4111yescharacteristic
152(C3⋊D3)⋊C3—24no—
153C24×C3—91no—
154C23×D3—31no—
155C23×D3—91no—
156C2×C3×Q8—24no—
157C2×C3×Q8—34no—
158C2×C3×Q8—34no—
159C2×C3×Q8—64no—
160C2×C3×Q8—64no—
161C2×C3×Q8—68no—
162C2×C3×Q8—64no—
163C2×C3×Q8C2×C3×D314yes—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—32no—
167C22×C3×C4—32no—
168C22×C3×C4—62no—
169C22×C3×C4—61no—
170C22×C3×C4—94no—
171C22×C3×C4—92no—
172C22×C3×C4—91no—
173C22×C3×C4—184no—
174C22×C3×C4C2×C3×D312yes—
175C22×C3×C4C2×C3×D311yescharacteristic
176C22×Dic3—31no—
177C22×Dic3—32no—
178C22×Dic3—91no—
179C22×Dic3—92no—
180C2×C4×D3—34no—
181C2×C4×D3—64no—
182C2×C4×D3—94no—
183C2×C4×D3—184no—
184C3×C4⋊C4—98no—
185C3×C4⋊C4—94no—
186C3×C22⋊C4—98no—
187(C2×D3)⋊C4—38no—
188(C2×D3)⋊C4—98no—
189C4⋊Dic3—34no—
190C4⋊Dic3—94no—
191Dic3⋊C4—38no—
192Dic3⋊C4—98no—
193C22×C32—21no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32D3×Q812yes—
198C22×C32#23912yes—
199C22×C32C2×#6712yes—
200C22×C32C2×C3×D412yes—
201C22×C32C2×C3×Q812yes—
202C22×C32C3×#4112yes—
203C22×C32C23×D311yescharacteristic
204C22×C32C24×C311yescharacteristic
205C2×C3⋊D3—28no—
206C2×C3⋊D3—22no—
207C2×C3⋊D3—22no—
208C2×C3⋊D3C2×C3×Q814yescomplemented
209C2×C3×D3—34no—
210C2×C3×D3—68no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C32×C4—28no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4C2×#6714yes—
218C32×C4C2×C3×D414yes—
219C3⋊Dic3—22no—
220C3⋊Dic3—24no—
221C3×Dic3—64no—
222C3×Dic3—62no—
223C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
224C23×C4—91no—
225C22⋊Q8—98no—
226C2×C4⋊C4—92no—
227C2×C4⋊C4—91no—
228C2×C22⋊C4—92no—
229H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
230C23×C3—31no—
231C23×C3—31no—
232C23×C3—61no—
233C23×C3—92no—
234C23×C3—94no—
235C23×C3—182no—
236C23×C3—182no—
237C23×C3C22×C3×D311yescharacteristic
238C22×D3—32no—
239C22×D3—34no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—94no—
243C22×D3—92no—
244C22×D3—182no—
245C22×D3—182no—
246C3×Q8—28no—
247C3×Q8—68no—
248C3×Q8—68no—
249C3×Q8—616no—
250C2×C3×C4—24no—
251C2×C3×C4—34no—
252C2×C3×C4—34no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—34no—
256C2×C3×C4—34no—
257C2×C3×C4—64no—
258C2×C3×C4—64no—
259C2×C3×C4—68no—
260C2×C3×C4—64no—
261C2×C3×C4—64no—
262C2×C3×C4—62no—
263C2×C3×C4—94no—
264C2×C3×C4—92no—
265C2×C3×C4—188no—
266C2×C3×C4—184no—
267C2×C3×C4—182no—
268C2×C3×C4C3×#6714yes—
269C2×C3×C4C22×C3×D314yes—
270C2×C3×C4C22×C3×D312yes—
271C2×Dic3—32no—
272C2×Dic3—34no—
273C2×Dic3—64no—
274C2×Dic3—62no—
275C2×Dic3—92no—
276C2×Dic3—94no—
277C2×Dic3—182no—
278C2×Dic3—184no—
279C4×D3—68no—
280C4×D3—188no—
281C2×C32—24no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32#94614yescomplemented
286C2×C32C3×C22⋊Q814yescomplemented
287C2×C32C2×D3×Q811yescharacteristic
288C2×C32C2×#23911yescharacteristic
289C2×C32C22×#6711yescharacteristic
290C2×C32C22×C3×D411yescharacteristic
291C2×C32C22×C3×Q811yescharacteristic
292C2×C32C2×C3×#4111yescharacteristic
293C3⋊D3—24no—
294C3×D3—64no—
295C24—91no—
296C2×Q8—24no—
297C2×Q8C2×(C3⋊D3)⋊C314yescomplemented
298C22×C4—92no—
299C22×C4—91no—
300C22×C4—94no—
301C22×C4—184no—
302C22×C4C2×(C3⋊D3)⋊C312yes—
303C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
304C4⋊C4—98no—
305C4⋊C4—94no—
306C22⋊C4—98no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—32no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—32no—
315C22×C3—61no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—62no—
319C22×C3—94no—
320C22×C3—182no—
321C22×C3—182no—
322C22×C3—188no—
323C22×C3C3×D3×Q812yes—
324C22×C3C3×#23912yes—
325C22×C3C2×C3×#6712yes—
326C22×C3C23×C3×D311yescharacteristic
327C2×D3—34no—
328C2×D3—68no—
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—94no—
332C2×D3—182no—
333C2×D3—182no—
334C2×D3—188no—
335C3×C4—24no—
336C3×C4—34no—
337C3×C4—34no—
338C3×C4—64no—
339C3×C4—64no—
340C3×C4—68no—
341C3×C4—64no—
342C3×C4—182no—
343C3×C4—184no—
344C3×C4C2×C3×#6714yes—
345Dic3—64no—
346Dic3—62no—
347Dic3—182no—
348Dic3—184no—
349C32—21no—
350C32C2×#94611yescharacteristic, complemented
351C32C2×C3×C22⋊Q811yescharacteristic, complemented
352C23—92no—
353C23—94no—
354C23—182no—
355C23—182no—
356C23C22×(C3⋊D3)⋊C311yescharacteristic
357Q8—28no—
358C2×C4—24no—
359C2×C4—94no—
360C2×C4—92no—
361C2×C4—184no—
362C2×C4—182no—
363C2×C4—188no—
364C2×C4#638⋊C314yes—
365C2×C4C22×(C3⋊D3)⋊C314yes—
366C2×C4C22×(C3⋊D3)⋊C312yes—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—34no—
374C2×C3—34no—
375C2×C3—61no—
376C2×C3—61no—
377C2×C3—61no—
378C2×C3—64no—
379C2×C3—184no—
380C2×C3C3×#94614yes—
381C2×C3C2×C3×D3×Q811yescharacteristic
382C2×C3C2×C3×#23911yescharacteristic
383C2×C3C22×C3×#6711yescharacteristic
384D3—64no—
385D3—184no—
386C22—94no—
387C22—182no—
388C22—182no—
389C22—188no—
390C22Q8×(C3⋊D3)⋊C312yes—
391C22#8872212yes—
392C22C2×#638⋊C312yes—
393C22C23×(C3⋊D3)⋊C311yescharacteristic
394C4—24no—
395C4—184no—
396C4—182no—
397C4C2×#638⋊C314yes—
398C3—31no—
399C3—31no—
400C3—61no—
401C3C2×C3×#94611yescharacteristic
402C2—184no—
403C2#64155⋊C314yesdirect product factor
404C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
405C2C2×#8872211yescharacteristic
406C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC22×Q8×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×Q8(C3⋊D3)⋊C3
hypocenterC32C2×C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×Q8×H(3,𝔽3)C2semidirect
3. #64155⋊C3C2direct (N × H)
5. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. C2×#3750⋊C3C2semidirect
7. C2×#64155C3semidirect
11. C2×Q8×H(3,𝔽3)C22semidirect
19. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
20. #3750⋊C3C22semidirect
22. C22×C32×Q8D3semidirect
23. C22×C32×Q8C2×C3semidirect
26. #64155C2×C3semidirect
28. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
29. C2×#3750C2×C3semidirect
37. C22×(C3⋊D3)⋊C3Q8semidirect
58. C2×C32×Q8C2×D3semidirect
59. C2×C32×Q8C22×C3semidirect
75. C4⋊(C3⋊Dic3)C22×C3semidirect
76. #3750C22×C3semidirect
87. C2×(C3⋊D3)⋊C3C2×Q8semidirect
116. C22×C3⋊D3C3×Q8semidirect
148. C2×H(3,𝔽3)C22⋊Q8semidirect
208. C2×C3⋊D3C2×C3×Q8semidirect
223. C22×Q8(C3⋊D3)⋊C3semidirect
229. H(3,𝔽3)C2×C22⋊Q8semidirect
285. C2×C32#946semidirect
286. C2×C32C3×C22⋊Q8semidirect
297. C2×Q8C2×(C3⋊D3)⋊C3semidirect
350. C32C2×#946semidirect
351. C32C2×C3×C22⋊Q8semidirect
403. C2#64155⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×H(3,𝔽3)2
C22×C4×(C3⋊D3)⋊C32
#64155⋊C32
C2×#3752⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
C2×#641553
C2×C3×#9463