Finite Group Databasebeta

← C2×#11721081 #13150626

Subgroups — C2×#11721081 #13150626

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⋊#1084C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal
5C2×#88447C212yesmaximal, complemented
6#11721081C218yesmaximal, direct product factor
7C2×#63849—31nomaximal
8C2×#63841—31nomaximal
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C418yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C4⋊#1084C2214yescomplemented
17C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
18#88450C2218yes—
19#88447C2218yescomplemented
20C22×C4×C3⋊D3D311yescharacteristic
21C22×C3×C4×D3—31no—
22C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
23C2×C3×C4⋊Dic3—31no—
24C2×C3×C4⋊Dic3—31no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—61no—
27C2×#3722—32no—
28C2×(C2×D3)⋊Dic3—32no—
29C2×Dic32—32no—
30C2×Dic32—32no—
31#63849—38no—
32#63841—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)C2×C414yes—
39C2×C4×H(3,𝔽3)C2312yes—
40C2×#1084—24no—
41C2×#1084C2314yes—
42C2×(C3⋊Dic3)⋊C3—24no—
43C2×(C3⋊Dic3)⋊C3C2×C414yes—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C4×(C3⋊D3)⋊C3C2×C4116yescomplemented
47C2×#926—91no—
48C2×#892—91no—
49C2×#892—91no—
50C23×C3⋊D3C2×D311yescharacteristic
51C23×C3×D3—31no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C22×C3⋊Dic3C2×D312yes—
56C22×C3⋊Dic3C2×D311yescharacteristic
57C2×C4×C3⋊D3Dic318yescomplemented
58C2×C4×C3⋊D3C2×D314yes—
59C22×C3×Dic3—31no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—62no—
64C2×C3×C4×D3—38no—
65C2×C3×C4×D3—34no—
66C4⋊(C3⋊Dic3)C2×D314yescomplemented
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3—34no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—64no—
71#3722—38no—
72(C2×D3)⋊Dic3—38no—
73Dic32—38no—
74Dic32—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)C22×C414yes—
84#1084—24no—
85(C3⋊Dic3)⋊C3—24no—
86(C3⋊Dic3)⋊C3C22×C414yes—
87C23×C3×C4—91no—
88C22×C4×D3—31no—
89C22×C4×D3—91no—
90C2×C3×C4⋊C4—91no—
91C2×C3×C4⋊C4—91no—
92C2×#217—92no—
93C2×(C2×D3)⋊C4—92no—
94C2×(C2×D3)⋊C4—92no—
95#926—98no—
96C2×C4⋊Dic3—31no—
97C2×C4⋊Dic3—31no—
98C2×C4⋊Dic3—31no—
99C2×C4⋊Dic3—61no—
100C2×C4⋊Dic3—91no—
101C2×C4×Dic3—92no—
102C2×C4×Dic3—92no—
103C2×C4×Dic3—92no—
104#892—98no—
105#892—98no—
106C23×C32—21no—
107C23×C32C22×D311yescharacteristic
108C23×C32C22×D311yescharacteristic
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3C2×Dic314yes—
112C22×C3⋊D3C22×D312yes—
113C22×C3×D3—34no—
114C22×C3×D3—32no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C2×C32×C4—24no—
118C2×C32×C4—22no—
119C2×C32×C4C4×D314yes—
120C2×C32×C4C2×Dic314yes—
121C2×C32×C4C22×D312yes—
122C2×C32×C4C22×D312yes—
123C2×C3⋊Dic3—24no—
124C2×C3⋊Dic3C2×Dic314yes—
125C2×C3⋊Dic3C22×D314yes—
126C2×C3⋊Dic3C22×D312yes—
127C4×C3⋊D3C2×Dic3116yescomplemented
128C2×C3×Dic3—34no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—124no—
138C3×C4×D3—316no—
139C2×#117—271noSylow
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—61no—
151C22×C3×C4—94no—
152C22×C3×C4—98no—
153C22×C3×C4—91no—
154C22×C3×C4—92no—
155C22×C3×C4—92no—
156C22×C3×C4D3211yescharacteristic
157C22×Dic3—31no—
158C22×Dic3—32no—
159C22×Dic3—32no—
160C22×Dic3—32no—
161C22×Dic3—62no—
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—94no—
169C3×C4⋊C4—94no—
170#217—98no—
171(C2×D3)⋊C4—98no—
172(C2×D3)⋊C4—98no—
173C4⋊Dic3—34no—
174C4⋊Dic3—34no—
175C4⋊Dic3—34no—
176C4⋊Dic3—64no—
177C4⋊Dic3—94no—
178C4×Dic3—98no—
179C4×Dic3—98no—
180C4×Dic3—98no—
181C22×C32—21no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32C2×C4×D312yes—
186C22×C32#23512yes—
187C22×C32#23512yes—
188C22×C32#23712yes—
189C22×C32#23912yes—
190C22×C32C22×Dic312yes—
191C22×C32C23×D311yescharacteristic
192C22×C32C23×D311yescharacteristic
193C2×C3⋊D3—28no—
194C2×C3⋊D3—22no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3C22×Dic314yes—
197C2×C3×D3—34no—
198C2×C3×D3—68no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201C32×C4—24no—
202C32×C4C2×C4×D314yes—
203C32×C4C22×Dic314yes—
204C3⋊Dic3—24no—
205C3⋊Dic3C22×Dic314yes—
206C3×Dic3—34no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C3×Dic3—124no—
211C23×C4—91no—
212#117—278no—
213C2×C4⋊C4—91no—
214C2×C4⋊C4—91no—
215C2×C22⋊C4—272no—
216C2×C42—272no—
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×C3C2×D3211yescharacteristic
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—34no—
235C2×C3×C4—34no—
236C2×C3×C4—32no—
237C2×C3×C4—32no—
238C2×C3×C4—64no—
239C2×C3×C4—62no—
240C2×C3×C4—916no—
241C2×C3×C4—92no—
242C2×C3×C4—94no—
243C2×C3×C4—94no—
244C2×C3×C4—94no—
245C2×C3×C4—184no—
246C2×C3×C4—184no—
247C2×C3×C4D3×Dic314yes—
248C2×C3×C4C2×D3212yes—
249C2×Dic3—32no—
250C2×Dic3—34no—
251C2×Dic3—34no—
252C2×Dic3—34no—
253C2×Dic3—34no—
254C2×Dic3—64no—
255C2×Dic3—64no—
256C2×Dic3—64no—
257C2×Dic3—64no—
258C2×Dic3—94no—
259C2×Dic3—92no—
260C2×Dic3—94no—
261C2×Dic3—124no—
262C2×Dic3—184no—
263C4×D3—316no—
264C4×D3—916no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—24no—
269C2×C32#89214yescomplemented
270C2×C32#92614yescomplemented
271C2×C32C22×C4×D311yescharacteristic
272C2×C32C2×#23511yescharacteristic
273C2×C32C2×#23511yescharacteristic
274C2×C32C2×#23711yescharacteristic
275C2×C32C2×#23911yescharacteristic
276C2×C32C23×Dic311yescharacteristic
277C3⋊D3—24no—
278C3×D3—64no—
279C24—91no—
280C22×C4—91no—
281C22×C4—98no—
282C22×C4—94no—
283C22×C4—92no—
284C22×C4—92no—
285C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C4⋊C4—94no—
287C4⋊C4—94no—
288C22⋊C4—278no—
289C42—278no—
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—188no—
304C22×C3—182no—
305C22×C3—182no—
306C22×C3#379512yes—
307C22×C3#379912yes—
308C22×C3C2×D3×Dic312yes—
309C22×C3C22×D3211yescharacteristic
310C2×D3—34no—
311C2×D3—62no—
312C2×D3—62no—
313C2×D3—68no—
314C2×D3—94no—
315C2×D3—188no—
316C2×D3—182no—
317C2×D3—182no—
318C3×C4—34no—
319C3×C4—34no—
320C3×C4—64no—
321C3×C4—94no—
322C3×C4—184no—
323C3×C4—184no—
324C3×C4C2×D3×Dic314yes—
325Dic3—34no—
326Dic3—64no—
327Dic3—64no—
328Dic3—64no—
329Dic3—94no—
330Dic3—124no—
331Dic3—184no—
332C32—21no—
333C32C2×#89211yescharacteristic, complemented
334C32C2×#92611yescharacteristic, complemented
335C23—92no—
336C23—94no—
337C23—182no—
338C23—182no—
339C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
340C2×C4—92no—
341C2×C4—94no—
342C2×C4—916no—
343C2×C4—94no—
344C2×C4—94no—
345C2×C4—184no—
346C2×C4—184no—
347C2×C4((C3⋊D3)⋊C3)⋊C414yes—
348C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—34no—
352C2×C3—34no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—64no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—184no—
362C2×C3#6384114yes—
363C2×C3C2×#379511yescharacteristic
364C2×C3C2×#379911yescharacteristic
365C2×C3C22×D3×Dic311yescharacteristic
366D3—64no—
367D3—184no—
368C22—94no—
369C22—188no—
370C22—182no—
371C22—182no—
372C22#8864812yes—
373C22#8865212yes—
374C22C2×((C3⋊D3)⋊C3)⋊C412yes—
375C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
376C4—94no—
377C4—184no—
378C4—184no—
379C4C2×((C3⋊D3)⋊C3)⋊C414yes—
380C3—31no—
381C3—31no—
382C3—61no—
383C3C2×#6384111yescharacteristic
384C2—184no—
385C2#1172108114yesdirect product factor
386C2C2×#8864811yescharacteristic
387C2C2×#8865211yescharacteristic
388C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#1084C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
5. C2×#88447C2semidirect
6. #11721081C2direct (N × H)
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
16. C4⋊#1084C22semidirect
17. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
19. #88447C22semidirect
22. C2×C4⋊(C3⋊Dic3)D3semidirect
46. C4×(C3⋊D3)⋊C3C2×C4semidirect
57. C2×C4×C3⋊D3Dic3semidirect
66. C4⋊(C3⋊Dic3)C2×D3semidirect
127. C4×C3⋊D3C2×Dic3semidirect
140. C2×H(3,𝔽3)#117semidirect
217. H(3,𝔽3)C2×#117semidirect
269. C2×C32#892semidirect
270. C2×C32#926semidirect
333. C32C2×#892semidirect
334. C32C2×#926semidirect
385. C2#11721081direct (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⋊#10842
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884502
C2×#884472
#117210812
C2×#638493
C2×#638413