Finite Group Databasebeta

← C2×#11723466 #13176903

Subgroups — C2×#11723466 #13176903

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic3×C3⋊D3C211yesmaximal, characteristic
2C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C3×#3752C211yesmaximal, characteristic
4C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C2×#88803C211yesmaximal, characteristic
7C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#11723466C218yesmaximal, direct product factor
9C2×#64101—31nomaximal
10C2×#63878—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C2×Dic3×C3⋊D3C418yescomplemented
19C2×Dic3×C3⋊D3C2214yes—
20C4×C3⋊(C3⋊Dic3)C2214yescomplemented
21C3×#3752C2214yes—
22C32×Dic3⋊C4C2214yescomplemented
23Dic3⋊(C3⋊Dic3)C2214yescomplemented
24#88803C2214yes—
25Dic3×C3⋊Dic3C2214yescomplemented
26C22×C4×C3⋊D3—31no—
27C22×D3×Dic3—34no—
28C2×C32×C4⋊C4—31no—
29C2×#3752—31no—
30C2×#3752D311yescharacteristic
31C2×C4⋊(C3⋊Dic3)—31no—
32C2×C4×C3⋊Dic3—31no—
33C2×C4×C3⋊Dic3—31no—
34C2×C4×C3⋊Dic3—34no—
35C2×C4×C3⋊Dic3—64no—
36#64101—38no—
37C2×C3×(C2×D3)⋊C4—34no—
38C2×C3×Dic3⋊C4D314yescomplemented
39C2×Dic3⋊Dic3—34no—
40C2×(C2×D3)⋊Dic3—34no—
41C2×Dic32—34no—
42#63878—332no—
43C23×C33C2311yescharacteristic
44C22×C3×C3⋊D3—22no—
45C22×C3×C3⋊D3—22no—
46C22×C3×C3⋊D3C2×C414yes—
47C22×C3×C3⋊D3C2312yes—
48C2×C33×C4—22no—
49C2×C33×C4C2312yes—
50C2×C3⋊(C3⋊Dic3)—22no—
51C2×C3⋊(C3⋊Dic3)C2×C414yes—
52C2×C3⋊(C3⋊Dic3)C2312yes—
53C2×C3⋊(C3⋊Dic3)C2312yes—
54C2×C3×C3⋊Dic3—22no—
55C2×C3×C3⋊Dic3C2312yes—
56C2×C32×Dic3—22no—
57C2×C32×Dic3C2×C414yes—
58C2×C32×Dic3C2312yes—
59C2×C32×Dic3C2312yes—
60Dic3×C3⋊D3C2×C4116yescomplemented
61C2×#892—94no—
62C2×#877—91no—
63C23×C3⋊D3C2×D311yescharacteristic
64C23×C3×D3—34no—
65C22×C32×C4—24no—
66C22×C32×C4—31no—
67C22×C32×C4—31no—
68C22×C32×C4C2×D314yes—
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—34no—
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—34no—
74C22×C3⋊Dic3—64no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3C2×D311yescharacteristic
77C2×C4×C3⋊D3—38no—
78C2×C4×C3⋊D3—34no—
79C22×C3×Dic3—34no—
80C22×C3×Dic3C2×D314yes—
81C22×C3×Dic3C2×D314yes—
82C2×D3×Dic3—332no—
83C2×D3×Dic3—316no—
84C32×C4⋊C4—34no—
85#3752—34no—
86#3752C2×D314yes—
87C4⋊(C3⋊Dic3)—34no—
88C4×C3⋊Dic3—34no—
89C4×C3⋊Dic3—316no—
90C4×C3⋊Dic3—34no—
91C4×C3⋊Dic3—616no—
92C3×(C2×D3)⋊C4—316no—
93C3×Dic3⋊C4C2×D3116yescomplemented
94Dic3⋊Dic3—316no—
95(C2×D3)⋊Dic3—316no—
96Dic32—316no—
97C22×C33C22×C412yes—
98C22×C33#4112yes—
99C22×C33#4112yes—
100C22×C33C2411yescharacteristic
101C2×C3×C3⋊D3—28no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—22no—
104C2×C3×C3⋊D3C22×C414yes—
105C33×C4—22no—
106C3⋊(C3⋊Dic3)—22no—
107C3⋊(C3⋊Dic3)C22×C414yes—
108C3×C3⋊Dic3—22no—
109C32×Dic3—22no—
110C32×Dic3C22×C414yes—
111C23×Dic3—91no—
112C22×C4×D3—94no—
113C2×C3×C4⋊C4—34no—
114C2×C3×C22⋊C4—91no—
115C2×#217—91no—
116C2×(C2×D3)⋊C4—34no—
117C2×(C2×D3)⋊C4—94no—
118C2×C4⋊Dic3—94no—
119C2×Dic3⋊C4—91no—
120C2×Dic3⋊C4C3⋊D311yescharacteristic, complemented
121C2×C4×Dic3—94no—
122C2×C4×Dic3—94no—
123C2×C4×Dic3—91no—
124C2×C4×Dic3—91no—
125C2×C4×Dic3—184no—
126#892—932no—
127#877—98no—
128C23×C32—24no—
129C23×C32C22×D314yes—
130C23×C32C22×D311yescharacteristic
131C22×C3⋊D3—22no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3C4×D314yes—
134C22×C3⋊D3C22×D312yes—
135C22×C3×D3—316no—
136C22×C3×D3—38no—
137C22×C3×D3—68no—
138C22×C3×D3—68no—
139C2×C32×C4—216no—
140C2×C32×C4—28no—
141C2×C32×C4—22no—
142C2×C32×C4—28no—
143C2×C32×C4—34no—
144C2×C32×C4—32no—
145C2×C32×C4—32no—
146C2×C32×C4—62no—
147C2×C32×C4C22×D318yes—
148C2×C32×C4C22×D312yes—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—34no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—32no—
153C2×C3⋊Dic3—316no—
154C2×C3⋊Dic3—38no—
155C2×C3⋊Dic3—38no—
156C2×C3⋊Dic3—68no—
157C2×C3⋊Dic3—68no—
158C2×C3⋊Dic3—616no—
159C2×C3⋊Dic3—616no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—68no—
162C2×C3⋊Dic3C22×D312yes—
163C4×C3⋊D3—316no—
164C2×C3×Dic3—28no—
165C2×C3×Dic3—38no—
166C2×C3×Dic3—68no—
167C2×C3×Dic3C4×D3116yes—
168C2×C3×Dic3C22×D318yes—
169C2×C3×Dic3C22×D318yes—
170D3×Dic3—364no—
171C2×#117—271noSylow
172C2×C33#11714yescomplemented
173C2×C33C23×C411yescharacteristic
174C2×C33C2×#4111yescharacteristic
175C2×C33C2×#4111yescharacteristic
176C3×C3⋊D3—24no—
177C24×C3—91no—
178C23×D3—34no—
179C22×C3×C4—24no—
180C22×C3×C4—34no—
181C22×C3×C4—34no—
182C22×C3×C4—91no—
183C22×C3×C4D3214yes—
184C22×C3×C4C2×C3⋊D311yescharacteristic
185C22×Dic3—34no—
186C22×Dic3—94no—
187C22×Dic3—91no—
188C22×Dic3—98no—
189C22×Dic3—94no—
190C22×Dic3—94no—
191C22×Dic3—91no—
192C22×Dic3—184no—
193C22×Dic3—184no—
194C22×Dic3C2×C3⋊D311yescharacteristic
195C22×Dic3C2×C3⋊D311yescharacteristic
196C2×C4×D3—916no—
197C2×C4×D3—932no—
198C3×C4⋊C4—316no—
199C3×C22⋊C4—94no—
200#217—94no—
201(C2×D3)⋊C4—316no—
202(C2×D3)⋊C4—916no—
203C4⋊Dic3—916no—
204Dic3⋊C4—94no—
205Dic3⋊C4C2×C3⋊D314yescomplemented
206C4×Dic3—916no—
207C4×Dic3—916no—
208C4×Dic3—94no—
209C4×Dic3—94no—
210C4×Dic3—1816no—
211C22×C32—24no—
212C22×C32—28no—
213C22×C32—28no—
214C22×C32—28no—
215C22×C32C2×C4×D318yes—
216C22×C32C2×C4×D312yes—
217C22×C32#23718yes—
218C22×C32#23712yes—
219C22×C32#23712yes—
220C22×C32#23918yes—
221C22×C32C23×D314yes—
222C22×C32C23×D311yescharacteristic
223C2×C3⋊D3—28no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—22no—
226C2×C3⋊D3C2×C4×D314yes—
227C2×C3×D3—316no—
228C2×C3×D3—632no—
229C2×C3×D3—68no—
230C2×C3×D3—68no—
231C32×C4—216no—
232C32×C4—28no—
233C32×C4—22no—
234C32×C4—34no—
235C32×C4—62no—
236C3⋊Dic3—22no—
237C3⋊Dic3—34no—
238C3⋊Dic3—316no—
239C3⋊Dic3—616no—
240C3⋊Dic3—616no—
241C3⋊Dic3—62no—
242C3⋊Dic3—68no—
243C3×Dic3—28no—
244C3×Dic3—68no—
245C3×Dic3C2×C4×D3116yes—
246C23×C4—271no—
247#117—278no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—271no—
250C2×C22⋊C4—91no—
251C2×C22⋊C4—271no—
252C2×C42—271no—
253C2×C42—271no—
254C33C2×#11711yescharacteristic, complemented, Sylow
255C23×C3—24no—
256C23×C3—92no—
257C23×C3—94no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3C2×D3214yes—
261C23×C3C22×C3⋊D311yescharacteristic
262C22×D3—38no—
263C22×D3—316no—
264C22×D3—68no—
265C22×D3—68no—
266C2×C3×C4—216no—
267C2×C3×C4—28no—
268C2×C3×C4—22no—
269C2×C3×C4—28no—
270C2×C3×C4—38no—
271C2×C3×C4—316no—
272C2×C3×C4—38no—
273C2×C3×C4—68no—
274C2×C3×C4—92no—
275C2×C3×C4—182no—
276C2×C3×C4C2×D3218yes—
277C2×C3×C4C22×C3⋊D312yes—
278C2×Dic3—22no—
279C2×Dic3—38no—
280C2×Dic3—68no—
281C2×Dic3—98no—
282C2×Dic3—92no—
283C2×Dic3—916no—
284C2×Dic3—94no—
285C2×Dic3—916no—
286C2×Dic3—98no—
287C2×Dic3—92no—
288C2×Dic3—188no—
289C2×Dic3—188no—
290C2×Dic3—1816no—
291C2×Dic3—1816no—
292C2×Dic3—188no—
293C2×Dic3—182no—
294C2×Dic3C4×C3⋊D314yes—
295C2×Dic3C22×C3⋊D312yes—
296C2×Dic3C22×C3⋊D312yes—
297C4×D3—964no—
298C2×C32—24no—
299C2×C32—24no—
300C2×C32—24no—
301C2×C32—216no—
302C2×C32#87714yescomplemented
303C2×C32#892116yescomplemented
304C2×C32C22×C4×D314yes—
305C2×C32C22×C4×D311yescharacteristic
306C2×C32C2×#23714yes—
307C2×C32C2×#23711yescharacteristic
308C2×C32C2×#23711yescharacteristic
309C2×C32C2×#23914yes—
310C3⋊D3—24no—
311C3×D3—616no—
312C24—91no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—91no—
316C22×C4—274no—
317C22×C4—278no—
318C22×C4—271no—
319C22×C4—271no—
320C22×C4D3×C3⋊D311yescharacteristic
321C4⋊C4—34no—
322C4⋊C4—274no—
323C22⋊C4—94no—
324C22⋊C4—274no—
325C42—274no—
326C42—274no—
327C22×C3—24no—
328C22×C3—28no—
329C22×C3—28no—
330C22×C3—28no—
331C22×C3—94no—
332C22×C3—188no—
333C22×C3—182no—
334C22×C3—182no—
335C22×C3#379618yes—
336C22×C3C4×D3218yes—
337C22×C3#380518yes—
338C22×C3C2×C4×C3⋊D312yes—
339C22×C3#383012yes—
340C22×C3#383212yes—
341C22×C3C22×D3214yes—
342C22×C3C23×C3⋊D311yescharacteristic
343C2×D3—316no—
344C2×D3—68no—
345C2×D3—68no—
346C2×D3—632no—
347C3×C4—216no—
348C3×C4—28no—
349C3×C4—22no—
350C3×C4—316no—
351C3×C4—68no—
352C3×C4—182no—
353Dic3—22no—
354Dic3—68no—
355Dic3—916no—
356Dic3—94no—
357Dic3—1816no—
358Dic3—1816no—
359Dic3—188no—
360Dic3—182no—
361Dic3C2×C4×C3⋊D314yes—
362C32—24no—
363C32C2×#87711yescharacteristic, complemented
364C32C2×#89214yescomplemented
365C23—92no—
366C23—94no—
367C23—182no—
368C23—182no—
369C23C2×D3×C3⋊D311yescharacteristic
370C2×C4—22no—
371C2×C4—32no—
372C2×C4—34no—
373C2×C4—32no—
374C2×C4—62no—
375C2×C4—92no—
376C2×C4—182no—
377C2×C4—2716no—
378C2×C4—272no—
379C2×C4—274no—
380C2×C4—272no—
381C2×C4—542no—
382C2×C4C2×D3×C3⋊D312yes—
383C2×C3—24no—
384C2×C3—24no—
385C2×C3—24no—
386C2×C3—216no—
387C2×C3—184no—
388C2×C3#63878116yescomplemented
389C2×C3#6410114yescomplemented
390C2×C3C2×#379614yes—
391C2×C3C2×C4×D3214yes—
392C2×C3C2×#380514yes—
393C2×C3C22×C4×C3⋊D311yescharacteristic
394C2×C3C2×#383011yescharacteristic
395C2×C3C2×#383211yescharacteristic
396D3—616no—
397C22—94no—
398C22—182no—
399C22—182no—
400C22—188no—
401C22#8901712yes—
402C22C4×D3×C3⋊D312yes—
403C22#8902912yes—
404C22C22×D3×C3⋊D311yescharacteristic
405C4—22no—
406C4—34no—
407C4—62no—
408C4—182no—
409C4—274no—
410C4—542no—
411C3—24no—
412C3C2×#6387814yescomplemented
413C3C2×#6410111yescharacteristic, complemented
414C2—184no—
415C2#1172346614yesdirect product factor
416C2C2×#8901711yescharacteristic
417C2C2×C4×D3×C3⋊D311yescharacteristic
418C2C2×#8902911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C33C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊(C3⋊Dic3)C2semidirect
4. C2×C32×Dic3⋊C4C2semidirect
5. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. C2×Dic3×C3⋊Dic3C2semidirect
8. #11723466C2direct (N × H)
18. C2×Dic3×C3⋊D3C4semidirect
20. C4×C3⋊(C3⋊Dic3)C22semidirect
22. C32×Dic3⋊C4C22semidirect
23. Dic3⋊(C3⋊Dic3)C22semidirect
25. Dic3×C3⋊Dic3C22semidirect
38. C2×C3×Dic3⋊C4D3semidirect
60. Dic3×C3⋊D3C2×C4semidirect
93. C3×Dic3⋊C4C2×D3semidirect
120. C2×Dic3⋊C4C3⋊D3semidirect
172. C2×C33#117semidirect
205. Dic3⋊C4C2×C3⋊D3semidirect
254. C33C2×#117semidirect
302. C2×C32#877semidirect
303. C2×C32#892semidirect
363. C32C2×#877semidirect
364. C32C2×#892semidirect
388. C2×C3#63878semidirect
389. C2×C3#64101semidirect
412. C3C2×#63878semidirect
413. C3C2×#64101semidirect
415. C2#11723466direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic3×C3⋊D32
C2×C4×C3⋊(C3⋊Dic3)2
C2×C3×#37522
C2×C32×Dic3⋊C42
C2×Dic3⋊(C3⋊Dic3)2
C2×#888032
C2×Dic3×C3⋊Dic32
#117234662
C2×#641013
C2×#638783