Finite Group Databasebeta

← C2×#11723453 #13176836

Subgroups — C2×#11723453 #13176836

Order 1728.

2004 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
1C2×#88859C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4C2×#88807C211yesmaximal, characteristic, complemented
5C2×#88806C211yesmaximal, characteristic, complemented
6C2×#88804C211yesmaximal, characteristic
7C2×#88804C211yesmaximal, characteristic
8#11723453C218yesmaximal, direct product factor
9C2×#64085—31nomaximal
10C2×#63868—34nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18#88859C2214yes—
19C3×#3750C2214yescomplemented
20C32×C4×Dic3C2214yescomplemented
21#88807C2214yescomplemented
22#88806C2214yescomplemented
23#88804C2214yes—
24#88804C2214yes—
25C2×C32×C42—31no—
26C2×#3752—31no—
27C2×#3752—31no—
28C2×#3752—31no—
29C2×#3752—34no—
30C2×#3752—64no—
31C2×#3750—31no—
32C2×#3750—31no—
33C2×#3750D311yescharacteristic, complemented
34#64085—38no—
35C2×C3×Dic3⋊C4—34no—
36C2×C3×C4×Dic3D314yescomplemented
37C2×#3724—34no—
38C2×Dic3⋊Dic3—34no—
39C2×#3722—34no—
40C2×#3722—34no—
41#63868—332no—
42C23×C33C2311yescharacteristic
43C22×C3⋊(C3⋊D3)—22no—
44C22×C3⋊(C3⋊D3)—22no—
45C22×C3⋊(C3⋊D3)—22no—
46C22×C3⋊(C3⋊D3)C2312yes—
47C2×C33×C4—22no—
48C2×C33×C4C2312yes—
49C2×C3⋊(C3⋊Dic3)—22no—
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2312yes—
55C2×C32×Dic3—22no—
56C2×C32×Dic3—22no—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C2×#898—91no—
60C2×#876—94no—
61C23×C3⋊D3—31no—
62C23×C3⋊D3—34no—
63C23×C3⋊D3—64no—
64C22×C32×C4—24no—
65C22×C32×C4—31no—
66C22×C32×C4—31no—
67C22×C32×C4C2×D314yes—
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—34no—
71C22×C3⋊Dic3—64no—
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3×Dic3—34no—
75C22×C3×Dic3—34no—
76C22×C3×Dic3C2×D314yes—
77C22×C3×Dic3C2×D314yes—
78C32×C42—34no—
79#3752—34no—
80#3752—316no—
81#3752—34no—
82#3752—34no—
83#3752—616no—
84#3750—34no—
85#3750—34no—
86#3750C2×D314yescomplemented
87C3×Dic3⋊C4—316no—
88C3×C4×Dic3C2×D3116yescomplemented
89#3724—316no—
90Dic3⋊Dic3—316no—
91#3722—316no—
92#3722—316no—
93C22×C33#4112yes—
94C22×C33#4112yes—
95C22×C33#4112yes—
96C22×C33C2411yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—22no—
100C2×C3⋊(C3⋊D3)—44no—
101C33×C4—22no—
102C3⋊(C3⋊Dic3)—22no—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3—22no—
105C32×Dic3—22no—
106C32×Dic3—22no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C42—34no—
109C2×(C2×D3)⋊C4—94no—
110C2×(C2×D3)⋊C4—94no—
111C2×(C2×D3)⋊C4—94no—
112C2×(C2×D3)⋊C4—91no—
113C2×(C2×D3)⋊C4—91no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—184no—
116C2×C4⋊Dic3—91no—
117C2×Dic3⋊C4—34no—
118C2×Dic3⋊C4—94no—
119C2×Dic3⋊C4—94no—
120C2×Dic3⋊C4—91no—
121C2×C4×Dic3C3⋊D311yescharacteristic, complemented
122#898—98no—
123#876—932no—
124C23×C32—24no—
125C23×C32C22×D314yes—
126C23×C32C22×D311yescharacteristic
127C22×C3⋊D3—32no—
128C22×C3⋊D3—38no—
129C22×C3⋊D3—68no—
130C22×C3⋊D3—616no—
131C22×C3⋊D3—68no—
132C22×C3⋊D3—62no—
133C22×C3⋊D3—62no—
134C22×C3⋊D3—62no—
135C22×C3⋊D3—68no—
136C22×C3⋊D3—68no—
137C22×C3⋊D3—128no—
138C22×C3⋊D3—128no—
139C2×C32×C4—22no—
140C2×C32×C4—28no—
141C2×C32×C4—28no—
142C2×C32×C4—32no—
143C2×C32×C4—32no—
144C2×C32×C4—48no—
145C2×C32×C4—62no—
146C2×C32×C4—62no—
147C2×C32×C4C22×D318yes—
148C2×C32×C4C22×D312yes—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—38no—
153C2×C3⋊Dic3—68no—
154C2×C3⋊Dic3—62no—
155C2×C3⋊Dic3—68no—
156C2×C3⋊Dic3—128no—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3×Dic3—28no—
160C2×C3×Dic3—28no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—38no—
163C2×C3×Dic3—68no—
164C2×C3×Dic3—68no—
165C2×C3×Dic3C22×D318yes—
166C2×C3×Dic3C22×D318yes—
167C2×#126—271noSylow
168C2×C33#12614yescomplemented
169C2×C33C2×#4111yescharacteristic
170C2×C33C2×#4111yescharacteristic
171C2×C33C2×#4111yescharacteristic
172C3⋊(C3⋊D3)—42no—
173C23×D3—94no—
174C23×D3—91no—
175C23×D3—184no—
176C22×C3×C4—24no—
177C22×C3×C4—34no—
178C22×C3×C4—34no—
179C22×C3×C4—91no—
180C22×C3×C4—91no—
181C22×C3×C4D3214yes—
182C22×C3×C4C2×C3⋊D311yescharacteristic
183C22×Dic3—34no—
184C22×Dic3—34no—
185C22×Dic3—94no—
186C22×Dic3—91no—
187C22×Dic3—184no—
188C22×Dic3C2×C3⋊D311yescharacteristic
189C22×Dic3C2×C3⋊D311yescharacteristic
190C3×C4⋊C4—94no—
191C3×C42—316no—
192(C2×D3)⋊C4—916no—
193(C2×D3)⋊C4—916no—
194(C2×D3)⋊C4—916no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—94no—
197(C2×D3)⋊C4—94no—
198(C2×D3)⋊C4—1816no—
199C4⋊Dic3—94no—
200Dic3⋊C4—316no—
201Dic3⋊C4—916no—
202Dic3⋊C4—916no—
203Dic3⋊C4—94no—
204C4×Dic3C2×C3⋊D314yescomplemented
205C22×C32—24no—
206C22×C32—28no—
207C22×C32—28no—
208C22×C32—28no—
209C22×C32#23518yes—
210C22×C32#23518yes—
211C22×C32#23518yes—
212C22×C32#23512yes—
213C22×C32#23712yes—
214C22×C32#23912yes—
215C22×C32C23×D314yes—
216C22×C32C23×D311yescharacteristic
217C2×C3⋊D3—616no—
218C2×C3⋊D3—68no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—62no—
221C2×C3⋊D3—62no—
222C2×C3⋊D3—68no—
223C2×C3⋊D3—68no—
224C2×C3⋊D3—128no—
225C2×C3⋊D3—128no—
226C2×C3⋊D3—1232no—
227C2×C3⋊D3—1216no—
228C2×C3⋊D3—124no—
229C32×C4—22no—
230C32×C4—28no—
231C32×C4—48no—
232C32×C4—62no—
233C32×C4—62no—
234C3⋊Dic3—22no—
235C3⋊Dic3—22no—
236C3⋊Dic3—62no—
237C3⋊Dic3—68no—
238C3⋊Dic3—128no—
239C3×Dic3—28no—
240C3×Dic3—28no—
241C3×Dic3—68no—
242C3×Dic3—68no—
243#126—278no—
244C2×C4⋊C4—91no—
245C2×C4⋊C4—271no—
246C2×C4⋊C4—271no—
247C2×C22⋊C4—271no—
248C2×C22⋊C4—271no—
249C2×C22⋊C4—271no—
250C2×C42—31no—
251C33C2×#12611yescharacteristic, complemented, Sylow
252C23×C3—24no—
253C23×C3C2×D3214yes—
254C23×C3C22×C3⋊D311yescharacteristic
255C22×D3—98no—
256C22×D3—92no—
257C22×D3—188no—
258C22×D3—1816no—
259C22×D3—188no—
260C22×D3—188no—
261C22×D3—188no—
262C22×D3—182no—
263C22×D3—182no—
264C22×D3—182no—
265C22×D3—368no—
266C22×D3—368no—
267C2×C3×C4—28no—
268C2×C3×C4—22no—
269C2×C3×C4—28no—
270C2×C3×C4—38no—
271C2×C3×C4—38no—
272C2×C3×C4—48no—
273C2×C3×C4—68no—
274C2×C3×C4—68no—
275C2×C3×C4—92no—
276C2×C3×C4—92no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279C2×C3×C4C2×D3218yes—
280C2×C3×C4C22×C3⋊D312yes—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—38no—
284C2×Dic3—38no—
285C2×Dic3—68no—
286C2×Dic3—68no—
287C2×Dic3—98no—
288C2×Dic3—92no—
289C2×Dic3—188no—
290C2×Dic3—188no—
291C2×Dic3—182no—
292C2×Dic3—368no—
293C2×Dic3C22×C3⋊D312yes—
294C2×Dic3C22×C3⋊D312yes—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—216no—
299C2×C32#876116yescomplemented
300C2×C32#89814yescomplemented
301C2×C32C2×#23514yes—
302C2×C32C2×#23514yes—
303C2×C32C2×#23514yes—
304C2×C32C2×#23511yescharacteristic
305C2×C32C2×#23711yescharacteristic
306C2×C32C2×#23911yescharacteristic
307C3⋊D3—1216no—
308C3⋊D3—128no—
309C3⋊D3—122no—
310C24—271no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—91no—
314C22×C4—91no—
315C22×C4—271no—
316C22×C4D3×C3⋊D311yescharacteristic
317C4⋊C4—94no—
318C4⋊C4—274no—
319C4⋊C4—274no—
320C22⋊C4—274no—
321C22⋊C4—274no—
322C22⋊C4—274no—
323C42—34no—
324C22×C3—24no—
325C22×C3—28no—
326C22×C3—28no—
327C22×C3—28no—
328C22×C3#379818yes—
329C22×C3#379918yes—
330C22×C3#380418yes—
331C22×C3#382812yes—
332C22×C3#382812yes—
333C22×C3#382812yes—
334C22×C3C22×D3214yes—
335C22×C3C23×C3⋊D311yescharacteristic
336C2×D3—1816no—
337C2×D3—188no—
338C2×D3—188no—
339C2×D3—188no—
340C2×D3—182no—
341C2×D3—182no—
342C2×D3—182no—
343C2×D3—368no—
344C2×D3—368no—
345C2×D3—3632no—
346C2×D3—364no—
347C2×D3—3616no—
348C3×C4—28no—
349C3×C4—22no—
350C3×C4—48no—
351C3×C4—68no—
352C3×C4—68no—
353C3×C4—182no—
354C3×C4—182no—
355Dic3—22no—
356Dic3—22no—
357Dic3—68no—
358Dic3—68no—
359Dic3—188no—
360Dic3—182no—
361Dic3—368no—
362C32—24no—
363C32C2×#87614yescomplemented
364C32C2×#89811yescharacteristic, complemented
365C23—272no—
366C23—542no—
367C23—542no—
368C23—542no—
369C23C2×D3×C3⋊D311yescharacteristic
370C2×C4—22no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—92no—
376C2×C4—92no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4—272no—
380C2×C4—542no—
381C2×C4C2×D3×C3⋊D312yes—
382C2×C3—24no—
383C2×C3—24no—
384C2×C3—24no—
385C2×C3—216no—
386C2×C3#63868116yescomplemented
387C2×C3#6408514yescomplemented
388C2×C3C2×#379814yes—
389C2×C3C2×#379914yes—
390C2×C3C2×#380414yes—
391C2×C3C2×#382811yescharacteristic
392C2×C3C2×#382811yescharacteristic
393C2×C3C2×#382811yescharacteristic
394D3—3616no—
395D3—368no—
396D3—362no—
397C22—542no—
398C22—542no—
399C22—542no—
400C22—1084no—
401C22#8902012yes—
402C22#8902212yes—
403C22#8903112yes—
404C22C22×D3×C3⋊D311yescharacteristic
405C4—22no—
406C4—62no—
407C4—62no—
408C4—182no—
409C4—182no—
410C4—542no—
411C3—24no—
412C3C2×#6386814yescomplemented
413C3C2×#6408511yescharacteristic, complemented
414C2—1082no—
415C2#1172345314yesdirect product factor
416C2C2×#8902011yescharacteristic
417C2C2×#8902211yescharacteristic
418C2C2×#8903111yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3750C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. C2×#88807C2semidirect
5. C2×#88806C2semidirect
8. #11723453C2direct (N × H)
19. C3×#3750C22semidirect
20. C32×C4×Dic3C22semidirect
21. #88807C22semidirect
22. #88806C22semidirect
33. C2×#3750D3semidirect
36. C2×C3×C4×Dic3D3semidirect
86. #3750C2×D3semidirect
88. C3×C4×Dic3C2×D3semidirect
121. C2×C4×Dic3C3⋊D3semidirect
168. C2×C33#126semidirect
204. C4×Dic3C2×C3⋊D3semidirect
251. C33C2×#126semidirect
299. C2×C32#876semidirect
300. C2×C32#898semidirect
363. C32C2×#876semidirect
364. C32C2×#898semidirect
386. C2×C3#63868semidirect
387. C2×C3#64085semidirect
412. C3C2×#63868semidirect
413. C3C2×#64085semidirect
415. C2#11723453direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#888592
C2×C3×#37502
C2×C32×C4×Dic32
C2×#888072
C2×#888062
C2×#888042
#117234532
C2×#640853
C2×#638683