Finite Group Databasebeta

← C2×#11722916 #13171983

Subgroups — C2×#11722916 #13171983

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88813C211yesmaximal, characteristic, complemented
3C2×#88806C212yesmaximal, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal
5C2×C3×Dic32C211yesmaximal, characteristic, complemented
6#11722916C218yesmaximal, direct product factor
7C2×#63957—31nomaximal
8C2×#63877—32nomaximal
9C23×C32×D3C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×D3×C3⋊Dic3C418yescomplemented
15C2×D3×C3⋊Dic3C2214yes—
16#88813C2214yescomplemented
17#88806C2218yescomplemented
18C3×(C2×D3)⋊Dic3C2218yes—
19C3×Dic32C2214yescomplemented
20C23×C3⋊Dic3—31no—
21C22×D3×Dic3—32no—
22C22×D3×Dic3—61no—
23C2×#3750—32no—
24C2×C3×#217—32no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×C3×C4×Dic3—32no—
27C2×#3724—32no—
28C2×#3724—32no—
29C2×#3724—62no—
30C2×(C2×D3)⋊Dic3D312yes—
31#63957—38no—
32C2×Dic32—32no—
33C2×Dic32—31no—
34C2×Dic32D311yescharacteristic, complemented
35#63877—316no—
36C23×C33C2311yescharacteristic
37C22×C32×D3—24no—
38C22×C32×D3C2×C414yes—
39C22×C32×D3C2312yes—
40C2×C3⋊(C3⋊Dic3)C2×C414yes—
41C2×C3⋊(C3⋊Dic3)C2312yes—
42C2×C3×C3⋊Dic3—24no—
43C2×C3×C3⋊Dic3C2×C414yes—
44C2×C3×C3⋊Dic3C2314yes—
45C2×C3×C3⋊Dic3C2312yes—
46C2×C32×Dic3—24no—
47C2×C32×Dic3C2314yes—
48D3×C3⋊Dic3C2×C4116yescomplemented
49C2×#877—92no—
50C2×#872—91no—
51C24×C32—31no—
52C23×C3×D3—21no—
53C23×C3×D3C2×D312yes—
54C22×C32×C4—32no—
55C22×C3⋊Dic3—32no—
56C22×C3⋊Dic3—31no—
57C22×C3⋊Dic3—38no—
58C22×C3⋊Dic3—34no—
59C22×C3⋊Dic3—61no—
60C22×C3⋊Dic3—62no—
61C22×C3⋊Dic3—121no—
62C22×C3⋊Dic3C2×D312yes—
63C22×C3⋊Dic3C2×D311yescharacteristic
64C22×C3×Dic3—22no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C22×C3×Dic3—62no—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D312yes—
72C2×D3×Dic3—316no—
73C2×D3×Dic3—38no—
74C2×D3×Dic3—68no—
75C2×D3×Dic3—64no—
76#3750—38no—
77C3×#217—38no—
78C3×(C2×D3)⋊C4—38no—
79C3×C4×Dic3—38no—
80#3724—38no—
81#3724—38no—
82#3724—68no—
83(C2×D3)⋊Dic3C2×D318yes—
84Dic32—38no—
85Dic32—34no—
86Dic32C2×D314yescomplemented
87C22×C33C22×C412yes—
88C22×C33#4114yes—
89C22×C33C2411yescharacteristic
90C2×C32×D3—28no—
91C2×C32×D3—24no—
92C2×C32×D3C22×C414yes—
93C3⋊(C3⋊Dic3)C22×C414yes—
94C3×C3⋊Dic3—24no—
95C3×C3⋊Dic3C22×C414yes—
96C32×Dic3—24no—
97C23×Dic3—92no—
98C23×Dic3—181no—
99C22×C4×D3—91no—
100C2×C3×C22⋊C4—92no—
101C2×C3×C42—91no—
102C2×#217—32no—
103C2×(C2×D3)⋊C4—32no—
104C2×Dic3⋊C4—92no—
105C2×Dic3⋊C4—92no—
106C2×Dic3⋊C4—92no—
107C2×Dic3⋊C4—182no—
108C2×C4×Dic3—32no—
109C2×C4×Dic3—91no—
110C2×C4×Dic3—92no—
111#877—916no—
112#872—98no—
113C23×C32—21no—
114C23×C32—22no—
115C23×C32—34no—
116C23×C32—32no—
117C23×C32—41no—
118C23×C32—64no—
119C23×C32C22×D311yescharacteristic
120C23×C32C22×D312yes—
121C22×C3×D3—28no—
122C22×C3×D3—24no—
123C22×C3×D3—24no—
124C22×C3×D3—24no—
125C22×C3×D3—22no—
126C22×C3×D3C4×D318yes—
127C22×C3×D3C22×D314yes—
128C2×C32×C4—34no—
129C2×C32×C4—64no—
130C2×C3⋊Dic3—24no—
131C2×C3⋊Dic3—38no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—32no—
134C2×C3⋊Dic3—34no—
135C2×C3⋊Dic3—316no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—68no—
139C2×C3⋊Dic3—62no—
140C2×C3⋊Dic3—124no—
141C2×C3⋊Dic3—122no—
142C2×C3⋊Dic3C4×D314yes—
143C2×C3⋊Dic3C22×D314yes—
144C2×C3⋊Dic3C22×D312yes—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—24no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—38no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—44no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—64no—
156C2×C3×Dic3—64no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—124no—
159C2×C3×Dic3C22×D314yes—
160C2×C3×Dic3C22×D314yes—
161D3×Dic3—332no—
162D3×Dic3—616no—
163C2×#117—271noSylow
164C2×C33#11714yescomplemented
165C2×C33C23×C411yescharacteristic
166C2×C33C2×#4112yes—
167C32×D3—24no—
168C24×C3—32no—
169C24×C3—61no—
170C23×D3D3211yescharacteristic
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—62no—
174C22×C3×C4—91no—
175C22×C3×C4—92no—
176C22×Dic3—32no—
177C22×Dic3—32no—
178C22×Dic3—32no—
179C22×Dic3—61no—
180C22×Dic3—62no—
181C22×Dic3—98no—
182C22×Dic3—916no—
183C22×Dic3—91no—
184C22×Dic3—92no—
185C22×Dic3—184no—
186C22×Dic3—188no—
187C22×Dic3—182no—
188C22×Dic3—181no—
189C22×Dic3—361no—
190C22×Dic3D3212yes—
191C2×C4×D3—98no—
192C2×C4×D3—94no—
193C3×C22⋊C4—98no—
194C3×C42—94no—
195#217—38no—
196(C2×D3)⋊C4—38no—
197Dic3⋊C4—98no—
198Dic3⋊C4—98no—
199Dic3⋊C4—98no—
200Dic3⋊C4—188no—
201C4×Dic3—38no—
202C4×Dic3—94no—
203C4×Dic3—98no—
204C22×C32—22no—
205C22×C32—21no—
206C22×C32—24no—
207C22×C32—24no—
208C22×C32—24no—
209C22×C32—24no—
210C22×C32—22no—
211C22×C32—34no—
212C22×C32—42no—
213C22×C32—44no—
214C22×C32—41no—
215C22×C32—68no—
216C22×C32—64no—
217C22×C32C2×C4×D312yes—
218C22×C32C2×C4×D314yes—
219C22×C32#23514yes—
220C22×C32#23714yes—
221C22×C32#23714yes—
222C22×C32C23×D311yescharacteristic
223C22×C32C23×D312yes—
224C2×C3×D3—28no—
225C2×C3×D3—216no—
226C2×C3×D3—216no—
227C2×C3×D3—24no—
228C2×C3×D3—24no—
229C2×C3×D3—24no—
230C2×C3×D3C2×C4×D318yes—
231C32×C4—64no—
232C3⋊Dic3—24no—
233C3⋊Dic3—38no—
234C3⋊Dic3—34no—
235C3⋊Dic3—64no—
236C3⋊Dic3—68no—
237C3⋊Dic3—124no—
238C3⋊Dic3C2×C4×D314yes—
239C3×Dic3—24no—
240C3×Dic3—24no—
241C3×Dic3—38no—
242C3×Dic3—44no—
243C3×Dic3—64no—
244C3×Dic3—64no—
245C3×Dic3—64no—
246C3×Dic3—124no—
247C23×C4—271no—
248#117—278no—
249C2×C4⋊C4—272no—
250C2×C22⋊C4—92no—
251C2×C42—91no—
252C2×C42—271no—
253C33C2×#11711yescharacteristic, complemented, Sylow
254C23×C3—22no—
255C23×C3—21no—
256C23×C3—34no—
257C23×C3—38no—
258C23×C3—41no—
259C23×C3—62no—
260C23×C3—64no—
261C23×C3—68no—
262C23×C3—64no—
263C23×C3—64no—
264C23×C3C2×D3212yes—
265C23×C3C2×D3211yescharacteristic
266C22×D3—24no—
267C22×D3#62414yes—
268C22×D3C2×D3212yes—
269C2×C3×C4—34no—
270C2×C3×C4—34no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—64no—
274C2×C3×C4—92no—
275C2×C3×C4—94no—
276C2×C3×C4—94no—
277C2×C3×C4—124no—
278C2×C3×C4—184no—
279C2×Dic3—24no—
280C2×Dic3—34no—
281C2×Dic3—38no—
282C2×Dic3—34no—
283C2×Dic3—34no—
284C2×Dic3—64no—
285C2×Dic3—62no—
286C2×Dic3—64no—
287C2×Dic3—64no—
288C2×Dic3—64no—
289C2×Dic3—932no—
290C2×Dic3—92no—
291C2×Dic3—94no—
292C2×Dic3—98no—
293C2×Dic3—94no—
294C2×Dic3—124no—
295C2×Dic3—1816no—
296C2×Dic3—184no—
297C2×Dic3—188no—
298C2×Dic3—182no—
299C2×Dic3—184no—
300C2×Dic3—362no—
301C2×Dic3—364no—
302C2×Dic3C2×D3214yes—
303C4×D3—916no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—28no—
310C2×C32—24no—
311C2×C32—44no—
312C2×C32—41no—
313C2×C32—42no—
314C2×C32—64no—
315C2×C32#87214yescomplemented
316C2×C32#87718yescomplemented
317C2×C32C22×C4×D311yescharacteristic
318C2×C32C22×C4×D312yes—
319C2×C32C2×#23512yes—
320C2×C32C2×#23712yes—
321C2×C32C2×#23712yes—
322C3×D3—28no—
323C3×D3—28no—
324C24—31no—
325C22×C4—32no—
326C22×C4—91no—
327C22×C4—92no—
328C22×C4—274no—
329C22×C4—278no—
330C22×C4—271no—
331C4⋊C4—278no—
332C22⋊C4—98no—
333C42—94no—
334C42—274no—
335C22×C3—22no—
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—24no—
339C22×C3—24no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—38no—
343C22×C3—41no—
344C22×C3—42no—
345C22×C3—44no—
346C22×C3—64no—
347C22×C3—68no—
348C22×C3—616no—
349C22×C3—64no—
350C22×C3—64no—
351C22×C3—616no—
352C22×C3#379514yes—
353C22×C3C4×D3214yes—
354C22×C3#380414yes—
355C22×C3#380514yes—
356C22×C3C2×#62412yes—
357C22×C3C22×D3212yes—
358C22×C3C22×D3211yescharacteristic
359C2×D3—24no—
360C2×D3—28no—
361C2×D3C2×#62414yes—
362C3×C4—64no—
363C3×C4—64no—
364C3×C4—94no—
365C3×C4—124no—
366C3×C4—184no—
367Dic3—24no—
368Dic3—38no—
369Dic3—64no—
370Dic3—64no—
371Dic3—64no—
372Dic3—98no—
373Dic3—94no—
374Dic3—124no—
375Dic3—184no—
376Dic3—188no—
377Dic3—364no—
378C32—22no—
379C32—21no—
380C32—41no—
381C32C2×#87211yescharacteristic, complemented
382C32C2×#87712yescomplemented
383C23—32no—
384C23—34no—
385C23—64no—
386C23D3311yescharacteristic
387C2×C4—34no—
388C2×C4—64no—
389C2×C4—92no—
390C2×C4—94no—
391C2×C4—94no—
392C2×C4—184no—
393C2×C4—2716no—
394C2×C4—272no—
395C2×C4—274no—
396C2×C3—21no—
397C2×C3—22no—
398C2×C3—22no—
399C2×C3—22no—
400C2×C3—22no—
401C2×C3—24no—
402C2×C3—28no—
403C2×C3—41no—
404C2×C3—42no—
405C2×C3—44no—
406C2×C3—68no—
407C2×C3—68no—
408C2×C3#6387718yescomplemented
409C2×C3#6395714yescomplemented
410C2×C3C2×#379512yes—
411C2×C3C2×C4×D3212yes—
412C2×C3C2×#380412yes—
413C2×C3C2×#380512yes—
414C2×C3C22×#62411yescharacteristic
415D3—24no—
416C22—34no—
417C22—64no—
418C22—68no—
419C22D3×#62412yes—
420C22#8896114yes—
421C22C2×D3311yescharacteristic
422C4—64no—
423C4—94no—
424C4—184no—
425C4—274no—
426C3—21no—
427C3—22no—
428C3—41no—
429C3C2×#6387712yescomplemented
430C3C2×#6395711yescharacteristic, complemented
431C2—64no—
432C2#1172291614yesdirect product factor
433C2C2×D3×#62411yescharacteristic
434C2C2×#8896112yes—
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socleC23×C33C23
prime radicalC23×C32×D3C22
cosocleC22×C33C24
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88813C2semidirect
3. C2×#88806C2semidirect
5. C2×C3×Dic32C2semidirect
6. #11722916C2direct (N × H)
14. C2×D3×C3⋊Dic3C4semidirect
16. #88813C22semidirect
17. #88806C22semidirect
19. C3×Dic32C22semidirect
34. C2×Dic32D3semidirect
48. D3×C3⋊Dic3C2×C4semidirect
86. Dic32C2×D3semidirect
164. C2×C33#117semidirect
253. C33C2×#117semidirect
315. C2×C32#872semidirect
316. C2×C32#877semidirect
381. C32C2×#872semidirect
382. C32C2×#877semidirect
408. C2×C3#63877semidirect
409. C2×C3#63957semidirect
429. C3C2×#63877semidirect
430. C3C2×#63957semidirect
432. C2#11722916direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C3⋊Dic32
C2×#888132
C2×#888062
C2×C3×(C2×D3)⋊Dic32
C2×C3×Dic322
#117229162
C2×#639573
C2×#638773