Finite Group Databasebeta

← C2×#11722917 #13171982

Subgroups — C2×#11722917 #13171982

Order 1728.

1826 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×C3⋊Dic3C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic, complemented
3C2×Dic3×C3⋊Dic3C212yesmaximal, complemented
4C2×C3×#3724C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D3)⋊Dic3C212yesmaximal
6#11722917C218yesmaximal, direct product factor
7C2×#63957—31nomaximal
8C2×#63878—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#88815C2214yescomplemented
17Dic3×C3⋊Dic3C2218yescomplemented
18C3×#3724C2214yescomplemented
19C3×(C2×D3)⋊Dic3C2218yes—
20C23×C3⋊Dic3—31no—
21C22×D3×Dic3—32no—
22C22×D3×Dic3—61no—
23C2×C4×C3⋊Dic3—32no—
24C2×C3×#217—32no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×C3×Dic3⋊C4—32no—
27C2×#3724—31no—
28C2×#3724D311yescharacteristic, complemented
29C2×Dic3⋊Dic3—32no—
30C2×(C2×D3)⋊Dic3D312yes—
31#63957—38no—
32C2×Dic32—32no—
33C2×Dic32—32no—
34C2×Dic32—62no—
35#63878—316no—
36C23×C33C2311yescharacteristic
37C22×C32×D3—22no—
38C22×C32×D3—22no—
39C22×C32×D3C2×C414yes—
40C22×C32×D3C2312yes—
41C2×C3⋊(C3⋊Dic3)C2×C414yes—
42C2×C3⋊(C3⋊Dic3)C2312yes—
43C2×C3×C3⋊Dic3—24no—
44C2×C3×C3⋊Dic3C2×C414yes—
45C2×C3×C3⋊Dic3C2314yes—
46C2×C3×C3⋊Dic3C2312yes—
47C2×C32×Dic3—24no—
48C2×C32×Dic3C2314yes—
49D3×C3⋊Dic3C2×C4116yescomplemented
50C2×#892—91no—
51C2×#877—92no—
52C24×C32—31no—
53C23×C3×D3—21no—
54C23×C3×D3C2×D312yes—
55C22×C32×C4—32no—
56C22×C3⋊Dic3—32no—
57C22×C3⋊Dic3—31no—
58C22×C3⋊Dic3—38no—
59C22×C3⋊Dic3—34no—
60C22×C3⋊Dic3—61no—
61C22×C3⋊Dic3—62no—
62C22×C3⋊Dic3—121no—
63C22×C3⋊Dic3C2×D312yes—
64C22×C3⋊Dic3C2×D311yescharacteristic
65C22×C3×Dic3—22no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3—62no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C2×D3×Dic3—316no—
74C2×D3×Dic3—38no—
75C2×D3×Dic3—68no—
76C2×D3×Dic3—64no—
77C4×C3⋊Dic3—38no—
78C3×#217—38no—
79C3×(C2×D3)⋊C4—38no—
80C3×Dic3⋊C4—38no—
81#3724—34no—
82#3724C2×D314yescomplemented
83Dic3⋊Dic3—38no—
84(C2×D3)⋊Dic3C2×D318yes—
85Dic32—38no—
86Dic32—38no—
87Dic32—68no—
88C22×C33C22×C412yes—
89C22×C33#4112yes—
90C22×C33#4112yes—
91C22×C33C2411yescharacteristic
92C2×C32×D3—28no—
93C2×C32×D3—22no—
94C2×C32×D3—22no—
95C2×C32×D3C22×C414yes—
96C3⋊(C3⋊Dic3)C22×C414yes—
97C3×C3⋊Dic3—24no—
98C3×C3⋊Dic3C22×C414yes—
99C32×Dic3—24no—
100C23×Dic3—92no—
101C23×Dic3—181no—
102C22×C4×D3—91no—
103C2×C3×C4⋊C4—91no—
104C2×C3×C22⋊C4—92no—
105C2×#217—32no—
106C2×(C2×D3)⋊C4—32no—
107C2×C4⋊Dic3—91no—
108C2×Dic3⋊C4—32no—
109C2×Dic3⋊C4—92no—
110C2×C4×Dic3—92no—
111C2×C4×Dic3—92no—
112C2×C4×Dic3—92no—
113C2×C4×Dic3—182no—
114#892—98no—
115#877—916no—
116C23×C32—21no—
117C23×C32—22no—
118C23×C32—34no—
119C23×C32—32no—
120C23×C32—41no—
121C23×C32—62no—
122C23×C32—62no—
123C23×C32C22×D312yes—
124C23×C32C22×D311yescharacteristic
125C22×C3×D3—24no—
126C22×C3×D3—24no—
127C22×C3×D3—24no—
128C22×C3×D3—24no—
129C22×C3×D3—24no—
130C22×C3×D3—22no—
131C22×C3×D3C4×D318yes—
132C22×C3×D3C22×D314yes—
133C2×C32×C4—34no—
134C2×C32×C4—64no—
135C2×C3⋊Dic3—24no—
136C2×C3⋊Dic3—38no—
137C2×C3⋊Dic3—34no—
138C2×C3⋊Dic3—32no—
139C2×C3⋊Dic3—34no—
140C2×C3⋊Dic3—316no—
141C2×C3⋊Dic3—64no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—68no—
144C2×C3⋊Dic3—62no—
145C2×C3⋊Dic3—124no—
146C2×C3⋊Dic3—122no—
147C2×C3⋊Dic3C4×D314yes—
148C2×C3⋊Dic3C22×D314yes—
149C2×C3⋊Dic3C22×D312yes—
150C2×C3×Dic3—28no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—38no—
155C2×C3×Dic3—34no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—34no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3—64no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—68no—
162C2×C3×Dic3—64no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3C22×D314yes—
165C2×C3×Dic3C22×D314yes—
166D3×Dic3—332no—
167D3×Dic3—616no—
168C2×#117—271noSylow
169C2×C33#11714yescomplemented
170C2×C33C23×C411yescharacteristic
171C2×C33C2×#4111yescharacteristic
172C2×C33C2×#4111yescharacteristic
173C32×D3—24no—
174C24×C3—32no—
175C24×C3—61no—
176C23×D3D3211yescharacteristic
177C22×C3×C4—32no—
178C22×C3×C4—32no—
179C22×C3×C4—62no—
180C22×C3×C4—91no—
181C22×C3×C4—92no—
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—32no—
185C22×Dic3—62no—
186C22×Dic3—61no—
187C22×Dic3—98no—
188C22×Dic3—916no—
189C22×Dic3—91no—
190C22×Dic3—92no—
191C22×Dic3—184no—
192C22×Dic3—188no—
193C22×Dic3—182no—
194C22×Dic3—181no—
195C22×Dic3—361no—
196C22×Dic3D3212yes—
197C2×C4×D3—98no—
198C2×C4×D3—94no—
199C3×C4⋊C4—94no—
200C3×C22⋊C4—98no—
201#217—38no—
202(C2×D3)⋊C4—38no—
203C4⋊Dic3—94no—
204Dic3⋊C4—38no—
205Dic3⋊C4—98no—
206C4×Dic3—98no—
207C4×Dic3—98no—
208C4×Dic3—98no—
209C4×Dic3—188no—
210C22×C32—22no—
211C22×C32—21no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—24no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—34no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32—68no—
224C22×C32—62no—
225C22×C32—62no—
226C22×C32C2×C4×D312yes—
227C22×C32C2×C4×D314yes—
228C22×C32#23714yes—
229C22×C32#23714yes—
230C22×C32#23712yes—
231C22×C32#23912yes—
232C22×C32C23×D312yes—
233C22×C32C23×D311yescharacteristic
234C2×C3×D3—24no—
235C2×C3×D3—24no—
236C2×C3×D3—216no—
237C2×C3×D3—216no—
238C2×C3×D3—24no—
239C2×C3×D3—24no—
240C2×C3×D3—24no—
241C2×C3×D3C2×C4×D318yes—
242C32×C4—64no—
243C3⋊Dic3—24no—
244C3⋊Dic3—38no—
245C3⋊Dic3—34no—
246C3⋊Dic3—64no—
247C3⋊Dic3—68no—
248C3⋊Dic3—124no—
249C3⋊Dic3C2×C4×D314yes—
250C3×Dic3—28no—
251C3×Dic3—24no—
252C3×Dic3—24no—
253C3×Dic3—38no—
254C3×Dic3—64no—
255C3×Dic3—68no—
256C3×Dic3—64no—
257C3×Dic3—64no—
258C23×C4—271no—
259#117—278no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—271no—
262C2×C22⋊C4—92no—
263C2×C42—272no—
264C33C2×#11711yescharacteristic, complemented, Sylow
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—34no—
268C23×C3—38no—
269C23×C3—41no—
270C23×C3—62no—
271C23×C3—64no—
272C23×C3—64no—
273C23×C3—64no—
274C23×C3—64no—
275C23×C3—64no—
276C23×C3C2×D3212yes—
277C23×C3C2×D3211yescharacteristic
278C22×D3—22no—
279C22×D3—22no—
280C22×D3#62414yes—
281C22×D3C2×D3212yes—
282C2×C3×C4—34no—
283C2×C3×C4—34no—
284C2×C3×C4—64no—
285C2×C3×C4—68no—
286C2×C3×C4—64no—
287C2×C3×C4—64no—
288C2×C3×C4—92no—
289C2×C3×C4—94no—
290C2×C3×C4—94no—
291C2×C3×C4—184no—
292C2×Dic3—24no—
293C2×Dic3—34no—
294C2×Dic3—38no—
295C2×Dic3—34no—
296C2×Dic3—34no—
297C2×Dic3—64no—
298C2×Dic3—64no—
299C2×Dic3—62no—
300C2×Dic3—68no—
301C2×Dic3—64no—
302C2×Dic3—64no—
303C2×Dic3—932no—
304C2×Dic3—92no—
305C2×Dic3—94no—
306C2×Dic3—98no—
307C2×Dic3—94no—
308C2×Dic3—1816no—
309C2×Dic3—184no—
310C2×Dic3—188no—
311C2×Dic3—182no—
312C2×Dic3—184no—
313C2×Dic3—362no—
314C2×Dic3—364no—
315C2×Dic3C2×D3214yes—
316C4×D3—916no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—28no—
324C2×C32—24no—
325C2×C32—44no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—64no—
330C2×C32#87718yescomplemented
331C2×C32#89214yescomplemented
332C2×C32C22×C4×D311yescharacteristic
333C2×C32C22×C4×D312yes—
334C2×C32C2×#23712yes—
335C2×C32C2×#23712yes—
336C2×C32C2×#23711yescharacteristic
337C2×C32C2×#23911yescharacteristic
338C3×D3—28no—
339C3×D3—28no—
340C24—31no—
341C22×C4—32no—
342C22×C4—91no—
343C22×C4—92no—
344C22×C4—274no—
345C22×C4—278no—
346C22×C4—271no—
347C4⋊C4—94no—
348C4⋊C4—274no—
349C22⋊C4—98no—
350C42—278no—
351C22×C3—22no—
352C22×C3—21no—
353C22×C3—22no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—22no—
357C22×C3—24no—
358C22×C3—22no—
359C22×C3—38no—
360C22×C3—41no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—64no—
365C22×C3—64no—
366C22×C3—64no—
367C22×C3—616no—
368C22×C3—64no—
369C22×C3—64no—
370C22×C3—616no—
371C22×C3#379614yes—
372C22×C3C4×D3214yes—
373C22×C3#380514yes—
374C22×C3#380512yes—
375C22×C3#380512yes—
376C22×C3C2×#62412yes—
377C22×C3C22×D3212yes—
378C22×C3C22×D3211yescharacteristic
379C2×D3—22no—
380C2×D3—22no—
381C2×D3—28no—
382C2×D3C2×#62414yes—
383C3×C4—68no—
384C3×C4—64no—
385C3×C4—64no—
386C3×C4—94no—
387C3×C4—184no—
388Dic3—24no—
389Dic3—38no—
390Dic3—64no—
391Dic3—68no—
392Dic3—64no—
393Dic3—64no—
394Dic3—98no—
395Dic3—94no—
396Dic3—184no—
397Dic3—188no—
398Dic3—364no—
399C32—22no—
400C32—21no—
401C32—41no—
402C32C2×#87712yescomplemented
403C32C2×#89211yescharacteristic, complemented
404C23—32no—
405C23—34no—
406C23—62no—
407C23—62no—
408C23D3311yescharacteristic
409C2×C4—34no—
410C2×C4—64no—
411C2×C4—92no—
412C2×C4—94no—
413C2×C4—94no—
414C2×C4—184no—
415C2×C4—2716no—
416C2×C4—272no—
417C2×C4—274no—
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—24no—
425C2×C3—28no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—44no—
430C2×C3—68no—
431C2×C3—68no—
432C2×C3#6387818yescomplemented
433C2×C3#6395714yescomplemented
434C2×C3C2×#379612yes—
435C2×C3C2×C4×D3212yes—
436C2×C3C2×#380512yes—
437C2×C3C2×#380511yescharacteristic
438C2×C3C2×#380511yescharacteristic
439C2×C3C22×#62411yescharacteristic
440D3—24no—
441C22—34no—
442C22—62no—
443C22—62no—
444C22—68no—
445C22D3×#62412yes—
446C22#8896312yes—
447C22#8896412yes—
448C22C2×D3311yescharacteristic
449C4—64no—
450C4—94no—
451C4—184no—
452C4—274no—
453C3—21no—
454C3—22no—
455C3—41no—
456C3C2×#6387812yescomplemented
457C3C2×#6395711yescharacteristic, complemented
458C2—64no—
459C2#1172291714yesdirect product factor
460C2C2×D3×#62411yescharacteristic
461C2C2×#8896311yescharacteristic
462C2C2×#8896411yescharacteristic
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×#88815C2semidirect
3. C2×Dic3×C3⋊Dic3C2semidirect
4. C2×C3×#3724C2semidirect
6. #11722917C2direct (N × H)
14. C2×D3×C3⋊Dic3C4semidirect
16. #88815C22semidirect
17. Dic3×C3⋊Dic3C22semidirect
18. C3×#3724C22semidirect
28. C2×#3724D3semidirect
49. D3×C3⋊Dic3C2×C4semidirect
82. #3724C2×D3semidirect
169. C2×C33#117semidirect
264. C33C2×#117semidirect
330. C2×C32#877semidirect
331. C2×C32#892semidirect
402. C32C2×#877semidirect
403. C32C2×#892semidirect
432. C2×C3#63878semidirect
433. C2×C3#63957semidirect
456. C3C2×#63878semidirect
457. C3C2×#63957semidirect
459. C2#11722917direct (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×#888152
C2×Dic3×C3⋊Dic32
C2×C3×#37242
C2×C3×(C2×D3)⋊Dic32
#117229172
C2×#639573
C2×#638783