Finite Group Databasebeta

← C2×#11722938 #13172198

Subgroups — C2×#11722938 #13172198

Order 1728.

2338 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6295C211yesmaximal, characteristic
2C22×Dic3×C3⋊D3C212yesmaximal, complemented
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3724C211yesmaximal, characteristic, complemented
5C2×C3×#3722C212yesmaximal, complemented
6#11722938C218yesmaximal, direct product factor
7C2×#63902—31nomaximal
8C2×#63881—32nomaximal
9C23×C3×C3⋊D3C2212yes—
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×#6295C418yescomplemented
15C2×#6295C2214yes—
16C2×Dic3×C3⋊D3—28no—
17C2×Dic3×C3⋊D3C2218yescomplemented
18Dic3×C3⋊Dic3C2214yescomplemented
19C3×#3724C2214yescomplemented
20C3×#3722C2218yescomplemented
21C22×C4×C3⋊D3—32no—
22C22×#626—32no—
23C22×#625—31no—
24C22×D3×Dic3—32no—
25C22×D3×Dic3—32no—
26C22×D3×Dic3—62no—
27C2×C4×C3⋊Dic3—31no—
28C2×C3×(C2×D3)⋊C4—32no—
29C2×C3×(C2×D3)⋊C4—32no—
30C2×C3×Dic3⋊C4—32no—
31C2×#3724D311yescharacteristic, complemented
32C2×#3722D312yescomplemented
33C2×Dic32—32no—
34C2×Dic32—61no—
35#63902—38no—
36#63881—316no—
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3—24no—
39C22×C3×C3⋊D3—24no—
40C22×C3×C3⋊D3C2×C418yescomplemented
41C22×C3×C3⋊D3C2314yes—
42C2×C3⋊(C3⋊Dic3)D414yescomplemented
43C2×C3⋊(C3⋊Dic3)C2312yes—
44C2×C3×C3⋊Dic3C2×C414yescomplemented
45C2×C3×C3⋊Dic3C2312yes—
46C2×C32×Dic3—24no—
47C2×C32×Dic3—22no—
48C2×C32×Dic3C2314yes—
49C2×C32×Dic3C2312yes—
50#6295C2×C4116yescomplemented
51Dic3×C3⋊D3—216no—
52C2×#893—91no—
53C2×#67⋊C4—92no—
54C23×C3⋊D3C2×D312yes—
55C23×C3×D3—32no—
56C23×C3×D3—32no—
57C23×C3×D3—62no—
58C22×C32×C4—32no—
59C22×C32×C4—31no—
60C22×C3⋊Dic3—31no—
61C22×C3⋊Dic3—32no—
62C22×C3⋊Dic3—62no—
63C22×C3⋊Dic3—61no—
64C22×C3⋊Dic3—121no—
65C22×C3⋊Dic3C2×D311yescharacteristic
66C2×C4×C3⋊D3—38no—
67C2×C4×C3⋊D3—68no—
68C22×C3×Dic3—22no—
69C22×C3×Dic3—21no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C2×D312yes—
74C22×C3×Dic3C2×D312yes—
75C2×#626—316no—
76C2×#626—38no—
77C2×#625—38no—
78C2×#625—34no—
79C2×D3×Dic3—38no—
80C2×D3×Dic3—38no—
81C2×D3×Dic3—68no—
82C2×D3×Dic3—68no—
83C2×D3×Dic3—68no—
84C2×D3×Dic3—616no—
85C4×C3⋊Dic3—34no—
86C3×(C2×D3)⋊C4—38no—
87C3×(C2×D3)⋊C4—38no—
88C3×Dic3⋊C4—38no—
89#3724C2×D314yescomplemented
90#3722C2×D318yescomplemented
91Dic32—38no—
92Dic32—64no—
93C22×C33C22×C412yes—
94C22×C33C2×D412yes—
95C22×C33#4112yes—
96C22×C33C2411yescharacteristic
97C2×C3×C3⋊D3—216no—
98C2×C3×C3⋊D3—24no—
99C2×C3×C3⋊D3—24no—
100C2×C3×C3⋊D3C22×C418yescomplemented
101C3⋊(C3⋊Dic3)C2×D414yescomplemented
102C3×C3⋊Dic3C22×C414yescomplemented
103C32×Dic3—24no—
104C32×Dic3—22no—
105C22×#67—92no—
106C23×Dic3—92no—
107C22×D12—91no—
108C22×C4×D3—92no—
109C22×C4×D3—92no—
110C22×C4×D3—182no—
111C2×C3×C4⋊C4—91no—
112C2×C3×C22⋊C4—92no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×Dic3⋊C4—32no—
116C2×C4×Dic3—92no—
117C2×C4×Dic3—91no—
118C2×C4×Dic3—181no—
119#893—98no—
120#67⋊C4—916no—
121C23×C32—22no—
122C23×C32—21no—
123C23×C32—41no—
124C23×C32C22×D312yes—
125C23×C32C22×D311yescharacteristic
126C22×C3⋊D3—24no—
127C22×C3⋊D3—24no—
128C22×C3⋊D3C4×D318yescomplemented
129C22×C3⋊D3C22×D314yes—
130C22×C3×D3—38no—
131C22×C3×D3—38no—
132C22×C3×D3—34no—
133C22×C3×D3—34no—
134C22×C3×D3—68no—
135C22×C3×D3—64no—
136C22×C3×D3—68no—
137C22×C3×D3—68no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C2×C32×C4—34no—
143C2×C32×C4—32no—
144C2×C32×C4—64no—
145C2×C32×C4—62no—
146C2×C3⋊Dic3—34no—
147C2×C3⋊Dic3—38no—
148C2×C3⋊Dic3—34no—
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—62no—
151C2×C3⋊Dic3—68no—
152C2×C3⋊Dic3—64no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—124no—
155C2×C3⋊Dic3—122no—
156C2×C3⋊Dic3C4×D314yescomplemented
157C2×C3⋊Dic3C22×D312yes—
158C4×C3⋊D3—616no—
159C2×C3×Dic3—28no—
160C2×C3×Dic3—24no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—24no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—38no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3C22×D314yes—
171C2×C3×Dic3C22×D314yes—
172C2×C3×Dic3C22×D314yes—
173#626—332no—
174#625—316no—
175D3×Dic3—616no—
176D3×Dic3—616no—
177D3×Dic3—632no—
178C2×C4×D4—271noSylow
179C2×C33C4×D414yescomplemented
180C2×C33C23×C411yescharacteristic
181C2×C33C22×D411yescharacteristic
182C2×C33C2×#4111yescharacteristic
183C3×C3⋊D3—28no—
184C24×C3—92no—
185C23×D3—32no—
186C23×D3—32no—
187C23×D3—62no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—61no—
192C22×C3×C4—62no—
193C22×C3×C4—91no—
194C2×#67—98no—
195C2×#67—916no—
196C22×Dic3—32no—
197C22×Dic3—61no—
198C22×Dic3—91no—
199C22×Dic3—92no—
200C22×Dic3—98no—
201C22×Dic3—181no—
202C22×Dic3—182no—
203C22×Dic3—188no—
204C22×Dic3—361no—
205C22×Dic3D3211yescharacteristic
206C22×Dic3D3212yes—
207C2×D12—94no—
208C2×D12—98no—
209C2×C4×D3—98no—
210C2×C4×D3—98no—
211C2×C4×D3—188no—
212C2×C4×D3—188no—
213C2×C4×D3—188no—
214C2×C4×D3—1816no—
215C3×C4⋊C4—94no—
216C3×C22⋊C4—98no—
217(C2×D3)⋊C4—38no—
218(C2×D3)⋊C4—38no—
219Dic3⋊C4—38no—
220C4×Dic3—98no—
221C4×Dic3—94no—
222C4×Dic3—184no—
223C22×C32—21no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—22no—
227C22×C32—24no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—24no—
231C22×C32—42no—
232C22×C32—42no—
233C22×C32—42no—
234C22×C32—41no—
235C22×C32C2×C4×D314yes—
236C22×C32C2×C4×D312yes—
237C22×C32D3×D414yes—
238C22×C32D3×D412yes—
239C22×C32#23714yes—
240C22×C32#23912yes—
241C22×C32C23×D312yes—
242C22×C32C23×D311yescharacteristic
243C2×C3⋊D3—216no—
244C2×C3⋊D3—24no—
245C2×C3⋊D3—24no—
246C2×C3⋊D3C2×C4×D318yescomplemented
247C2×C3×D3—38no—
248C2×C3×D3—38no—
249C2×C3×D3—68no—
250C2×C3×D3—68no—
251C2×C3×D3—68no—
252C2×C3×D3—632no—
253C2×C3×D3—616no—
254C2×C3×D3—616no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C32×C4—64no—
260C32×C4—62no—
261C3⋊Dic3—34no—
262C3⋊Dic3—38no—
263C3⋊Dic3—68no—
264C3⋊Dic3—64no—
265C3⋊Dic3—124no—
266C3⋊Dic3C2×C4×D314yescomplemented
267C3×Dic3—28no—
268C3×Dic3—24no—
269C3×Dic3—24no—
270C3×Dic3—24no—
271C3×Dic3—24no—
272C3×Dic3—38no—
273C3×Dic3—64no—
274C22×D4—271no—
275C23×C4—272no—
276C4×D4—278no—
277C2×C4⋊C4—91no—
278C2×C22⋊C4—92no—
279C2×C42—271no—
280C33C2×C4×D411yescharacteristic, complemented, Sylow
281C23×C3—21no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3—94no—
285C23×C3—98no—
286C23×C3—184no—
287C23×C3—184no—
288C23×C3C2×D3212yes—
289C23×C3C2×D3211yescharacteristic
290C22×D3—34no—
291C22×D3—34no—
292C22×D3—38no—
293C22×D3—38no—
294C22×D3—68no—
295C22×D3—64no—
296C22×D3—68no—
297C22×D3—68no—
298C22×D3—64no—
299C22×D3—64no—
300C22×D3—64no—
301C22×D3—64no—
302C2×C3×C4—34no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—62no—
306C2×C3×C4—64no—
307C2×C3×C4—68no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—64no—
311C2×C3×C4—64no—
312C2×C3×C4—92no—
313C2×C3×C4—94no—
314#67—932no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—34no—
318C2×Dic3—38no—
319C2×Dic3—64no—
320C2×Dic3—62no—
321C2×Dic3—92no—
322C2×Dic3—94no—
323C2×Dic3—94no—
324C2×Dic3—98no—
325C2×Dic3—182no—
326C2×Dic3—184no—
327C2×Dic3—188no—
328C2×Dic3—184no—
329C2×Dic3—1816no—
330C2×Dic3—362no—
331C2×Dic3—364no—
332C2×Dic3C2×D3212yes—
333C2×Dic3C2×D3214yes—
334D12—916no—
335C4×D3—1816no—
336C4×D3—1816no—
337C4×D3—1832no—
338C2×C32—22no—
339C2×C32—21no—
340C2×C32—22no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—22no—
344C2×C32—28no—
345C2×C32—24no—
346C2×C32—44no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32#67⋊C418yescomplemented
351C2×C32#89314yescomplemented
352C2×C32C22×C4×D312yes—
353C2×C32C22×C4×D311yescharacteristic
354C2×C32C2×D3×D412yes—
355C2×C32C2×D3×D411yescharacteristic
356C2×C32C2×#23712yes—
357C2×C32C2×#23911yescharacteristic
358C3⋊D3—28no—
359C3×D3—616no—
360C3×D3—68no—
361C3×D3—68no—
362C24—92no—
363C2×D4—274no—
364C2×D4—278no—
365C22×C4—31no—
366C22×C4—32no—
367C22×C4—91no—
368C22×C4—278no—
369C22×C4—271no—
370C22×C4—548no—
371C4⋊C4—94no—
372C22⋊C4—98no—
373C42—274no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—24no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—98no—
387C22×C3—184no—
388C22×C3—184no—
389C22×C3—1816no—
390C22×C3#379614yes—
391C22×C3C4×D3214yes—
392C22×C3C4×D3212yes—
393C22×C3#380212yes—
394C22×C3#380512yes—
395C22×C3#381114yes—
396C22×C3C22×D3212yes—
397C22×C3C22×D3211yescharacteristic
398C2×D3—38no—
399C2×D3—38no—
400C2×D3—68no—
401C2×D3—68no—
402C2×D3—68no—
403C2×D3—632no—
404C2×D3—64no—
405C2×D3—64no—
406C2×D3—64no—
407C2×D3—64no—
408C2×D3—616no—
409C2×D3—616no—
410C3×C4—68no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4—64no—
415C3×C4—94no—
416Dic3—22no—
417Dic3—24no—
418Dic3—38no—
419Dic3—64no—
420Dic3—94no—
421Dic3—98no—
422Dic3—188no—
423Dic3—184no—
424Dic3—364no—
425C32—22no—
426C32—21no—
427C32—41no—
428C32C2×#67⋊C412yescomplemented
429C32C2×#89311yescharacteristic, complemented
430C23—94no—
431C23—98no—
432C23—184no—
433C23—184no—
434C23D3311yescharacteristic
435D4—2716no—
436C2×C4—32no—
437C2×C4—34no—
438C2×C4—64no—
439C2×C4—62no—
440C2×C4—92no—
441C2×C4—94no—
442C2×C4—272no—
443C2×C4—274no—
444C2×C4—5416no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—22no—
451C2×C3—28no—
452C2×C3—24no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—44no—
457C2×C3—188no—
458C2×C3#6388118yescomplemented
459C2×C3#6390214yescomplemented
460C2×C3C2×#379612yes—
461C2×C3C2×C4×D3212yes—
462C2×C3C2×C4×D3211yescharacteristic
463C2×C3C2×#380211yescharacteristic
464C2×C3C2×#380511yescharacteristic
465C2×C3C2×#381112yes—
466D3—616no—
467D3—68no—
468D3—68no—
469C22—98no—
470C22—184no—
471C22—184no—
472C22—1816no—
473C22#8894912yes—
474C22#8895412yes—
475C22#8896412yes—
476C22C2×D3311yescharacteristic
477C4—64no—
478C4—62no—
479C4—94no—
480C4—274no—
481C3—22no—
482C3—21no—
483C3—41no—
484C3C2×#6388112yescomplemented
485C3C2×#6390211yescharacteristic, complemented
486C2—188no—
487C2#1172293814yesdirect product factor
488C2C2×#8894911yescharacteristic
489C2C2×#8895411yescharacteristic
490C2C2×#8896411yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socleC23×C33C23
prime radicalC22×#6295C2
cosocleC22×C33C24
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×C3⋊D3C2semidirect
3. C2×Dic3×C3⋊Dic3C2semidirect
4. C2×C3×#3724C2semidirect
5. C2×C3×#3722C2semidirect
6. #11722938C2direct (N × H)
14. C2×#6295C4semidirect
17. C2×Dic3×C3⋊D3C22semidirect
18. Dic3×C3⋊Dic3C22semidirect
19. C3×#3724C22semidirect
20. C3×#3722C22semidirect
31. C2×#3724D3semidirect
32. C2×#3722D3semidirect
40. C22×C3×C3⋊D3C2×C4semidirect
42. C2×C3⋊(C3⋊Dic3)D4semidirect
44. C2×C3×C3⋊Dic3C2×C4semidirect
50. #6295C2×C4semidirect
89. #3724C2×D3semidirect
90. #3722C2×D3semidirect
100. C2×C3×C3⋊D3C22×C4semidirect
101. C3⋊(C3⋊Dic3)C2×D4semidirect
102. C3×C3⋊Dic3C22×C4semidirect
128. C22×C3⋊D3C4×D3semidirect
156. C2×C3⋊Dic3C4×D3semidirect
179. C2×C33C4×D4semidirect
246. C2×C3⋊D3C2×C4×D3semidirect
266. C3⋊Dic3C2×C4×D3semidirect
280. C33C2×C4×D4semidirect
350. C2×C32#67⋊C4semidirect
351. C2×C32#893semidirect
428. C32C2×#67⋊C4semidirect
429. C32C2×#893semidirect
458. C2×C3#63881semidirect
459. C2×C3#63902semidirect
484. C3C2×#63881semidirect
485. C3C2×#63902semidirect
487. C2#11722938direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62952
C22×Dic3×C3⋊D32
C2×Dic3×C3⋊Dic32
C2×C3×#37242
C2×C3×#37222
#117229382
C2×#639023
C2×#638813