Finite Group Databasebeta

← C2×#11722931 #13172150

Subgroups — C2×#11722931 #13172150

Order 1728.

2130 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6295C211yesmaximal, characteristic
2C22×C3×#624C212yesmaximal, complemented
3C2×#88803C212yesmaximal, complemented
4C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C2×C3×#3724C211yesmaximal, characteristic, complemented
6#11722931C218yesmaximal, direct product factor
7C2×#63904—32nomaximal
8C2×C4×#625—31nomaximal
9C23×C3×C3⋊D3C2212yes—
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C2×#6295C418yescomplemented
15C2×#6295C2214yes—
16C2×C3×#624—28no—
17C2×C3×#624C2218yescomplemented
18#88803C2218yescomplemented
19Dic3×C3⋊Dic3C2214yescomplemented
20C3×#3724C2214yescomplemented
21C22×C3×C4×D3—32no—
22C22×C3×C4×D3—32no—
23C22×#626—32no—
24C22×#625—31no—
25C22×#624D312yescomplemented
26C2×#3752—32no—
27C2×C4×C3⋊Dic3—31no—
28C2×C3×Dic3⋊C4—32no—
29C2×#3724D311yescharacteristic, complemented
30C2×(C2×D3)⋊Dic3—32no—
31C2×(C2×D3)⋊Dic3—32no—
32C2×(C2×D3)⋊Dic3—62no—
33C2×Dic32—32no—
34C2×Dic32—61no—
35#63904—316no—
36C4×#625—38no—
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)—22no—
43C2×C3⋊(C3⋊Dic3)C2312yes—
44C2×C3×C3⋊Dic3C2×C414yescomplemented
45C2×C3×C3⋊Dic3C2312yes—
46C2×C32×Dic3—24no—
47C2×C32×Dic3D414yescomplemented
48C2×C32×Dic3C2314yes—
49C2×C32×Dic3C2312yes—
50#6295C2×C4116yescomplemented
51C3×#624—216no—
52C2×C4×#67—92no—
53C2×#893—91no—
54C23×C3⋊D3C2×D312yes—
55C23×C3×D3—32no—
56C23×C3×D3—32no—
57C23×C3×D3—62no—
58C22×C32×C4—31no—
59C22×C32×C4—32no—
60C22×C3⋊Dic3—31no—
61C22×C3⋊Dic3—32no—
62C22×C3⋊Dic3—61no—
63C22×C3⋊Dic3—62no—
64C22×C3⋊Dic3—121no—
65C22×C3⋊Dic3C2×D311yescharacteristic
66C22×C3×Dic3—21no—
67C22×C3×Dic3—22no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3C2×D312yes—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C2×C3×C4×D3—38no—
74C2×C3×C4×D3—38no—
75C2×C3×C4×D3—68no—
76C2×C3×C4×D3—68no—
77C2×#626—316no—
78C2×#626—38no—
79C2×#625—38no—
80C2×#625—34no—
81C2×#624—28no—
82C2×#624C2×D318yescomplemented
83#3752—38no—
84C4×C3⋊Dic3—34no—
85C3×Dic3⋊C4—38no—
86#3724C2×D314yescomplemented
87(C2×D3)⋊Dic3—38no—
88(C2×D3)⋊Dic3—38no—
89(C2×D3)⋊Dic3—68no—
90Dic32—38no—
91Dic32—64no—
92C22×C33C22×C412yes—
93C22×C33C2×D412yes—
94C22×C33#4112yes—
95C22×C33C2411yescharacteristic
96C2×C3×C3⋊D3—216no—
97C2×C3×C3⋊D3—24no—
98C2×C3×C3⋊D3—24no—
99C2×C3×C3⋊D3C22×C418yescomplemented
100C3⋊(C3⋊Dic3)—22no—
101C3×C3⋊Dic3C22×C414yescomplemented
102C32×Dic3—24no—
103C32×Dic3C2×D414yescomplemented
104C23×C3×C4—92no—
105C22×#67—92no—
106C22×D12—91no—
107C22×C4×D3—32no—
108C22×C4×D3—32no—
109C2×C3×C4⋊C4—91no—
110C2×#217—92no—
111C4×#67—916no—
112C2×(C2×D3)⋊C4—92no—
113C2×(C2×D3)⋊C4—92no—
114C2×(C2×D3)⋊C4—182no—
115C2×Dic3⋊C4—32no—
116C2×C4×Dic3—92no—
117C2×C4×Dic3—91no—
118C2×C4×Dic3—181no—
119#893—98no—
120C23×C32—22no—
121C23×C32—21no—
122C23×C32—41no—
123C23×C32C22×D312yes—
124C23×C32C22×D311yescharacteristic
125C22×C3⋊D3—24no—
126C22×C3⋊D3—24no—
127C22×C3⋊D3C4×D318yescomplemented
128C22×C3⋊D3C22×D314yes—
129C22×C3×D3—38no—
130C22×C3×D3—38no—
131C22×C3×D3—34no—
132C22×C3×D3—34no—
133C22×C3×D3—64no—
134C22×C3×D3—68no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C22×C3×D3—64no—
138C22×C3×D3—64no—
139C22×C3×D3—124no—
140C22×C3×D3—124no—
141C2×C32×C4—34no—
142C2×C32×C4—32no—
143C2×C32×C4—34no—
144C2×C32×C4—64no—
145C2×C3⋊Dic3—34no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—64no—
149C2×C3⋊Dic3—62no—
150C2×C3⋊Dic3—64no—
151C2×C3⋊Dic3—64no—
152C2×C3⋊Dic3—122no—
153C2×C3⋊Dic3—124no—
154C2×C3⋊Dic3—124no—
155C2×C3⋊Dic3C4×D314yescomplemented
156C2×C3⋊Dic3C22×D312yes—
157C2×C3×Dic3—24no—
158C2×C3×Dic3—24no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—24no—
162C2×C3×Dic3—38no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—44no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—64no—
167C2×C3×Dic3#6718yescomplemented
168C2×C3×Dic3C22×D314yes—
169C2×C3×Dic3C22×D314yes—
170C2×C3×Dic3C22×D314yes—
171C3×C4×D3—616no—
172C3×C4×D3—616no—
173#626—332no—
174#625—316no—
175#624—216no—
176C2×C4×D4—271noSylow
177C2×C33C4×D414yescomplemented
178C2×C33C23×C411yescharacteristic
179C2×C33C22×D411yescharacteristic
180C2×C33C2×#4111yescharacteristic
181C3×C3⋊D3—28no—
182C24×C3—92no—
183C23×D3—32no—
184C23×D3—32no—
185C23×D3—62no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—61no—
190C22×C3×C4—62no—
191C22×C3×C4—98no—
192C22×C3×C4—91no—
193C22×C3×C4—188no—
194C2×#67—98no—
195C2×#67—916no—
196C22×Dic3—32no—
197C22×Dic3—61no—
198C22×Dic3—91no—
199C22×Dic3—92no—
200C22×Dic3—181no—
201C22×Dic3—182no—
202C22×Dic3—361no—
203C22×Dic3D3212yes—
204C22×Dic3D3211yescharacteristic
205C2×D12—94no—
206C2×D12—98no—
207C2×C4×D3—38no—
208C2×C4×D3—38no—
209C2×C4×D3—68no—
210C2×C4×D3—68no—
211C3×C4⋊C4—94no—
212#217—98no—
213(C2×D3)⋊C4—98no—
214(C2×D3)⋊C4—98no—
215(C2×D3)⋊C4—188no—
216Dic3⋊C4—38no—
217C4×Dic3—98no—
218C4×Dic3—94no—
219C4×Dic3—184no—
220C22×C32—21no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—24no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—22no—
227C22×C32—24no—
228C22×C32—42no—
229C22×C32—42no—
230C22×C32—42no—
231C22×C32—41no—
232C22×C32C2×C4×D312yes—
233C22×C32C2×C4×D314yes—
234C22×C32#23514yes—
235C22×C32D3×D412yes—
236C22×C32#23912yes—
237C22×C32C2×#6714yes—
238C22×C32C23×D312yes—
239C22×C32C23×D311yescharacteristic
240C2×C3⋊D3—216no—
241C2×C3⋊D3—24no—
242C2×C3⋊D3—24no—
243C2×C3⋊D3C2×C4×D318yescomplemented
244C2×C3×D3—38no—
245C2×C3×D3—38no—
246C2×C3×D3—68no—
247C2×C3×D3—616no—
248C2×C3×D3—616no—
249C2×C3×D3—64no—
250C2×C3×D3—64no—
251C2×C3×D3—64no—
252C2×C3×D3—64no—
253C2×C3×D3—124no—
254C2×C3×D3—124no—
255C2×C3×D3—1216no—
256C32×C4—34no—
257C32×C4—64no—
258C3⋊Dic3—62no—
259C3⋊Dic3—64no—
260C3⋊Dic3—64no—
261C3⋊Dic3—124no—
262C3⋊Dic3—124no—
263C3⋊Dic3C2×C4×D314yescomplemented
264C3×Dic3—24no—
265C3×Dic3—24no—
266C3×Dic3—24no—
267C3×Dic3—38no—
268C3×Dic3—44no—
269C3×Dic3—64no—
270C3×Dic3C2×#6718yescomplemented
271C22×D4—271no—
272C23×C4—92no—
273C4×D4—278no—
274C2×C4⋊C4—91no—
275C2×C22⋊C4—272no—
276C2×C42—271no—
277C33C2×C4×D411yescharacteristic, complemented, Sylow
278C23×C3—21no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—94no—
282C23×C3—98no—
283C23×C3—184no—
284C23×C3—184no—
285C23×C3C2×D3211yescharacteristic
286C23×C3C2×D3212yes—
287C22×D3—34no—
288C22×D3—34no—
289C22×D3—38no—
290C22×D3—38no—
291C22×D3—68no—
292C22×D3—64no—
293C22×D3—64no—
294C22×D3—64no—
295C22×D3—64no—
296C22×D3—64no—
297C22×D3—124no—
298C22×D3—124no—
299C2×C3×C4—34no—
300C2×C3×C4—38no—
301C2×C3×C4—34no—
302C2×C3×C4—34no—
303C2×C3×C4—62no—
304C2×C3×C4—64no—
305C2×C3×C4—64no—
306C2×C3×C4—64no—
307C2×C3×C4—64no—
308C2×C3×C4—92no—
309C2×C3×C4—94no—
310C2×C3×C4—124no—
311C2×C3×C4—1816no—
312#67—932no—
313C2×Dic3—24no—
314C2×Dic3—34no—
315C2×Dic3—38no—
316C2×Dic3—64no—
317C2×Dic3—62no—
318C2×Dic3—92no—
319C2×Dic3—94no—
320C2×Dic3—182no—
321C2×Dic3—184no—
322C2×Dic3—184no—
323C2×Dic3—182no—
324C2×Dic3—184no—
325C2×Dic3—362no—
326C2×Dic3—364no—
327C2×Dic3—364no—
328C2×Dic3#62514yescomplemented
329C2×Dic3C2×D3214yes—
330C2×Dic3C2×D3212yes—
331D12—916no—
332C4×D3—616no—
333C4×D3—616no—
334C2×C32—21no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32—24no—
341C2×C32—28no—
342C2×C32—44no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32#89314yescomplemented
347C2×C32C4×#6718yescomplemented
348C2×C32C22×C4×D311yescharacteristic
349C2×C32C22×C4×D312yes—
350C2×C32C2×#23512yes—
351C2×C32C2×D3×D411yescharacteristic
352C2×C32C2×#23911yescharacteristic
353C2×C32C22×#6712yes—
354C3⋊D3—28no—
355C3×D3—68no—
356C3×D3—68no—
357C3×D3—128no—
358C24—92no—
359C2×D4—274no—
360C2×D4—278no—
361C22×C4—31no—
362C22×C4—32no—
363C22×C4—98no—
364C22×C4—91no—
365C22×C4—188no—
366C22×C4—271no—
367C4⋊C4—94no—
368C22⋊C4—278no—
369C42—274no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—22no—
375C22×C3—24no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—41no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—98no—
383C22×C3—184no—
384C22×C3—184no—
385C22×C3—1816no—
386C22×C3#379812yes—
387C22×C3#379914yes—
388C22×C3C4×D3212yes—
389C22×C3C4×D3214yes—
390C22×C3C2×#62512yes—
391C22×C3D3×#6714yes—
392C22×C3C22×D3211yescharacteristic
393C22×C3C22×D3212yes—
394C2×D3—38no—
395C2×D3—38no—
396C2×D3—68no—
397C2×D3—64no—
398C2×D3—64no—
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—616no—
402C2×D3—616no—
403C2×D3—124no—
404C2×D3—124no—
405C2×D3—1216no—
406C3×C4—38no—
407C3×C4—64no—
408C3×C4—64no—
409C3×C4—64no—
410C3×C4—94no—
411C3×C4—124no—
412Dic3—24no—
413Dic3—38no—
414Dic3—64no—
415Dic3—184no—
416Dic3—182no—
417Dic3—184no—
418Dic3—364no—
419Dic3—364no—
420Dic3C2×#62514yescomplemented
421C32—21no—
422C32—22no—
423C32—41no—
424C32C2×#89311yescharacteristic, complemented
425C32C2×C4×#6712yescomplemented
426C23—94no—
427C23—98no—
428C23—184no—
429C23—184no—
430C23D3311yescharacteristic
431D4—2716no—
432C2×C4—32no—
433C2×C4—34no—
434C2×C4—34no—
435C2×C4—64no—
436C2×C4—92no—
437C2×C4—94no—
438C2×C4—1816no—
439C2×C4—272no—
440C2×C4—542no—
441C2×C3—21no—
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—28no—
448C2×C3—24no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—44no—
453C2×C3—188no—
454C2×C3C4×#62514yescomplemented
455C2×C3#6390418yescomplemented
456C2×C3C2×#379811yescharacteristic
457C2×C3C2×#379912yes—
458C2×C3C2×C4×D3211yescharacteristic
459C2×C3C2×C4×D3212yes—
460C2×C3C22×#62511yescharacteristic
461C2×C3C2×D3×#6712yes—
462D3—68no—
463D3—68no—
464D3—128no—
465C22—98no—
466C22—184no—
467C22—184no—
468C22—1816no—
469C22#8894912yes—
470C22D3×#62512yes—
471C22#8896512yes—
472C22C2×D3311yescharacteristic
473C4—34no—
474C4—64no—
475C4—94no—
476C4—542no—
477C3—22no—
478C3—21no—
479C3—41no—
480C3C2×C4×#62511yescharacteristic, complemented
481C3C2×#6390412yescomplemented
482C2—188no—
483C2#1172293114yesdirect product factor
484C2C2×#8894911yescharacteristic
485C2C2×D3×#62511yescharacteristic
486C2C2×#8896511yescharacteristic
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×C3×#624C2semidirect
3. C2×#88803C2semidirect
4. C2×Dic3×C3⋊Dic3C2semidirect
5. C2×C3×#3724C2semidirect
6. #11722931C2direct (N × H)
14. C2×#6295C4semidirect
17. C2×C3×#624C22semidirect
18. #88803C22semidirect
19. Dic3×C3⋊Dic3C22semidirect
20. C3×#3724C22semidirect
25. C22×#624D3semidirect
29. C2×#3724D3semidirect
40. C22×C3×C3⋊D3C2×C4semidirect
44. C2×C3×C3⋊Dic3C2×C4semidirect
47. C2×C32×Dic3D4semidirect
50. #6295C2×C4semidirect
82. C2×#624C2×D3semidirect
86. #3724C2×D3semidirect
99. C2×C3×C3⋊D3C22×C4semidirect
101. C3×C3⋊Dic3C22×C4semidirect
103. C32×Dic3C2×D4semidirect
127. C22×C3⋊D3C4×D3semidirect
155. C2×C3⋊Dic3C4×D3semidirect
167. C2×C3×Dic3#67semidirect
177. C2×C33C4×D4semidirect
243. C2×C3⋊D3C2×C4×D3semidirect
263. C3⋊Dic3C2×C4×D3semidirect
270. C3×Dic3C2×#67semidirect
277. C33C2×C4×D4semidirect
328. C2×Dic3#625semidirect
346. C2×C32#893semidirect
347. C2×C32C4×#67semidirect
420. Dic3C2×#625semidirect
424. C32C2×#893semidirect
425. C32C2×C4×#67semidirect
454. C2×C3C4×#625semidirect
455. C2×C3#63904semidirect
480. C3C2×C4×#625semidirect
481. C3C2×#63904semidirect
483. C2#11722931direct (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×C3×#6242
C2×#888032
C2×Dic3×C3⋊Dic32
C2×C3×#37242
#117229312
C2×#639043
C2×C4×#6253