Finite Group Databasebeta

← C22×#88954 #13181330

Subgroups — C22×#88954 #13181330

Order 1728.

7314 conjugacy classes of subgroups; only the 756 normal subgroups are shown.

normal subgroups only

This group has 7314 conjugacy classes of subgroups — too many to draw. Only its 756 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×C3⋊D3C212yesmaximal, complemented
2C22×#6295C211yesmaximal, characteristic, complemented
3C22×#6289C211yesmaximal, characteristic, complemented
4C22×C3×#626C212yesmaximal, complemented
5C22×C3×#625C211yesmaximal, characteristic, complemented
6C2×#88954C2124yesmaximal, direct product factor
7C22×#3811—32nomaximal
8C22×#3802—31nomaximal
9C23×C3⋊(C3⋊D3)C2211yescharacteristic
10C23×C3×C3⋊D3C2212yescomplemented
11C23×C32×D3C2212yescomplemented
12C22×D3×C3⋊D3—216no—
13C22×D3×C3⋊D3C22124yescomplemented
14C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
15C22×C32×Dic3C2211yescharacteristic, complemented
16C2×#6295C22112yescomplemented
17C2×#6289C22112yescomplemented
18C2×C3×#626C22124yescomplemented
19C2×C3×#625C22112yescomplemented
20#88954C22164yesdirect product factor
21C24×C3⋊D3—32no—
22C23×D32—32no—
23C23×D32—32no—
24C23×D32—62no—
25C22×C4×C3⋊D3—31no—
26C22×C3×#67—32no—
27C22×C3×#67—32no—
28C22×C3×D12—32no—
29C2×#3811—348no—
30C22×#626—32no—
31C22×#626D312yescomplemented
32C22×#625—31no—
33C22×#625D311yescharacteristic, complemented
34C22×#624—32no—
35C22×#624—61no—
36C2×#3802—324no—
37C23×C33C2311yescharacteristic
38C22×C3⋊(C3⋊D3)D418yescomplemented
39C22×C3⋊(C3⋊D3)C2316yes—
40C22×C3×C3⋊D3—28no—
41C22×C3×C3⋊D3C23112yescomplemented
42C22×C32×D3—28no—
43C22×C32×D3C23112yescomplemented
44C2×D3×C3⋊D3—296no—
45C2×D3×C3⋊D3C23132yescomplemented
46C2×C3×C3⋊Dic3C2316yescomplemented
47C2×C32×Dic3C2316yescomplemented
48#6295C23116yescomplemented
49#6289C23116yescomplemented
50C3×#626C23132yescomplemented
51C3×#625C23116yescomplemented
52C22×D3×D4—92no—
53C22×D3×D4—91no—
54C24×C32—32no—
55C23×C3⋊D3—32no—
56C23×C3⋊D3—31no—
57C23×C3⋊D3—324no—
58C23×C3⋊D3—616no—
59C23×C3⋊D3—61no—
60C23×C3⋊D3—62no—
61C23×C3⋊D3—121no—
62C23×C3⋊D3C2×D312yescomplemented
63C23×C3×D3—22no—
64C23×C3×D3—32no—
65C23×C3×D3—32no—
66C23×C3×D3—62no—
67C23×C3×D3C2×D312yescomplemented
68C23×C3×D3C2×D312yescomplemented
69C22×D32—324no—
70C22×D32—324no—
71C22×D32—616no—
72C22×D32—616no—
73C22×D32—624no—
74C22×D32—632no—
75C22×C32×C4—31no—
76C22×C3⋊Dic3C2×D311yescharacteristic, complemented
77C2×C4×C3⋊D3—312no—
78C2×C3×#67—324no—
79C2×C3×#67—324no—
80C22×C3×Dic3—21no—
81C22×C3×Dic3—32no—
82C22×C3×Dic3—61no—
83C22×C3×Dic3C2×D312yescomplemented
84C2×C3×D12—324no—
85#3811—3128no—
86C2×#626—324no—
87C2×#626C2×D3124yescomplemented
88C2×#625—312no—
89C2×#625C2×D3112yescomplemented
90C2×#624—324no—
91C2×#624—612no—
92#3802—364no—
93C22×C33C2×D414yescomplemented
94C22×C33C2413yes—
95C2×C3⋊(C3⋊D3)C2×D4124yescomplemented
96C2×C3⋊(C3⋊D3)C2414yes—
97C2×C3×C3⋊D3—224no—
98C2×C3×C3⋊D3C2418yescomplemented
99C2×C32×D3—224no—
100C2×C32×D3C2418yescomplemented
101D3×C3⋊D3—264no—
102C3×C3⋊Dic3C2414yescomplemented
103C32×Dic3C2414yescomplemented
104C24×D3—92no—
105C24×D3—92no—
106C24×D3—92no—
107C24×D3—182no—
108C22×C3×D4—92no—
109C22×C3×D4—91no—
110C22×#67—32no—
111C22×#67—32no—
112C22×#67—92no—
113C2×D3×D4—948no—
114C2×D3×D4—924no—
115C22×D12—32no—
116C22×D12—91no—
117C22×C4×D3—92no—
118C22×C4×D3—91no—
119C22×C4×D3—181no—
120C23×C32—22no—
121C23×C32—21no—
122C23×C32—312no—
123C23×C32—41no—
124C23×C32—68no—
125C23×C32C22×D311yescharacteristic
126C23×C32C22×D312yes—
127C22×C3⋊D3—28no—
128C22×C3⋊D3—38no—
129C22×C3⋊D3—316no—
130C22×C3⋊D3—36no—
131C22×C3⋊D3—312no—
132C22×C3⋊D3—332no—
133C22×C3⋊D3—616no—
134C22×C3⋊D3—612no—
135C22×C3⋊D3—68no—
136C22×C3⋊D3—66no—
137C22×C3⋊D3—696no—
138C22×C3⋊D3—128no—
139C22×C3⋊D3—126no—
140C22×C3⋊D3C22×D3112yescomplemented
141C22×C3×D3—216no—
142C22×C3×D3—28no—
143C22×C3×D3—28no—
144C22×C3×D3—212no—
145C22×C3×D3—312no—
146C22×C3×D3—312no—
147C22×C3×D3—612no—
148C22×C3×D3—616no—
149C22×C3×D3—68no—
150C22×C3×D3—68no—
151C22×C3×D3C22×D3112yescomplemented
152C22×C3×D3C22×D3112yescomplemented
153C2×D32—332no—
154C2×D32—332no—
155C2×D32—632no—
156C2×D32—696no—
157C2×D32—696no—
158C2×D32—6192no—
159C2×C32×C4—36no—
160C2×C3⋊Dic3C22×D316yescomplemented
161C4×C3⋊D3—316no—
162C3×#67—332no—
163C3×#67—332no—
164C2×C3×Dic3—26no—
165C2×C3×Dic3—312no—
166C2×C3×Dic3—66no—
167C2×C3×Dic3C22×D3112yescomplemented
168C3×D12—332no—
169#626—332no—
170#626C22×D3132yescomplemented
171#625—316no—
172#625C22×D3116yescomplemented
173#624—332no—
174#624—616no—
175C23×D4—271noSylow
176C2×C33C22×D416yescomplemented
177C2×C33C2511yescharacteristic
178C3⋊(C3⋊D3)C22×D418yescomplemented
179C3×C3⋊D3—28no—
180C32×D3—28no—
181C24×C3—32no—
182C24×C3—32no—
183C24×C3—62no—
184C24×C3—92no—
185C23×D3—32no—
186C23×D3—32no—
187C23×D3—62no—
188C23×D3—924no—
189C23×D3—924no—
190C23×D3—91no—
191C23×D3—92no—
192C23×D3—924no—
193C23×D3—1824no—
194C23×D3—181no—
195C23×D3—182no—
196C23×D3—1816no—
197C23×D3—1816no—
198C23×D3—1816no—
199C23×D3—1832no—
200C23×D3—361no—
201C23×D3D3212yescomplemented
202C2×C3×D4—924no—
203C2×C3×D4—912no—
204C22×C3×C4—32no—
205C22×C3×C4—61no—
206C22×C3×C4—91no—
207C2×#67—324no—
208C2×#67—324no—
209C2×#67—924no—
210C22×Dic3—32no—
211C22×Dic3—61no—
212C22×Dic3D3211yescharacteristic, complemented
213D3×D4—9128no—
214D3×D4—964no—
215C2×D12—324no—
216C2×D12—912no—
217C2×C4×D3—924no—
218C2×C4×D3—912no—
219C2×C4×D3—1812no—
220C22×C32—23no—
221C22×C32—26no—
222C22×C32—24no—
223C22×C32—28no—
224C22×C32—38no—
225C22×C32—44no—
226C22×C32—43no—
227C22×C32—624no—
228C22×C32D3×D418yescomplemented
229C22×C32D3×D414yescomplemented
230C22×C32C23×D316yes—
231C22×C32C23×D313yes—
232C2×C3⋊D3—224no—
233C2×C3⋊D3—324no—
234C2×C3⋊D3—348no—
235C2×C3⋊D3—38no—
236C2×C3⋊D3—34no—
237C2×C3⋊D3—648no—
238C2×C3⋊D3—624no—
239C2×C3⋊D3—64no—
240C2×C3⋊D3—68no—
241C2×C3⋊D3—664no—
242C2×C3⋊D3—1224no—
243C2×C3⋊D3—124no—
244C2×C3⋊D3C23×D318yescomplemented
245C2×C3×D3—248no—
246C2×C3×D3—224no—
247C2×C3×D3—224no—
248C2×C3×D3—28no—
249C2×C3×D3—38no—
250C2×C3×D3—38no—
251C2×C3×D3—68no—
252C2×C3×D3—648no—
253C2×C3×D3—624no—
254C2×C3×D3—624no—
255C2×C3×D3C23×D318yescomplemented
256C2×C3×D3C23×D318yescomplemented
257D32—664no—
258D32—664no—
259D32—6128no—
260C32×C4—34no—
261C3⋊Dic3C23×D314yescomplemented
262C3×Dic3—24no—
263C3×Dic3—38no—
264C3×Dic3—64no—
265C3×Dic3C23×D318yescomplemented
266C25—272no—
267C22×D4—92no—
268C22×D4—91no—
269C22×D4—271no—
270C22×D4—2724no—
271C23×C4—271no—
272C33C23×D411yescharacteristic, complemented, Sylow
273C23×C3—21no—
274C23×C3—22no—
275C23×C3—312no—
276C23×C3—312no—
277C23×C3—41no—
278C23×C3—612no—
279C23×C3—616no—
280C23×C3—68no—
281C23×C3—68no—
282C23×C3—912no—
283C23×C3—188no—
284C23×C3C2×D3211yescharacteristic
285C23×C3C2×D3212yes—
286C22×D3—28no—
287C22×D3—312no—
288C22×D3—312no—
289C22×D3—612no—
290C22×D3—616no—
291C22×D3—68no—
292C22×D3—68no—
293C22×D3—932no—
294C22×D3—932no—
295C22×D3—96no—
296C22×D3—912no—
297C22×D3—98no—
298C22×D3—916no—
299C22×D3—932no—
300C22×D3—1832no—
301C22×D3—186no—
302C22×D3—188no—
303C22×D3—1812no—
304C22×D3—1816no—
305C22×D3—1896no—
306C22×D3—1896no—
307C22×D3—1896no—
308C22×D3—18192no—
309C22×D3—366no—
310C22×D3—368no—
311C22×D3C2×D32112yescomplemented
312C3×D4—932no—
313C3×D4—916no—
314C2×C3×C4—312no—
315C2×C3×C4—66no—
316C2×C3×C4—96no—
317#67—332no—
318#67—332no—
319#67—932no—
320C2×Dic3—312no—
321C2×Dic3—66no—
322C2×Dic3C2×D3216yescomplemented
323D12—332no—
324D12—916no—
325C4×D3—932no—
326C4×D3—916no—
327C4×D3—1816no—
328C2×C32—26no—
329C2×C32—212no—
330C2×C32—21no—
331C2×C32—22no—
332C2×C32—46no—
333C2×C32—41no—
334C2×C32—68no—
335C2×C32C2×D3×D4112yescomplemented
336C2×C32C2×D3×D416yescomplemented
337C2×C32C24×D312yes—
338C2×C32C24×D311yescharacteristic
339C3⋊D3—28no—
340C3⋊D3—38no—
341C3⋊D3—316no—
342C3⋊D3—616no—
343C3⋊D3—68no—
344C3⋊D3—128no—
345C3×D3—216no—
346C3×D3—28no—
347C3×D3—28no—
348C3×D3—616no—
349C3×D3—68no—
350C3×D3—68no—
351C24—32no—
352C24—92no—
353C24—2724no—
354C24—271no—
355C24—5416no—
356C2×D4—924no—
357C2×D4—912no—
358C2×D4—2712no—
359C2×D4—2764no—
360C22×C4—31no—
361C22×C4—91no—
362C22×C4—2712no—
363C22×C3—23no—
364C22×C3—26no—
365C22×C3—28no—
366C22×C3—24no—
367C22×C3—38no—
368C22×C3—38no—
369C22×C3—43no—
370C22×C3—44no—
371C22×C3—68no—
372C22×C3—648no—
373C22×C3—624no—
374C22×C3—624no—
375C22×C3—98no—
376C22×C3—1824no—
377C22×C3#380214yescomplemented
378C22×C3#381118yescomplemented
379C22×C3C22×D3213yes—
380C22×C3C22×D3216yes—
381C2×D3—224no—
382C2×D3—38no—
383C2×D3—38no—
384C2×D3—68no—
385C2×D3—648no—
386C2×D3—624no—
387C2×D3—624no—
388C2×D3—924no—
389C2×D3—948no—
390C2×D3—94no—
391C2×D3—98no—
392C2×D3—184no—
393C2×D3—1848no—
394C2×D3—188no—
395C2×D3—1824no—
396C2×D3—1864no—
397C2×D3—1864no—
398C2×D3—1864no—
399C2×D3—18128no—
400C2×D3—3624no—
401C2×D3—364no—
402C2×D3C22×D3218yescomplemented
403C3×C4—38no—
404C3×C4—64no—
405C3×C4—94no—
406Dic3—38no—
407Dic3—64no—
408Dic3C22×D3214yescomplemented
409C32—21no—
410C32—22no—
411C32—41no—
412C32C22×D3×D412yescomplemented
413C32C22×D3×D411yescharacteristic, complemented
414C23—312no—
415C23—68no—
416C23—912no—
417C23—188no—
418C23—2732no—
419C23—276no—
420C23—278no—
421C23—5496no—
422C23D3311yescharacteristic
423D4—932no—
424D4—916no—
425D4—2716no—
426C2×C4—36no—
427C2×C4—96no—
428C2×C4—2716no—
429C2×C3—212no—
430C2×C3—26no—
431C2×C3—21no—
432C2×C3—22no—
433C2×C3—46no—
434C2×C3—41no—
435C2×C3—616no—
436C2×C3—68no—
437C2×C3—68no—
438C2×C3—188no—
439C2×C3C2×#380216yescomplemented
440C2×C3C2×#3811112yescomplemented
441C2×C3C23×D3211yescharacteristic
442C2×C3C23×D3212yes—
443D3—28no—
444D3—616no—
445D3—68no—
446D3—68no—
447D3—98no—
448D3—916no—
449D3—1816no—
450D3—188no—
451D3—368no—
452C22—38no—
453C22—624no—
454C22—98no—
455C22—1824no—
456C22—2724no—
457C22—274no—
458C22—5464no—
459C22#8895414yesdirect product factor
460C22C2×D3313yes—
461C4—34no—
462C4—94no—
463C3—22no—
464C3—21no—
465C3—41no—
466C3C22×#380211yescharacteristic, complemented
467C3C22×#381112yescomplemented
468C2—68no—
469C2—188no—
470C2—278no—
471C2C2×#8895416yesdirect product factor
472C2C22×D3311yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC2×C33C25
FrattiniC2C22×D33
Fitting, socleC23×C33C23
hypocenterC33C23×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×D3×C3⋊D3C2semidirect
2. C22×#6295C2semidirect
3. C22×#6289C2semidirect
4. C22×C3×#626C2semidirect
5. C22×C3×#625C2semidirect
6. C2×#88954C2direct (N × H)
10. C23×C3×C3⋊D3C22semidirect
11. C23×C32×D3C22semidirect
13. C22×D3×C3⋊D3C22semidirect
14. C22×C3×C3⋊Dic3C22semidirect
15. C22×C32×Dic3C22semidirect
16. C2×#6295C22semidirect
17. C2×#6289C22semidirect
18. C2×C3×#626C22semidirect
19. C2×C3×#625C22semidirect
20. #88954C22direct (N × H)
31. C22×#626D3semidirect
33. C22×#625D3semidirect
38. C22×C3⋊(C3⋊D3)D4semidirect
41. C22×C3×C3⋊D3C23semidirect
43. C22×C32×D3C23semidirect
45. C2×D3×C3⋊D3C23semidirect
46. C2×C3×C3⋊Dic3C23semidirect
47. C2×C32×Dic3C23semidirect
48. #6295C23semidirect
49. #6289C23semidirect
50. C3×#626C23semidirect
51. C3×#625C23semidirect
62. C23×C3⋊D3C2×D3semidirect
67. C23×C3×D3C2×D3semidirect
68. C23×C3×D3C2×D3semidirect
76. C22×C3⋊Dic3C2×D3semidirect
83. C22×C3×Dic3C2×D3semidirect
87. C2×#626C2×D3semidirect
89. C2×#625C2×D3semidirect
93. C22×C33C2×D4semidirect
95. C2×C3⋊(C3⋊D3)C2×D4semidirect
98. C2×C3×C3⋊D3C24semidirect
100. C2×C32×D3C24semidirect
102. C3×C3⋊Dic3C24semidirect
103. C32×Dic3C24semidirect
140. C22×C3⋊D3C22×D3semidirect
151. C22×C3×D3C22×D3semidirect
152. C22×C3×D3C22×D3semidirect
160. C2×C3⋊Dic3C22×D3semidirect
167. C2×C3×Dic3C22×D3semidirect
170. #626C22×D3semidirect
172. #625C22×D3semidirect
176. C2×C33C22×D4semidirect
178. C3⋊(C3⋊D3)C22×D4semidirect
201. C23×D3D32semidirect
212. C22×Dic3D32semidirect
228. C22×C32D3×D4semidirect
229. C22×C32D3×D4semidirect
244. C2×C3⋊D3C23×D3semidirect
255. C2×C3×D3C23×D3semidirect
256. C2×C3×D3C23×D3semidirect
261. C3⋊Dic3C23×D3semidirect
265. C3×Dic3C23×D3semidirect
272. C33C23×D4semidirect
311. C22×D3C2×D32semidirect
322. C2×Dic3C2×D32semidirect
335. C2×C32C2×D3×D4semidirect
336. C2×C32C2×D3×D4semidirect
377. C22×C3#3802semidirect
378. C22×C3#3811semidirect
402. C2×D3C22×D32semidirect
408. Dic3C22×D32semidirect
412. C32C22×D3×D4semidirect
413. C32C22×D3×D4semidirect
439. C2×C3C2×#3802semidirect
440. C2×C3C2×#3811semidirect
459. C22#88954direct (N × H)
466. C3C22×#3802semidirect
467. C3C22×#3811semidirect
471. C2C2×#88954direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×C3⋊D32
C22×#62952
C22×#62892
C22×C3×#6262
C22×C3×#6252
C2×#889542
C22×#38113
C22×#38023