Finite Group Databasebeta

← C2×D12×C22⋊C9 #13164605

Subgroups — C2×D12×C22⋊C9 #13164605

Order 1728.

1226 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×D3×C22⋊C9C212yesmaximal, complemented
2C2×C3×C4×C22⋊C9C211yesmaximal, characteristic, complemented
3D12×C22⋊C9C214yesmaximal, direct product factor
4C23×C3×D12C311yesmaximal, characteristic
5C2×D4×C22⋊C9—31nomaximal
6C22×C3×C22⋊C9C2211yescharacteristic
7C2×D3×C22⋊C9—24no—
8C2×D3×C22⋊C9C2214yescomplemented
9C2×C9×D12—41nomaximal
10C3×C4×C22⋊C9C2212yescomplemented
11C24×C3×D3C2×C312yes—
12C23×C32×C4C2×C311yescharacteristic
13C22×C3×D12—31no—
14C22×C3×D12—32no—
15C22×C3×D12—31no—
16C22×C3×D12—34no—
17C22×C3×D12C2×C314yes—
18C23×C22⋊C9—32no—
19D4×C22⋊C9—34no—
20C2×C4×C22⋊C9—21no—
21C2×C4×C22⋊C9D311yescharacteristic, complemented
22C22×D3×C9—42no—
23C2×C3×C22⋊C9D412yescomplemented
24C2×C3×C22⋊C9C2311yescharacteristic
25D3×C22⋊C9—24no—
26C2×C3×C4×C9—41no—
27C9×D12—44no—
28C23×C3×D4—31no—
29C23×D12C911yescharacteristic, complemented
30C24×C32C22×C311yescharacteristic
31C23×C3×D3—24no—
32C23×C3×D3—32no—
33C23×C3×D3—32no—
34C23×C3×D3—34no—
35C23×C3×D3—64no—
36C23×C3×D3C22×C314yes—
37C22×C32×C4—31no—
38C22×C32×C4—31no—
39C22×C32×C4—32no—
40C22×C32×C4C22×C312yes—
41C2×C3×D12—32no—
42C2×C3×D12—38no—
43C2×C3×D12—34no—
44C2×C3×D12—32no—
45C2×C3×D12—31no—
46C2×C3×D12—34no—
47C2×C3×D12—34no—
48C2×C3×D12—38no—
49C2×C3×D12—34no—
50C2×C3×D12A411yescharacteristic
51C22×C22⋊C9—21no—
52C22×C22⋊C9—34no—
53C22×C22⋊C9—64no—
54C22×C22⋊C9C2×D311yescharacteristic
55C2×D4×C9—121no—
56C4×C22⋊C9—22no—
57C4×C22⋊C9C2×D312yescomplemented
58C22×C3×C9—41no—
59C2×D3×C9—44no—
60C2×D3×C9—84no—
61C3×C22⋊C9C2×D411yescharacteristic, complemented
62C3×C4×C9—42no—
63C25×C3—32no—
64C24×D3C2×C912yescomplemented
65C22×C3×D4—34no—
66C22×C3×D4—91no—
67C22×C3×D4—91no—
68C22×C3×D4—94no—
69C22×C3×D4—92no—
70C23×C3×C4—21no—
71C23×C3×C4C2×C911yescharacteristic, complemented
72C23×C3×C4C3×D311yescharacteristic
73C22×D12—31no—
74C22×D12—32no—
75C22×D12—31no—
76C22×D12—34no—
77C22×D12C2×C914yescomplemented
78C23×C32—31no—
79C23×C32—32no—
80C23×C32—31no—
81C23×C32C3×D412yes—
82C23×C32C23×C311yescharacteristic
83C22×C3×D3—24no—
84C22×C3×D3—34no—
85C22×C3×D3—32no—
86C22×C3×D3—34no—
87C22×C3×D3—34no—
88C22×C3×D3—34no—
89C22×C3×D3—64no—
90C22×C3×D3—64no—
91C22×C3×D3—68no—
92C22×C3×D3—68no—
93C22×C3×D3—64no—
94C22×C3×D3—64no—
95C22×C3×D3—64no—
96C22×C3×D3C2×A412yes—
97C2×C32×C4—32no—
98C2×C32×C4—31no—
99C2×C32×C4—32no—
100C2×C32×C4—32no—
101C2×C32×C4—32no—
102C2×C32×C4C2×A411yescharacteristic
103C3×D12—38no—
104C3×D12—38no—
105C3×D12—34no—
106C3×D12C2×A414yes—
107C23×C9—122no—
108C2×C22⋊C9—22no—
109C2×C22⋊C9—21no—
110C2×C22⋊C9—64no—
111C2×C22⋊C9D1212yesdirect product factor
112C2×C22⋊C9C22×D311yescharacteristic
113D4×C9—124no—
114C2×C4×C9—41no—
115C2×C4×C9—81no—
116C23×D4—31noSylow
117C2×C3×C9—42no—
118C2×C3×C9—41no—
119D3×C9—84no—
120C24×C3—21no—
121C24×C3—34no—
122C24×C3—64no—
123C24×C3—92no—
124C24×C3—94no—
125C24×C3—92no—
126C24×C3—184no—
127C24×C3C22×C911yescharacteristic
128C24×C3C2×C3×D311yescharacteristic
129C23×D3—24no—
130C23×D3—32no—
131C23×D3—32no—
132C23×D3—34no—
133C23×D3—64no—
134C23×D3C22×C914yescomplemented
135C2×C3×D4—31no—
136C2×C3×D4—94no—
137C2×C3×D4—94no—
138C2×C3×D4—92no—
139C2×C3×D4—98no—
140C2×C3×D4—92no—
141C2×C3×D4—94no—
142C2×C3×D4—94no—
143C2×C3×D4—98no—
144C2×C3×D4—91no—
145C22×C3×C4—22no—
146C22×C3×C4—31no—
147C22×C3×C4—31no—
148C22×C3×C4—31no—
149C22×C3×C4—32no—
150C22×C3×C4—31no—
151C22×C3×C4—32no—
152C22×C3×C4—61no—
153C22×C3×C4—62no—
154C22×C3×C4—61no—
155C22×C3×C4C22×C912yescomplemented
156C22×C3×C4C2×C3×D312yes—
157C2×D12—32no—
158C2×D12—38no—
159C2×D12—34no—
160C2×D12—32no—
161C2×D12—31no—
162C2×D12—34no—
163C2×D12—34no—
164C2×D12—38no—
165C2×D12—34no—
166C2×D12C22⋊C911yescharacteristic, direct product factor
167C22×C32—32no—
168C22×C32—31no—
169C22×C32—32no—
170C22×C32—32no—
171C22×C32—32no—
172C22×C32—31no—
173C22×C32—31no—
174C22×C32C2×C3×D411yescharacteristic
175C22×C32C22×A411yescharacteristic
176C2×C3×D3—24no—
177C2×C3×D3—34no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—68no—
181C2×C3×D3—68no—
182C2×C3×D3—64no—
183C2×C3×D3—64no—
184C2×C3×D3—64no—
185C2×C3×D3C22×A414yes—
186C32×C4—32no—
187C32×C4C22×A412yes—
188C22×C9—41no—
189C22×C9—81no—
190C22×C9—124no—
191C22×C9—244no—
192C22⋊C9—21no—
193C22⋊C9C2×D1211yescharacteristic, direct product factor
194C4×C9—42no—
195C4×C9—82no—
196C25—32no—
197C22×D4—34no—
198C22×D4—91no—
199C22×D4—91no—
200C22×D4—94no—
201C22×D4—92no—
202C23×C4D3×C911yescharacteristic, complemented
203C3×C9—41noSylow
204C23×C3—22no—
205C23×C3—21no—
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—32no—
209C23×C3—32no—
210C23×C3—31no—
211C23×C3—31no—
212C23×C3—32no—
213C23×C3—61no—
214C23×C3—61no—
215C23×C3—62no—
216C23×C3—64no—
217C23×C3—94no—
218C23×C3—94no—
219C23×C3—94no—
220C23×C3—94no—
221C23×C3—92no—
222C23×C3—184no—
223C23×C3—184no—
224C23×C3—184no—
225C23×C3—188no—
226C23×C3—184no—
227C23×C3—184no—
228C23×C3—188no—
229C23×C3D4×C912yescomplemented
230C23×C3C23×C911yescharacteristic
231C23×C3C3×D1212yes—
232C23×C3C22×C3×D311yescharacteristic
233C22×D3—24no—
234C22×D3—34no—
235C22×D3—32no—
236C22×D3—34no—
237C22×D3—34no—
238C22×D3—34no—
239C22×D3—64no—
240C22×D3—64no—
241C22×D3—68no—
242C22×D3—68no—
243C22×D3—64no—
244C22×D3—64no—
245C22×D3—64no—
246C22×D3C2×C22⋊C912yescomplemented
247C3×D4—34no—
248C3×D4—98no—
249C3×D4—98no—
250C3×D4—94no—
251C2×C3×C4—21no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—32no—
257C2×C3×C4—32no—
258C2×C3×C4—32no—
259C2×C3×C4—32no—
260C2×C3×C4—32no—
261C2×C3×C4—32no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4C2×C22⋊C911yescharacteristic, complemented
268C2×C3×C4D3×A411yescharacteristic
269D12—38no—
270D12—38no—
271D12—34no—
272D12C2×C22⋊C914yesdirect product factor
273C2×C32—32no—
274C2×C32—31no—
275C2×C32—31no—
276C2×C32D4×A412yes—
277C2×C32C23×A411yescharacteristic
278C3×D3—24no—
279C3×D3—64no—
280C2×C9—42no—
281C2×C9—41no—
282C2×C9—82no—
283C2×C9—81no—
284C2×C9—244no—
285C24—34no—
286C24—64no—
287C24—92no—
288C24—94no—
289C24—92no—
290C24—184no—
291C24C2×D3×C911yescharacteristic
292C2×D4—31no—
293C2×D4—94no—
294C2×D4—94no—
295C2×D4—92no—
296C2×D4—98no—
297C2×D4—92no—
298C2×D4—94no—
299C2×D4—94no—
300C2×D4—98no—
301C2×D4—91no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—32no—
305C22×C4C2×D3×C912yescomplemented
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—32no—
309C22×C3—31no—
310C22×C3—32no—
311C22×C3—31no—
312C22×C3—32no—
313C22×C3—32no—
314C22×C3—32no—
315C22×C3—32no—
316C22×C3—31no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—32no—
321C22×C3—32no—
322C22×C3—34no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—62no—
326C22×C3—62no—
327C22×C3—62no—
328C22×C3—61no—
329C22×C3—62no—
330C22×C3—64no—
331C22×C3—94no—
332C22×C3—184no—
333C22×C3—184no—
334C22×C3—188no—
335C22×C3—184no—
336C22×C3—184no—
337C22×C3—188no—
338C22×C3—184no—
339C22×C3C2×D4×C911yescharacteristic, complemented
340C22×C3C22×C22⋊C911yescharacteristic
341C22×C3C2×C3×D1211yescharacteristic
342C22×C3C2×D3×A411yescharacteristic
343C2×D3—24no—
344C2×D3—34no—
345C2×D3—64no—
346C2×D3—68no—
347C2×D3—64no—
348C2×D3—68no—
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—64no—
352C2×D3C22×C22⋊C914yescomplemented
353C3×C4—22no—
354C3×C4—32no—
355C3×C4—32no—
356C3×C4—62no—
357C3×C4C22×C22⋊C912yescomplemented
358C3×C4C2×D3×A412yes—
359C32C2×D4×A411yescharacteristic
360C9—41no—
361C9—81no—
362C23—31no—
363C23—32no—
364C23—31no—
365C23—32no—
366C23—64no—
367C23—94no—
368C23—94no—
369C23—94no—
370C23—94no—
371C23—92no—
372C23—184no—
373C23—184no—
374C23—184no—
375C23—188no—
376C23—184no—
377C23—184no—
378C23—188no—
379C23C9×D1212yescomplemented
380C23C22×D3×C911yescharacteristic
381D4—34no—
382D4—98no—
383D4—98no—
384D4—94no—
385C2×C4—32no—
386C2×C4—31no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4D3×C22⋊C911yescharacteristic, complemented
391C2×C3—21no—
392C2×C3—22no—
393C2×C3—31no—
394C2×C3—32no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—32no—
398C2×C3—31no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—62no—
402C2×C3—64no—
403C2×C3—184no—
404C2×C3D4×C22⋊C912yescomplemented
405C2×C3C23×C22⋊C911yescharacteristic
406C2×C3A4×D1212yes—
407C2×C3C22×D3×A411yescharacteristic
408D3—24no—
409D3—64no—
410C22—32no—
411C22—31no—
412C22—32no—
413C22—31no—
414C22—31no—
415C22—32no—
416C22—32no—
417C22—34no—
418C22—64no—
419C22—94no—
420C22—184no—
421C22—184no—
422C22—188no—
423C22—184no—
424C22—184no—
425C22—188no—
426C22—184no—
427C22C2×C9×D1211yescharacteristic, complemented
428C22C2×D3×C22⋊C911yescharacteristic
429C4—32no—
430C4C2×D3×C22⋊C912yescomplemented
431C3—21no—
432C3C2×D4×C22⋊C911yescharacteristic, complemented
433C3C2×A4×D1211yescharacteristic
434C2—31no—
435C2—32no—
436C2—31no—
437C2—64no—
438C2—184no—
439C2D12×C22⋊C912yesdirect product factor
440C2C22×D3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C3C2×D3×A4
commutatorC23×C3C23×C9
FrattiniC2×C3C22×D3×A4
FittingC23×C32×C4C2×C3
socleC24×C32C22×C3
prime radicalC23×C3×D12C3
cosocleC23×C32C23×C3
hypercenterC2×C3×C4D3×A4
hypocenterC22×C3C2×D4×C9
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×D3×C22⋊C9C2semidirect
2. C2×C3×C4×C22⋊C9C2semidirect
3. D12×C22⋊C9C2direct (N × H)
8. C2×D3×C22⋊C9C22semidirect
10. C3×C4×C22⋊C9C22semidirect
21. C2×C4×C22⋊C9D3semidirect
23. C2×C3×C22⋊C9D4semidirect
29. C23×D12C9semidirect
57. C4×C22⋊C9C2×D3semidirect
61. C3×C22⋊C9C2×D4semidirect
64. C24×D3C2×C9semidirect
71. C23×C3×C4C2×C9semidirect
77. C22×D12C2×C9semidirect
111. C2×C22⋊C9D12direct (N × H)
134. C23×D3C22×C9semidirect
155. C22×C3×C4C22×C9semidirect
166. C2×D12C22⋊C9direct (N × H)
193. C22⋊C9C2×D12direct (N × H)
202. C23×C4D3×C9semidirect
229. C23×C3D4×C9semidirect
246. C22×D3C2×C22⋊C9semidirect
267. C2×C3×C4C2×C22⋊C9semidirect
272. D12C2×C22⋊C9direct (N × H)
305. C22×C4C2×D3×C9semidirect
339. C22×C3C2×D4×C9semidirect
352. C2×D3C22×C22⋊C9semidirect
357. C3×C4C22×C22⋊C9semidirect
379. C23C9×D12semidirect
390. C2×C4D3×C22⋊C9semidirect
404. C2×C3D4×C22⋊C9semidirect
427. C22C2×C9×D12semidirect
430. C4C2×D3×C22⋊C9semidirect
432. C3C2×D4×C22⋊C9semidirect
439. C2D12×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×C22⋊C92
C2×C3×C4×C22⋊C92
D12×C22⋊C92
C23×C3×D123
C2×D4×C22⋊C93
C2×C9×D124