Finite Group Databasebeta

← C2×D3×(C22⋊C9)⋊C4 #13164243

Subgroups — C2×D3×(C22⋊C9)⋊C4 #13164243

Order 1728.

1063 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1063 conjugacy classes of subgroups — too many to draw. Only its 119 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⋊C9C211yesmaximal, characteristic
2C2×C3⋊((C22⋊C9)⋊C4)C211yesmaximal, characteristic, complemented
3C2×C3×(C22⋊C9)⋊C4C211yesmaximal, characteristic, complemented
4D3×(C22⋊C9)⋊C4C214yesmaximal, direct product factor
5C2×D3×#217—31nomaximal
6C22×(C22⋊C9)⋊C4—31nomaximal
7C22×C3×C22⋊C9C2211yescharacteristic
8C2×D3×C22⋊C9C414yescomplemented
9C2×D3×C22⋊C9C2212yes—
10C2×D3×Dic9—41nomaximal
11C3⋊((C22⋊C9)⋊C4)C2212yescomplemented
12C3×(C22⋊C9)⋊C4C2212yescomplemented
13C24×C3×D3D311yescharacteristic
14C22×D3×Dic3—31no—
15C22×D3×Dic3—31no—
16C23×C22⋊C9—31no—
17C2×#3757—31no—
18C2×C3×#217—31no—
19D3×#217—34no—
20D3×#217—34no—
21C2×(C2×D3)⋊Dic3—31no—
22C2×(C2×D3)⋊Dic3—31no—
23C2×(C22⋊C9)⋊C4—31no—
24C2×(C22⋊C9)⋊C4—34no—
25C2×(C22⋊C9)⋊C4—61no—
26C2×(C22⋊C9)⋊C4D311yescharacteristic, direct product factor
27C22×D3×C9—41no—
28C2×C3×C22⋊C9C2×C412yescomplemented
29C2×C3×C22⋊C9C2311yescharacteristic
30D3×C22⋊C9C2×C414yescomplemented
31C2×C3⋊Dic9—41no—
32C2×C3×Dic9—41no—
33D3×Dic9—44no—
34C22×#217—91no—
35C2×D3×C22⋊C4—91no—
36C24×C32C2×D311yescharacteristic
37C23×C3×D3—31no—
38C23×C3×D3—34no—
39C23×C3×D3—31no—
40C23×C3×D3—32no—
41C23×C3×D3Dic314yes—
42C23×C3×D3C2×D312yes—
43C22×C3⋊Dic3—31no—
44C22×C3⋊Dic3—31no—
45C22×C3×Dic3—31no—
46C22×C3×Dic3—31no—
47C2×D3×Dic3—34no—
48C2×D3×Dic3—34no—
49C2×D3×Dic3—64no—
50C2×D3×Dic3—61no—
51C2×D3×Dic3—62no—
52C2×D3×Dic3—61no—
53C22×C22⋊C9—21no—
54C22×C22⋊C9—34no—
55C22×C22⋊C9—32no—
56C22×C22⋊C9C2×D311yescharacteristic
57#3757—32no—
58#3757—32no—
59C3×#217—32no—
60C3×#217—32no—
61(C2×D3)⋊Dic3—34no—
62(C2×D3)⋊Dic3—34no—
63C22×Dic9—121no—
64(C22⋊C9)⋊C4—32no—
65(C22⋊C9)⋊C4—62no—
66(C22⋊C9)⋊C4C2×D312yesdirect product factor
67C22×C3×C9—41no—
68C2×D3×C9—44no—
69C2×D3×C9—42no—
70C3×C22⋊C9C22×C411yescharacteristic, complemented
71C3⋊Dic9—42no—
72C3×Dic9—42no—
73C25×C3—31no—
74C24×D3D911yescharacteristic
75C23×Dic3—91no—
76C23×Dic3—91no—
77C22×C4×D3—91no—
78C22×C4×D3—91no—
79C2×C3×C22⋊C4—91no—
80C2×#217—31no—
81C2×#217—94no—
82C2×#217—94no—
83C2×#217—91no—
84C2×#217—91no—
85C2×#217—91no—
86C2×#217—91no—
87C2×#217—181no—
88C2×(C2×D3)⋊C4—91no—
89C2×(C2×D3)⋊C4—91no—
90D3×C22⋊C4—94no—
91D3×C22⋊C4—94no—
92C23×C32—31no—
93C23×C32—32no—
94C23×C32—31no—
95C23×C32C2×Dic312yes—
96C23×C32C22×D311yescharacteristic
97C22×C3×D3—31no—
98C22×C3×D3—34no—
99C22×C3×D3—34no—
100C22×C3×D3—32no—
101C22×C3×D3—32no—
102C22×C3×D3—34no—
103C22×C3×D3—38no—
104C22×C3×D3—34no—
105C22×C3×D3—32no—
106C22×C3×D3—64no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3C2×Dic314yes—
110C22×C3×D3S411yescharacteristic
111C2×C3⋊Dic3—32no—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—62no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—61no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121D3×Dic3—68no—
122D3×Dic3—64no—
123D3×Dic3—64no—
124C23×C9—121no—
125C2×C22⋊C9—22no—
126C2×C22⋊C9—21no—
127C2×C22⋊C9—34no—
128C2×C22⋊C9C4×D312yescomplemented
129C2×C22⋊C9C22×D311yescharacteristic
130C2×Dic9—41no—
131C2×Dic9—124no—
132C2×Dic9—121no—
133C2×Dic9—241no—
134C22×C22⋊C4—271noSylow
135C2×C3×C9—42no—
136C2×C3×C9—41no—
137D3×C9—44no—
138C24×C3—21no—
139C24×C3—34no—
140C24×C3—32no—
141C24×C3—91no—
142C24×C3—94no—
143C24×C3—91no—
144C24×C3—92no—
145C24×C3C2×D911yescharacteristic
146C24×C3D3211yescharacteristic
147C23×D3—31no—
148C23×D3—32no—
149C23×D3—31no—
150C23×D3—34no—
151C23×D3Dic914yescomplemented
152C23×D3C2×D912yes—
153C22×C3×C4—91no—
154C22×C3×C4—91no—
155C22×Dic3—31no—
156C22×Dic3—31no—
157C22×Dic3—94no—
158C22×Dic3—94no—
159C22×Dic3—91no—
160C22×Dic3—91no—
161C22×Dic3—91no—
162C22×Dic3—91no—
163C22×Dic3—181no—
164C22×Dic3—181no—
165C22×Dic3—181no—
166C22×Dic3—182no—
167C22×Dic3—184no—
168C22×Dic3—181no—
169C2×C4×D3—94no—
170C2×C4×D3—94no—
171C2×C4×D3—184no—
172C2×C4×D3—181no—
173C2×C4×D3—182no—
174C2×C4×D3—181no—
175C3×C22⋊C4—92no—
176C3×C22⋊C4—92no—
177#217—32no—
178#217—32no—
179#217—94no—
180#217—94no—
181#217—92no—
182#217—92no—
183#217—92no—
184#217—92no—
185#217—182no—
186#217—182no—
187(C2×D3)⋊C4—94no—
188(C2×D3)⋊C4—94no—
189C22×C32—32no—
190C22×C32—31no—
191C22×C32—32no—
192C22×C32—32no—
193C22×C32—31no—
194C22×C32—31no—
195C22×C32—61no—
196C22×C32C22×Dic311yescharacteristic
197C22×C32C2×S411yescharacteristic
198C2×C3×D3—34no—
199C2×C3×D3—32no—
200C2×C3×D3—34no—
201C2×C3×D3—34no—
202C2×C3×D3—38no—
203C2×C3×D3—34no—
204C2×C3×D3—62no—
205C2×C3×D3—64no—
206C2×C3×D3SL(2,ℤ/4)14yes—
207C2×C3×D3C2×S412yes—
208C3⋊Dic3—62no—
209C3⋊Dic3—62no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C22×C9—41no—
213C22×C9—81no—
214C22×C9—122no—
215C22×C9—124no—
216C22⋊C9—21no—
217C22⋊C9C2×C4×D311yescharacteristic, complemented
218Dic9—42no—
219Dic9—122no—
220Dic9—242no—
221C25—31no—
222C23×C4—271no—
223C23×C4—271no—
224C2×C22⋊C4—91no—
225C2×C22⋊C4—274no—
226C2×C22⋊C4—274no—
227C2×C22⋊C4—271no—
228C2×C22⋊C4—271no—
229C2×C22⋊C4—271no—
230C3×C9—41noSylow
231C23×C3—22no—
232C23×C3—21no—
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—31no—
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—32no—
239C23×C3—32no—
240C23×C3—34no—
241C23×C3—61no—
242C23×C3—61no—
243C23×C3—62no—
244C23×C3—92no—
245C23×C3—92no—
246C23×C3—94no—
247C23×C3—94no—
248C23×C3—92no—
249C23×C3—98no—
250C23×C3—94no—
251C23×C3—94no—
252C23×C3—91no—
253C23×C3—181no—
254C23×C3—182no—
255C23×C3—184no—
256C23×C3C2×Dic912yescomplemented
257C23×C3C22×D911yescharacteristic
258C23×C3D3×Dic312yes—
259C23×C3C2×D3211yescharacteristic
260C22×D3—31no—
261C22×D3—34no—
262C22×D3—34no—
263C22×D3—32no—
264C22×D3—32no—
265C22×D3—32no—
266C22×D3—34no—
267C22×D3—38no—
268C22×D3—34no—
269C22×D3—64no—
270C22×D3—61no—
271C22×D3—62no—
272C22×D3C2×Dic914yescomplemented
273C22×D3(C22⋊C9)⋊C211yescharacteristic
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—182no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×Dic3—32no—
280C2×Dic3—32no—
281C2×Dic3—62no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—92no—
285C2×Dic3—92no—
286C2×Dic3—92no—
287C2×Dic3—92no—
288C2×Dic3—182no—
289C2×Dic3—182no—
290C2×Dic3—184no—
291C2×Dic3—188no—
292C2×Dic3—184no—
293C2×Dic3—182no—
294C2×Dic3—181no—
295C2×Dic3—182no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C2×Dic3—361no—
300C2×Dic3—361no—
301C2×Dic3—362no—
302C4×D3—188no—
303C4×D3—184no—
304C4×D3—184no—
305C2×C32—31no—
306C2×C32—31no—
307C2×C32—32no—
308C2×C32C2×SL(2,ℤ/4)12yes—
309C2×C32C22×S411yescharacteristic
310C3×D3—34no—
311C3×D3C2×SL(2,ℤ/4)14yes—
312C2×C9—41no—
313C2×C9—42no—
314C2×C9—81no—
315C2×C9—82no—
316C2×C9—124no—
317C24—34no—
318C24—32no—
319C24—91no—
320C24—91no—
321C24—92no—
322C24—94no—
323C24D3×D911yescharacteristic
324C22×C4—91no—
325C22×C4—91no—
326C22×C4—274no—
327C22×C4—274no—
328C22×C4—271no—
329C22×C4—271no—
330C22×C4—541no—
331C22×C4—542no—
332C22×C4—544no—
333C22×C4—541no—
334C22⋊C4—92no—
335C22⋊C4—92no—
336C22⋊C4—274no—
337C22⋊C4—274no—
338C22⋊C4—272no—
339C22⋊C4—272no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—32no—
343C22×C3—31no—
344C22×C3—32no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—32no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—32no—
352C22×C3—32no—
353C22×C3—32no—
354C22×C3—32no—
355C22×C3—34no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—62no—
359C22×C3—62no—
360C22×C3—62no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—94no—
365C22×C3—94no—
366C22×C3—98no—
367C22×C3—94no—
368C22×C3—92no—
369C22×C3—94no—
370C22×C3—121no—
371C22×C3—182no—
372C22×C3—184no—
373C22×C3C22×Dic911yescharacteristic, complemented
374C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
375C22×C3C2×D3×Dic311yescharacteristic
376C22×C3D3×S411yescharacteristic
377C2×D3—32no—
378C2×D3—34no—
379C2×D3—34no—
380C2×D3—34no—
381C2×D3—38no—
382C2×D3—34no—
383C2×D3—62no—
384C2×D3—64no—
385C2×D3(C22⋊C9)⋊C414yesdirect product factor
386C2×D3C2×(C22⋊C9)⋊C212yes—
387C3×C4—182no—
388C3×C4—182no—
389Dic3—62no—
390Dic3—62no—
391Dic3—182no—
392Dic3—182no—
393Dic3—182no—
394Dic3—182no—
395Dic3—362no—
396Dic3—362no—
397C32C22×SL(2,ℤ/4)11yescharacteristic
398C9—41no—
399C9—81no—
400C23—31no—
401C23—31no—
402C23—31no—
403C23—32no—
404C23—34no—
405C23—92no—
406C23—92no—
407C23—94no—
408C23—94no—
409C23—92no—
410C23—98no—
411C23—94no—
412C23—94no—
413C23—91no—
414C23—181no—
415C23—182no—
416C23—184no—
417C23D3×Dic912yescomplemented
418C23C2×D3×D911yescharacteristic
419C2×C4—92no—
420C2×C4—92no—
421C2×C4—182no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4—272no—
425C2×C4—272no—
426C2×C4—544no—
427C2×C4—548no—
428C2×C4—544no—
429C2×C4—542no—
430C2×C4—541no—
431C2×C4—541no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—32no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—32no—
440C2×C3—34no—
441C2×C3—61no—
442C2×C3—61no—
443C2×C3—62no—
444C2×C3—94no—
445C2×C3C2×(C22⋊C9)⋊C412yescomplemented
446C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
447C2×C3D3×SL(2,ℤ/4)12yes—
448C2×C3C2×D3×S411yescharacteristic
449D3—34no—
450D3C2×(C22⋊C9)⋊C414yesdirect product factor
451C22—32no—
452C22—31no—
453C22—31no—
454C22—31no—
455C22—32no—
456C22—32no—
457C22—32no—
458C22—34no—
459C22—61no—
460C22—94no—
461C22—94no—
462C22—98no—
463C22—94no—
464C22—92no—
465C22—94no—
466C22—182no—
467C22—184no—
468C22C2×D3×Dic911yescharacteristic, complemented
469C22D3×(C22⋊C9)⋊C211yescharacteristic
470C4—182no—
471C4—182no—
472C4—542no—
473C4—542no—
474C3—21no—
475C3C22×(C22⋊C9)⋊C411yescharacteristic, complemented
476C3C2×D3×SL(2,ℤ/4)11yescharacteristic
477C2—31no—
478C2—31no—
479C2—32no—
480C2—34no—
481C2—94no—
482C2D3×(C22⋊C9)⋊C412yesdirect product factor
483C2C2×D3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC22D3×(C22⋊C9)⋊C2
commutator, hypocenterC3×C22⋊C9C22×C4
FrattiniC2×C3C2×D3×S4
Fitting, socleC24×C32C2×D3
prime radicalC24×C3×D3D3
cosocleC2×C3×C22⋊C9C23
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊((C22⋊C9)⋊C4)C2semidirect
3. C2×C3×(C22⋊C9)⋊C4C2semidirect
4. D3×(C22⋊C9)⋊C4C2direct (N × H)
8. C2×D3×C22⋊C9C4semidirect
11. C3⋊((C22⋊C9)⋊C4)C22semidirect
12. C3×(C22⋊C9)⋊C4C22semidirect
26. C2×(C22⋊C9)⋊C4D3direct (N × H)
28. C2×C3×C22⋊C9C2×C4semidirect
30. D3×C22⋊C9C2×C4semidirect
66. (C22⋊C9)⋊C4C2×D3direct (N × H)
70. C3×C22⋊C9C22×C4semidirect
128. C2×C22⋊C9C4×D3semidirect
151. C23×D3Dic9semidirect
217. C22⋊C9C2×C4×D3semidirect
256. C23×C3C2×Dic9semidirect
272. C22×D3C2×Dic9semidirect
373. C22×C3C22×Dic9semidirect
385. C2×D3(C22⋊C9)⋊C4direct (N × H)
417. C23D3×Dic9semidirect
445. C2×C3C2×(C22⋊C9)⋊C4semidirect
450. D3C2×(C22⋊C9)⋊C4direct (N × H)
468. C22C2×D3×Dic9semidirect
475. C3C22×(C22⋊C9)⋊C4semidirect
482. C2D3×(C22⋊C9)⋊C4direct (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⋊((C22⋊C9)⋊C4)2
C2×C3×(C22⋊C9)⋊C42
D3×(C22⋊C9)⋊C42
C2×D3×#2173
C22×(C22⋊C9)⋊C43
C2×D3×Dic94