Finite Group Databasebeta

← C2×(C3⋊((C22⋊C9)⋊C2))⋊C4 #13164241

Subgroups — C2×(C3⋊((C22⋊C9)⋊C2))⋊C4 #13164241

Order 1728.

955 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 955 conjugacy classes of subgroups — too many to draw. Only its 87 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×C3⋊((C22⋊C9)⋊C2)C211yesmaximal, characteristic
2C2×Dic3×C22⋊C9C211yesmaximal, characteristic, complemented
3C2×C3×(C22⋊C9)⋊C4C211yesmaximal, characteristic, complemented
4(C3⋊((C22⋊C9)⋊C2))⋊C4C214yesmaximal, direct product factor
5C2×#63952—31nomaximal
6C2×C4×(C22⋊C9)⋊C2—31nomaximal
7C22×C3×C22⋊C9C2211yescharacteristic
8C2×C3⋊((C22⋊C9)⋊C2)C414yescomplemented
9C2×C3⋊((C22⋊C9)⋊C2)C2212yes—
10C2×(C3⋊D9)⋊C4—41nomaximal
11Dic3×C22⋊C9C2212yescomplemented
12C3×(C22⋊C9)⋊C4C2212yescomplemented
13C22×#638—31no—
14C23×C3×Dic3D311yescharacteristic
15C22×#624—31no—
16C22×(C22⋊C9)⋊C2—31no—
17C22×(C22⋊C9)⋊C2—61no—
18C2×C3×#217—31no—
19C2×#3724—31no—
20C2×#3722—31no—
21C2×Dic32—31no—
22#63952—34no—
23#63952—34no—
24C2×C4×C22⋊C9—31no—
25C2×(C22⋊C9)⋊C4D311yescharacteristic, complemented
26C4×(C22⋊C9)⋊C2—34no—
27C22×C3⋊D9—41no—
28C2×C3×C22⋊C9C2×C412yescomplemented
29C2×C3×C22⋊C9C2311yescharacteristic
30C3⋊((C22⋊C9)⋊C2)C2×C414yescomplemented
31C2×C9×Dic3—41no—
32C2×C3×Dic9—41no—
33(C3⋊D9)⋊C4—44no—
34C2×C4×#67—91no—
35C2×#67⋊C4—91no—
36C24×C32C2×D311yescharacteristic
37C23×C3⋊D3—31no—
38C2×#638—34no—
39C2×#638—32no—
40C2×#638—34no—
41C2×#638—32no—
42C22×C3⋊Dic3—31no—
43C22×C3×Dic3—31no—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—31no—
46C22×C3×Dic3—31no—
47C22×C3×Dic3—31no—
48C22×C3×Dic3C2×D312yes—
49C2×#624—34no—
50C2×#624—61no—
51C2×#624—62no—
52C2×#624—61no—
53C22×C22⋊C9—21no—
54C22×C22⋊C9C2×D311yescharacteristic
55C2×(C22⋊C9)⋊C2—34no—
56C2×(C22⋊C9)⋊C2—32no—
57C2×(C22⋊C9)⋊C2—64no—
58C2×(C22⋊C9)⋊C2—62no—
59C3×#217—32no—
60C3×#217—32no—
61#3724—34no—
62#3722—34no—
63Dic32—34no—
64C2×C4×D9—121no—
65C4×C22⋊C9—32no—
66(C22⋊C9)⋊C4C2×D312yescomplemented
67C22×C3×C9—41no—
68C2×C3⋊D9—42no—
69C2×C3⋊D9—44no—
70C3×C22⋊C9C22×C411yescharacteristic, complemented
71C9×Dic3—42no—
72C3×Dic9—42no—
73C23×C3×C4—31no—
74C22×#67—91no—
75C22×#67—91no—
76C22×#67—181no—
77C23×Dic3D911yescharacteristic, complemented
78C22×C4×D3—91no—
79C22×C4×D3—91no—
80C2×C3×C22⋊C4—91no—
81C2×#217—31no—
82C4×#67—94no—
83C4×#67—94no—
84C2×(C2×D3)⋊C4—91no—
85C2×(C2×D3)⋊C4—91no—
86C2×Dic3⋊C4—91no—
87C2×Dic3⋊C4—91no—
88C2×C4×Dic3—91no—
89C2×C4×Dic3—91no—
90#67⋊C4—94no—
91#67⋊C4—94no—
92C23×C32—31no—
93C23×C32—31no—
94C23×C32—32no—
95C23×C32C4×D312yes—
96C23×C32C22×D311yescharacteristic
97C22×C3⋊D3—32no—
98C22×C3⋊D3—34no—
99C22×C3⋊D3—62no—
100C22×C3⋊D3—61no—
101C22×C3⋊D3—61no—
102#638—38no—
103#638—34no—
104#638—34no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—34no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—32no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3S411yescharacteristic
118#624—64no—
119#624—64no—
120C22×D9—121no—
121C22×D9—241no—
122C2×C22⋊C9—22no—
123C2×C22⋊C9—21no—
124C2×C22⋊C9C4×D312yescomplemented
125C2×C22⋊C9C22×D311yescharacteristic
126(C22⋊C9)⋊C2—34no—
127(C22⋊C9)⋊C2—64no—
128C2×C4×C9—121no—
129C2×Dic9—41no—
130C4×D9—124no—
131C2×C4×D4—271noSylow
132C2×C3×C9—42no—
133C2×C3×C9—41no—
134C3⋊D9—44no—
135C24×C3—21no—
136C24×C3C2×D911yescharacteristic
137C24×C3D3211yescharacteristic
138C23×D3—91no—
139C23×D3—91no—
140C23×D3—181no—
141C22×C3×C4—32no—
142C22×C3×C4—91no—
143C22×C3×C4—91no—
144C22×C3×C4—92no—
145C22×C3×C4—91no—
146C22×C3×C4—91no—
147C2×#67—92no—
148C2×#67—92no—
149C2×#67—94no—
150C2×#67—94no—
151C2×#67—94no—
152C2×#67—92no—
153C2×#67—94no—
154C2×#67—92no—
155C2×#67—184no—
156C2×#67—182no—
157C2×#67—182no—
158C2×#67—184no—
159C22×Dic3—31no—
160C22×Dic3—32no—
161C22×Dic3—31no—
162C22×Dic3—31no—
163C22×Dic3—31no—
164C22×Dic3—91no—
165C22×Dic3—91no—
166C22×Dic3—181no—
167C22×Dic3C2×D912yescomplemented
168C2×C4×D3—94no—
169C2×C4×D3—94no—
170C2×C4×D3—181no—
171C2×C4×D3—182no—
172C2×C4×D3—181no—
173C2×C4×D3—182no—
174C2×C4×D3—181no—
175C2×C4×D3—181no—
176C3×C22⋊C4—92no—
177C3×C22⋊C4—92no—
178#217—32no—
179#217—32no—
180(C2×D3)⋊C4—94no—
181(C2×D3)⋊C4—94no—
182Dic3⋊C4—94no—
183Dic3⋊C4—94no—
184C4×Dic3—94no—
185C4×Dic3—94no—
186C22×C32—32no—
187C22×C32—31no—
188C22×C32—32no—
189C22×C32—32no—
190C22×C32—31no—
191C22×C32—31no—
192C22×C32—61no—
193C22×C32C2×C4×D311yescharacteristic
194C22×C32C2×S411yescharacteristic
195C2×C3⋊D3—34no—
196C2×C3⋊D3—62no—
197C2×C3⋊D3—64no—
198C2×C3⋊D3—64no—
199C2×C3⋊D3—62no—
200C3⋊Dic3—34no—
201C3×Dic3—32no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3C2×S412yes—
205C22×C9—41no—
206C22×C9—81no—
207C2×D9—122no—
208C2×D9—124no—
209C2×D9—242no—
210C2×D9—244no—
211C22⋊C9—21no—
212C22⋊C9C2×C4×D311yescharacteristic, complemented
213C4×C9—122no—
214Dic9—42no—
215C22×D4—271no—
216C23×C4—31no—
217C23×C4—271no—
218C4×D4—274no—
219C4×D4—274no—
220C2×C4⋊C4—271no—
221C2×C22⋊C4—91no—
222C2×C22⋊C4—271no—
223C2×C42—271no—
224C3×C9—41noSylow
225C23×C3—22no—
226C23×C3—21no—
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—31no—
231C23×C3—32no—
232C23×C3—32no—
233C23×C3—61no—
234C23×C3—61no—
235C23×C3—62no—
236C23×C3C4×D912yescomplemented
237C23×C3C22×D911yescharacteristic
238C23×C3#62412yes—
239C23×C3C2×D3211yescharacteristic
240C22×D3—92no—
241C22×D3—92no—
242C22×D3—94no—
243C22×D3—94no—
244C22×D3—182no—
245C22×D3—184no—
246C22×D3—181no—
247C22×D3—181no—
248C22×D3—182no—
249C22×D3—182no—
250C22×D3—181no—
251C22×D3—181no—
252C22×D3—361no—
253C22×D3—361no—
254C22×D3—362no—
255C2×C3×C4—31no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—91no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—92no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4—182no—
265C2×C3×C4—181no—
266#67—98no—
267#67—94no—
268#67—98no—
269#67—94no—
270#67—94no—
271#67—94no—
272#67—184no—
273#67—188no—
274#67—184no—
275C2×Dic3—31no—
276C2×Dic3—32no—
277C2×Dic3—32no—
278C2×Dic3—32no—
279C2×Dic3—32no—
280C2×Dic3—32no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—62no—
284C2×Dic3—61no—
285C2×Dic3—92no—
286C2×Dic3—92no—
287C2×Dic3—94no—
288C2×Dic3—94no—
289C2×Dic3—182no—
290C2×Dic3—184no—
291C2×Dic3(C22⋊C9)⋊C211yescharacteristic, complemented
292C4×D3—184no—
293C4×D3—184no—
294C4×D3—184no—
295C4×D3—184no—
296C2×C32—31no—
297C2×C32—31no—
298C2×C32—32no—
299C2×C32C4×S412yes—
300C2×C32C22×S411yescharacteristic
301C3⋊D3—64no—
302C2×C9—41no—
303C2×C9—42no—
304C2×C9—81no—
305C2×C9—82no—
306D9—124no—
307D9—244no—
308C24—271no—
309C24D3×D911yescharacteristic
310C2×D4—272no—
311C2×D4—274no—
312C2×D4—274no—
313C2×D4—272no—
314C22×C4—32no—
315C22×C4—91no—
316C22×C4—91no—
317C22×C4—92no—
318C22×C4—91no—
319C22×C4—91no—
320C22×C4—274no—
321C22×C4—271no—
322C22×C4—541no—
323C22×C4—542no—
324C22×C4—541no—
325C4⋊C4—274no—
326C22⋊C4—92no—
327C22⋊C4—92no—
328C22⋊C4—274no—
329C42—274no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—32no—
333C22×C3—31no—
334C22×C3—32no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—32no—
341C22×C3—32no—
342C22×C3—32no—
343C22×C3—32no—
344C22×C3—61no—
345C22×C3—62no—
346C22×C3—61no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—121no—
353C22×C3C2×C4×D911yescharacteristic, complemented
354C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
355C22×C3C2×#62411yescharacteristic
356C22×C3D3×S411yescharacteristic
357C2×D3—94no—
358C2×D3—94no—
359C2×D3—184no—
360C2×D3—182no—
361C2×D3—184no—
362C2×D3—182no—
363C2×D3—184no—
364C2×D3—182no—
365C2×D3—182no—
366C2×D3—184no—
367C2×D3—184no—
368C2×D3—362no—
369C2×D3—364no—
370C2×D3—362no—
371C2×D3—364no—
372C3×C4—32no—
373C3×C4—92no—
374C3×C4—182no—
375C3×C4—182no—
376Dic3—32no—
377Dic3—62no—
378Dic3—62no—
379Dic3—94no—
380Dic3—94no—
381Dic3—184no—
382Dic3C2×(C22⋊C9)⋊C212yescomplemented
383C32C2×C4×S411yescharacteristic
384C9—41no—
385C9—81no—
386C23—31no—
387C23—31no—
388C23—32no—
389C23—272no—
390C23—274no—
391C23—541no—
392C23—542no—
393C23—541no—
394C23(C3⋊D9)⋊C412yescomplemented
395C23C2×D3×D911yescharacteristic
396D4—278no—
397D4—274no—
398D4—274no—
399C2×C4—31no—
400C2×C4—92no—
401C2×C4—92no—
402C2×C4—91no—
403C2×C4—92no—
404C2×C4—92no—
405C2×C4—92no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—182no—
409C2×C4—181no—
410C2×C4—272no—
411C2×C4—274no—
412C2×C4—544no—
413C2×C4—544no—
414C2×C3—21no—
415C2×C3—22no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—32no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—32no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—62no—
425C2×C3C4×(C22⋊C9)⋊C212yescomplemented
426C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
427C2×C3(C3⋊S4)⋊C412yes—
428C2×C3C2×D3×S411yescharacteristic
429D3—184no—
430D3—184no—
431D3—364no—
432C22—32no—
433C22—31no—
434C22—31no—
435C22—31no—
436C22—32no—
437C22—32no—
438C22—61no—
439C22—274no—
440C22—542no—
441C22—544no—
442C22—542no—
443C22—544no—
444C22C2×(C3⋊D9)⋊C411yescharacteristic, complemented
445C22D3×(C22⋊C9)⋊C211yescharacteristic
446C4—32no—
447C4—92no—
448C4—182no—
449C4—182no—
450C4—274no—
451C3—21no—
452C3C2×C4×(C22⋊C9)⋊C211yescharacteristic, complemented
453C3C2×(C3⋊S4)⋊C411yescharacteristic
454C2—31no—
455C2—31no—
456C2—32no—
457C2—544no—
458C2(C3⋊((C22⋊C9)⋊C2))⋊C412yesdirect product factor
459C2C2×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 radicalC22×C3⋊((C22⋊C9)⋊C2)C2
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×Dic3×C22⋊C9C2semidirect
3. C2×C3×(C22⋊C9)⋊C4C2semidirect
4. (C3⋊((C22⋊C9)⋊C2))⋊C4C2direct (N × H)
8. C2×C3⋊((C22⋊C9)⋊C2)C4semidirect
11. Dic3×C22⋊C9C22semidirect
12. C3×(C22⋊C9)⋊C4C22semidirect
25. C2×(C22⋊C9)⋊C4D3semidirect
28. C2×C3×C22⋊C9C2×C4semidirect
30. C3⋊((C22⋊C9)⋊C2)C2×C4semidirect
66. (C22⋊C9)⋊C4C2×D3semidirect
70. C3×C22⋊C9C22×C4semidirect
77. C23×Dic3D9semidirect
124. C2×C22⋊C9C4×D3semidirect
167. C22×Dic3C2×D9semidirect
212. C22⋊C9C2×C4×D3semidirect
236. C23×C3C4×D9semidirect
291. C2×Dic3(C22⋊C9)⋊C2semidirect
353. C22×C3C2×C4×D9semidirect
382. Dic3C2×(C22⋊C9)⋊C2semidirect
394. C23(C3⋊D9)⋊C4semidirect
425. C2×C3C4×(C22⋊C9)⋊C2semidirect
444. C22C2×(C3⋊D9)⋊C4semidirect
452. C3C2×C4×(C22⋊C9)⋊C2semidirect
458. C2(C3⋊((C22⋊C9)⋊C2))⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊((C22⋊C9)⋊C2)2
C2×Dic3×C22⋊C92
C2×C3×(C22⋊C9)⋊C42
(C3⋊((C22⋊C9)⋊C2))⋊C42
C2×#639523
C2×C4×(C22⋊C9)⋊C23
C2×(C3⋊D9)⋊C44