Finite Group Databasebeta

← C2×#11720941 #13150328

Subgroups — C2×#11720941 #13150328

Order 1728.

1262 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C22×#6192C211yesmaximal, characteristic
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720941C218yesmaximal, direct product factor
9C2×#63853—31nomaximal
10C2×#63456—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C4×C3⋊D9—24no—
16C2×C4×C3⋊D9C2214yescomplemented
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6192—24no—
21C2×#6192C2214yes—
22C3×(C2×D9)⋊C4C2214yescomplemented
23C9×Dic3⋊C4C2214yescomplemented
24#88445C2214yescomplemented
25(C2×D9)⋊Dic3C2214yescomplemented
26Dic9⋊Dic3C2214yescomplemented
27C22×C4×C3⋊D3—31no—
28C22×#626—31no—
29C2×C3×(C2×D3)⋊C4—31no—
30C2×C3×Dic3⋊C4D311yescharacteristic
31C2×Dic3⋊Dic3—31no—
32C2×#3722—31no—
33C2×(C2×D3)⋊Dic3—31no—
34#63853—38no—
35C22×D36—31no—
36C22×C4×D9—31no—
37C22×C4×D9—61no—
38C2×C9×C4⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4—31no—
41C2×(C2×D9)⋊C4D311yescharacteristic, complemented
42C2×Dic9⋊C4—31no—
43#63456—38no—
44C23×C3×C9C2311yescharacteristic
45C22×C3⋊D9—22no—
46C22×C3⋊D9—22no—
47C22×C3⋊D9D414yes—
48C22×C3⋊D9C2312yes—
49C22×C3×D9—22no—
50C22×C3×D9—22no—
51C22×C3×D9—22no—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9—22no—
54C2×C3×C4×C9C2312yes—
55C2×C3⋊Dic9—22no—
56C2×C3⋊Dic9C2312yes—
57C4×C3⋊D9—28no—
58C2×C9×Dic3—22no—
59C2×C9×Dic3—22no—
60C2×C9×Dic3C2312yes—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9—22no—
63C2×C3×Dic9C2312yes—
64#6192—28no—
65C2×#894—91no—
66C2×#883—91no—
67C23×C3⋊D3—31no—
68C23×C3×D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C2×C4×C3⋊D3—34no—
72C2×C4×C3⋊D3—64no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#626—34no—
77C2×#626—64no—
78C23×D9—31no—
79C23×D9—61no—
80C23×D9C2×D311yescharacteristic
81C3×(C2×D3)⋊C4—34no—
82C3×Dic3⋊C4C2×D314yes—
83Dic3⋊Dic3—34no—
84#3722—34no—
85(C2×D3)⋊Dic3—34no—
86C22×C4×C9—21no—
87C22×C4×C9—31no—
88C22×C4×C9—31no—
89C22×C4×C9C2×D311yescharacteristic
90C22×Dic9—31no—
91C22×Dic9—61no—
92C22×Dic9C2×D311yescharacteristic
93C2×D36—34no—
94C2×D36—64no—
95C2×C4×D9—34no—
96C2×C4×D9—64no—
97C2×C4×D9—64no—
98C2×C4×D9—68no—
99C9×C4⋊C4—34no—
100(C2×D9)⋊C4—34no—
101(C2×D9)⋊C4—34no—
102(C2×D9)⋊C4C2×D314yescomplemented
103Dic9⋊C4—34no—
104C22×C3×C9C2×D412yes—
105C22×C3×C9#4112yes—
106C22×C3×C9#4112yes—
107C22×C3×C9C2411yescharacteristic
108C2×C3⋊D9—28no—
109C2×C3⋊D9—22no—
110C2×C3⋊D9—22no—
111C2×C3⋊D9C2×D414yes—
112C2×C3×D9—22no—
113C2×C3×D9—22no—
114C2×C3×D9—22no—
115C2×C3×D9—44no—
116C3×C4×C9—22no—
117C3⋊Dic9—22no—
118C9×Dic3—22no—
119C9×Dic3—22no—
120C3×Dic9—22no—
121C22×#67—91no—
122C22×D12—91no—
123C22×C4×D3—91no—
124C22×C4×D3—91no—
125C22×C4×D3—181no—
126C2×C3×C4⋊C4—31no—
127C2×C3×C22⋊C4—91no—
128C2×#217—91no—
129C2×(C2×D3)⋊C4—31no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×(C2×D3)⋊C4—91no—
133C2×C4⋊Dic3—91no—
134C2×Dic3⋊C4—91no—
135C2×Dic3⋊C4D911yescharacteristic, complemented
136#894—98no—
137#883—98no—
138C23×C32C22×D311yescharacteristic
139C22×C3⋊D3—34no—
140C22×C3⋊D3—32no—
141C22×C3⋊D3—62no—
142C22×C3⋊D3—62no—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C2×C32×C4—22no—
148C2×C32×C4C22×D312yes—
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—62no—
151C4×C3⋊D3—68no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3C22×D312yes—
157C2×C3×Dic3C22×D312yes—
158#626—68no—
159C23×C9—21no—
160C23×C9C22×D311yescharacteristic
161C22×D9—22no—
162C22×D9—22no—
163C22×D9—22no—
164C22×D9—34no—
165C22×D9—32no—
166C22×D9—64no—
167C22×D9—62no—
168C22×D9—64no—
169C22×D9—64no—
170C22×D9—62no—
171C22×D9—62no—
172C22×D9C22×D312yes—
173C2×C4×C9—24no—
174C2×C4×C9—22no—
175C2×C4×C9—22no—
176C2×C4×C9—32no—
177C2×C4×C9—32no—
178C2×C4×C9—62no—
179C2×C4×C9—62no—
180C2×C4×C9C22×D312yes—
181C2×Dic9—22no—
182C2×Dic9—32no—
183C2×Dic9—62no—
184C2×Dic9—64no—
185C2×Dic9—62no—
186C2×Dic9C22×D312yes—
187D36—68no—
188C4×D9—68no—
189C4×D9—616no—
190C2×#123—271noSylow
191C2×C3×C9#12314yescomplemented
192C2×C3×C9C22×D411yescharacteristic
193C2×C3×C9C2×#4111yescharacteristic
194C2×C3×C9C2×#4111yescharacteristic
195C3⋊D9—24no—
196C3×D9—42no—
197C24×C3—91no—
198C23×D3—31no—
199C23×D3—91no—
200C23×D3—91no—
201C23×D3—181no—
202C22×C3×C4—21no—
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—91no—
206C22×C3×C4C2×D911yescharacteristic
207C22×C3×C4D3211yescharacteristic
208C2×#67—94no—
209C2×#67—184no—
210C22×Dic3—31no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—181no—
214C22×Dic3C2×D911yescharacteristic
215C22×Dic3C2×D911yescharacteristic
216C2×D12—94no—
217C2×D12—184no—
218C2×C4×D3—94no—
219C2×C4×D3—94no—
220C2×C4×D3—184no—
221C2×C4×D3—184no—
222C2×C4×D3—184no—
223C2×C4×D3—188no—
224C3×C4⋊C4—34no—
225C3×C22⋊C4—94no—
226#217—94no—
227(C2×D3)⋊C4—34no—
228(C2×D3)⋊C4—94no—
229(C2×D3)⋊C4—94no—
230(C2×D3)⋊C4—94no—
231C4⋊Dic3—94no—
232Dic3⋊C4—94no—
233Dic3⋊C4C2×D914yescomplemented
234C22×C32#23512yes—
235C22×C32D3×D412yes—
236C22×C32#23912yes—
237C22×C32C23×D311yescharacteristic
238C2×C3⋊D3—34no—
239C2×C3⋊D3—68no—
240C2×C3⋊D3—62no—
241C2×C3⋊D3—62no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—124no—
246C32×C4—22no—
247C3⋊Dic3—62no—
248C3×Dic3—22no—
249C3×Dic3—22no—
250C3×Dic3—62no—
251C22×C9—21no—
252C22×C9—22no—
253C22×C9—22no—
254C22×C9—22no—
255C22×C9#23512yes—
256C22×C9D3×D412yes—
257C22×C9#23712yes—
258C22×C9C23×D311yescharacteristic
259C2×D9—22no—
260C2×D9—22no—
261C2×D9—22no—
262C2×D9—34no—
263C2×D9—44no—
264C2×D9—64no—
265C2×D9—64no—
266C2×D9—64no—
267C2×D9—616no—
268C2×D9—68no—
269C2×D9—62no—
270C2×D9—62no—
271C4×C9—24no—
272C4×C9—22no—
273C4×C9—62no—
274C4×C9—62no—
275Dic9—22no—
276Dic9—64no—
277Dic9—62no—
278C22×D4—271no—
279C23×C4—271no—
280#123—278no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—271no—
283C2×C22⋊C4—91no—
284C2×C22⋊C4—271no—
285C2×C22⋊C4—271no—
286C3×C9C2×#12311yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—92no—
289C23×C3—182no—
290C23×C3—182no—
291C23×C3—182no—
292C23×C3C22×D911yescharacteristic
293C23×C3C2×D3211yescharacteristic
294C22×D3—32no—
295C22×D3—62no—
296C22×D3—62no—
297C22×D3—62no—
298C22×D3—92no—
299C22×D3—94no—
300C22×D3—92no—
301C22×D3—94no—
302C22×D3—182no—
303C22×D3—184no—
304C22×D3—184no—
305C22×D3—184no—
306C22×D3—182no—
307C22×D3—182no—
308C22×D3—182no—
309C22×D3—182no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—32no—
315C2×C3×C4—32no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—92no—
319C2×C3×C4—182no—
320C2×C3×C4C22×D912yes—
321C2×C3×C4C2×D3212yes—
322#67—188no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—32no—
326C2×Dic3—62no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—182no—
330C2×Dic3—184no—
331C2×Dic3—182no—
332C2×Dic3—182no—
333C2×Dic3C22×D912yes—
334C2×Dic3C22×D912yes—
335D12—188no—
336C4×D3—188no—
337C4×D3—188no—
338C4×D3—1816no—
339C2×C32#89414yes—
340C2×C32C2×#23511yescharacteristic
341C2×C32C2×D3×D411yescharacteristic
342C2×C32C2×#23911yescharacteristic
343C3⋊D3—64no—
344C3×D3—122no—
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—24no—
349C2×C9#88314yescomplemented
350C2×C9C2×#23511yescharacteristic
351C2×C9C2×D3×D411yescharacteristic
352C2×C9C2×#23711yescharacteristic
353D9—42no—
354D9—68no—
355D9—64no—
356C24—91no—
357C24—271no—
358C2×D4—274no—
359C2×D4—544no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—91no—
363C22×C4—271no—
364C22×C4—274no—
365C22×C4—544no—
366C22×C4D3×D911yescharacteristic
367C4⋊C4—34no—
368C4⋊C4—274no—
369C22⋊C4—94no—
370C22⋊C4—274no—
371C22⋊C4—274no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—182no—
377C22×C3—182no—
378C22×C3—182no—
379C22×C3—364no—
380C22×C3#369712yes—
381C22×C3D4×D912yes—
382C22×C3#370112yes—
383C22×C3C23×D911yescharacteristic
384C22×C3#379612yes—
385C22×C3#379812yes—
386C22×C3#381112yes—
387C22×C3C22×D3211yescharacteristic
388C2×D3—62no—
389C2×D3—62no—
390C2×D3—62no—
391C2×D3—94no—
392C2×D3—94no—
393C2×D3—124no—
394C2×D3—184no—
395C2×D3—184no—
396C2×D3—184no—
397C2×D3—1816no—
398C2×D3—182no—
399C2×D3—182no—
400C2×D3—188no—
401C2×D3—182no—
402C2×D3—188no—
403C2×D3—182no—
404C3×C4—24no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—62no—
408C3×C4—62no—
409C3×C4—182no—
410Dic3—22no—
411Dic3—22no—
412Dic3—62no—
413Dic3—184no—
414Dic3—182no—
415Dic3—182no—
416C32C2×#89411yescharacteristic
417C9—21no—
418C9C2×#88311yescharacteristic, complemented
419C23—92no—
420C23—182no—
421C23—182no—
422C23—182no—
423C23—272no—
424C23—274no—
425C23—542no—
426C23—542no—
427C23C2×D3×D911yescharacteristic
428D4—548no—
429C2×C4—22no—
430C2×C4—32no—
431C2×C4—32no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—92no—
435C2×C4—182no—
436C2×C4—272no—
437C2×C4—548no—
438C2×C4—542no—
439C2×C4C2×D3×D912yes—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—24no—
444C2×C3—362no—
445C2×C3#6345614yescomplemented
446C2×C3C2×#369711yescharacteristic
447C2×C3C2×D4×D911yescharacteristic
448C2×C3C2×#370111yescharacteristic
449C2×C3#6385314yes—
450C2×C3C2×#379611yescharacteristic
451C2×C3C2×#379811yescharacteristic
452C2×C3C2×#381111yescharacteristic
453D3—122no—
454D3—188no—
455D3—184no—
456D3—184no—
457C22—182no—
458C22—182no—
459C22—182no—
460C22—274no—
461C22—364no—
462C22—542no—
463C22—542no—
464C22—548no—
465C22#8863412yes—
466C22#8864012yes—
467C22#8866812yes—
468C22C22×D3×D911yescharacteristic
469C4—22no—
470C4—62no—
471C4—62no—
472C4—182no—
473C4—542no—
474C3—21no—
475C3C2×#6345611yescharacteristic, complemented
476C3C2×#6385311yescharacteristic
477C2—362no—
478C2—544no—
479C2#1172094114yesdirect product factor
480C2C2×#8863411yescharacteristic
481C2C2×#8864011yescharacteristic
482C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6192C2
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C3⋊D9C2semidirect
3. C2×C3×(C2×D9)⋊C4C2semidirect
4. C2×C9×Dic3⋊C4C2semidirect
5. C2×#88445C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720941C2direct (N × H)
16. C2×C4×C3⋊D9C22semidirect
22. C3×(C2×D9)⋊C4C22semidirect
23. C9×Dic3⋊C4C22semidirect
24. #88445C22semidirect
25. (C2×D9)⋊Dic3C22semidirect
26. Dic9⋊Dic3C22semidirect
41. C2×(C2×D9)⋊C4D3semidirect
102. (C2×D9)⋊C4C2×D3semidirect
135. C2×Dic3⋊C4D9semidirect
191. C2×C3×C9#123semidirect
233. Dic3⋊C4C2×D9semidirect
286. C3×C9C2×#123semidirect
349. C2×C9#883semidirect
418. C9C2×#883semidirect
445. C2×C3#63456semidirect
475. C3C2×#63456semidirect
479. C2#11720941direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C22×#61922
C2×C3×(C2×D9)⋊C42
C2×C9×Dic3⋊C42
C2×#884452
C2×(C2×D9)⋊Dic32
C2×Dic9⋊Dic32
#117209412
C2×#638533
C2×#634563