Finite Group Databasebeta

← C2×#11720977 #13150317

Subgroups — C2×#11720977 #13150317

Order 1728.

1410 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6192C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C22×Dic3×D9C211yesmaximal, characteristic, complemented
4C2×#88536C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
7C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
8#11720977C218yesmaximal, direct product factor
9C2×#63880—31nomaximal
10C2×C4×D36—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×#6192C418yescomplemented
19C2×#6192C2214yes—
20C2×(C3⋊D9)⋊C4—24no—
21C2×(C3⋊D9)⋊C4C2214yescomplemented
22C2×Dic3×D9—24no—
23C2×Dic3×D9C2214yescomplemented
24#88536C2214yescomplemented
25C3×(C2×D9)⋊C4C2214yescomplemented
26C4×C9×Dic3C2214yescomplemented
27Dic3⋊Dic9C2214yescomplemented
28C22×#626—31no—
29C22×#624—31no—
30C22×D3×Dic3—31no—
31C2×#3752—31no—
32C2×C3×(C2×D3)⋊C4—31no—
33C2×C3×C4×Dic3D311yescharacteristic
34C2×Dic3⋊Dic3—31no—
35#63880—38no—
36C22×D36—31no—
37C22×C4×D9—31no—
38C22×C4×D9—31no—
39C2×C42×C9—31no—
40C2×(C2×D9)⋊C4—31no—
41C2×(C2×D9)⋊C4—61no—
42C2×(C2×D9)⋊C4D311yescharacteristic, complemented
43C2×C4⋊Dic9—31no—
44C4×D36—38no—
45C23×C3×C9C2311yescharacteristic
46C22×C3⋊D9—22no—
47C22×C3⋊D9—22no—
48C22×C3⋊D9C2×C414yescomplemented
49C22×C3⋊D9C2312yes—
50C22×C3×D9—22no—
51C22×C3×D9—22no—
52C22×C3×D9C2×C414yescomplemented
53C22×C3×D9C2312yes—
54C2×C3×C4×C9—22no—
55C2×C3×C4×C9C2312yes—
56C2×C3⋊Dic9—22no—
57C2×C3⋊Dic9C2312yes—
58C2×C9×Dic3C2×C414yescomplemented
59C2×C9×Dic3D414yescomplemented
60C2×C9×Dic3C2312yes—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9—22no—
63C2×C3×Dic9C2312yes—
64#6192C2×C4116yescomplemented
65(C3⋊D9)⋊C4—28no—
66Dic3×D9—28no—
67C2×#67⋊C4—91no—
68C2×C4×D12—91no—
69C23×C3⋊D3—31no—
70C23×C3×D3—31no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#626—38no—
77C2×#626—34no—
78C2×#624—34no—
79C2×#624—64no—
80C2×D3×Dic3—34no—
81C2×D3×Dic3—64no—
82C23×D9—31no—
83C23×D9—61no—
84C23×D9C2×D311yescharacteristic
85#3752—34no—
86C3×(C2×D3)⋊C4—34no—
87C3×C4×Dic3C2×D314yes—
88Dic3⋊Dic3—34no—
89C22×C4×C9—21no—
90C22×C4×C9—31no—
91C22×C4×C9—31no—
92C22×C4×C9C2×D311yescharacteristic
93C22×Dic9—31no—
94C22×Dic9—61no—
95C22×Dic9C2×D311yescharacteristic
96C2×D36—38no—
97C2×D36—34no—
98C2×C4×D9—34no—
99C2×C4×D9—34no—
100C2×C4×D9—64no—
101C2×C4×D9—64no—
102C42×C9—34no—
103(C2×D9)⋊C4—34no—
104(C2×D9)⋊C4—64no—
105(C2×D9)⋊C4C2×D314yescomplemented
106C4⋊Dic9—34no—
107C22×C3×C9C22×C412yes—
108C22×C3×C9C2×D412yes—
109C22×C3×C9#4112yes—
110C22×C3×C9C2411yescharacteristic
111C2×C3⋊D9—28no—
112C2×C3⋊D9—22no—
113C2×C3⋊D9—22no—
114C2×C3⋊D9C22×C414yescomplemented
115C2×C3×D9—28no—
116C2×C3×D9—22no—
117C2×C3×D9—22no—
118C2×C3×D9C22×C414yescomplemented
119C3×C4×C9—22no—
120C3⋊Dic9—22no—
121C9×Dic3C22×C414yescomplemented
122C9×Dic3C2×D414yescomplemented
123C3×Dic9—22no—
124C22×#67—91no—
125C23×Dic3—91no—
126C22×D12—91no—
127C22×C4×D3—91no—
128C22×C4×D3—91no—
129C22×C4×D3—91no—
130C2×C3×C22⋊C4—91no—
131C2×C3×C42—31no—
132C2×(C2×D3)⋊C4—31no—
133C2×(C2×D3)⋊C4—91no—
134C2×(C2×D3)⋊C4—91no—
135C2×(C2×D3)⋊C4—181no—
136C2×C4⋊Dic3—91no—
137C2×Dic3⋊C4—91no—
138C2×C4×Dic3D911yescharacteristic, complemented
139#67⋊C4—98no—
140C4×D12—98no—
141C23×C32C22×D311yescharacteristic
142C22×C3⋊D3—34no—
143C22×C3⋊D3—32no—
144C22×C3⋊D3—62no—
145C22×C3⋊D3—62no—
146C22×C3×D3—34no—
147C22×C3×D3—32no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C2×C32×C4—22no—
151C2×C32×C4C22×D312yes—
152C2×C3⋊Dic3—32no—
153C2×C3⋊Dic3—62no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3C4×D314yes—
157C2×C3×Dic3D1214yes—
158C2×C3×Dic3C22×D312yes—
159C2×C3×Dic3C22×D312yes—
160#626—316no—
161#624—68no—
162D3×Dic3—68no—
163C23×C9—21no—
164C23×C9C22×D311yescharacteristic
165C22×D9—22no—
166C22×D9—22no—
167C22×D9—34no—
168C22×D9—32no—
169C22×D9—62no—
170C22×D9—64no—
171C22×D9—62no—
172C22×D9—62no—
173C22×D9—122no—
174C22×D9—122no—
175C22×D9C4×D314yescomplemented
176C22×D9C22×D312yes—
177C2×C4×C9—22no—
178C2×C4×C9—22no—
179C2×C4×C9—34no—
180C2×C4×C9—34no—
181C2×C4×C9—32no—
182C2×C4×C9—32no—
183C2×C4×C9—42no—
184C2×C4×C9C22×D312yes—
185C2×Dic9—22no—
186C2×Dic9—32no—
187C2×Dic9—62no—
188C2×Dic9—62no—
189C2×Dic9—122no—
190C2×Dic9C22×D312yes—
191D36—316no—
192C4×D9—68no—
193C4×D9—68no—
194C2×C4×D4—271noSylow
195C2×C3×C9C4×D414yescomplemented
196C2×C3×C9C23×C411yescharacteristic
197C2×C3×C9C22×D411yescharacteristic
198C2×C3×C9C2×#4111yescharacteristic
199C3⋊D9—24no—
200C3×D9—24no—
201C24×C3—91no—
202C23×D3—31no—
203C23×D3—91no—
204C23×D3—91no—
205C23×D3—181no—
206C22×C3×C4—21no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—91no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4D3211yescharacteristic
212C2×#67—98no—
213C2×#67—94no—
214C22×Dic3—31no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—94no—
218C22×Dic3—181no—
219C22×Dic3—184no—
220C22×Dic3C2×D911yescharacteristic
221C22×Dic3C2×D911yescharacteristic
222C2×D12—94no—
223C2×D12—98no—
224C2×C4×D3—94no—
225C2×C4×D3—94no—
226C2×C4×D3—94no—
227C2×C4×D3—184no—
228C2×C4×D3—184no—
229C2×C4×D3—184no—
230C3×C22⋊C4—94no—
231C3×C42—34no—
232(C2×D3)⋊C4—34no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—94no—
235(C2×D3)⋊C4—184no—
236C4⋊Dic3—94no—
237Dic3⋊C4—94no—
238C4×Dic3C2×D914yescomplemented
239C22×C32C2×C4×D312yes—
240C22×C32C2×D1212yes—
241C22×C32#23512yes—
242C22×C32C23×D311yescharacteristic
243C2×C3⋊D3—34no—
244C2×C3⋊D3—68no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3—62no—
247C2×C3×D3—34no—
248C2×C3×D3—68no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C32×C4—22no—
252C3⋊Dic3—62no—
253C3×Dic3—62no—
254C3×Dic3C2×C4×D314yes—
255C3×Dic3C2×D1214yes—
256C22×C9—21no—
257C22×C9—22no—
258C22×C9—22no—
259C22×C9—22no—
260C22×C9C2×C4×D312yes—
261C22×C9D3×D412yes—
262C22×C9#23712yes—
263C22×C9C23×D311yescharacteristic
264C2×D9—28no—
265C2×D9—22no—
266C2×D9—22no—
267C2×D9—34no—
268C2×D9—64no—
269C2×D9—68no—
270C2×D9—62no—
271C2×D9—62no—
272C2×D9—122no—
273C2×D9—122no—
274C2×D9—128no—
275C2×D9C2×C4×D314yescomplemented
276C4×C9—22no—
277C4×C9—34no—
278C4×C9—34no—
279C4×C9—42no—
280Dic9—22no—
281Dic9—62no—
282Dic9—122no—
283C22×D4—271no—
284C23×C4—271no—
285C23×C4—271no—
286C4×D4—278no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—271no—
290C2×C42—31no—
291C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
292C23×C3—21no—
293C23×C3—94no—
294C23×C3—92no—
295C23×C3—182no—
296C23×C3—182no—
297C23×C3C22×D911yescharacteristic
298C23×C3C2×D3211yescharacteristic
299C22×D3—32no—
300C22×D3—34no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—92no—
304C22×D3—94no—
305C22×D3—92no—
306C22×D3—94no—
307C22×D3—182no—
308C22×D3—184no—
309C22×D3—182no—
310C22×D3—182no—
311C22×D3—182no—
312C22×D3—182no—
313C22×D3—362no—
314C22×D3—362no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—32no—
319C2×C3×C4—32no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4—42no—
323C2×C3×C4—92no—
324C2×C3×C4—182no—
325C2×C3×C4C22×D912yes—
326C2×C3×C4C2×D3212yes—
327#67—916no—
328C2×Dic3—32no—
329C2×Dic3—62no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—182no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×Dic3—188no—
336C2×Dic3—362no—
337C2×Dic3C4×D914yescomplemented
338C2×Dic3D3614yescomplemented
339C2×Dic3C22×D912yes—
340C2×Dic3C22×D912yes—
341D12—916no—
342C4×D3—188no—
343C4×D3—188no—
344C4×D3—188no—
345C2×C32C4×D1214yes—
346C2×C32C22×C4×D311yescharacteristic
347C2×C32C22×D1211yescharacteristic
348C2×C32C2×#23511yescharacteristic
349C3⋊D3—64no—
350C3×D3—64no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—24no—
355C2×C9#67⋊C414yescomplemented
356C2×C9C22×C4×D311yescharacteristic
357C2×C9C2×D3×D411yescharacteristic
358C2×C9C2×#23711yescharacteristic
359D9—24no—
360D9—64no—
361D9—124no—
362C24—91no—
363C24—271no—
364C2×D4—274no—
365C2×D4—278no—
366C22×C4—31no—
367C22×C4—31no—
368C22×C4—91no—
369C22×C4—274no—
370C22×C4—274no—
371C22×C4—271no—
372C22×C4—544no—
373C22×C4—544no—
374C22×C4D3×D911yescharacteristic
375C4⋊C4—274no—
376C22⋊C4—94no—
377C22⋊C4—274no—
378C42—34no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—94no—
384C22×C3—188no—
385C22×C3—182no—
386C22×C3—182no—
387C22×C3C2×C4×D912yes—
388C22×C3C2×D3612yes—
389C22×C3#369712yes—
390C22×C3C23×D911yescharacteristic
391C22×C3C4×D3212yes—
392C22×C3D3×D1212yes—
393C22×C3#380412yes—
394C22×C3C22×D3211yescharacteristic
395C2×D3—34no—
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—68no—
399C2×D3—94no—
400C2×D3—94no—
401C2×D3—184no—
402C2×D3—182no—
403C2×D3—182no—
404C2×D3—188no—
405C2×D3—188no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—368no—
409C2×D3—362no—
410C2×D3—362no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—34no—
414C3×C4—34no—
415C3×C4—42no—
416C3×C4—182no—
417Dic3—62no—
418Dic3—182no—
419Dic3—182no—
420Dic3—362no—
421Dic3C2×C4×D914yescomplemented
422Dic3C2×D3614yescomplemented
423C32C2×C4×D1211yescharacteristic
424C9—21no—
425C9C2×#67⋊C411yescharacteristic, complemented
426C23—92no—
427C23—94no—
428C23—182no—
429C23—182no—
430C23—272no—
431C23—274no—
432C23—542no—
433C23—542no—
434C23C2×D3×D911yescharacteristic
435D4—2716no—
436C2×C4—22no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—34no—
440C2×C4—34no—
441C2×C4—92no—
442C2×C4—182no—
443C2×C4—272no—
444C2×C4—548no—
445C2×C4—548no—
446C2×C4—542no—
447C2×C4C2×D3×D912yes—
448C2×C3—24no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—184no—
453C2×C3C4×D3614yescomplemented
454C2×C3C22×C4×D911yescharacteristic
455C2×C3C22×D3611yescharacteristic
456C2×C3C2×#369711yescharacteristic
457C2×C3#6388014yes—
458C2×C3C2×C4×D3211yescharacteristic
459C2×C3C2×D3×D1211yescharacteristic
460C2×C3C2×#380411yescharacteristic
461D3—64no—
462D3—184no—
463D3—184no—
464D3—364no—
465C22—94no—
466C22—188no—
467C22—182no—
468C22—182no—
469C22—274no—
470C22—548no—
471C22—542no—
472C22—542no—
473C22C4×D3×D912yes—
474C22D3×D3612yes—
475C22#8865812yes—
476C22C22×D3×D911yescharacteristic
477C4—22no—
478C4—34no—
479C4—34no—
480C4—182no—
481C4—542no—
482C3—21no—
483C3C2×C4×D3611yescharacteristic, complemented
484C3C2×#6388011yescharacteristic
485C2—184no—
486C2—544no—
487C2#1172097714yesdirect product factor
488C2C2×C4×D3×D911yescharacteristic
489C2C2×D3×D3611yescharacteristic
490C2C2×#8865811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6192C2
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C4×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊D9)⋊C4C2semidirect
3. C22×Dic3×D9C2semidirect
4. C2×#88536C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C2×C4×C9×Dic3C2semidirect
7. C2×Dic3⋊Dic9C2semidirect
8. #11720977C2direct (N × H)
18. C2×#6192C4semidirect
21. C2×(C3⋊D9)⋊C4C22semidirect
23. C2×Dic3×D9C22semidirect
24. #88536C22semidirect
25. C3×(C2×D9)⋊C4C22semidirect
26. C4×C9×Dic3C22semidirect
27. Dic3⋊Dic9C22semidirect
42. C2×(C2×D9)⋊C4D3semidirect
48. C22×C3⋊D9C2×C4semidirect
52. C22×C3×D9C2×C4semidirect
58. C2×C9×Dic3C2×C4semidirect
59. C2×C9×Dic3D4semidirect
64. #6192C2×C4semidirect
105. (C2×D9)⋊C4C2×D3semidirect
114. C2×C3⋊D9C22×C4semidirect
118. C2×C3×D9C22×C4semidirect
121. C9×Dic3C22×C4semidirect
122. C9×Dic3C2×D4semidirect
138. C2×C4×Dic3D9semidirect
175. C22×D9C4×D3semidirect
195. C2×C3×C9C4×D4semidirect
238. C4×Dic3C2×D9semidirect
275. C2×D9C2×C4×D3semidirect
291. C3×C9C2×C4×D4semidirect
337. C2×Dic3C4×D9semidirect
338. C2×Dic3D36semidirect
355. C2×C9#67⋊C4semidirect
421. Dic3C2×C4×D9semidirect
422. Dic3C2×D36semidirect
425. C9C2×#67⋊C4semidirect
453. C2×C3C4×D36semidirect
483. C3C2×C4×D36semidirect
487. C2#11720977direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61922
C22×(C3⋊D9)⋊C42
C22×Dic3×D92
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×C4×C9×Dic32
C2×Dic3⋊Dic92
#117209772
C2×#638803
C2×C4×D363