Finite Group Databasebeta

← C2×#11723420 #13177190

Subgroups — C2×#11723420 #13177190

Order 1728.

2996 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⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C22×#6291C211yesmaximal, characteristic
3C2×C3×#3750C211yesmaximal, characteristic, complemented
4C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C2×#88804C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8#11723420C218yesmaximal, direct product factor
9C2×#64092—31nomaximal
10C2×#63853—34nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C23×C32×D3C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—24no—
16C2×C4×C3⋊(C3⋊D3)C2214yescomplemented
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×#6291—24no—
21C2×#6291C2214yes—
22C3×#3750C2214yescomplemented
23C32×(C2×D3)⋊C4C2214yescomplemented
24Dic3⋊(C3⋊Dic3)C2214yescomplemented
25#88804C2214yescomplemented
26(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
27C22×#638—31no—
28C22×C4×C3⋊D3—31no—
29C22×C4×C3⋊D3—34no—
30C22×C4×C3⋊D3—64no—
31C22×#626—34no—
32C2×C32×C22⋊C4—31no—
33C2×#3757—31no—
34C2×#3752—31no—
35C2×C4⋊(C3⋊Dic3)—31no—
36C2×#3750D311yescharacteristic, complemented
37#64092—38no—
38C2×C3×(C2×D3)⋊C4D314yescomplemented
39C2×C3×Dic3⋊C4—34no—
40C2×Dic3⋊Dic3—34no—
41C2×#3722—34no—
42C2×(C2×D3)⋊Dic3—34no—
43#63853—332no—
44C23×C33C2311yescharacteristic
45C22×C3⋊(C3⋊D3)—22no—
46C22×C3⋊(C3⋊D3)—22no—
47C22×C3⋊(C3⋊D3)D414yes—
48C22×C3⋊(C3⋊D3)C2312yes—
49C22×C32×D3—22no—
50C22×C32×D3—22no—
51C22×C32×D3—22no—
52C22×C32×D3C2312yes—
53C2×C33×C4—22no—
54C2×C33×C4C2312yes—
55C2×C3⋊(C3⋊Dic3)—22no—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C4×C3⋊(C3⋊D3)—28no—
58C2×C3×C3⋊Dic3—22no—
59C2×C3×C3⋊Dic3—22no—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C32×Dic3—22no—
63C2×C32×Dic3C2312yes—
64#6291—28no—
65C2×#894—91no—
66C2×#883—94no—
67C24×C32—31no—
68C23×C3⋊D3—31no—
69C23×C3⋊D3—34no—
70C23×C3⋊D3—64no—
71C23×C3×D3C2×D314yes—
72C22×C32×C4—24no—
73C22×C32×C4—31no—
74C22×C32×C4C2×D314yes—
75C22×C32×C4C2×D311yescharacteristic
76C2×#638—34no—
77C2×#638—64no—
78C22×C3⋊Dic3—31no—
79C22×C3⋊Dic3—34no—
80C22×C3⋊Dic3—64no—
81C22×C3⋊Dic3C2×D311yescharacteristic
82C22×C3⋊Dic3C2×D311yescharacteristic
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—316no—
85C2×C4×C3⋊D3—64no—
86C2×C4×C3⋊D3—616no—
87C2×C4×C3⋊D3—616no—
88C2×C4×C3⋊D3—632no—
89C22×C3×Dic3—34no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3C2×D314yes—
92C2×#626—316no—
93C2×#626—616no—
94C32×C22⋊C4—34no—
95#3757—34no—
96#3752—34no—
97C4⋊(C3⋊Dic3)—34no—
98#3750C2×D314yescomplemented
99C3×(C2×D3)⋊C4C2×D3116yescomplemented
100C3×Dic3⋊C4—316no—
101Dic3⋊Dic3—316no—
102#3722—316no—
103(C2×D3)⋊Dic3—316no—
104C22×C33C2×D412yes—
105C22×C33#4112yes—
106C22×C33#4112yes—
107C22×C33C2411yescharacteristic
108C2×C3⋊(C3⋊D3)—28no—
109C2×C3⋊(C3⋊D3)—22no—
110C2×C3⋊(C3⋊D3)—22no—
111C2×C3⋊(C3⋊D3)C2×D414yes—
112C2×C32×D3—22no—
113C2×C32×D3—22no—
114C2×C32×D3—22no—
115C2×C32×D3—44no—
116C33×C4—22no—
117C3⋊(C3⋊Dic3)—22no—
118C3×C3⋊Dic3—22no—
119C3×C3⋊Dic3—22no—
120C32×Dic3—22no—
121C22×#67—94no—
122C22×D12—91no—
123C22×C4×D3—94no—
124C22×C4×D3—91no—
125C22×C4×D3—184no—
126C2×C3×C4⋊C4—91no—
127C2×C3×C22⋊C4—34no—
128C2×#217—94no—
129C2×(C2×D3)⋊C4—94no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×(C2×D3)⋊C4C3⋊D311yescharacteristic, complemented
133C2×C4⋊Dic3—94no—
134C2×Dic3⋊C4—34no—
135C2×Dic3⋊C4—91no—
136#894—98no—
137#883—932no—
138C23×C32—24no—
139C23×C32—32no—
140C23×C32—62no—
141C23×C32—62no—
142C23×C32—62no—
143C23×C32C22×D314yes—
144C23×C32C22×D311yescharacteristic
145C22×C3⋊D3—34no—
146C22×C3⋊D3—32no—
147C22×C3⋊D3—316no—
148C22×C3⋊D3—38no—
149C22×C3⋊D3—616no—
150C22×C3⋊D3—68no—
151C22×C3⋊D3—616no—
152C22×C3⋊D3—616no—
153C22×C3⋊D3—68no—
154C22×C3⋊D3—62no—
155C22×C3⋊D3—68no—
156C22×C3⋊D3—62no—
157C22×C3×D3—28no—
158C22×C3×D3—28no—
159C22×C3×D3—28no—
160C22×C3×D3C22×D318yes—
161C2×C32×C4—216no—
162C2×C32×C4—28no—
163C2×C32×C4—22no—
164C2×C32×C4—28no—
165C2×C32×C4—32no—
166C2×C32×C4—62no—
167C2×C32×C4C22×D318yes—
168C2×C32×C4C22×D312yes—
169#638—68no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—22no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—38no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—616no—
176C2×C3⋊Dic3—68no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3C22×D312yes—
179C2×C3⋊Dic3C22×D312yes—
180C4×C3⋊D3—68no—
181C4×C3⋊D3—632no—
182C4×C3⋊D3—664no—
183C2×C3×Dic3—28no—
184C2×C3×Dic3—38no—
185C2×C3×Dic3—38no—
186C2×C3×Dic3—68no—
187C2×C3×Dic3—68no—
188C2×C3×Dic3C22×D318yes—
189#626—632no—
190C2×#123—271noSylow
191C2×C33#12314yescomplemented
192C2×C33C22×D411yescharacteristic
193C2×C33C2×#4111yescharacteristic
194C2×C33C2×#4111yescharacteristic
195C3⋊(C3⋊D3)—24no—
196C32×D3—42no—
197C24×C3—34no—
198C23×D3—94no—
199C23×D3—91no—
200C23×D3—184no—
201C23×D3C2×C3⋊D311yescharacteristic
202C22×C3×C4—24no—
203C22×C3×C4—34no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4D3214yes—
207C22×C3×C4C2×C3⋊D311yescharacteristic
208C2×#67—916no—
209C2×#67—1816no—
210C22×Dic3—34no—
211C22×Dic3—34no—
212C22×Dic3—94no—
213C22×Dic3—91no—
214C22×Dic3—184no—
215C22×Dic3C2×C3⋊D311yescharacteristic
216C2×D12—94no—
217C2×D12—184no—
218C2×C4×D3—916no—
219C2×C4×D3—94no—
220C2×C4×D3—1816no—
221C2×C4×D3—1816no—
222C2×C4×D3—184no—
223C2×C4×D3—1832no—
224C3×C4⋊C4—94no—
225C3×C22⋊C4—316no—
226#217—916no—
227(C2×D3)⋊C4—916no—
228(C2×D3)⋊C4—94no—
229(C2×D3)⋊C4—94no—
230(C2×D3)⋊C4C2×C3⋊D314yescomplemented
231C4⋊Dic3—916no—
232Dic3⋊C4—316no—
233Dic3⋊C4—94no—
234C22×C32—24no—
235C22×C32—28no—
236C22×C32—28no—
237C22×C32—28no—
238C22×C32—62no—
239C22×C32—62no—
240C22×C32—62no—
241C22×C32—124no—
242C22×C32#23518yes—
243C22×C32#23512yes—
244C22×C32D3×D418yes—
245C22×C32D3×D412yes—
246C22×C32#23718yes—
247C22×C32#23912yes—
248C22×C32C23×D314yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—34no—
251C2×C3⋊D3—316no—
252C2×C3⋊D3—616no—
253C2×C3⋊D3—616no—
254C2×C3⋊D3—616no—
255C2×C3⋊D3—664no—
256C2×C3⋊D3—632no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—68no—
261C2×C3⋊D3—62no—
262C2×C3×D3—28no—
263C2×C3×D3—28no—
264C2×C3×D3—28no—
265C2×C3×D3—416no—
266C32×C4—216no—
267C32×C4—28no—
268C32×C4—22no—
269C32×C4—62no—
270C3⋊Dic3—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3—616no—
273C3⋊Dic3—68no—
274C3⋊Dic3—62no—
275C3×Dic3—28no—
276C3×Dic3—68no—
277C3×Dic3—68no—
278C22×D4—271no—
279C23×C4—271no—
280#123—278no—
281C2×C4⋊C4—91no—
282C2×C4⋊C4—271no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—271no—
285C2×C22⋊C4—271no—
286C33C2×#12311yescharacteristic, complemented, Sylow
287C23×C3—24no—
288C23×C3—38no—
289C23×C3—68no—
290C23×C3—68no—
291C23×C3—68no—
292C23×C3C2×D3214yes—
293C23×C3C22×C3⋊D311yescharacteristic
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—98no—
298C22×D3—92no—
299C22×D3—916no—
300C22×D3—94no—
301C22×D3—188no—
302C22×D3—1816no—
303C22×D3—1816no—
304C22×D3—1816no—
305C22×D3—188no—
306C22×D3—188no—
307C22×D3—182no—
308C22×D3—182no—
309C22×D3C22×C3⋊D312yes—
310C2×C3×C4—216no—
311C2×C3×C4—28no—
312C2×C3×C4—22no—
313C2×C3×C4—28no—
314C2×C3×C4—38no—
315C2×C3×C4—68no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
318C2×C3×C4—182no—
319C2×C3×C4—182no—
320C2×C3×C4C2×D3218yes—
321C2×C3×C4C22×C3⋊D312yes—
322#67—1832no—
323C2×Dic3—22no—
324C2×Dic3—38no—
325C2×Dic3—38no—
326C2×Dic3—68no—
327C2×Dic3—68no—
328C2×Dic3—98no—
329C2×Dic3—92no—
330C2×Dic3—188no—
331C2×Dic3—1816no—
332C2×Dic3—188no—
333C2×Dic3—182no—
334C2×Dic3C22×C3⋊D312yes—
335D12—188no—
336C4×D3—1832no—
337C4×D3—188no—
338C4×D3—1864no—
339C2×C32—24no—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32—216no—
343C2×C32—122no—
344C2×C32#883116yescomplemented
345C2×C32#89414yescomplemented
346C2×C32C2×#23514yes—
347C2×C32C2×#23511yescharacteristic
348C2×C32C2×D3×D414yes—
349C2×C32C2×D3×D411yescharacteristic
350C2×C32C2×#23714yes—
351C2×C32C2×#23911yescharacteristic
352C3⋊D3—632no—
353C3⋊D3—616no—
354C3⋊D3—64no—
355C3×D3—48no—
356C24—31no—
357C24—271no—
358C2×D4—274no—
359C2×D4—544no—
360C22×C4—31no—
361C22×C4—91no—
362C22×C4—91no—
363C22×C4—271no—
364C22×C4—274no—
365C22×C4—544no—
366C22×C4D3×C3⋊D311yescharacteristic
367C4⋊C4—94no—
368C4⋊C4—274no—
369C22⋊C4—34no—
370C22⋊C4—274no—
371C22⋊C4—274no—
372C22×C3—24no—
373C22×C3—28no—
374C22×C3—28no—
375C22×C3—28no—
376C22×C3—68no—
377C22×C3—68no—
378C22×C3—68no—
379C22×C3—1216no—
380C22×C3#379618yes—
381C22×C3#379818yes—
382C22×C3#381118yes—
383C22×C3#382812yes—
384C22×C3D4×C3⋊D312yes—
385C22×C3#383012yes—
386C22×C3C22×D3214yes—
387C22×C3C23×C3⋊D311yescharacteristic
388C2×D3—22no—
389C2×D3—22no—
390C2×D3—22no—
391C2×D3—44no—
392C2×D3—916no—
393C2×D3—94no—
394C2×D3—1816no—
395C2×D3—1816no—
396C2×D3—1816no—
397C2×D3—1864no—
398C2×D3—188no—
399C2×D3—188no—
400C2×D3—188no—
401C2×D3—182no—
402C2×D3—182no—
403C2×D3—1832no—
404C3×C4—216no—
405C3×C4—28no—
406C3×C4—22no—
407C3×C4—68no—
408C3×C4—182no—
409C3×C4—182no—
410Dic3—22no—
411Dic3—68no—
412Dic3—68no—
413Dic3—1816no—
414Dic3—188no—
415Dic3—182no—
416C32—24no—
417C32C2×#88314yescomplemented
418C32C2×#89411yescharacteristic, complemented
419C23—32no—
420C23—62no—
421C23—62no—
422C23—62no—
423C23—272no—
424C23—274no—
425C23—542no—
426C23—542no—
427C23C2×D3×C3⋊D311yescharacteristic
428D4—548no—
429C2×C4—22no—
430C2×C4—32no—
431C2×C4—62no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—182no—
435C2×C4—182no—
436C2×C4—272no—
437C2×C4—548no—
438C2×C4—542no—
439C2×C4C2×D3×C3⋊D312yes—
440C2×C3—24no—
441C2×C3—24no—
442C2×C3—24no—
443C2×C3—216no—
444C2×C3—128no—
445C2×C3#63853116yescomplemented
446C2×C3#6409214yescomplemented
447C2×C3C2×#379614yes—
448C2×C3C2×#379814yes—
449C2×C3C2×#381114yes—
450C2×C3C2×#382811yescharacteristic
451C2×C3C2×D4×C3⋊D311yescharacteristic
452C2×C3C2×#383011yescharacteristic
453D3—42no—
454D3—1832no—
455D3—1816no—
456D3—184no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—124no—
461C22—274no—
462C22—542no—
463C22—542no—
464C22—548no—
465C22#8901512yes—
466C22#8902012yes—
467C22#8903912yes—
468C22C22×D3×C3⋊D311yescharacteristic
469C4—22no—
470C4—62no—
471C4—182no—
472C4—182no—
473C4—542no—
474C3—24no—
475C3C2×#6385314yescomplemented
476C3C2×#6409211yescharacteristic, complemented
477C2—122no—
478C2—544no—
479C2#1172342014yesdirect product factor
480C2C2×#8901511yescharacteristic
481C2C2×#8902011yescharacteristic
482C2C2×#8903911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×#6291C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C3⋊(C3⋊D3)C2semidirect
3. C2×C3×#3750C2semidirect
4. C2×C32×(C2×D3)⋊C4C2semidirect
5. C2×Dic3⋊(C3⋊Dic3)C2semidirect
6. C2×#88804C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. #11723420C2direct (N × H)
16. C2×C4×C3⋊(C3⋊D3)C22semidirect
22. C3×#3750C22semidirect
23. C32×(C2×D3)⋊C4C22semidirect
24. Dic3⋊(C3⋊Dic3)C22semidirect
25. #88804C22semidirect
26. (C2×D3)⋊(C3⋊Dic3)C22semidirect
36. C2×#3750D3semidirect
38. C2×C3×(C2×D3)⋊C4D3semidirect
98. #3750C2×D3semidirect
99. C3×(C2×D3)⋊C4C2×D3semidirect
132. C2×(C2×D3)⋊C4C3⋊D3semidirect
191. C2×C33#123semidirect
230. (C2×D3)⋊C4C2×C3⋊D3semidirect
286. C33C2×#123semidirect
344. C2×C32#883semidirect
345. C2×C32#894semidirect
417. C32C2×#883semidirect
418. C32C2×#894semidirect
445. C2×C3#63853semidirect
446. C2×C3#64092semidirect
475. C3C2×#63853semidirect
476. C3C2×#64092semidirect
479. C2#11723420direct (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⋊(C3⋊D3)2
C22×#62912
C2×C3×#37502
C2×C32×(C2×D3)⋊C42
C2×Dic3⋊(C3⋊Dic3)2
C2×#888042
C2×(C2×D3)⋊(C3⋊Dic3)2
#117234202
C2×#640923
C2×#638533