Finite Group Databasebeta

← C2×#11723530 #13177192

Subgroups — C2×#11723530 #13177192

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6293C211yesmaximal, characteristic, complemented
2C22×#6290C211yesmaximal, characteristic
3C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C2×C3×#3752C211yesmaximal, characteristic, complemented
5C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C2×#88803C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8#11723530C218yesmaximal, direct product factor
9C2×#64094—31nomaximal
10C2×#63915—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C23×C32×D3C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C2×#6293—24no—
19C2×#6293C2214yescomplemented
20C2×#6290—24no—
21C2×#6290C2214yes—
22C4×C3⋊(C3⋊Dic3)C2214yescomplemented
23C3×#3752C2214yescomplemented
24C32×(C2×D3)⋊C4C2214yescomplemented
25#88803C2214yescomplemented
26(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
27C22×#638—31no—
28C22×#634—31no—
29C22×#627—34no—
30C22×#625—34no—
31C2×C32×C22⋊C4—31no—
32C2×#3757—31no—
33C2×#3752—31no—
34C2×#3752D311yescharacteristic, complemented
35C2×C4×C3⋊Dic3—31no—
36C2×C4×C3⋊Dic3—34no—
37C2×C4×C3⋊Dic3—64no—
38#64094—38no—
39C2×C3×(C2×D3)⋊C4—34no—
40C2×C3×(C2×D3)⋊C4D314yescomplemented
41C2×(C2×D3)⋊Dic3—34no—
42C2×(C2×D3)⋊Dic3—34no—
43#63915—332no—
44C23×C33C2311yescharacteristic
45C22×C3×C3⋊D3—22no—
46C22×C3×C3⋊D3—22no—
47C22×C3×C3⋊D3—22no—
48C22×C3×C3⋊D3C2312yes—
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)D414yes—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3×C3⋊Dic3—22no—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C32×Dic3—22no—
62C2×C32×Dic3C2312yes—
63#6293—28no—
64#6290—28no—
65C2×#885—94no—
66C2×#885—91no—
67C24×C32—31no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—34no—
70C23×C3×D3C2×D314yes—
71C22×C32×C4—24no—
72C22×C32×C4—31no—
73C22×C32×C4C2×D314yes—
74C22×C32×C4C2×D311yescharacteristic
75C2×#638—34no—
76C2×#638—64no—
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—34no—
79C22×C3⋊Dic3—31no—
80C22×C3⋊Dic3—34no—
81C22×C3⋊Dic3—64no—
82C22×C3⋊Dic3—64no—
83C22×C3⋊Dic3C2×D311yescharacteristic
84C2×#634—34no—
85C2×#634—64no—
86C22×C3×Dic3—34no—
87C22×C3×Dic3C2×D314yes—
88C2×#627—316no—
89C2×#627—616no—
90C2×#625—316no—
91C2×#625—616no—
92C32×C22⋊C4—34no—
93#3757—34no—
94#3752—34no—
95#3752C2×D314yescomplemented
96C4×C3⋊Dic3—34no—
97C4×C3⋊Dic3—316no—
98C4×C3⋊Dic3—616no—
99C3×(C2×D3)⋊C4—316no—
100C3×(C2×D3)⋊C4C2×D3116yescomplemented
101(C2×D3)⋊Dic3—316no—
102(C2×D3)⋊Dic3—316no—
103C22×C33C2×D412yes—
104C22×C33#4112yes—
105C22×C33#4112yes—
106C22×C33C2411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—44no—
111C2×C32×D3—22no—
112C2×C32×D3—22no—
113C2×C32×D3—22no—
114C2×C32×D3—44no—
115C33×C4—22no—
116C3⋊(C3⋊Dic3)—22no—
117C3⋊(C3⋊Dic3)C2×D414yes—
118C3×C3⋊Dic3—22no—
119C32×Dic3—22no—
120C22×#67—94no—
121C22×#67—91no—
122C22×Dic6—94no—
123C22×Dic6—91no—
124C2×C3×C22⋊C4—34no—
125C2×C3×C22⋊C4—91no—
126C2×#217—94no—
127C2×#217—91no—
128C2×(C2×D3)⋊C4—34no—
129C2×(C2×D3)⋊C4—94no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4C3⋊D311yescharacteristic, complemented
132C2×C4×Dic3—94no—
133C2×C4×Dic3—91no—
134C2×C4×Dic3—184no—
135#885—932no—
136#885—98no—
137C23×C32—24no—
138C23×C32—32no—
139C23×C32—62no—
140C23×C32—62no—
141C23×C32—62no—
142C23×C32C22×D314yes—
143C23×C32C22×D311yescharacteristic
144C22×C3⋊D3—22no—
145C22×C3⋊D3—22no—
146C22×C3⋊D3—22no—
147C22×C3⋊D3C22×D312yes—
148C22×C3×D3—28no—
149C22×C3×D3—28no—
150C22×C3×D3—28no—
151C22×C3×D3—38no—
152C22×C3×D3—68no—
153C22×C3×D3—68no—
154C22×C3×D3—68no—
155C22×C3×D3C22×D318yes—
156C2×C32×C4—216no—
157C2×C32×C4—28no—
158C2×C32×C4—22no—
159C2×C32×C4—28no—
160C2×C32×C4—32no—
161C2×C32×C4—62no—
162C2×C32×C4C22×D318yes—
163C2×C32×C4C22×D312yes—
164#638—68no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—316no—
170C2×C3⋊Dic3—38no—
171C2×C3⋊Dic3—38no—
172C2×C3⋊Dic3—68no—
173C2×C3⋊Dic3—68no—
174C2×C3⋊Dic3—616no—
175C2×C3⋊Dic3—616no—
176C2×C3⋊Dic3—68no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3C22×D312yes—
179#634—68no—
180C2×C3×Dic3—28no—
181C2×C3×Dic3—38no—
182C2×C3×Dic3—68no—
183C2×C3×Dic3C22×D318yes—
184#627—632no—
185#625—632no—
186C2×#124—271noSylow
187C2×C33#12414yescomplemented
188C2×C33C22×D411yescharacteristic
189C2×C33C2×#4111yescharacteristic
190C2×C33C2×#4111yescharacteristic
191C3×C3⋊D3—42no—
192C32×D3—42no—
193C24×C3—34no—
194C24×C3—91no—
195C23×D3—34no—
196C23×D3C2×C3⋊D311yescharacteristic
197C22×C3×C4—24no—
198C22×C3×C4—34no—
199C22×C3×C4—91no—
200C22×C3×C4D3214yes—
201C22×C3×C4C2×C3⋊D311yescharacteristic
202C2×#67—916no—
203C2×#67—94no—
204C2×#67—184no—
205C2×#67—1816no—
206C22×Dic3—34no—
207C22×Dic3—94no—
208C22×Dic3—91no—
209C22×Dic3—94no—
210C22×Dic3—91no—
211C22×Dic3—184no—
212C22×Dic3—184no—
213C22×Dic3C2×C3⋊D311yescharacteristic
214C2×Dic6—916no—
215C2×Dic6—94no—
216C2×Dic6—184no—
217C2×Dic6—1816no—
218C3×C22⋊C4—316no—
219C3×C22⋊C4—94no—
220#217—916no—
221#217—94no—
222(C2×D3)⋊C4—316no—
223(C2×D3)⋊C4—916no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4C2×C3⋊D314yescomplemented
226C4×Dic3—916no—
227C4×Dic3—94no—
228C4×Dic3—1816no—
229C22×C32—24no—
230C22×C32—28no—
231C22×C32—28no—
232C22×C32—28no—
233C22×C32—62no—
234C22×C32—62no—
235C22×C32—62no—
236C22×C32—124no—
237C22×C32#23518yes—
238C22×C32#23512yes—
239C22×C32D3×D418yes—
240C22×C32D3×D412yes—
241C22×C32#23718yes—
242C22×C32#23712yes—
243C22×C32C23×D314yes—
244C22×C32C23×D311yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—44no—
249C2×C3×D3—28no—
250C2×C3×D3—28no—
251C2×C3×D3—28no—
252C2×C3×D3—416no—
253C2×C3×D3—68no—
254C2×C3×D3—68no—
255C2×C3×D3—68no—
256C2×C3×D3—1216no—
257C32×C4—216no—
258C32×C4—28no—
259C32×C4—22no—
260C32×C4—62no—
261C3⋊Dic3—22no—
262C3⋊Dic3—34no—
263C3⋊Dic3—316no—
264C3⋊Dic3—616no—
265C3⋊Dic3—616no—
266C3⋊Dic3—68no—
267C3⋊Dic3—62no—
268C3×Dic3—28no—
269C3×Dic3—68no—
270C22×Q8—271no—
271C22×D4—271no—
272#124—278no—
273C2×C22⋊C4—31no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—271no—
276C2×C22⋊C4—271no—
277C2×C42—271no—
278C33C2×#12411yescharacteristic, complemented, Sylow
279C23×C3—24no—
280C23×C3—38no—
281C23×C3—68no—
282C23×C3—68no—
283C23×C3—68no—
284C23×C3—92no—
285C23×C3—182no—
286C23×C3—182no—
287C23×C3—182no—
288C23×C3C2×D3214yes—
289C23×C3C22×C3⋊D311yescharacteristic
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—38no—
294C22×D3—68no—
295C22×D3—68no—
296C22×D3—68no—
297C22×D3C22×C3⋊D312yes—
298C2×C3×C4—216no—
299C2×C3×C4—28no—
300C2×C3×C4—22no—
301C2×C3×C4—28no—
302C2×C3×C4—38no—
303C2×C3×C4—68no—
304C2×C3×C4—92no—
305C2×C3×C4—182no—
306C2×C3×C4C2×D3218yes—
307C2×C3×C4C22×C3⋊D312yes—
308#67—188no—
309#67—1832no—
310C2×Dic3—22no—
311C2×Dic3—38no—
312C2×Dic3—68no—
313C2×Dic3—98no—
314C2×Dic3—92no—
315C2×Dic3—916no—
316C2×Dic3—94no—
317C2×Dic3—98no—
318C2×Dic3—92no—
319C2×Dic3—188no—
320C2×Dic3—188no—
321C2×Dic3—1816no—
322C2×Dic3—1816no—
323C2×Dic3—188no—
324C2×Dic3—182no—
325C2×Dic3C22×C3⋊D312yes—
326Dic6—188no—
327Dic6—1832no—
328C2×C32—24no—
329C2×C32—24no—
330C2×C32—24no—
331C2×C32—216no—
332C2×C32—122no—
333C2×C32#885116yescomplemented
334C2×C32#88514yescomplemented
335C2×C32C2×#23514yes—
336C2×C32C2×#23511yescharacteristic
337C2×C32C2×D3×D414yes—
338C2×C32C2×D3×D411yescharacteristic
339C2×C32C2×#23714yes—
340C2×C32C2×#23711yescharacteristic
341C3⋊D3—42no—
342C3×D3—48no—
343C3×D3—128no—
344C24—31no—
345C24—91no—
346C2×Q8—274no—
347C2×Q8—544no—
348C2×D4—274no—
349C2×D4—544no—
350C22×C4—31no—
351C22×C4—91no—
352C22×C4—271no—
353C22×C4—271no—
354C22×C4D3×C3⋊D311yescharacteristic
355C22⋊C4—34no—
356C22⋊C4—94no—
357C22⋊C4—274no—
358C22⋊C4—274no—
359C42—274no—
360C22×C3—24no—
361C22×C3—28no—
362C22×C3—28no—
363C22×C3—28no—
364C22×C3—68no—
365C22×C3—68no—
366C22×C3—68no—
367C22×C3—1216no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3—182no—
371C22×C3—364no—
372C22×C3#379818yes—
373C22×C3#380218yes—
374C22×C3#380518yes—
375C22×C3#382812yes—
376C22×C3D4×C3⋊D312yes—
377C22×C3#383012yes—
378C22×C3C22×D3214yes—
379C22×C3C23×C3⋊D311yescharacteristic
380C2×D3—22no—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—44no—
384C2×D3—68no—
385C2×D3—68no—
386C2×D3—68no—
387C2×D3—1216no—
388C3×C4—216no—
389C3×C4—28no—
390C3×C4—22no—
391C3×C4—68no—
392C3×C4—182no—
393Dic3—22no—
394Dic3—68no—
395Dic3—916no—
396Dic3—94no—
397Dic3—1816no—
398Dic3—1816no—
399Dic3—188no—
400Dic3—182no—
401C32—24no—
402C32C2×#88514yescomplemented
403C32C2×#88511yescharacteristic, complemented
404C23—32no—
405C23—62no—
406C23—62no—
407C23—62no—
408C23—92no—
409C23—182no—
410C23—182no—
411C23—182no—
412C23C2×D3×C3⋊D311yescharacteristic
413Q8—548no—
414D4—548no—
415C2×C4—22no—
416C2×C4—32no—
417C2×C4—62no—
418C2×C4—92no—
419C2×C4—182no—
420C2×C4—272no—
421C2×C4—274no—
422C2×C4—272no—
423C2×C4—542no—
424C2×C4C2×D3×C3⋊D312yes—
425C2×C3—216no—
426C2×C3—24no—
427C2×C3—24no—
428C2×C3—24no—
429C2×C3—128no—
430C2×C3—362no—
431C2×C3#63915116yescomplemented
432C2×C3#6409414yescomplemented
433C2×C3C2×#379814yes—
434C2×C3C2×#380214yes—
435C2×C3C2×#380514yes—
436C2×C3C2×#382811yescharacteristic
437C2×C3C2×D4×C3⋊D311yescharacteristic
438C2×C3C2×#383011yescharacteristic
439D3—42no—
440D3—128no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—124no—
445C22—182no—
446C22—182no—
447C22—182no—
448C22—364no—
449C22#8902012yes—
450C22#8902612yes—
451C22#8902912yes—
452C22C22×D3×C3⋊D311yescharacteristic
453C4—22no—
454C4—62no—
455C4—182no—
456C4—274no—
457C4—542no—
458C3—24no—
459C3C2×#6391514yescomplemented
460C3C2×#6409411yescharacteristic, complemented
461C2—122no—
462C2—362no—
463C2#1172353014yesdirect product factor
464C2C2×#8902011yescharacteristic
465C2C2×#8902611yescharacteristic
466C2C2×#8902911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×#6290C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6293C2semidirect
3. C2×C4×C3⋊(C3⋊Dic3)C2semidirect
4. C2×C3×#3752C2semidirect
5. C2×C32×(C2×D3)⋊C4C2semidirect
6. C2×#88803C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. #11723530C2direct (N × H)
19. C2×#6293C22semidirect
22. C4×C3⋊(C3⋊Dic3)C22semidirect
23. C3×#3752C22semidirect
24. C32×(C2×D3)⋊C4C22semidirect
25. #88803C22semidirect
26. (C2×D3)⋊(C3⋊Dic3)C22semidirect
34. C2×#3752D3semidirect
40. C2×C3×(C2×D3)⋊C4D3semidirect
95. #3752C2×D3semidirect
100. C3×(C2×D3)⋊C4C2×D3semidirect
131. C2×(C2×D3)⋊C4C3⋊D3semidirect
187. C2×C33#124semidirect
225. (C2×D3)⋊C4C2×C3⋊D3semidirect
278. C33C2×#124semidirect
333. C2×C32#885semidirect
334. C2×C32#885semidirect
402. C32C2×#885semidirect
403. C32C2×#885semidirect
431. C2×C3#63915semidirect
432. C2×C3#64094semidirect
459. C3C2×#63915semidirect
460. C3C2×#64094semidirect
463. C2#11723530direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62932
C22×#62902
C2×C4×C3⋊(C3⋊Dic3)2
C2×C3×#37522
C2×C32×(C2×D3)⋊C42
C2×#888032
C2×(C2×D3)⋊(C3⋊Dic3)2
#117235302
C2×#640943
C2×#639153