Finite Group Databasebeta

← #13178344

Subgroups — #13178344

Order 1728.

1062 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64101C211yesmaximal, characteristic
2C3×#64097C212yesmaximal, complemented
3#11723689C211yesmaximal, characteristic
4#11723688C214yesmaximal
5#11723680C212yesmaximal, complemented
6#11723679C212yesmaximal, complemented
7#11723678C212yesmaximal, complemented
8#11723673C211yesmaximal, characteristic
9#10594502—32nomaximal
10#10594351—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2214yes—
13C2×#6294C2212yes—
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752C2212yes—
16C3×C4⋊(C3⋊Dic3)C2212yes—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750C2214yes—
19C3×C4×C3⋊Dic3C2212yes—
20C3×C4×C3⋊Dic3C2212yes—
21#88815C2211yescharacteristic
22#88815C2211yescharacteristic
23#88815C2212yes—
24#88815C2214yes—
25#88815C2214yes—
26#88814C2212yes—
27#64101D311yescharacteristic
28#64097D312yescomplemented
29C3×#892—32no—
30C3×#892—61no—
31C3×#888—32no—
32C3×#888—32no—
33C3×#888—62no—
34#63900—31no—
35#63896—34no—
36D3×C4⋊Dic3—32no—
37#63888—34no—
38#63887—34no—
39#63874—31no—
40#63873—32no—
41#63872—32no—
42#63870—32no—
43#63869—32no—
44#63869—32no—
45#63867—34no—
46#63866—34no—
47#63855—32no—
48C22×C3×C3⋊D3C2311yescharacteristic
49C2×C33×C4C2312yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2314yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×C4×C3⋊D3Q814yescomplemented
57C3×#634—44no—
58#6294—42no—
59#5729—92no—
60#5519—91no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×#634C2×D314yes—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—34no—
66C2×C3×Dic6—34no—
67C2×C3×Dic6—64no—
68C2×#624—32no—
69C2×D3×Dic3—34no—
70C32×C4⋊C4—22no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—41no—
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C2×D312yes—
75#3752C2×D312yes—
76C4⋊(C3⋊Dic3)C2×D312yes—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78#3750C2×D314yes—
79C4×C3⋊Dic3C2×D312yes—
80C4×C3⋊Dic3C2×D312yes—
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4—62no—
83C3×C4⋊Dic3—32no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—62no—
87C3×C4⋊Dic3—61no—
88C3×Dic3⋊C4—34no—
89C3×Dic3⋊C4—34no—
90C3×Dic3⋊C4—64no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—34no—
94C3×C4×Dic3—62no—
95C3×C4×Dic3—62no—
96#3724—34no—
97#3724—34no—
98#3724—31no—
99#3724—32no—
100#3724—31no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—34no—
106Dic3⋊Dic3—34no—
107Dic3⋊Dic3—34no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—32no—
111C22×C33C2411yescharacteristic
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3—21no—
114C2×C3×C3⋊D3C2×Q812yes—
115C33×C4—22no—
116C33×C4C2×Q812yes—
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—24no—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3C2×Q812yes—
122C3×C4⋊Q8—92no—
123C3×#117—91no—
124#946—94no—
125#944—94no—
126#926—91no—
127C2×C4⋊Dic3—91no—
128#896—94no—
129#892—32no—
130#892—61no—
131D3×C4⋊C4—92no—
132#890—92no—
133#890—92no—
134#889—94no—
135#888—32no—
136#888—32no—
137#888—62no—
138#878—94no—
139#872—92no—
140#870—92no—
141#870—92no—
142#869—92no—
143C22×C3⋊D3C22×D311yescharacteristic
144C22×C3×D3—32no—
145C22×C3×D3—61no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—24no—
150C2×C32×C4—41no—
151C2×C32×C4—42no—
152C2×C32×C4C22×D314yes—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D312yes—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3⋊Dic3C22×D314yes—
160C2×C3⋊Dic3C22×D311yescharacteristic
161C4×C3⋊D3Dic614yescomplemented
162#634—44no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—34no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—64no—
176C3×C4×D3—38no—
177C3×C4×D3—64no—
178C3×Dic6—124no—
179C3×Dic6—124no—
180C3×Dic6—128no—
181#624—122no—
182D3×Dic3—124no—
183#544—271noSylow
184C2×C33C22×Q811yescharacteristic
185C2×C33H(5,𝔽2)11yescharacteristic
186C2×C332−1+411yescharacteristic
187C3×C3⋊D3—22no—
188C2×C3×Q8—94no—
189C22×C3×C4—91no—
190C22×Dic3—92no—
191C2×C4×D3—32no—
192C2×C4×D3—61no—
193C2×C4×D3—94no—
194C2×Dic6—34no—
195C2×Dic6—34no—
196C2×Dic6—64no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4—94no—
201C3×C4⋊C4—91no—
202C3×C4⋊C4—92no—
203C3×C4⋊C4D3211yescharacteristic
204C3×C4⋊C4D3212yes—
205C3×C22⋊C4—92no—
206C3×C42—92no—
207C3×C42—92no—
208#217—92no—
209(C2×D3)⋊C4—34no—
210(C2×D3)⋊C4—62no—
211(C2×D3)⋊C4—92no—
212(C2×D3)⋊C4—92no—
213C4⋊Dic3—32no—
214C4⋊Dic3—32no—
215C4⋊Dic3—32no—
216C4⋊Dic3—62no—
217C4⋊Dic3—61no—
218C4⋊Dic3—91no—
219C4⋊Dic3—92no—
220C4⋊Dic3—94no—
221C4⋊Dic3—91no—
222C4⋊Dic3—94no—
223Dic3⋊C4—34no—
224Dic3⋊C4—34no—
225Dic3⋊C4—64no—
226Dic3⋊C4—94no—
227Dic3⋊C4—94no—
228Dic3⋊C4—94no—
229Dic3⋊C4—92no—
230Dic3⋊C4—92no—
231Dic3⋊C4—94no—
232Dic3⋊C4—94no—
233C4×Dic3—32no—
234C4×Dic3—32no—
235C4×Dic3—34no—
236C4×Dic3—62no—
237C4×Dic3—62no—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32C23×D311yescharacteristic
242C22×C32C23×D312yes—
243C2×C3⋊D3—21no—
244C2×C3⋊D3—21no—
245C2×C3⋊D3C2×Dic612yes—
246C2×C3×D3—34no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C32×C4—24no—
253C32×C4—22no—
254C32×C4—24no—
255C32×C4—22no—
256C32×C4—42no—
257C32×C4—44no—
258C32×C4—42no—
259C32×C4—82no—
260C32×C4C2×Dic612yes—
261C32×C4D3×Q814yes—
262C3⋊Dic3—22no—
263C3⋊Dic3—22no—
264C3⋊Dic3—24no—
265C3⋊Dic3—22no—
266C3⋊Dic3C2×Dic612yes—
267C3×Dic3—34no—
268C3×Dic3—62no—
269C3×Dic3—64no—
270C3×Dic3—64no—
271C3×Dic3—64no—
272C3×Dic3—62no—
273C3×Dic3—64no—
274C3×Dic3—62no—
275C3×Dic3—64no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—124no—
279C3×Dic3—122no—
280C4⋊Q8—92no—
281C4⋊Q8—272no—
282#125—272no—
283#125—272no—
284C22⋊Q8—274no—
285#117—91no—
286#117—271no—
287C2×C4⋊C4—271no—
288C33#54411yescharacteristic, complemented, Sylow
289C23×C3—91no—
290C22×D3—32no—
291C22×D3—61no—
292C3×Q8—364no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—91no—
300C2×C3×C4—94no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—94no—
305C2×C3×C4C2×D3214yes—
306C2×C3×C4C2×D3212yes—
307C2×C3×C4C2×D3212yes—
308C2×C3×C4C2×D3211yescharacteristic
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—34no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—34no—
316C2×Dic3—34no—
317C2×Dic3—61no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—64no—
322C2×Dic3—362no—
323C4×D3—38no—
324C4×D3—64no—
325C4×D3—364no—
326Dic6—124no—
327Dic6—124no—
328Dic6—128no—
329C2×C32—22no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—21no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32C22×Dic611yescharacteristic
339C2×C32#100412yes—
340C2×C32C2×D3×Q812yes—
341C2×C32#100712yes—
342C2×C32#100911yescharacteristic
343C2×C32#101011yescharacteristic
344C3⋊D3—22no—
345C3×D3—64no—
346C3×D3—122no—
347C2×Q8—94no—
348C22×C4—91no—
349C22×C4—272no—
350C4⋊C4—92no—
351C4⋊C4—91no—
352C4⋊C4—94no—
353C4⋊C4—272no—
354C4⋊C4—271no—
355C4⋊C4—271no—
356C4⋊C4—274no—
357C4⋊C4—274no—
358C4⋊C4#111311yescharacteristic
359C22⋊C4—92no—
360C22⋊C4—272no—
361C42—92no—
362C42—92no—
363C22×C3—22no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—92no—
367C22×C3—181no—
368C22×C3—181no—
369C22×C3C22×D3212yes—
370C22×C3C22×D3211yescharacteristic
371C2×D3—34no—
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—121no—
376C2×D3—121no—
377C3×C4—22no—
378C3×C4—24no—
379C3×C4—22no—
380C3×C4—24no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4—42no—
384C3×C4—82no—
385C3×C4—92no—
386C3×C4—184no—
387C3×C4—182no—
388C3×C4—182no—
389C3×C4—182no—
390C3×C4D3×Dic614yes—
391C3×C4#379712yes—
392Dic3—34no—
393Dic3—62no—
394Dic3—64no—
395Dic3—64no—
396Dic3—62no—
397Dic3—64no—
398Dic3—62no—
399Dic3—64no—
400Dic3—62no—
401Dic3—64no—
402Dic3—62no—
403Dic3—124no—
404Dic3—122no—
405C32—21no—
406C32—22no—
407C32—41no—
408C32#551911yescharacteristic, complemented
409C32#572912yescomplemented
410C23—91no—
411Q8—364no—
412C2×C4—91no—
413C2×C4—94no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4—92no—
417C2×C4—94no—
418C2×C4—1082no—
419C2×C4C2×#111312yes—
420C2×C4C2×#111311yescharacteristic
421C2×C3—21no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—21no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—182no—
431C2×C3C2×D3×Dic612yes—
432C2×C3C2×#379711yescharacteristic
433C2×C3#6431311yescharacteristic
434C2×C3#6431412yes—
435C2×C3#6431512yes—
436C2×C3#6431611yescharacteristic
437D3—64no—
438D3—122no—
439C22—92no—
440C22—181no—
441C22—181no—
442C22C22×#111311yescharacteristic
443C4—22no—
444C4—92no—
445C4—184no—
446C4—182no—
447C4—182no—
448C4—182no—
449C4#8904012yes—
450C3—21no—
451C3—22no—
452C3—41no—
453C3#1059435111yescharacteristic, complemented
454C3#1059450212yescomplemented
455C2—182no—
456C2C2×#8904011yescharacteristic
457C2#1172462311yescharacteristic
458C2#1172462411yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC4⋊C4#1113
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64097C2semidirect
5. #11723680C2semidirect
6. #11723679C2semidirect
7. #11723678C2semidirect
28. #64097D3semidirect
56. C3×C4×C3⋊D3Q8semidirect
161. C4×C3⋊D3Dic6semidirect
288. C33#544semidirect
408. C32#5519semidirect
409. C32#5729semidirect
453. C3#10594351semidirect
454. C3#10594502semidirect

Maximal subgroups

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

subgroupindex
C3×#641012
C3×#640972
#117236892
#117236882
#117236802
#117236792
#117236782
#117236732
#105945023
#105943513