Finite Group Databasebeta

← #13170471

Subgroups — #13170471

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
1#11723689C211yesmaximal, characteristic
2#11723534C212yesmaximal, complemented
3#11723404C211yesmaximal, characteristic
4C3×#63874C211yesmaximal, characteristic
5C3×#63869C212yesmaximal, complemented
6#11723037C212yesmaximal, complemented
7#11723018C212yesmaximal, complemented
8#11722984C214yesmaximal
9#10594508—31nomaximal
10#10593959—32nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6294C2212yes—
13C2×#6293C2214yes—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2212yes—
18#88815C2212yes—
19Dic3⋊(C3⋊Dic3)C2214yes—
20#88803C2212yes—
21Dic3×C3⋊Dic3C2212yes—
22C3×#3724C2211yescharacteristic
23C3×#3724C2214yes—
24C3×#3724C2211yescharacteristic
25C3×Dic3⋊Dic3C2214yes—
26C3×#3722C2212yes—
27#64101—31no—
28#64078—32no—
29C3×#890—32no—
30C3×#872—32no—
31C3×#870—32no—
32#63919—32no—
33#63918—32no—
34#63918—32no—
35#63918—62no—
36#63900—31no—
37#63896—34no—
38#63895—34no—
39#63893—34no—
40D3×C4⋊Dic3—32no—
41#63874D311yescharacteristic
42#63870—32no—
43#63869D312yescomplemented
44#63845—34no—
45#63841—32no—
46#63841—61no—
47#63840—34no—
48C22×C3×C3⋊D3C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3C2314yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×C4×C3⋊D3Q814yescomplemented
54C2×C32×Dic3C2312yes—
55C2×C32×Dic3C2312yes—
56C2×C32×Dic3C2312yes—
57#6294—42no—
58#6293—44no—
59#5689—92no—
60#5519—91no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×#634—34no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—61no—
65C2×#627—34no—
66C2×#627—34no—
67C2×#627—64no—
68C2×#624—32no—
69C2×D3×Dic3—34no—
70C32×C4⋊C4—31no—
71C32×C42—32no—
72#3752—32no—
73C4⋊(C3⋊Dic3)—32no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—34no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—62no—
78C4⋊(C3⋊Dic3)—121no—
79C4⋊(C3⋊Dic3)C2×D312yes—
80C4×C3⋊Dic3—32no—
81C3×(C2×D3)⋊C4—32no—
82C3×(C2×D3)⋊C4—32no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—32no—
85C3×C4⋊Dic3—32no—
86C3×C4⋊Dic3—34no—
87C3×C4⋊Dic3—62no—
88C3×C4⋊Dic3C2×D312yes—
89C3×Dic3⋊C4—34no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—34no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—34no—
94C3×C4×Dic3—22no—
95C3×C4×Dic3C2×D312yes—
96C3×C4×Dic3C2×D312yes—
97#3724—32no—
98#3724C2×D314yes—
99#3724C2×D311yescharacteristic
100#3724C2×D311yescharacteristic
101Dic3⋊Dic3—34no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—64no—
105Dic3⋊Dic3C2×D314yes—
106#3722C2×D312yes—
107(C2×D3)⋊Dic3—34no—
108(C2×D3)⋊Dic3—62no—
109Dic32—34no—
110Dic32—62no—
111C22×C33C2411yescharacteristic
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3—21no—
114C2×C3×C3⋊D3C2×Q812yes—
115C33×C4C2×Q812yes—
116C3⋊(C3⋊Dic3)—22no—
117C3×C3⋊Dic3—24no—
118C3×C3⋊Dic3C2×Q812yes—
119C32×Dic3—22no—
120C32×Dic3—22no—
121C32×Dic3—22no—
122C3×#125—92no—
123C3×#117—91no—
124#926—91no—
125C2×C4⋊Dic3—91no—
126#897—94no—
127#896—94no—
128#892—92no—
129#892—181no—
130D3×C4⋊C4—92no—
131#890—32no—
132#890—92no—
133#889—94no—
134#888—92no—
135#888—94no—
136#888—92no—
137#878—94no—
138#872—32no—
139#870—32no—
140#869—92no—
141#869—92no—
142#869—182no—
143C22×C3⋊D3C22×D311yescharacteristic
144C22×C3×D3—32no—
145C22×C3×D3—61no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—32no—
149C2×C32×C4—32no—
150C2×C32×C4—32no—
151C2×C32×C4—41no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D312yes—
154C2×C3⋊Dic3—34no—
155C2×C3⋊Dic3—32no—
156C2×C3⋊Dic3—62no—
157C2×C3⋊Dic3—64no—
158C2×C3⋊Dic3—122no—
159C2×C3⋊Dic3C22×D314yes—
160C2×C3⋊Dic3C22×D311yescharacteristic
161C4×C3⋊D3Dic614yescomplemented
162#634—124no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3C22×D314yes—
172C2×C3×Dic3C22×D312yes—
173C2×C3×Dic3C22×D312yes—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D312yes—
176C3×C4×D3—38no—
177C3×C4×D3—64no—
178#627—124no—
179#627—124no—
180#627—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—
188C22×C3×C4—91no—
189C22×Dic3—92no—
190C2×C4×D3—32no—
191C2×C4×D3—61no—
192C2×C4×D3—94no—
193C2×Dic6—94no—
194C2×Dic6—94no—
195C2×Dic6—94no—
196C2×Dic6—184no—
197C3×C4⋊C4—32no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—91no—
200C3×C4⋊C4—94no—
201C3×C4⋊C4—94no—
202C3×C4⋊C4—91no—
203C3×C4⋊C4—92no—
204C3×C22⋊C4—92no—
205C3×C42—32no—
206C3×C42—32no—
207C3×C42—62no—
208#217—92no—
209(C2×D3)⋊C4—32no—
210(C2×D3)⋊C4—32no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—182no—
213C4⋊Dic3—32no—
214C4⋊Dic3—32no—
215C4⋊Dic3—34no—
216C4⋊Dic3—62no—
217C4⋊Dic3—92no—
218C4⋊Dic3—94no—
219C4⋊Dic3—94no—
220C4⋊Dic3—91no—
221C4⋊Dic3—92no—
222C4⋊Dic3—184no—
223C4⋊Dic3—182no—
224C4⋊Dic3—181no—
225C4⋊Dic3—361no—
226C4⋊Dic3D3211yescharacteristic
227Dic3⋊C4—34no—
228Dic3⋊C4—34no—
229Dic3⋊C4—32no—
230Dic3⋊C4—32no—
231Dic3⋊C4—34no—
232Dic3⋊C4—94no—
233Dic3⋊C4—94no—
234C4×Dic3—94no—
235C4×Dic3—92no—
236C4×Dic3—182no—
237C4×Dic3D3212yes—
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—42no—
255C32×C4—62no—
256C32×C4—62no—
257C32×C4—62no—
258C32×C4C2×Dic612yes—
259C32×C4D3×Q814yes—
260C3⋊Dic3—24no—
261C3⋊Dic3—62no—
262C3⋊Dic3—64no—
263C3⋊Dic3—124no—
264C3⋊Dic3—122no—
265C3⋊Dic3—242no—
266C3⋊Dic3C2×Dic612yes—
267C3×Dic3—24no—
268C3×Dic3—24no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—22no—
272C3×Dic3—22no—
273C3×Dic3—24no—
274C3×Dic3—34no—
275C3×Dic3—42no—
276C3×Dic3—62no—
277C3×Dic3—64no—
278C3×Dic3—64no—
279C3×Dic3—124no—
280C4⋊Q8—272no—
281C4⋊Q8—272no—
282#125—92no—
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—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—34no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—41no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—62no—
303C2×C3×C4—91no—
304C2×C3×C4—94no—
305C2×C3×C4—94no—
306C2×C3×C4C2×D3212yes—
307C2×C3×C4C2×D3211yescharacteristic
308C2×Dic3—32no—
309C2×Dic3—34no—
310C2×Dic3—34no—
311C2×Dic3—61no—
312C2×Dic3—64no—
313C2×Dic3—94no—
314C2×Dic3—92no—
315C2×Dic3—182no—
316C2×Dic3—184no—
317C2×Dic3—362no—
318C2×Dic3—362no—
319C2×Dic3C2×D3212yes—
320C2×Dic3C2×D3212yes—
321C2×Dic3C2×D3212yes—
322C4×D3—38no—
323C4×D3—64no—
324C4×D3—364no—
325Dic6—364no—
326Dic6—364no—
327Dic6—364no—
328Dic6—368no—
329C2×C32—22no—
330C2×C32—22no—
331C2×C32—21no—
332C2×C32—22no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32C22×Dic611yescharacteristic
339C2×C32C2×D3×Q812yes—
340C2×C32#100911yescharacteristic
341C2×C32#100912yes—
342C2×C32#101011yescharacteristic
343C2×C32#101012yes—
344C3⋊D3—22no—
345C3×D3—64no—
346C3×D3—122no—
347C2×Q8—274no—
348C22×C4—91no—
349C22×C4—272no—
350C4⋊C4—31no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—94no—
354C4⋊C4—94no—
355C4⋊C4—92no—
356C4⋊C4—274no—
357C4⋊C4—272no—
358C4⋊C4—271no—
359C22⋊C4—92no—
360C22⋊C4—272no—
361C42—32no—
362C42—272no—
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—42no—
380C3×C4—64no—
381C3×C4—64no—
382C3×C4—62no—
383C3×C4—64no—
384C3×C4—62no—
385C3×C4—62no—
386C3×C4—62no—
387C3×C4—92no—
388C3×C4—122no—
389C3×C4—184no—
390C3×C4D3×Dic614yes—
391C3×C4#379712yes—
392Dic3—22no—
393Dic3—22no—
394Dic3—22no—
395Dic3—34no—
396Dic3—62no—
397Dic3—64no—
398Dic3—64no—
399Dic3—124no—
400Dic3—184no—
401Dic3—182no—
402Dic3—364no—
403Dic3—362no—
404Dic3—722no—
405C32—22no—
406C32—21no—
407C32—41no—
408C32#551911yescharacteristic, complemented
409C32#568912yescomplemented
410C23—91no—
411Q8—1084no—
412C2×C4—32no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—91no—
416C2×C4—94no—
417C2×C4—94no—
418C2×C4—272no—
419C2×C4—1082no—
420C2×C4D3311yescharacteristic
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—22no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—182no—
431C2×C3C2×D3×Dic612yes—
432C2×C3#6429812yes—
433C2×C3C2×#379711yescharacteristic
434C2×C3#6430812yes—
435C2×C3#6430911yescharacteristic
436C2×C3#6432011yescharacteristic
437D3—64no—
438D3—122no—
439C22—92no—
440C22—181no—
441C22—181no—
442C22C2×D3311yescharacteristic
443C4—62no—
444C4—62no—
445C4—62no—
446C4—92no—
447C4—184no—
448C4—542no—
449C4#8895712yes—
450C3—21no—
451C3—22no—
452C3—41no—
453C3#1059395912yescomplemented
454C3#1059450811yescharacteristic, complemented
455C2—182no—
456C2#1172439011yescharacteristic
457C2#1172439211yescharacteristic
458C2C2×#8895711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723534C2semidirect
5. C3×#63869C2semidirect
6. #11723037C2semidirect
7. #11723018C2semidirect
43. #63869D3semidirect
53. C3×C4×C3⋊D3Q8semidirect
161. C4×C3⋊D3Dic6semidirect
288. C33#544semidirect
408. C32#5519semidirect
409. C32#5689semidirect
453. C3#10593959semidirect
454. C3#10594508semidirect

Maximal subgroups

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

subgroupindex
#117236892
#117235342
#117234042
C3×#638742
C3×#638692
#117230372
#117230182
#117229842
#105945083
#105939593