Finite Group Databasebeta

← #13156711

Subgroups — #13156711

Order 1728.

1122 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88806C212yesmaximal
2C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×Dic32C212yesmaximal
4#11722082C211yesmaximal, characteristic
5C3×#63579C211yesmaximal, characteristic
6C4×#3724—31nomaximal
7#10591243—32nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13#88806C418yescomplemented
14Dic3×C3⋊Dic3C414yescomplemented
15C3×Dic32—24no—
16C2×#3750—32no—
17C2×C4×C3⋊Dic3—31no—
18C2×C3×C4×Dic3—32no—
19C2×C3×C4×Dic3—32no—
20C2×#3724—32no—
21C2×#3724—32no—
22C2×#3724—62no—
23C2×Dic32—32no—
24C2×Dic32—61no—
25C2×Dic32D312yes—
26C3×C3⋊#95—32no—
27#63579—32no—
28#63579—31no—
29#63579D311yescharacteristic
30C23×C33C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)C2×C412yes—
32C2×C3⋊(C3⋊Dic3)C2×C414yes—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C3×C3⋊Dic3C2×C414yes—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3C2×C414yes—
41C2×C32×Dic3C2×C412yes—
42C2×C32×Dic3D412yes—
43C2×C32×Dic3Q812yes—
44C4×Dic3⋊C4—92no—
45#4649—91no—
46C22×C32×C4—31no—
47C22×C32×C4—32no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—32no—
50C22×C3⋊Dic3—61no—
51C22×C3⋊Dic3—62no—
52C22×C3⋊Dic3—121no—
53C22×C3⋊Dic3C2×D311yescharacteristic
54C22×C3⋊Dic3C2×D312yes—
55C22×C3×Dic3—21no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3—62no—
62C22×C3×Dic3C2×D312yes—
63C22×C3×Dic3C2×D312yes—
64C22×C3×Dic3C2×D312yes—
65#3750—38no—
66C4×C3⋊Dic3—34no—
67C3×C4×Dic3—64no—
68C3×C4×Dic3—64no—
69#3724—38no—
70#3724—38no—
71#3724—68no—
72Dic32—24no—
73Dic32—38no—
74Dic32—64no—
75C22×C33C22×C411yescharacteristic
76C22×C33C22×C412yes—
77C22×C33C2×D411yescharacteristic
78C22×C33C2×Q811yescharacteristic
79C22×C33#4112yes—
80C3⋊(C3⋊Dic3)C4214yescomplemented
81C3×C3⋊Dic3—22no—
82C3×C3⋊Dic3—24no—
83C32×Dic3—24no—
84C32×Dic3C4⋊C414yescomplemented
85C2×C3×C42—92no—
86C2×Dic3⋊C4—92no—
87C2×Dic3⋊C4—92no—
88C2×Dic3⋊C4—92no—
89C2×Dic3⋊C4—182no—
90C2×C4×Dic3—32no—
91C2×C4×Dic3—32no—
92C2×C4×Dic3—92no—
93C2×C4×Dic3—91no—
94C2×C4×Dic3—181no—
95C3×#95—91no—
96C3⋊#95—32no—
97C3⋊#95—91no—
98C3⋊#95—92no—
99C23×C32—22no—
100C23×C32—21no—
101C23×C32—41no—
102C23×C32C22×D312yes—
103C23×C32C22×D311yescharacteristic
104C2×C32×C4—32no—
105C2×C32×C4—32no—
106C2×C32×C4—32no—
107C2×C32×C4—34no—
108C2×C32×C4—62no—
109C2×C32×C4—62no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3—34no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—32no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3—124no—
125C2×C3⋊Dic3C4×D312yes—
126C2×C3⋊Dic3C4×D314yes—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—42no—
139C2×C3×Dic3—42no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—122no—
150C2×C3×Dic3—122no—
151C2×C3×Dic3Dic614yes—
152C2×C3×Dic3C4×D314yes—
153C2×C3×Dic3C4×D314yes—
154C2×C3×Dic3C4×D314yes—
155C2×C3×Dic3#6714yes—
156C4×C4⋊C4—271noSylow
157C2×C33C2×C4211yescharacteristic
158C2×C33C2×C4⋊C411yescharacteristic
159C2×C33#11711yescharacteristic
160C2×C33C4×D412yes—
161C2×C33C4×Q812yes—
162C22×C3×C4—32no—
163C22×C3×C4—32no—
164C22×C3×C4—32no—
165C22×C3×C4—61no—
166C22×C3×C4—62no—
167C22×C3×C4—91no—
168C22×C3×C4—92no—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—32no—
172C22×Dic3—61no—
173C22×Dic3—62no—
174C22×Dic3—92no—
175C22×Dic3—91no—
176C22×Dic3—181no—
177C22×Dic3—182no—
178C22×Dic3—361no—
179C22×Dic3D3212yes—
180C22×Dic3D3211yescharacteristic
181C3×C42—184no—
182Dic3⋊C4—98no—
183Dic3⋊C4—98no—
184Dic3⋊C4—98no—
185Dic3⋊C4—188no—
186C4×Dic3—64no—
187C4×Dic3—64no—
188C4×Dic3—98no—
189C4×Dic3—94no—
190C4×Dic3—184no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—22no—
203C22×C32—41no—
204C22×C32—42no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—42no—
208C22×C32C2×Dic612yes—
209C22×C32C2×C4×D312yes—
210C22×C32C2×C4×D311yescharacteristic
211C22×C32C2×C4×D312yes—
212C22×C32C2×C4×D312yes—
213C22×C32C2×C4×D312yes—
214C22×C32#23512yes—
215C22×C32#23512yes—
216C22×C32D3×D411yescharacteristic
217C22×C32#23712yes—
218C22×C32D3×Q811yescharacteristic
219C22×C32C2×#6712yes—
220C32×C4—34no—
221C32×C4—64no—
222C3⋊Dic3—22no—
223C3⋊Dic3—24no—
224C3⋊Dic3—34no—
225C3⋊Dic3—38no—
226C3⋊Dic3—64no—
227C3⋊Dic3—68no—
228C3⋊Dic3—124no—
229C3×Dic3—24no—
230C3×Dic3—24no—
231C3×Dic3—24no—
232C3×Dic3—44no—
233C3×Dic3—64no—
234C3×Dic3—64no—
235C3×Dic3—64no—
236C3×Dic3—64no—
237C3×Dic3—124no—
238C3×Dic3Dic3⋊C418yescomplemented
239C2×C4⋊C4—272no—
240C2×C42—92no—
241C2×C42—271no—
242#95—91no—
243#95—271no—
244C33C4×C4⋊C411yescharacteristic, complemented, Sylow
245C23×C3—22no—
246C23×C3—21no—
247C23×C3—41no—
248C23×C3C2×D3212yes—
249C23×C3C2×D3211yescharacteristic
250C2×C3×C4—34no—
251C2×C3×C4—34no—
252C2×C3×C4—34no—
253C2×C3×C4—34no—
254C2×C3×C4—34no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—64no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—92no—
264C2×C3×C4—94no—
265C2×C3×C4—122no—
266C2×C3×C4—122no—
267C2×C3×C4—182no—
268C2×C3×C4—182no—
269C2×C3×C4—182no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—34no—
273C2×Dic3—34no—
274C2×Dic3—34no—
275C2×Dic3—62no—
276C2×Dic3—64no—
277C2×Dic3—64no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—94no—
285C2×Dic3—94no—
286C2×Dic3—94no—
287C2×Dic3—92no—
288C2×Dic3—94no—
289C2×Dic3—122no—
290C2×Dic3—122no—
291C2×Dic3—182no—
292C2×Dic3—184no—
293C2×Dic3—184no—
294C2×Dic3—184no—
295C2×Dic3—184no—
296C2×Dic3—362no—
297C2×Dic3—364no—
298C2×Dic3#62414yes—
299C2×Dic3#62412yes—
300C2×Dic3#62512yes—
301C2×Dic3#62712yes—
302C2×C32—22no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—42no—
318C2×C32—42no—
319C2×C32C4×Dic612yes—
320C2×C32C4×Dic612yes—
321C2×C32C42×D311yescharacteristic
322C2×C32C42×D312yes—
323C2×C32#87212yes—
324C2×C32#87711yescharacteristic
325C2×C32#67⋊C412yes—
326C2×C32#88712yes—
327C2×C32D3×C4⋊C411yescharacteristic
328C2×C32C2×Dic3⋊C412yes—
329C2×C32C4×#6712yes—
330C2×C32C4×#6712yes—
331C22×C4—31no—
332C22×C4—32no—
333C22×C4—91no—
334C22×C4—92no—
335C22×C4—271no—
336C4⋊C4—278no—
337C42—184no—
338C42—274no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—42no—
356C22×C3D3×Dic612yes—
357C22×C3#379512yes—
358C22×C3#379812yes—
359C22×C3C4×D3212yes—
360C22×C3C4×D3212yes—
361C22×C3C4×D3212yes—
362C22×C3#380412yes—
363C22×C3C2×#62412yes—
364C22×C3C2×#62411yescharacteristic
365C22×C3C2×#62511yescharacteristic
366C22×C3C2×#62711yescharacteristic
367C22×C3D3×#6712yes—
368C3×C4—38no—
369C3×C4—64no—
370C3×C4—64no—
371C3×C4—64no—
372C3×C4—124no—
373C3×C4—182no—
374C3×C4—184no—
375Dic3—24no—
376Dic3—64no—
377Dic3—64no—
378Dic3—64no—
379Dic3—64no—
380Dic3—98no—
381Dic3—94no—
382Dic3—124no—
383Dic3—184no—
384Dic3—188no—
385Dic3—364no—
386Dic3#372414yescomplemented
387C32—22no—
388C32—21no—
389C32—41no—
390C32#464911yescharacteristic, complemented
391C32C4×Dic3⋊C412yescomplemented
392C23D3311yescharacteristic
393C2×C4—32no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—34no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—92no—
400C2×C4—94no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4—272no—
405C2×C4—274no—
406C2×C3—21no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—22no—
414C2×C3—22no—
415C2×C3—22no—
416C2×C3—22no—
417C2×C3—22no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—42no—
422C2×C3—42no—
423C2×C3#6383712yes—
424C2×C3#6384412yes—
425C2×C3#6387411yescharacteristic
426C2×C3D3×Dic3⋊C412yes—
427C2×C3#6387712yes—
428C2×C3#6387912yes—
429C2×C3C4×#62412yes—
430C2×C3C4×#62411yescharacteristic
431C2×C3C4×#62512yes—
432C2×C3C4×#62712yes—
433C2×C3#6395212yes—
434C2×C3C2×#372411yescharacteristic
435C22D3×#62411yescharacteristic
436C22D3×#62412yes—
437C22D3×#62511yescharacteristic
438C22D3×#62711yescharacteristic
439C22#8896012yes—
440C4—34no—
441C4—64no—
442C4—182no—
443C4—184no—
444C4—274no—
445C3—21no—
446C3—22no—
447C3—41no—
448C3#1059124312yescomplemented
449C3C4×#372411yescharacteristic, complemented
450C2#1172290011yescharacteristic
451C2D3×#372411yescharacteristic
452C2#1172291911yescharacteristic
453C2#1172293012yes—
454C2#1172294212yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC2×C33C2×C42
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33C4×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
13. #88806C4semidirect
14. Dic3×C3⋊Dic3C4semidirect
80. C3⋊(C3⋊Dic3)C42semidirect
84. C32×Dic3C4⋊C4semidirect
238. C3×Dic3Dic3⋊C4semidirect
244. C33C4×C4⋊C4semidirect
386. Dic3#3724semidirect
390. C32#4649semidirect
391. C32C4×Dic3⋊C4semidirect
448. C3#10591243semidirect
449. C3C4×#3724semidirect

Maximal subgroups

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

subgroupindex
C2×#888062
C2×Dic3×C3⋊Dic32
C2×C3×Dic322
#117220822
C3×#635792
C4×#37243
#105912433