Finite Group Databasebeta

← #13162056

Subgroups — #13162056

Order 1728.

1570 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal
3C2×#88815C212yesmaximal, complemented
4#11722082C212yesmaximal, complemented
5#10591720—31nomaximal
6#10591716—32nomaximal
7C24×C33C2211yescharacteristic
8C22×C3×C3⋊Dic3—24no—
9C22×C3×C3⋊Dic3C414yescomplemented
10C22×C3×C3⋊Dic3C2214yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C3×#3757—24no—
14#88815—24no—
15C23×C3⋊Dic3D311yescharacteristic
16C23×C3×Dic3—32no—
17C23×C3×Dic3—61no—
18C2×#3757D312yes—
19C2×C3×#217—32no—
20C2×C3×#217—32no—
21C2×C3×#217—62no—
22C2×#3724—32no—
23C2×Dic3⋊Dic3—34no—
24#63579—34no—
25#63579—32no—
26C23×C33—22no—
27C23×C33—22no—
28C23×C33C2×C412yes—
29C23×C33D412yes—
30C23×C33Q812yes—
31C23×C33C2311yescharacteristic
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—28no—
34C2×C3×C3⋊Dic3—24no—
35C2×C3×C3⋊Dic3—24no—
36C2×C3×C3⋊Dic3—24no—
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3D414yes—
40#5212—91no—
41#4942—92no—
42C24×C32—21no—
43C24×C32—22no—
44C24×C32—41no—
45C24×C32C2×D312yes—
46C24×C32C2×D311yescharacteristic
47C22×C3⋊Dic3—24no—
48C22×C3⋊Dic3Dic314yescomplemented
49C22×C3⋊Dic3C2×D314yes—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3×Dic3—32no—
53C22×C3×Dic3—38no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—34no—
56C22×C3×Dic3—34no—
57C22×C3×Dic3—68no—
58C22×C3×Dic3—64no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3—68no—
62C22×C3×Dic3—64no—
63#3757—24no—
64C3×#217—68no—
65C3×#217—64no—
66C3×#217—64no—
67#3724—64no—
68Dic3⋊Dic3—68no—
69C22×C33—22no—
70C22×C33—22no—
71C22×C33—24no—
72C22×C33—24no—
73C22×C33C4⋊C414yescomplemented
74C22×C33C22×C411yescharacteristic
75C22×C33C2×D412yes—
76C22×C33C2×D411yescharacteristic
77C22×C33C2×Q811yescharacteristic
78C22×C33#4112yes—
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3—44no—
81C3×C3⋊Dic3C22⋊C414yescomplemented
82C23×C3×C4—91no—
83C23×Dic3—32no—
84C23×Dic3—61no—
85C2×C3×C22⋊C4—92no—
86C2×#217—32no—
87C2×#217—32no—
88C2×#217—62no—
89C2×C4⋊Dic3—92no—
90C2×Dic3⋊C4—94no—
91C3⋊#95—94no—
92C3⋊#95—92no—
93C23×C32—22no—
94C23×C32—22no—
95C23×C32—24no—
96C23×C32—24no—
97C23×C32—24no—
98C23×C32—24no—
99C23×C32—24no—
100C23×C32—22no—
101C23×C32—22no—
102C23×C32—24no—
103C23×C32—22no—
104C23×C32—24no—
105C23×C32—22no—
106C23×C32—21no—
107C23×C32—42no—
108C23×C32—42no—
109C23×C32—42no—
110C23×C32—41no—
111C23×C32—44no—
112C23×C32—44no—
113C23×C32—44no—
114C23×C32—44no—
115C23×C32Dic614yes—
116C23×C32Dic612yes—
117C23×C32C4×D314yes—
118C23×C32D1212yes—
119C23×C32C2×Dic312yes—
120C23×C32#6714yes—
121C23×C32C22×D312yes—
122C23×C32C22×D311yescharacteristic
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—28no—
125C2×C3⋊Dic3—24no—
126C2×C3⋊Dic3—24no—
127C2×C3⋊Dic3—24no—
128C2×C3⋊Dic3C2×Dic312yes—
129C2×C3⋊Dic3C2×Dic312yes—
130C2×C3⋊Dic3#6714yes—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—38no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—62no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—616no—
138C2×C3×Dic3—616no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—68no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—124no—
149C2×C3×Dic3—124no—
150#380—271noSylow
151C2×C33—24no—
152C2×C33C2×C22⋊C411yescharacteristic
153C2×C33C2×C4⋊C411yescharacteristic
154C2×C33#11711yescharacteristic
155C2×C33#12112yes—
156C2×C33C22⋊Q812yes—
157C24×C3—21no—
158C24×C3—22no—
159C24×C3—41no—
160C24×C3D3212yes—
161C24×C3D3211yescharacteristic
162C22×C3×C4—91no—
163C22×C3×C4—94no—
164C22×C3×C4—91no—
165C22×C3×C4—94no—
166C22×C3×C4—184no—
167C22×Dic3—32no—
168C22×Dic3—38no—
169C22×Dic3—32no—
170C22×Dic3—34no—
171C22×Dic3—34no—
172C22×Dic3—64no—
173C22×Dic3—61no—
174C22×Dic3—61no—
175C22×Dic3—64no—
176C22×Dic3—68no—
177C22×Dic3—68no—
178C3×C22⋊C4—184no—
179#217—68no—
180#217—64no—
181#217—64no—
182C4⋊Dic3—184no—
183Dic3⋊C4—188no—
184C22×C32—24no—
185C22×C32—22no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—28no—
191C22×C32—28no—
192C22×C32—24no—
193C22×C32—28no—
194C22×C32—24no—
195C22×C32—28no—
196C22×C32—28no—
197C22×C32—24no—
198C22×C32—22no—
199C22×C32—24no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—21no—
206C22×C32—24no—
207C22×C32—22no—
208C22×C32—44no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32—44no—
215C22×C32—48no—
216C22×C32—44no—
217C22×C32—48no—
218C22×C32—44no—
219C22×C32—44no—
220C22×C32—48no—
221C22×C32—48no—
222C22×C32Dic3⋊C418yescomplemented
223C22×C32C4⋊Dic314yescomplemented
224C22×C32C2×Dic612yes—
225C22×C32C2×Dic611yescharacteristic
226C22×C32C2×C4×D312yes—
227C22×C32C2×D1211yescharacteristic
228C22×C32#23512yes—
229C22×C32D3×D414yes—
230C22×C32#23714yes—
231C22×C32C22×Dic311yescharacteristic
232C22×C32C2×#6712yes—
233C22×C32C2×#6712yes—
234C3⋊Dic3—22no—
235C3⋊Dic3—44no—
236C3⋊Dic3#21714yescomplemented
237C3×Dic3—38no—
238C3×Dic3—64no—
239C3×Dic3—64no—
240C3×Dic3—64no—
241C3×Dic3—128no—
242C3×Dic3—124no—
243C3×Dic3—124no—
244C23×C4—91no—
245C2×C4⋊C4—272no—
246C2×C22⋊C4—92no—
247#95—272no—
248C33#38011yescharacteristic, complemented, Sylow
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—24no—
257C23×C3—22no—
258C23×C3—24no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—44no—
264C23×C3—44no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—41no—
271C23×C3D3×Dic314yes—
272C23×C3#62412yes—
273C23×C3#62512yes—
274C23×C3#62614yes—
275C23×C3#62714yes—
276C23×C3#62712yes—
277C23×C3C2×D3212yes—
278C23×C3C2×D3211yescharacteristic
279C2×C3×C4—92no—
280C2×C3×C4—94no—
281C2×C3×C4—92no—
282C2×C3×C4—188no—
283C2×C3×C4—182no—
284C2×C3×C4—184no—
285C2×C3×C4—184no—
286C2×C3×C4—184no—
287C2×Dic3—34no—
288C2×Dic3—38no—
289C2×Dic3—34no—
290C2×Dic3—62no—
291C2×Dic3—62no—
292C2×Dic3—64no—
293C2×Dic3—616no—
294C2×Dic3—616no—
295C2×Dic3—64no—
296C2×Dic3—64no—
297C2×Dic3—68no—
298C2×Dic3—64no—
299C2×Dic3—64no—
300C2×Dic3—64no—
301C2×Dic3—64no—
302C2×Dic3—64no—
303C2×Dic3—64no—
304C2×Dic3—124no—
305C2×Dic3—124no—
306C2×C32—24no—
307C2×C32—28no—
308C2×C32—28no—
309C2×C32—22no—
310C2×C32—21no—
311C2×C32—24no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—24no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—42no—
323C2×C32—42no—
324C2×C32—48no—
325C2×C32—48no—
326C2×C32#87712yes—
327C2×C32#87814yes—
328C2×C32D3×C22⋊C412yes—
329C2×C32C2×Dic3⋊C412yes—
330C2×C32#92412yes—
331C2×C32C2×C4⋊Dic311yescharacteristic
332C2×C32#92611yescharacteristic
333C2×C32#93012yes—
334C2×C32#93914yes—
335C2×C32C2×#21711yescharacteristic
336C24#111311yescharacteristic
337C22×C4—91no—
338C22×C4—94no—
339C22×C4—91no—
340C22×C4—94no—
341C22×C4—184no—
342C4⋊C4—544no—
343C22⋊C4—184no—
344C22×C3—24no—
345C22×C3—24no—
346C22×C3—28no—
347C22×C3—28no—
348C22×C3—24no—
349C22×C3—22no—
350C22×C3—28no—
351C22×C3—24no—
352C22×C3—22no—
353C22×C3—28no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—24no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—24no—
366C22×C3—24no—
367C22×C3—28no—
368C22×C3—44no—
369C22×C3—48no—
370C22×C3—48no—
371C22×C3—44no—
372C22×C3—44no—
373C22×C3—48no—
374C22×C3—48no—
375C22×C3—44no—
376C22×C3—41no—
377C22×C3—42no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—42no—
381C22×C3—44no—
382C22×C3Dic3⋊Dic318yescomplemented
383C22×C3#372414yescomplemented
384C22×C3C2×D3×Dic312yes—
385C22×C3#380414yes—
386C22×C3#380512yes—
387C22×C3C2×#62411yescharacteristic
388C22×C3C2×#62511yescharacteristic
389C22×C3C2×#62612yes—
390C22×C3C2×#62712yes—
391C22×C3C2×#62711yescharacteristic
392C22×C3D3×#6714yes—
393C22×C3#381112yes—
394C3×C4—94no—
395C3×C4—182no—
396C3×C4—364no—
397Dic3—38no—
398Dic3—64no—
399Dic3—64no—
400Dic3—64no—
401Dic3—128no—
402Dic3—124no—
403Dic3—124no—
404C32—22no—
405C32—21no—
406C32—41no—
407C32#494212yescomplemented
408C32#521211yescharacteristic, complemented
409C23—22no—
410C23—22no—
411C23#629412yes—
412C23#629512yes—
413C23#629612yes—
414C23C2×#111311yescharacteristic
415C2×C4—92no—
416C2×C4—94no—
417C2×C4—92no—
418C2×C4—188no—
419C2×C4—182no—
420C2×C4—184no—
421C2×C4—184no—
422C2×C4—184no—
423C2×C3—28no—
424C2×C3—28no—
425C2×C3—24no—
426C2×C3—21no—
427C2×C3—22no—
428C2×C3—24no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—24no—
432C2×C3—22no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—22no—
436C2×C3—48no—
437C2×C3—48no—
438C2×C3—41no—
439C2×C3—42no—
440C2×C3—42no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3#6395512yes—
444C2×C3#6395711yescharacteristic
445C2×C3#6396414yes—
446C2×C3C2×Dic3⋊Dic312yes—
447C2×C3C2×#372411yescharacteristic
448C2×C3D3×#21712yes—
449C2×C3#6397411yescharacteristic
450C2×C3#6397814yes—
451C2×C3#6398012yes—
452C2×C3#6398212yes—
453C22—22no—
454C22—24no—
455C22—24no—
456C22—22no—
457C22#8881514yescomplemented
458C22C2×#629411yescharacteristic
459C22#8904612yes—
460C22C2×#629511yescharacteristic
461C22C2×#629611yescharacteristic
462C22#8904912yes—
463C4—94no—
464C4—182no—
465C4—364no—
466C3—21no—
467C3—22no—
468C3—41no—
469C3#1059171612yescomplemented
470C3#1059172011yescharacteristic, complemented
471C2—24no—
472C2#1172371511yescharacteristic
473C2#1172371712yes—
474C2#1172372112yes—
475C2C2×#8881511yescharacteristic
476C2#1172372311yescharacteristic
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC22×C33C22×C4
Fitting, prime radicalC24×C33C22
socle, cosocleC23×C33C23
hypercenterC24#1113
hypocenterC33#380
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#88815C2semidirect
4. #11722082C2semidirect
9. C22×C3×C3⋊Dic3C4semidirect
48. C22×C3⋊Dic3Dic3semidirect
73. C22×C33C4⋊C4semidirect
81. C3×C3⋊Dic3C22⋊C4semidirect
222. C22×C32Dic3⋊C4semidirect
223. C22×C32C4⋊Dic3semidirect
236. C3⋊Dic3#217semidirect
248. C33#380semidirect
382. C22×C3Dic3⋊Dic3semidirect
383. C22×C3#3724semidirect
407. C32#4942semidirect
408. C32#5212semidirect
457. C22#88815semidirect
469. C3#10591716semidirect
470. C3#10591720semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#37572
C2×#888152
#117220822
#105917203
#105917163