Finite Group Databasebeta

← #13156712

Subgroups — #13156712

Order 1728.

1122 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
1C2×#88813C211yesmaximal, characteristic
2C2×#88806C212yesmaximal
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4C2×C3×Dic32C211yesmaximal, characteristic
5C3×#63579C212yesmaximal
6#10591257—31nomaximal
7#10591243—32nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13#88813—22no—
14#88806C418yescomplemented
15Dic3×C3⋊Dic3C414yescomplemented
16C3×Dic32—22no—
17C2×#3750—32no—
18C2×C4×C3⋊Dic3—31no—
19C2×C3×C4×Dic3—32no—
20C2×#3724—32no—
21C2×#3724—32no—
22C2×#3724—62no—
23C2×Dic32—32no—
24C2×Dic32—32no—
25C2×Dic32—31no—
26C2×Dic32—61no—
27C2×Dic32D311yescharacteristic
28C3×C3⋊#95—32no—
29C3×C3⋊#95—32no—
30#63579D312yes—
31C23×C33C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)C2×C412yes—
33C2×C3⋊(C3⋊Dic3)C2×C414yes—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3C2×C414yes—
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3D412yes—
39C2×C3×C3⋊Dic3Q812yes—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3C2×C412yes—
44C2×C32×Dic3C2×C414yes—
45C4×Dic3⋊C4—91no—
46#4649—92no—
47C22×C32×C4—32no—
48C22×C32×C4—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—32no—
51C22×C3⋊Dic3—61no—
52C22×C3⋊Dic3—62no—
53C22×C3⋊Dic3—121no—
54C22×C3⋊Dic3C2×D312yes—
55C22×C3⋊Dic3C2×D311yescharacteristic
56C22×C3×Dic3—21no—
57C22×C3×Dic3—22no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—62no—
63C22×C3×Dic3C2×D312yes—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C2×D312yes—
66#3750—38no—
67C4×C3⋊Dic3—34no—
68C3×C4×Dic3—64no—
69#3724—38no—
70#3724—38no—
71#3724—68no—
72Dic32—22no—
73Dic32—38no—
74Dic32—64no—
75Dic32—64no—
76Dic32—62no—
77C22×C33C22×C411yescharacteristic
78C22×C33C22×C412yes—
79C22×C33C2×D411yescharacteristic
80C22×C33C2×Q811yescharacteristic
81C22×C33#4112yes—
82C3⋊(C3⋊Dic3)C4214yescomplemented
83C3×C3⋊Dic3—24no—
84C3×C3⋊Dic3C4⋊C414yescomplemented
85C32×Dic3—22no—
86C32×Dic3—24no—
87C2×C3×C42—91no—
88C2×Dic3⋊C4—92no—
89C2×Dic3⋊C4—92no—
90C2×Dic3⋊C4—92no—
91C2×Dic3⋊C4—182no—
92C2×C4×Dic3—32no—
93C2×C4×Dic3—92no—
94C2×C4×Dic3—91no—
95C2×C4×Dic3—92no—
96C2×C4×Dic3—91no—
97C2×C4×Dic3—181no—
98C3×#95—92no—
99C3⋊#95—32no—
100C3⋊#95—32no—
101C23×C32—22no—
102C23×C32—21no—
103C23×C32—41no—
104C23×C32C22×D311yescharacteristic
105C23×C32C22×D312yes—
106C2×C32×C4—32no—
107C2×C32×C4—34no—
108C2×C32×C4—62no—
109C2×C32×C4—62no—
110C2×C32×C4—62no—
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⋊Dic3Dic612yes—
126C2×C3⋊Dic3C4×D314yes—
127C2×C3⋊Dic3C4×D312yes—
128C2×C3⋊Dic3#6712yes—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—42no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—64no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—122no—
154C2×C3×Dic3—122no—
155C2×C3×Dic3C4×D314yes—
156C2×C3×Dic3C4×D314yes—
157C2×C3×Dic3C4×D314yes—
158C4×C4⋊C4—271noSylow
159C2×C33C2×C4211yescharacteristic
160C2×C33C2×C4⋊C411yescharacteristic
161C2×C33#11711yescharacteristic
162C2×C33C4×D412yes—
163C2×C33C4×Q812yes—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4—32no—
167C22×C3×C4—61no—
168C22×C3×C4—62no—
169C22×C3×C4—91no—
170C22×C3×C4—92no—
171C22×Dic3—32no—
172C22×Dic3—32no—
173C22×Dic3—32no—
174C22×Dic3—61no—
175C22×Dic3—62no—
176C22×Dic3—92no—
177C22×Dic3—91no—
178C22×Dic3—181no—
179C22×Dic3—182no—
180C22×Dic3—361no—
181C22×Dic3D3211yescharacteristic
182C22×Dic3D3212yes—
183C3×C42—182no—
184Dic3⋊C4—98no—
185Dic3⋊C4—98no—
186Dic3⋊C4—98no—
187Dic3⋊C4—188no—
188C4×Dic3—64no—
189C4×Dic3—98no—
190C4×Dic3—94no—
191C4×Dic3—184no—
192C4×Dic3—184no—
193C4×Dic3—182no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—22no—
206C22×C32—41no—
207C22×C32—42no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—42no—
211C22×C32C2×Dic611yescharacteristic
212C22×C32C2×C4×D311yescharacteristic
213C22×C32C2×C4×D312yes—
214C22×C32C2×C4×D312yes—
215C22×C32C2×C4×D312yes—
216C22×C32C2×C4×D312yes—
217C22×C32#23512yes—
218C22×C32D3×D412yes—
219C22×C32#23712yes—
220C22×C32#23712yes—
221C22×C32D3×Q812yes—
222C22×C32C2×#6711yescharacteristic
223C32×C4—62no—
224C32×C4—64no—
225C3⋊Dic3—24no—
226C3⋊Dic3—38no—
227C3⋊Dic3—34no—
228C3⋊Dic3—64no—
229C3⋊Dic3—68no—
230C3⋊Dic3—124no—
231C3⋊Dic3Dic3⋊C414yescomplemented
232C3×Dic3—24no—
233C3×Dic3—24no—
234C3×Dic3—24no—
235C3×Dic3—24no—
236C3×Dic3—38no—
237C3×Dic3—44no—
238C3×Dic3—64no—
239C3×Dic3—64no—
240C3×Dic3—64no—
241C3×Dic3—124no—
242C2×C4⋊C4—272no—
243C2×C42—91no—
244C2×C42—271no—
245C2×C42—271no—
246#95—92no—
247C33C4×C4⋊C411yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—41no—
251C23×C3C2×D3212yes—
252C23×C3C2×D3211yescharacteristic
253C2×C3×C4—34no—
254C2×C3×C4—34no—
255C2×C3×C4—34no—
256C2×C3×C4—62no—
257C2×C3×C4—64no—
258C2×C3×C4—64no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—92no—
268C2×C3×C4—94no—
269C2×C3×C4—122no—
270C2×C3×C4—122no—
271C2×C3×C4—182no—
272C2×C3×C4—182no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—34no—
277C2×Dic3—34no—
278C2×Dic3—34no—
279C2×Dic3—34no—
280C2×Dic3—34no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—64no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—94no—
290C2×Dic3—94no—
291C2×Dic3—94no—
292C2×Dic3—92no—
293C2×Dic3—94no—
294C2×Dic3—122no—
295C2×Dic3—122no—
296C2×Dic3—182no—
297C2×Dic3—184no—
298C2×Dic3—184no—
299C2×Dic3—184no—
300C2×Dic3—184no—
301C2×Dic3—362no—
302C2×Dic3—364no—
303C2×Dic3#62412yes—
304C2×Dic3#62414yes—
305C2×C32—22no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—42no—
321C2×C32—42no—
322C2×C32C4×Dic612yes—
323C2×C32C42×D312yes—
324C2×C32C42×D311yescharacteristic
325C2×C32#87211yescharacteristic
326C2×C32#87712yes—
327C2×C32#67⋊C412yes—
328C2×C32#67⋊C412yes—
329C2×C32#88712yes—
330C2×C32#88712yes—
331C2×C32D3×C4⋊C412yes—
332C2×C32C2×Dic3⋊C411yescharacteristic
333C2×C32C4×#6712yes—
334C22×C4—31no—
335C22×C4—32no—
336C22×C4—91no—
337C22×C4—92no—
338C22×C4—271no—
339C4⋊C4—278no—
340C42—182no—
341C42—274no—
342C42—542no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—22no—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—41no—
356C22×C3—42no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—42no—
360C22×C3D3×Dic612yes—
361C22×C3#379512yes—
362C22×C3#379711yescharacteristic
363C22×C3C4×D3212yes—
364C22×C3C4×D3212yes—
365C22×C3C4×D3212yes—
366C22×C3#380211yescharacteristic
367C22×C3#380412yes—
368C22×C3#380512yes—
369C22×C3C2×#62412yes—
370C22×C3C2×#62411yescharacteristic
371C22×C3D3×#6712yes—
372C3×C4—64no—
373C3×C4—64no—
374C3×C4—64no—
375C3×C4—64no—
376C3×C4—94no—
377C3×C4—124no—
378C3×C4—184no—
379Dic3—24no—
380Dic3—22no—
381Dic3—38no—
382Dic3—64no—
383Dic3—64no—
384Dic3—64no—
385Dic3—98no—
386Dic3—94no—
387Dic3—124no—
388Dic3—184no—
389Dic3—188no—
390Dic3—364no—
391C32—22no—
392C32—21no—
393C32—41no—
394C32#464912yescomplemented
395C32C4×Dic3⋊C411yescharacteristic, complemented
396C23D3311yescharacteristic
397C2×C4—32no—
398C2×C4—34no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—92no—
403C2×C4—92no—
404C2×C4—92no—
405C2×C4—94no—
406C2×C4—182no—
407C2×C4—182no—
408C2×C4—272no—
409C2×C4—274no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—22no—
414C2×C3—21no—
415C2×C3—22no—
416C2×C3—22no—
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—41no—
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—42no—
426C2×C3—42no—
427C2×C3#6383712yes—
428C2×C3#6383812yes—
429C2×C3#6384412yes—
430C2×C3D3×Dic3⋊C412yes—
431C2×C3#6387712yes—
432C2×C3#6387912yes—
433C2×C3C4×#62412yes—
434C2×C3C4×#62411yescharacteristic
435C2×C3#6390011yescharacteristic
436C2×C3#6390212yes—
437C2×C3#6395212yes—
438C2×C3#6395711yescharacteristic
439C22D3×#62411yescharacteristic
440C22D3×#62412yes—
441C22#8895311yescharacteristic
442C22#8895711yescharacteristic
443C22#8896112yes—
444C4—62no—
445C4—64no—
446C4—94no—
447C4—184no—
448C4—274no—
449C3—22no—
450C3—21no—
451C3—41no—
452C3#1059124312yescomplemented
453C3#1059125711yescharacteristic, complemented
454C2#1172290011yescharacteristic
455C2#1172291111yescharacteristic
456C2#1172291611yescharacteristic
457C2#1172293612yes—
458C2#1172294312yes—
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
14. #88806C4semidirect
15. Dic3×C3⋊Dic3C4semidirect
82. C3⋊(C3⋊Dic3)C42semidirect
84. C3×C3⋊Dic3C4⋊C4semidirect
231. C3⋊Dic3Dic3⋊C4semidirect
247. C33C4×C4⋊C4semidirect
394. C32#4649semidirect
395. C32C4×Dic3⋊C4semidirect
452. C3#10591243semidirect
453. C3#10591257semidirect

Maximal subgroups

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

subgroupindex
C2×#888132
C2×#888062
C2×Dic3×C3⋊Dic32
C2×C3×Dic322
C3×#635792
#105912573
#105912433