Finite Group Databasebeta

← #13156862

Subgroups — #13156862

Order 1728.

1042 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic
2C2×#88813C211yesmaximal, characteristic
3C2×#88807C211yesmaximal, characteristic
4#11722061C212yesmaximal
5C3×#63579C212yesmaximal
6#10591261—32nomaximal
7#10591236—31nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×C3×#627—22no—
14C2×C3×#627—22no—
15C2×C3×#627C414yescomplemented
16#88813—22no—
17#88807—22no—
18C22×C3×Dic6—32no—
19C22×#627D311yescharacteristic
20C2×#3750—31no—
21C2×Dic3⋊Dic3—32no—
22C2×Dic3⋊Dic3—61no—
23C2×Dic32—31no—
24C2×Dic32—32no—
25#63658—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28#63579—32no—
29#63579—32no—
30#63579—62no—
31#63579D312yes—
32C23×C33C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—22no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3D412yes—
39C2×C3×C3⋊Dic3D412yes—
40C2×C3×C3⋊Dic3Q814yescomplemented
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C414yes—
46C3×#627—28no—
47#5229—91no—
48#4646—92no—
49C22×C32×C4—32no—
50C22×C32×C4—31no—
51C22×C3⋊Dic3—31no—
52C22×C3⋊Dic3—32no—
53C22×C3⋊Dic3—62no—
54C22×C3⋊Dic3—61no—
55C22×C3⋊Dic3—121no—
56C22×C3⋊Dic3C2×D312yes—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3×Dic3—22no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—62no—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C2×C3×Dic6—38no—
69C2×C3×Dic6—64no—
70C2×C3×Dic6—64no—
71C2×#627—22no—
72C2×#627—22no—
73C2×#627Dic314yescomplemented
74#3750—62no—
75Dic3⋊Dic3—64no—
76Dic3⋊Dic3—64no—
77Dic32—64no—
78Dic32—62no—
79C22×C33C22×C411yescharacteristic
80C22×C33C2×D411yescharacteristic
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q812yes—
83C22×C33#4112yes—
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—24no—
86C3×C3⋊Dic3C22⋊C414yes—
87C32×Dic3—24no—
88C32×Dic3—41no—
89C22×C3×Q8—91no—
90C22×Dic6—32no—
91C2×C4⋊Dic3—91no—
92C2×Dic3⋊C4—92no—
93C2×Dic3⋊C4—181no—
94C2×C4×Dic3—92no—
95C2×C4×Dic3—91no—
96C3×#95—92no—
97C3⋊#95—32no—
98C3⋊#95—32no—
99C3⋊#95—92no—
100C3⋊#95—92no—
101C3⋊#95—92no—
102C3⋊#95—182no—
103C23×C32—21no—
104C23×C32—22no—
105C23×C32—41no—
106C23×C32C22×D312yes—
107C23×C32C22×D311yescharacteristic
108C2×C32×C4—34no—
109C2×C32×C4—61no—
110C2×C32×C4—62no—
111C2×C32×C4—62no—
112C2×C32×C4—62no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—62no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—62no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—122no—
123C2×C3⋊Dic3—122no—
124C2×C3⋊Dic3—122no—
125C2×C3⋊Dic3—122no—
126C2×C3⋊Dic3—124no—
127C2×C3⋊Dic3Dic614yescomplemented
128C2×C3⋊Dic3C2×Dic312yes—
129C2×C3⋊Dic3#6712yes—
130C2×C3⋊Dic3#6712yes—
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—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3—42no—
147C2×C3×Dic3—42no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—122no—
157C2×C3×Dic3—122no—
158C2×C3×Dic3C4×D314yes—
159C2×C3×Dic3C2×Dic314yes—
160C3×Dic6—616no—
161#627—28no—
162#391—271noSylow
163C2×C33C2×C22⋊C411yescharacteristic
164C2×C33C4×Q812yes—
165C2×C33C22⋊Q812yes—
166C2×C33#12411yescharacteristic
167C2×C33C4⋊Q811yescharacteristic
168C2×C3×Q8—94no—
169C2×C3×Q8—182no—
170C2×C3×Q8—182no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—62no—
175C22×C3×C4—61no—
176C22×C3×C4—91no—
177C22×C3×C4—92no—
178C22×Dic3—32no—
179C22×Dic3—32no—
180C22×Dic3—32no—
181C22×Dic3—61no—
182C22×Dic3—62no—
183C22×Dic3—92no—
184C22×Dic3—91no—
185C22×Dic3—182no—
186C22×Dic3—181no—
187C22×Dic3—361no—
188C22×Dic3D3211yescharacteristic
189C22×Dic3D3212yes—
190C2×Dic6—38no—
191C2×Dic6—64no—
192C2×Dic6—64no—
193C4⋊Dic3—182no—
194Dic3⋊C4—184no—
195Dic3⋊C4—184no—
196C4×Dic3—184no—
197C4×Dic3—182no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—21no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32—42no—
215C22×C32C2×Dic612yes—
216C22×C32C2×C4×D312yes—
217C22×C32#23512yes—
218C22×C32D3×D412yes—
219C22×C32D3×D412yes—
220C22×C32#23712yes—
221C22×C32D3×Q812yes—
222C22×C32D3×Q812yes—
223C22×C32#23912yes—
224C22×C32C22×Dic311yescharacteristic
225C22×C32C2×#6711yescharacteristic
226C22×C32C2×#6711yescharacteristic
227C32×C4—64no—
228C32×C4—121no—
229C3⋊Dic3—24no—
230C3⋊Dic3—124no—
231C3⋊Dic3—122no—
232C3⋊Dic3—124no—
233C3⋊Dic3—122no—
234C3⋊Dic3—121no—
235C3⋊Dic3#21714yes—
236C3×Dic3—24no—
237C3×Dic3—24no—
238C3×Dic3—38no—
239C3×Dic3—44no—
240C3×Dic3—42no—
241C3×Dic3—42no—
242C3×Dic3—64no—
243C3×Dic3—64no—
244C3×Dic3—64no—
245C3×Dic3—124no—
246C22×Q8—91no—
247C2×C4⋊C4—271no—
248C2×C42—271no—
249#95—92no—
250#95—272no—
251C33#39111yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—22no—
254C23×C3—41no—
255C23×C3C2×D3211yescharacteristic
256C23×C3C2×D3212yes—
257C3×Q8—188no—
258C2×C3×C4—34no—
259C2×C3×C4—34no—
260C2×C3×C4—64no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—94no—
273C2×C3×C4—122no—
274C2×C3×C4—122no—
275C2×C3×C4—122no—
276C2×C3×C4—182no—
277C2×C3×C4—182no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—21no—
282C2×Dic3—34no—
283C2×Dic3—34no—
284C2×Dic3—34no—
285C2×Dic3—34no—
286C2×Dic3—34no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—64no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3—62no—
295C2×Dic3—122no—
296C2×Dic3—122no—
297C2×Dic3—182no—
298C2×Dic3—184no—
299C2×Dic3—182no—
300C2×Dic3—182no—
301C2×Dic3—182no—
302C2×Dic3—182no—
303C2×Dic3—181no—
304C2×Dic3—362no—
305C2×Dic3—362no—
306C2×Dic3—362no—
307C2×Dic3—362no—
308C2×Dic3—364no—
309C2×Dic3D3×Dic314yes—
310Dic6—616no—
311C2×C32—22no—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—22no—
323C2×C32—41no—
324C2×C32—42no—
325C2×C32—42no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32C4×Dic612yes—
329C2×C32#87812yes—
330C2×C32D3×C22⋊C412yes—
331C2×C32#88512yes—
332C2×C32#88712yes—
333C2×C32#88812yes—
334C2×C32#89612yes—
335C2×C32#94411yescharacteristic
336C2×C32Q8×Dic312yes—
337C2×C32#94612yes—
338C2×C32#94711yescharacteristic
339C2×C32C2×#21711yescharacteristic
340C2×Q8—94no—
341C2×Q8—182no—
342C2×Q8—182no—
343C22×C4—32no—
344C22×C4—31no—
345C22×C4—91no—
346C22×C4—92no—
347C22×C4—271no—
348C4⋊C4—542no—
349C42—542no—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3—21no—
353C22×C3—22no—
354C22×C3—21no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3D3×Dic612yes—
368C22×C3D3×Dic612yes—
369C22×C3#379612yes—
370C22×C3#379712yes—
371C22×C3#379912yes—
372C22×C3C4×D3211yescharacteristic
373C22×C3#380211yescharacteristic
374C22×C3C2×D3×Dic312yes—
375C22×C3#380412yes—
376C22×C3D3×#6712yes—
377C22×C3D3×#6712yes—
378C22×C3#381111yescharacteristic
379C3×C4—64no—
380C3×C4—64no—
381C3×C4—94no—
382C3×C4—124no—
383C3×C4—122no—
384C3×C4—122no—
385C3×C4—184no—
386Dic3—24no—
387Dic3—38no—
388Dic3—41no—
389Dic3—64no—
390Dic3—64no—
391Dic3—64no—
392Dic3—124no—
393Dic3—362no—
394Dic3—364no—
395Dic3—364no—
396Dic3—362no—
397Dic3—361no—
398C32—21no—
399C32—22no—
400C32—41no—
401C32#464612yescomplemented
402C32#522911yescharacteristic, complemented
403C23D3311yescharacteristic
404Q8—188no—
405C2×C4—34no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—61no—
409C2×C4—62no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—92no—
413C2×C4—94no—
414C2×C4—182no—
415C2×C4—182no—
416C2×C4—541no—
417C2×C4—542no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—42no—
433C2×C3—42no—
434C2×C3—41no—
435C2×C3#6383712yes—
436C2×C3#6383912yes—
437C2×C3#6384011yescharacteristic
438C2×C3Dic3×Dic612yes—
439C2×C3#6384312yes—
440C2×C3#6388712yes—
441C2×C3#6388812yes—
442C2×C3#6389912yes—
443C2×C3#6390712yes—
444C2×C3#6392111yescharacteristic
445C2×C3#6395311yescharacteristic
446C2×C3D3×#21712yes—
447C22D32×Dic311yescharacteristic
448C22#8895211yescharacteristic
449C22#8895311yescharacteristic
450C22#8895712yes—
451C22#8896212yes—
452C4—64no—
453C4—94no—
454C4—121no—
455C4—184no—
456C4—1081no—
457C3—22no—
458C3—21no—
459C3—41no—
460C3#1059123611yescharacteristic, complemented
461C3#1059126112yescomplemented
462C2#1172290211yescharacteristic
463C2#1172294012yes—
464C2#1172297912yes—
465C2#1172300511yescharacteristic
466C2#1172303411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#391
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
15. C2×C3×#627C4semidirect
40. C2×C3×C3⋊Dic3Q8semidirect
73. C2×#627Dic3semidirect
127. C2×C3⋊Dic3Dic6semidirect
251. C33#391semidirect
401. C32#4646semidirect
402. C32#5229semidirect
460. C3#10591236semidirect
461. C3#10591261semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×#888132
C2×#888072
#117220612
C3×#635792
#105912613
#105912363