Finite Group Databasebeta

← #13156742

Subgroups — #13156742

Order 1728.

1074 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
1C2×#88815C211yesmaximal, characteristic
2C2×#88807C211yesmaximal, characteristic
3C2×Dic3⋊(C3⋊Dic3)C212yesmaximal
4C2×C3×Dic3⋊Dic3C212yesmaximal
5C2×C3×Dic32C211yesmaximal, characteristic
6#10591349—32nomaximal
7#10591244—31nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13#88815—22no—
14#88807—22no—
15Dic3⋊(C3⋊Dic3)—24no—
16C3×Dic3⋊Dic3—24no—
17C3×Dic32C414yescomplemented
18C2×C4⋊(C3⋊Dic3)—32no—
19C2×#3750—31no—
20C2×C3×C4⋊Dic3—32no—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×C4×Dic3—32no—
23C2×#3724—31no—
24C2×Dic3⋊Dic3—32no—
25C2×Dic3⋊Dic3—32no—
26C2×Dic3⋊Dic3—32no—
27C2×Dic3⋊Dic3—32no—
28C2×Dic3⋊Dic3—62no—
29C2×Dic3⋊Dic3—61no—
30C2×Dic3⋊Dic3D312yes—
31C2×Dic32D311yescharacteristic
32C23×C33C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—22no—
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3D412yes—
40C2×C3×C3⋊Dic3Q812yes—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3C2×C414yes—
44C2×C32×Dic3D414yes—
45C2×C32×Dic3Q814yes—
46#4975—92no—
47#4935—91no—
48C22×C32×C4—32no—
49C22×C32×C4—31no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—32no—
52C22×C3⋊Dic3—62no—
53C22×C3⋊Dic3—61no—
54C22×C3⋊Dic3—121no—
55C22×C3⋊Dic3C2×D312yes—
56C22×C3⋊Dic3C2×D311yescharacteristic
57C22×C3×Dic3—22no—
58C22×C3×Dic3—21no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3—62no—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C2×D312yes—
66C22×C3×Dic3C2×D312yes—
67C4⋊(C3⋊Dic3)—64no—
68#3750—62no—
69C3×C4⋊Dic3—64no—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—38no—
72#3724—62no—
73Dic3⋊Dic3—24no—
74Dic3⋊Dic3—64no—
75Dic3⋊Dic3—68no—
76Dic3⋊Dic3—64no—
77Dic3⋊Dic3—64no—
78Dic3⋊Dic3—64no—
79Dic3⋊Dic3—64no—
80Dic32Dic314yescomplemented
81C22×C33C22×C411yescharacteristic
82C22×C33C2×D411yescharacteristic
83C22×C33C2×D412yes—
84C22×C33C2×Q812yes—
85C22×C33C2×Q811yescharacteristic
86C3⋊(C3⋊Dic3)—41no—
87C3×C3⋊Dic3—42no—
88C3×C3⋊Dic3C4⋊C414yescomplemented
89C32×Dic3—41no—
90C32×Dic3C4⋊C418yescomplemented
91C2×C3×C4⋊C4—92no—
92C2×C3×C42—91no—
93C2×C4⋊Dic3—32no—
94C2×C4⋊Dic3—91no—
95C2×C4⋊Dic3—92no—
96C2×C4⋊Dic3—92no—
97C2×C4⋊Dic3—91no—
98C2×C4⋊Dic3—182no—
99C2×Dic3⋊C4—32no—
100C2×Dic3⋊C4—92no—
101C2×Dic3⋊C4—92no—
102C2×Dic3⋊C4—92no—
103C2×Dic3⋊C4—181no—
104C2×C4×Dic3—32no—
105C23×C32—21no—
106C23×C32—22no—
107C23×C32—41no—
108C23×C32C22×D311yescharacteristic
109C23×C32C22×D312yes—
110C2×C32×C4—34no—
111C2×C32×C4—34no—
112C2×C32×C4—34no—
113C2×C32×C4—61no—
114C2×C32×C4—62no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—62no—
120C2×C3⋊Dic3—62no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—62no—
124C2×C3⋊Dic3—62no—
125C2×C3⋊Dic3—122no—
126C2×C3⋊Dic3—122no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3Dic612yes—
131C2×C3⋊Dic3D1212yes—
132C2×C3⋊Dic3C2×Dic312yes—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—34no—
143C2×C3×Dic3—42no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—122no—
155C2×C3×Dic3—122no—
156C2×C3×Dic3Dic614yes—
157C2×C3×Dic3Dic614yes—
158C2×C3×Dic3C4×D314yes—
159C2×C3×Dic3D1214yes—
160C2×C3×Dic3C2×Dic314yes—
161C2×C3×Dic3#6714yes—
162#384—271noSylow
163C2×C33C2×C4⋊C411yescharacteristic
164C2×C33C2×C4⋊C412yes—
165C2×C33#12711yescharacteristic
166C2×C33C4⋊Q811yescharacteristic
167C2×C33C4⋊Q812yes—
168C22×C3×C4—32no—
169C22×C3×C4—32no—
170C22×C3×C4—32no—
171C22×C3×C4—62no—
172C22×C3×C4—61no—
173C22×C3×C4—91no—
174C22×C3×C4—92no—
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—32no—
178C22×Dic3—61no—
179C22×Dic3—62no—
180C22×Dic3—92no—
181C22×Dic3—91no—
182C22×Dic3—181no—
183C22×Dic3—182no—
184C22×Dic3—361no—
185C22×Dic3D3211yescharacteristic
186C22×Dic3D3212yes—
187C3×C4⋊C4—184no—
188C3×C42—94no—
189C4⋊Dic3—64no—
190C4⋊Dic3—182no—
191C4⋊Dic3—182no—
192C4⋊Dic3—184no—
193C4⋊Dic3—184no—
194C4⋊Dic3—188no—
195Dic3⋊C4—64no—
196Dic3⋊C4—184no—
197Dic3⋊C4—184no—
198Dic3⋊C4—184no—
199Dic3⋊C4—184no—
200C4×Dic3—38no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—22no—
207C22×C32—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—21no—
213C22×C32—41no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—42no—
217C22×C32—41no—
218C22×C32C2×Dic611yescharacteristic
219C22×C32C2×Dic612yes—
220C22×C32C2×Dic612yes—
221C22×C32C2×C4×D312yes—
222C22×C32C2×D1211yescharacteristic
223C22×C32C2×D1212yes—
224C22×C32D3×D412yes—
225C22×C32D3×D412yes—
226C22×C32D3×Q812yes—
227C22×C32D3×Q812yes—
228C22×C32C22×Dic311yescharacteristic
229C22×C32C2×#6712yes—
230C32×C4—38no—
231C32×C4—121no—
232C3⋊Dic3—42no—
233C3⋊Dic3—124no—
234C3⋊Dic3—122no—
235C3⋊Dic3—124no—
236C3⋊Dic3—121no—
237C3⋊Dic3—122no—
238C3⋊Dic3C4⋊Dic314yescomplemented
239C3×Dic3—28no—
240C3×Dic3—38no—
241C3×Dic3—42no—
242C3×Dic3—42no—
243C3×Dic3—64no—
244C3×Dic3—124no—
245C3×Dic3—122no—
246C3×Dic3—122no—
247C3×Dic3Dic3⋊C418yescomplemented
248C3×Dic3C4⋊Dic318yescomplemented
249C2×C4⋊C4—92no—
250C2×C4⋊C4—271no—
251C2×C4⋊C4—272no—
252C2×C4⋊C4—271no—
253C2×C42—91no—
254C33#38411yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—22no—
257C23×C3—41no—
258C23×C3C2×D3211yescharacteristic
259C23×C3C2×D3212yes—
260C2×C3×C4—34no—
261C2×C3×C4—34no—
262C2×C3×C4—34no—
263C2×C3×C4—34no—
264C2×C3×C4—34no—
265C2×C3×C4—34no—
266C2×C3×C4—64no—
267C2×C3×C4—64no—
268C2×C3×C4—64no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—92no—
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—122no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279C2×C3×C4—182no—
280C2×Dic3—22no—
281C2×Dic3—21no—
282C2×Dic3—34no—
283C2×Dic3—34no—
284C2×Dic3—34no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—64no—
289C2×Dic3—62no—
290C2×Dic3—62no—
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—364no—
308C2×Dic3—362no—
309C2×Dic3D3×Dic314yes—
310C2×Dic3#62614yes—
311C2×Dic3#62714yes—
312C2×C32—22no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—22no—
322C2×C32—22no—
323C2×C32—21no—
324C2×C32—42no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—42no—
328C2×C32—41no—
329C2×C32#86912yes—
330C2×C32#86911yescharacteristic
331C2×C32#87411yescharacteristic
332C2×C32#88812yes—
333C2×C32#88812yes—
334C2×C32D3×C4⋊C412yes—
335C2×C32D3×C4⋊C412yes—
336C2×C32C2×Dic3⋊C412yes—
337C2×C32C2×C4⋊Dic311yescharacteristic
338C2×C32C2×C4⋊Dic312yes—
339C2×C32#94012yes—
340C2×C32#94412yes—
341C22×C4—32no—
342C22×C4—31no—
343C22×C4—91no—
344C22×C4—92no—
345C22×C4—271no—
346C4⋊C4—184no—
347C4⋊C4—544no—
348C4⋊C4—542no—
349C4⋊C4—542no—
350C42—94no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—41no—
364C22×C3—42no—
365C22×C3—41no—
366C22×C3—42no—
367C22×C3—41no—
368C22×C3D3×Dic612yes—
369C22×C3D3×Dic612yes—
370C22×C3D3×Dic612yes—
371C22×C3#379711yescharacteristic
372C22×C3C4×D3211yescharacteristic
373C22×C3D3×D1212yes—
374C22×C3D3×D1212yes—
375C22×C3C2×D3×Dic312yes—
376C22×C3C2×#62612yes—
377C22×C3C2×#62712yes—
378C22×C3D3×#6712yes—
379C22×C3#381111yescharacteristic
380C3×C4—38no—
381C3×C4—38no—
382C3×C4—68no—
383C3×C4—94no—
384C3×C4—122no—
385C3×C4—122no—
386C3×C4—362no—
387Dic3—38no—
388Dic3—41no—
389Dic3—64no—
390Dic3—124no—
391Dic3—122no—
392Dic3—122no—
393Dic3—364no—
394Dic3—362no—
395Dic3—364no—
396Dic3—362no—
397Dic3—361no—
398Dic3Dic3⋊Dic318yescomplemented
399C32—21no—
400C32—22no—
401C32—41no—
402C32#493511yescharacteristic, complemented
403C32#497512yescomplemented
404C23D3311yescharacteristic
405C2×C4—34no—
406C2×C4—34no—
407C2×C4—34no—
408C2×C4—62no—
409C2×C4—61no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—92no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—182no—
416C2×C4—542no—
417C2×C4—541no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—21no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—22no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3—42no—
432C2×C3—41no—
433C2×C3—42no—
434C2×C3—41no—
435C2×C3#6383912yes—
436C2×C3#6384011yescharacteristic
437C2×C3#6386612yes—
438C2×C3D3×Dic3⋊C412yes—
439C2×C3#6388311yescharacteristic
440C2×C3D3×C4⋊Dic312yes—
441C2×C3D3×C4⋊Dic312yes—
442C2×C3#6391112yes—
443C2×C3#6391812yes—
444C2×C3#6391812yes—
445C2×C3C2×Dic3⋊Dic312yes—
446C2×C3#6397911yescharacteristic
447C22D32×Dic311yescharacteristic
448C22D3×#62612yes—
449C22#8895511yescharacteristic
450C22D3×#62712yes—
451C22#8895711yescharacteristic
452C4—38no—
453C4—94no—
454C4—121no—
455C4—362no—
456C4—1081no—
457C3—21no—
458C3—22no—
459C3—41no—
460C3#1059124411yescharacteristic, complemented
461C3#1059134912yescomplemented
462C2D3×Dic3⋊Dic312yes—
463C2#1172290811yescharacteristic
464C2#1172303011yescharacteristic
465C2#1172303312yes—
466C2#1172303611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#384
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
17. C3×Dic32C4semidirect
80. Dic32Dic3semidirect
88. C3×C3⋊Dic3C4⋊C4semidirect
90. C32×Dic3C4⋊C4semidirect
238. C3⋊Dic3C4⋊Dic3semidirect
247. C3×Dic3Dic3⋊C4semidirect
248. C3×Dic3C4⋊Dic3semidirect
254. C33#384semidirect
398. Dic3Dic3⋊Dic3semidirect
402. C32#4935semidirect
403. C32#4975semidirect
460. C3#10591244semidirect
461. C3#10591349semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
C2×#888072
C2×Dic3⋊(C3⋊Dic3)2
C2×C3×Dic3⋊Dic32
C2×C3×Dic322
#105913493
#105912443