Finite Group Databasebeta

← #13156736

Subgroups — #13156736

Order 1728.

930 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88815C211yesmaximal, characteristic
2C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3724C211yesmaximal, characteristic
4#11722061C212yesmaximal
5C3×#63579C212yesmaximal
6#10591360—31nomaximal
7#10591245—32nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13#88815—22no—
14Dic3×C3⋊Dic3C414yescomplemented
15C3×#3724—22no—
16C2×C4×C3⋊Dic3—31no—
17C2×C3×Dic3⋊C4—32no—
18C2×#3724—31no—
19C2×#3724D311yescharacteristic
20C2×Dic3⋊Dic3—32no—
21C2×Dic32—32no—
22C2×Dic32—61no—
23#63658—32no—
24C3×C3⋊#95—32no—
25C3×C3⋊#95—32no—
26#63579—32no—
27#63579—32no—
28#63579—62no—
29#63579D312yes—
30C23×C33C2311yescharacteristic
31C2×C3⋊(C3⋊Dic3)—21no—
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)C2×C412yes—
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⋊Dic3Q812yes—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C412yes—
46#4977—91no—
47#4650—92no—
48C22×C32×C4—31no—
49C22×C32×C4—32no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—32no—
52C22×C3⋊Dic3—61no—
53C22×C3⋊Dic3—62no—
54C22×C3⋊Dic3—121no—
55C22×C3⋊Dic3C2×D312yes—
56C22×C3⋊Dic3C2×D311yescharacteristic
57C22×C3×Dic3—21no—
58C22×C3×Dic3—22no—
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—34no—
68C3×Dic3⋊C4—64no—
69#3724—22no—
70#3724—62no—
71Dic3⋊Dic3—64no—
72Dic32—38no—
73Dic32—64no—
74C22×C33C22×C411yescharacteristic
75C22×C33C2×D411yescharacteristic
76C22×C33C2×Q811yescharacteristic
77C22×C33#4111yescharacteristic
78C22×C33#4111yescharacteristic
79C22×C33#4111yescharacteristic
80C22×C33#4111yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C3×C3⋊Dic3—42no—
83C3×C3⋊Dic3C4⋊C414yes—
84C32×Dic3—22no—
85C32×Dic3—42no—
86C2×C3×C4⋊C4—91no—
87C2×C4⋊Dic3—91no—
88C2×Dic3⋊C4—32no—
89C2×Dic3⋊C4—92no—
90C2×C4×Dic3—92no—
91C2×C4×Dic3—91no—
92C2×C4×Dic3—181no—
93C3×#95—92no—
94C3⋊#95—32no—
95C3⋊#95—32no—
96C3⋊#95—92no—
97C3⋊#95—92no—
98C3⋊#95—92no—
99C3⋊#95—182no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32C22×D311yescharacteristic
104C23×C32C22×D312yes—
105C2×C32×C4—32no—
106C2×C32×C4—61no—
107C2×C32×C4—61no—
108C2×C32×C4—62no—
109C2×C32×C4—62no—
110C2×C32×C4—62no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3—34no—
115C2×C3⋊Dic3—32no—
116C2×C3⋊Dic3—62no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—62no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—62no—
124C2×C3⋊Dic3—122no—
125C2×C3⋊Dic3—122no—
126C2×C3⋊Dic3—122no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3Dic612yes—
130C2×C3⋊Dic3C4×D312yes—
131C2×C3⋊Dic3#6712yes—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—34no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—42no—
148C2×C3×Dic3—42no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—62no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—122no—
160C2×C3×Dic3—122no—
161C2×C3×Dic3C4×D314yes—
162#382—271noSylow
163C2×C33C2×C4⋊C411yescharacteristic
164C2×C33#11711yescharacteristic
165C2×C33#11711yescharacteristic
166C2×C33#12411yescharacteristic
167C2×C33#12411yescharacteristic
168C2×C33#12511yescharacteristic
169C2×C33#12511yescharacteristic
170C22×C3×C4—32no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—61no—
174C22×C3×C4—62no—
175C22×C3×C4—91no—
176C22×C3×C4—92no—
177C22×Dic3—32no—
178C22×Dic3—32no—
179C22×Dic3—32no—
180C22×Dic3—61no—
181C22×Dic3—62no—
182C22×Dic3—92no—
183C22×Dic3—91no—
184C22×Dic3—181no—
185C22×Dic3—182no—
186C22×Dic3—361no—
187C22×Dic3D3212yes—
188C22×Dic3D3211yescharacteristic
189C3×C4⋊C4—182no—
190C4⋊Dic3—182no—
191Dic3⋊C4—64no—
192Dic3⋊C4—184no—
193C4×Dic3—98no—
194C4×Dic3—94no—
195C4×Dic3—184no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—22no—
205C22×C32—22no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—22no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32C2×Dic611yescharacteristic
218C22×C32C2×C4×D311yescharacteristic
219C22×C32C2×C4×D312yes—
220C22×C32#23512yes—
221C22×C32#23512yes—
222C22×C32D3×D412yes—
223C22×C32#23712yes—
224C22×C32#23711yescharacteristic
225C22×C32#23712yes—
226C22×C32#23711yescharacteristic
227C22×C32D3×Q812yes—
228C22×C32#23911yescharacteristic
229C22×C32#23911yescharacteristic
230C22×C32C2×#6711yescharacteristic
231C32×C4—62no—
232C32×C4—122no—
233C3⋊Dic3—42no—
234C3⋊Dic3—62no—
235C3⋊Dic3—64no—
236C3⋊Dic3—64no—
237C3⋊Dic3—124no—
238C3⋊Dic3—124no—
239C3⋊Dic3Dic3⋊C414yes—
240C3×Dic3—24no—
241C3×Dic3—24no—
242C3×Dic3—38no—
243C3×Dic3—44no—
244C3×Dic3—42no—
245C3×Dic3—42no—
246C3×Dic3—64no—
247C3×Dic3—124no—
248C3×Dic3—122no—
249C3×Dic3—122no—
250C2×C4⋊C4—91no—
251C2×C4⋊C4—271no—
252C2×C42—271no—
253#95—92no—
254#95—272no—
255C33#38211yescharacteristic, complemented, Sylow
256C23×C3—21no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3C2×D3212yes—
260C23×C3C2×D3211yescharacteristic
261C2×C3×C4—34no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—64no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—62no—
273C2×C3×C4—62no—
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—92no—
277C2×C3×C4—122no—
278C2×C3×C4—122no—
279C2×C3×C4—182no—
280C2×C3×C4—182no—
281C2×C3×C4—182no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—34no—
288C2×Dic3—34no—
289C2×Dic3—34no—
290C2×Dic3—62no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—64no—
294C2×Dic3—62no—
295C2×Dic3—62no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—94no—
301C2×Dic3—92no—
302C2×Dic3—122no—
303C2×Dic3—122no—
304C2×Dic3—182no—
305C2×Dic3—184no—
306C2×Dic3—182no—
307C2×Dic3—182no—
308C2×Dic3—181no—
309C2×Dic3—181no—
310C2×Dic3—182no—
311C2×Dic3—182no—
312C2×Dic3—362no—
313C2×Dic3—362no—
314C2×Dic3—362no—
315C2×Dic3—362no—
316C2×Dic3—362no—
317C2×Dic3#62412yes—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—22no—
325C2×C32—22no—
326C2×C32—21no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—22no—
332C2×C32—41no—
333C2×C32—41no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32#87212yes—
340C2×C32#87712yes—
341C2×C32#88512yes—
342C2×C32#88512yes—
343C2×C32#88912yes—
344C2×C32#88912yes—
345C2×C32#89011yescharacteristic
346C2×C32#89011yescharacteristic
347C2×C32D3×C4⋊C412yes—
348C2×C32#89211yescharacteristic
349C2×C32#89211yescharacteristic
350C2×C32C2×Dic3⋊C411yescharacteristic
351C2×C32#93611yescharacteristic
352C2×C32#94711yescharacteristic
353C22×C4—31no—
354C22×C4—32no—
355C22×C4—91no—
356C22×C4—92no—
357C22×C4—271no—
358C4⋊C4—182no—
359C4⋊C4—542no—
360C42—274no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—21no—
364C22×C3—22no—
365C22×C3—21no—
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3D3×Dic612yes—
383C22×C3#379612yes—
384C22×C3#379711yescharacteristic
385C22×C3#379811yescharacteristic
386C22×C3#379811yescharacteristic
387C22×C3#379912yes—
388C22×C3C4×D3212yes—
389C22×C3#380211yescharacteristic
390C22×C3#380412yes—
391C22×C3#380512yes—
392C22×C3#380511yescharacteristic
393C22×C3#380511yescharacteristic
394C22×C3C2×#62411yescharacteristic
395C22×C3D3×#6712yes—
396C3×C4—64no—
397C3×C4—64no—
398C3×C4—94no—
399C3×C4—124no—
400C3×C4—122no—
401C3×C4—122no—
402C3×C4—362no—
403Dic3—22no—
404Dic3—38no—
405Dic3—42no—
406Dic3—64no—
407Dic3—124no—
408Dic3—122no—
409Dic3—122no—
410Dic3—184no—
411Dic3—184no—
412Dic3—182no—
413Dic3—364no—
414Dic3—364no—
415C32—22no—
416C32—21no—
417C32—41no—
418C32#465012yescomplemented
419C32#497711yescharacteristic, complemented
420C23D3311yescharacteristic
421C2×C4—32no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—182no—
433C2×C4—272no—
434C2×C4—541no—
435C2×C4—541no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—22no—
442C2×C3—21no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—22no—
447C2×C3—21no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3#6383612yes—
458C2×C3#6384512yes—
459C2×C3#6384612yes—
460C2×C3#6387011yescharacteristic
461C2×C3#6387011yescharacteristic
462C2×C3#6387411yescharacteristic
463C2×C3D3×Dic3⋊C412yes—
464C2×C3#6387812yes—
465C2×C3#6390011yescharacteristic
466C2×C3#6390712yes—
467C2×C3#6391511yescharacteristic
468C2×C3#6391511yescharacteristic
469C2×C3#6395711yescharacteristic
470C2×C3#6395912yes—
471C22D3×#62411yescharacteristic
472C22#8895311yescharacteristic
473C22#8895711yescharacteristic
474C22#8895911yescharacteristic
475C22#8896311yescharacteristic
476C22#8896411yescharacteristic
477C22#8896511yescharacteristic
478C4—62no—
479C4—94no—
480C4—122no—
481C4—362no—
482C4—542no—
483C3—22no—
484C3—21no—
485C3—41no—
486C3#1059124512yescomplemented
487C3#1059136011yescharacteristic, complemented
488C2#1172291111yescharacteristic
489C2#1172291711yescharacteristic
490C2#1172291811yescharacteristic
491C2#1172300611yescharacteristic
492C2#1172300711yescharacteristic
493C2#1172301511yescharacteristic
494C2#1172301811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#382
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
14. Dic3×C3⋊Dic3C4semidirect
255. C33#382semidirect
418. C32#4650semidirect
419. C32#4977semidirect
486. C3#10591245semidirect
487. C3#10591360semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
C2×Dic3×C3⋊Dic32
C2×C3×#37242
#117220612
C3×#635792
#105913603
#105912453