Finite Group Databasebeta

← #13156837

Subgroups — #13156837

Order 1728.

994 conjugacy classes of subgroups.

The subgroup lattice has 504 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
4C2×C3×Dic3⋊Dic3C212yesmaximal
5#11722061C212yesmaximal
6#10591337—31nomaximal
7#10591265—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⋊Dic3—22no—
15C3×#3724C414yescomplemented
16C3×Dic3⋊Dic3—24no—
17C2×C4×C3⋊Dic3—31no—
18C2×C3×C4⋊Dic3—32no—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×Dic3⋊C4—32no—
21C2×#3724—31no—
22C2×#3724D311yescharacteristic
23C2×Dic3⋊Dic3—32no—
24C2×Dic3⋊Dic3D312yes—
25C2×Dic32—32no—
26C2×Dic32—61no—
27#63658—32no—
28#63579—32no—
29#63579—32no—
30#63579—62no—
31C23×C33C2311yescharacteristic
32C2×C3⋊(C3⋊Dic3)—21no—
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)Q812yes—
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×Dic3—21no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C414yes—
46C2×C32×Dic3D412yes—
47#4984—91no—
48#4976—92no—
49C22×C32×C4—31no—
50C22×C32×C4—32no—
51C22×C3⋊Dic3—31no—
52C22×C3⋊Dic3—32no—
53C22×C3⋊Dic3—61no—
54C22×C3⋊Dic3—62no—
55C22×C3⋊Dic3—121no—
56C22×C3⋊Dic3C2×D312yes—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C22×C3×Dic3—21no—
59C22×C3×Dic3—22no—
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—
68C4×C3⋊Dic3—62no—
69C3×C4⋊Dic3—64no—
70C3×Dic3⋊C4—38no—
71C3×Dic3⋊C4—64no—
72#3724—62no—
73#3724Dic314yescomplemented
74Dic3⋊Dic3—24no—
75Dic3⋊Dic3—64no—
76Dic32—64no—
77Dic32—64no—
78C22×C33C22×C411yescharacteristic
79C22×C33C2×D411yescharacteristic
80C22×C33C2×D411yescharacteristic
81C22×C33C2×Q811yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4111yescharacteristic
85C3⋊(C3⋊Dic3)—22no—
86C3×C3⋊Dic3—42no—
87C3×C3⋊Dic3C4⋊C414yescomplemented
88C32×Dic3—24no—
89C32×Dic3—22no—
90C2×C3×C4⋊C4—91no—
91C2×C3×C4⋊C4—92no—
92C2×C4⋊Dic3—32no—
93C2×C4⋊Dic3—91no—
94C2×Dic3⋊C4—32no—
95C2×Dic3⋊C4—32no—
96C2×Dic3⋊C4—92no—
97C2×C4×Dic3—92no—
98C2×C4×Dic3—91no—
99C2×C4×Dic3—181no—
100C3⋊#95—92no—
101C3⋊#95—92no—
102C3⋊#95—92no—
103C3⋊#95—182no—
104C23×C32—22no—
105C23×C32—21no—
106C23×C32—41no—
107C23×C32C22×D311yescharacteristic
108C23×C32C22×D312yes—
109C2×C32×C4—34no—
110C2×C32×C4—32no—
111C2×C32×C4—61no—
112C2×C32×C4—61no—
113C2×C32×C4—62no—
114C2×C32×C4—62no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—32no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—62no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—62no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—62no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3—122no—
130C2×C3⋊Dic3—122no—
131C2×C3⋊Dic3—122no—
132C2×C3⋊Dic3—122no—
133C2×C3⋊Dic3Dic612yes—
134C2×C3⋊Dic3D1212yes—
135C2×C3⋊Dic3C2×Dic312yes—
136C2×C3×Dic3—22no—
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—24no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—34no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—42no—
150C2×C3×Dic3—42no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—122no—
162C2×C3×Dic3—122no—
163C2×C3×Dic3C4×D314yes—
164C2×C3×Dic3C2×Dic314yes—
165C2×C3×Dic3#6714yes—
166#389—271noSylow
167C2×C33C2×C4⋊C411yescharacteristic
168C2×C33C4×D411yescharacteristic
169C2×C33C4×Q811yescharacteristic
170C2×C33#12111yescharacteristic
171C2×C33C22⋊Q811yescharacteristic
172C2×C33#12511yescharacteristic
173C2×C33C4⋊Q811yescharacteristic
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—32no—
177C22×C3×C4—61no—
178C22×C3×C4—62no—
179C22×C3×C4—91no—
180C22×C3×C4—92no—
181C22×Dic3—32no—
182C22×Dic3—32no—
183C22×Dic3—32no—
184C22×Dic3—61no—
185C22×Dic3—62no—
186C22×Dic3—91no—
187C22×Dic3—92no—
188C22×Dic3—181no—
189C22×Dic3—182no—
190C22×Dic3—361no—
191C22×Dic3D3212yes—
192C22×Dic3D3211yescharacteristic
193C3×C4⋊C4—94no—
194C3×C4⋊C4—184no—
195C4⋊Dic3—64no—
196C4⋊Dic3—182no—
197Dic3⋊C4—38no—
198Dic3⋊C4—64no—
199Dic3⋊C4—184no—
200C4×Dic3—182no—
201C4×Dic3—184no—
202C4×Dic3—184no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—22no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32C2×Dic611yescharacteristic
225C22×C32C2×C4×D312yes—
226C22×C32C2×D1211yescharacteristic
227C22×C32#23512yes—
228C22×C32D3×D412yes—
229C22×C32D3×D411yescharacteristic
230C22×C32#23712yes—
231C22×C32#23711yescharacteristic
232C22×C32D3×Q812yes—
233C22×C32D3×Q812yes—
234C22×C32D3×Q811yescharacteristic
235C22×C32#23911yescharacteristic
236C22×C32C22×Dic311yescharacteristic
237C22×C32C2×#6712yes—
238C32×C4—62no—
239C32×C4—64no—
240C3⋊Dic3—42no—
241C3⋊Dic3—62no—
242C3⋊Dic3—64no—
243C3⋊Dic3—64no—
244C3⋊Dic3—124no—
245C3⋊Dic3—124no—
246C3⋊Dic3C4⋊Dic314yescomplemented
247C3×Dic3—24no—
248C3×Dic3—24no—
249C3×Dic3—24no—
250C3×Dic3—24no—
251C3×Dic3—38no—
252C3×Dic3—44no—
253C3×Dic3—64no—
254C3×Dic3—124no—
255C3×Dic3—122no—
256C3×Dic3—122no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—92no—
259C2×C4⋊C4—271no—
260C2×C42—271no—
261#95—272no—
262C33#38911yescharacteristic, complemented, Sylow
263C23×C3—21no—
264C23×C3—22no—
265C23×C3—41no—
266C23×C3C2×D3212yes—
267C23×C3C2×D3211yescharacteristic
268C2×C3×C4—34no—
269C2×C3×C4—34no—
270C2×C3×C4—34no—
271C2×C3×C4—62no—
272C2×C3×C4—64no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—92no—
284C2×C3×C4—122no—
285C2×C3×C4—122no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4—182no—
289C2×Dic3—22no—
290C2×Dic3—21no—
291C2×Dic3—22no—
292C2×Dic3—21no—
293C2×Dic3—34no—
294C2×Dic3—34no—
295C2×Dic3—34no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—64no—
300C2×Dic3—62no—
301C2×Dic3—62no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—92no—
307C2×Dic3—94no—
308C2×Dic3—122no—
309C2×Dic3—122no—
310C2×Dic3—182no—
311C2×Dic3—184no—
312C2×Dic3—182no—
313C2×Dic3—182no—
314C2×Dic3—181no—
315C2×Dic3—182no—
316C2×Dic3—182no—
317C2×Dic3—181no—
318C2×Dic3—362no—
319C2×Dic3—362no—
320C2×Dic3—362no—
321C2×Dic3—362no—
322C2×Dic3—362no—
323C2×Dic3D3×Dic314yes—
324C2×Dic3#62512yes—
325C2×C32—21no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—22no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32#88712yes—
347C2×C32#88811yescharacteristic
348C2×C32#88912yes—
349C2×C32#89011yescharacteristic
350C2×C32D3×C4⋊C412yes—
351C2×C32#89511yescharacteristic
352C2×C32#89612yes—
353C2×C32#89711yescharacteristic
354C2×C32C2×C4⋊Dic311yescharacteristic
355C2×C32C4×#6712yes—
356C2×C32D4×Dic311yescharacteristic
357C2×C32#93912yes—
358C2×C32#94412yes—
359C2×C32Q8×Dic311yescharacteristic
360C22×C4—32no—
361C22×C4—31no—
362C22×C4—91no—
363C22×C4—92no—
364C22×C4—271no—
365C4⋊C4—94no—
366C4⋊C4—184no—
367C4⋊C4—542no—
368C42—542no—
369C22×C3—21no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—22no—
379C22×C3—21no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3D3×Dic612yes—
391C22×C3#379612yes—
392C22×C3#379711yescharacteristic
393C22×C3#379711yescharacteristic
394C22×C3#379712yes—
395C22×C3#379811yescharacteristic
396C22×C3C4×D3211yescharacteristic
397C22×C3D3×D1212yes—
398C22×C3C2×D3×Dic312yes—
399C22×C3#380412yes—
400C22×C3#380511yescharacteristic
401C22×C3C2×#62511yescharacteristic
402C22×C3D3×#6712yes—
403C22×C3#381111yescharacteristic
404C3×C4—64no—
405C3×C4—64no—
406C3×C4—64no—
407C3×C4—64no—
408C3×C4—94no—
409C3×C4—124no—
410C3×C4—362no—
411Dic3—24no—
412Dic3—22no—
413Dic3—38no—
414Dic3—64no—
415Dic3—124no—
416Dic3—122no—
417Dic3—122no—
418Dic3—184no—
419Dic3—182no—
420Dic3—184no—
421Dic3—364no—
422Dic3—364no—
423C32—22no—
424C32—21no—
425C32—41no—
426C32#497612yescomplemented
427C32#498411yescharacteristic, complemented
428C23D3311yescharacteristic
429C2×C4—34no—
430C2×C4—32no—
431C2×C4—62no—
432C2×C4—61no—
433C2×C4—62no—
434C2×C4—61no—
435C2×C4—92no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—182no—
439C2×C4—182no—
440C2×C4—182no—
441C2×C4—272no—
442C2×C4—541no—
443C2×C4—541no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—22no—
451C2×C3—21no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3#6383811yescharacteristic
466C2×C3#6383912yes—
467C2×C3#6384312yes—
468C2×C3#6384612yes—
469C2×C3#6386511yescharacteristic
470C2×C3#6387011yescharacteristic
471C2×C3#6387311yescharacteristic
472C2×C3#6388311yescharacteristic
473C2×C3#6388612yes—
474C2×C3D3×C4⋊Dic312yes—
475C2×C3#6389512yes—
476C2×C3C4×#62511yescharacteristic
477C2×C3Dic3×#6712yes—
478C2×C3#6398011yescharacteristic
479C22D32×Dic311yescharacteristic
480C22D3×#62511yescharacteristic
481C22#8895511yescharacteristic
482C22#8895711yescharacteristic
483C22#8895811yescharacteristic
484C22#8895911yescharacteristic
485C22#8896411yescharacteristic
486C4—64no—
487C4—62no—
488C4—94no—
489C4—362no—
490C4—542no—
491C3—22no—
492C3—21no—
493C3—41no—
494C3#1059126512yescomplemented
495C3#1059133711yescharacteristic, complemented
496C2#1172290811yescharacteristic
497C2Dic3×#62511yescharacteristic
498C2#1172294111yescharacteristic
499C2#1172295511yescharacteristic
500C2#1172298211yescharacteristic
501C2#1172301511yescharacteristic
502C2#1172303211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#389
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
15. C3×#3724C4semidirect
73. #3724Dic3semidirect
87. C3×C3⋊Dic3C4⋊C4semidirect
246. C3⋊Dic3C4⋊Dic3semidirect
262. C33#389semidirect
426. C32#4976semidirect
427. C32#4984semidirect
494. C3#10591265semidirect
495. C3#10591337semidirect

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
C2×C3×Dic3⋊Dic32
#117220612
#105913373
#105912653