Finite Group Databasebeta

← #13156326

Subgroups — #13156326

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723536C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722050C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591631—31nomaximal
9#10590817—31nomaximal
10#10590496—31nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C2×#6293—21no—
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22Q8×C3⋊Dic3—31no—
23#64097—31no—
24#63919—31no—
25#63919—31no—
26#63919—61no—
27#63843—31no—
28#63843—31no—
29#63843—61no—
30#63641—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#818—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#807—31no—
37#63576—31no—
38#63575—31no—
39#63575—31no—
40#63575D311yescharacteristic
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63566D311yescharacteristic
45#63565D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×Dic6—21no—
54C32×Dic6Q812yescomplemented
55#6293—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#4875—91no—
59#4793—91no—
60#4793—91no—
61C2×C32×Q8—31no—
62C2×#634—61no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×D311yescharacteristic
65C2×C3×Dic6C2×D311yescharacteristic
66C2×#627—61no—
67C2×#627—61no—
68C2×#627—62no—
69C2×C32×C8—31no—
70C32×C4⋊C4—31no—
71C4⋊(C3⋊Dic3)—61no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74#3750—61no—
75C4×C3⋊Dic3—31no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—61no—
79C4×C3⋊Dic3—61no—
80C4×C3⋊Dic3—61no—
81C4×C3⋊Dic3—61no—
82C4×C3⋊Dic3—121no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—61no—
100#3724—61no—
101#3724—62no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—62no—
105Dic32—61no—
106Dic32—61no—
107Dic32—62no—
108C22×C33C2×D411yescharacteristic
109C33×C4C2×Q811yescharacteristic
110C33×C4#4111yescharacteristic
111C3⋊(C3⋊Dic3)—21no—
112C3⋊(C3⋊Dic3)Q1612yes—
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—22no—
116C32×Dic3—41no—
117Q8×Dic3—91no—
118Q8×Dic3—91no—
119Q8×Dic3—181no—
120#888—91no—
121#888—91no—
122#888—91no—
123#888—181no—
124#887—91no—
125C3×#107—91no—
126C3×#103—91no—
127C3×#103—91no—
128#835—31no—
129#835—31no—
130#818—31no—
131#818—91no—
132#816—31no—
133Dic3⋊C8—91no—
134Dic3⋊C8—91no—
135Dic3⋊C8—91no—
136Dic3⋊C8—181no—
137#810—31no—
138#807—31no—
139#807—91no—
140#807—91no—
141C32×Q8—32no—
142C32×Q8—61no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4—61no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—31no—
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—31no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—121no—
167C2×C3⋊Dic3—121no—
168#634—122no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C3×Dic6—22no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—22no—
185C3×Dic6Dic612yescomplemented
186C3×Dic6Dic612yescomplemented
187#627—124no—
188#627—122no—
189#627—122no—
190C32×C8—61no—
191C3⋊(C3⋊C8)—21no—
192C3×C3⋊C8—21no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—41no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—121no—
198#477—271noSylow
199C2×C33C22⋊Q811yescharacteristic
200C2×C33C2×Q1611yescharacteristic
201C2×C33AGL(1,ℤ/8)11yescharacteristic
202C2×C3×Q8—31no—
203C2×C3×Q8—31no—
204C2×C3×Q8—61no—
205C2×Dic6—181no—
206C2×Dic6—181no—
207C2×Dic6—181no—
208C2×Dic6—182no—
209C2×Dic6D3211yescharacteristic
210C2×C3×C8—31no—
211C2×C3×C8—31no—
212C2×C3×C8—61no—
213C2×C3×C8—91no—
214C3×C4⋊C4—31no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C4⋊Dic3—31no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—31no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3—181no—
226C4⋊Dic3—181no—
227C4⋊Dic3—182no—
228C4⋊Dic3D3211yescharacteristic
229Dic3⋊C4—181no—
230Dic3⋊C4—181no—
231Dic3⋊C4—181no—
232Dic3⋊C4—181no—
233Dic3⋊C4—182no—
234C4×Dic3—91no—
235C4×Dic3—91no—
236C4×Dic3—91no—
237C4×Dic3—181no—
238C4×Dic3—181no—
239C4×Dic3—181no—
240C4×Dic3—181no—
241C4×Dic3—181no—
242C4×Dic3—181no—
243C4×Dic3—182no—
244C4×Dic3—361no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8D3211yescharacteristic
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32D3×D411yescharacteristic
254C22×C32D3×D411yescharacteristic
255C22×C32D3×D411yescharacteristic
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—62no—
265C32×C4—121no—
266C32×C4C2×Dic611yescharacteristic
267C32×C4C2×Dic611yescharacteristic
268C32×C4#23511yescharacteristic
269C32×C4#23711yescharacteristic
270C32×C4#23711yescharacteristic
271C32×C4D3×Q811yescharacteristic
272C3⋊Dic3—32no—
273C3⋊Dic3—32no—
274C3⋊Dic3—32no—
275C3⋊Dic3—41no—
276C3⋊Dic3—41no—
277C3⋊Dic3—62no—
278C3⋊Dic3—62no—
279C3⋊Dic3—62no—
280C3⋊Dic3—61no—
281C3⋊Dic3—62no—
282C3⋊Dic3—61no—
283C3⋊Dic3—61no—
284C3⋊Dic3—122no—
285C3⋊Dic3—121no—
286C3⋊Dic3—121no—
287C3⋊Dic3—122no—
288C3×Dic3—24no—
289C3×Dic3—22no—
290C3×Dic3—22no—
291C3×Dic3—42no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—122no—
295C3×Dic3—122no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C4⋊Q8—271no—
301C4×Q8—271no—
302#107—91no—
303#107—271no—
304C4⋊C8—271no—
305#103—91no—
306#103—91no—
307C33#47711yescharacteristic, complemented, Sylow
308C3×Q8—32no—
309C3×Q8—32no—
310C3×Q8—62no—
311C3×Q8—62no—
312C3×Q8—61no—
313C3×Q8—61no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—62no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—62no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341C2×Dic3—91no—
342C2×Dic3—91no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—361no—
350C2×Dic3—361no—
351Dic6—21no—
352Dic6—362no—
353Dic6—364no—
354Dic6—362no—
355Dic6—362no—
356Dic6#62712yescomplemented
357C3×C8—61no—
358C3×C8—61no—
359C3×C8—121no—
360C3×C8—181no—
361C3⋊C8—21no—
362C3⋊C8—61no—
363C3⋊C8—61no—
364C3⋊C8—121no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32#87811yescharacteristic
378C2×C32#87811yescharacteristic
379C2×C32#89611yescharacteristic
380C2×C32#91111yescharacteristic
381C2×C32#91411yescharacteristic
382C2×C32#91411yescharacteristic
383C2×C32D3×Q1611yescharacteristic
384C2×C32D3×Q1611yescharacteristic
385C2×C32D3×Q1611yescharacteristic
386C2×Q8—31no—
387C2×Q8—541no—
388C2×C8—31no—
389C2×C8—91no—
390C4⋊C4—31no—
391C4⋊C4—91no—
392C4⋊C4—91no—
393C4⋊C4—541no—
394C4⋊C4—541no—
395C42—271no—
396C42—541no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3#380211yescharacteristic
402C22×C3#381111yescharacteristic
403C22×C3#381111yescharacteristic
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—64no—
413C3×C4—62no—
414C3×C4—62no—
415C3×C4—122no—
416C3×C4—121no—
417C3×C4—121no—
418C3×C4—361no—
419C3×C4—361no—
420C3×C4D3×Dic611yescharacteristic
421C3×C4D3×Dic611yescharacteristic
422C3×C4#379511yescharacteristic
423C3×C4#380411yescharacteristic
424C3×C4#380511yescharacteristic
425C3×C4C2×#62711yescharacteristic
426Dic3—22no—
427Dic3—41no—
428Dic3—92no—
429Dic3—92no—
430Dic3—92no—
431Dic3—122no—
432Dic3—122no—
433Dic3—121no—
434Dic3—121no—
435Dic3—121no—
436Dic3—121no—
437Dic3—182no—
438Dic3—182no—
439Dic3—182no—
440Dic3—182no—
441Dic3—181no—
442Dic3—181no—
443Dic3—181no—
444Dic3—362no—
445Dic3—361no—
446Dic3—361no—
447Dic3—362no—
448C32—21no—
449C32—21no—
450C32—21no—
451C32—41no—
452C32#479311yescharacteristic, complemented
453C32#479311yescharacteristic, complemented
454C32#487511yescharacteristic, complemented
455Q8—32no—
456Q8—61no—
457Q8—1082no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—271no—
463C2×C4—271no—
464C2×C4D3311yescharacteristic
465C8—61no—
466C8—181no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—41no—
477C2×C3—41no—
478C2×C3—41no—
479C2×C3#6379811yescharacteristic
480C2×C3#6380211yescharacteristic
481C2×C3#6389611yescharacteristic
482C2×C3#6389911yescharacteristic
483C2×C3#6393711yescharacteristic
484C2×C3#6394111yescharacteristic
485C2×C3#6394411yescharacteristic
486C2×C3#6394411yescharacteristic
487C2×C3#6398211yescharacteristic
488C22#8895411yescharacteristic
489C4—62no—
490C4—121no—
491C4—272no—
492C4—361no—
493C4—361no—
494C4—541no—
495C4D3×#62711yescharacteristic
496C4#8896111yescharacteristic
497C3—21no—
498C3—21no—
499C3—21no—
500C3—41no—
501C3#1059049611yescharacteristic, complemented
502C3#1059081711yescharacteristic, complemented
503C3#1059163111yescharacteristic, complemented
504C2#1172284811yescharacteristic
505C2#1172287811yescharacteristic
506C2#1172298011yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
54. C32×Dic6Q8semidirect
185. C3×Dic6Dic6semidirect
186. C3×Dic6Dic6semidirect
307. C33#477semidirect
356. Dic6#627semidirect
452. C32#4793semidirect
453. C32#4793semidirect
454. C32#4875semidirect
501. C3#10590496semidirect
502. C3#10590817semidirect
503. C3#10591631semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117233972
#117220802
#117220502
C3×#635752
C3×#635662
C3×#635652
#105916313
#105908173
#105904963