Finite Group Databasebeta

← #13155700

Subgroups — #13155700

Order 1728.

606 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic
4#11722052C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic
8#10591582—31nomaximal
9#10590812—31nomaximal
10#10590794—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88813—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22C3×#3719C414yescomplemented
23Q8×C3⋊Dic3—31no—
24C2×C3×#216—31no—
25C2×C3×Dic12—31no—
26#63844—31no—
27#63843—31no—
28Dic3×Dic6—31no—
29Dic3×Dic6—31no—
30Dic3×Dic6—31no—
31Dic3×Dic6—61no—
32C2×#3719D311yescharacteristic
33#63644—31no—
34#63643—31no—
35C3×#835—31no—
36C3×#816—31no—
37C3×#815—31no—
38C3×#815—31no—
39#63573—31no—
40#63573—31no—
41#63573—61no—
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63566D311yescharacteristic
46#63555D311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×#634—21no—
54C3×#634C2×C412yes—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yes—
58C32×C3⋊C8—21no—
59C32×C3⋊C8C2×C412yes—
60#5184—91no—
61#4789—91no—
62#4716—91no—
63C2×C32×Q8—31no—
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C32×C8—31no—
72C2×C32×C8—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—121no—
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C4×C3⋊Dic3—61no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C3×#216—34no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×C4×Dic3—31no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—61no—
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97C3×Dic12—34no—
98#3724—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic32—61no—
105Dic32—61no—
106Dic32—62no—
107Dic32—61no—
108Dic32—61no—
109Dic32—61no—
110#3719Dic314yescomplemented
111C22×C33C2×D411yescharacteristic
112C33×C4C22×C411yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—21no—
118C3×C3⋊Dic3—41no—
119C32×Dic3—22no—
120C2×C3×Q16—91no—
121Q8×Dic3—91no—
122Q8×Dic3—91no—
123Q8×Dic3—91no—
124Q8×Dic3—181no—
125C2×#216—31no—
126C2×Dic12—31no—
127#887—91no—
128C4×Dic6—91no—
129C4×Dic6—91no—
130C3×#103—91no—
131C3×#97—91no—
132#835—31no—
133#835—91no—
134#817—91no—
135#817—91no—
136#817—181no—
137#816—31no—
138#816—91no—
139#816—91no—
140#816—181no—
141#815—31no—
142#815—31no—
143#808—91no—
144C32×Q8—32no—
145C32×Q8—61no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—62no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—121no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—122no—
163C2×C3⋊Dic3D1211yescharacteristic
164C2×C3⋊Dic3D1211yescharacteristic
165#634—21no—
166#634C2×Dic312yes—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—121no—
182C3×Dic6—22no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—22no—
186C3×Dic6—32no—
187C3×Dic6—32no—
188C3×Dic6—62no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—121no—
192C3×Dic6C4×D312yes—
193C3×Dic6C2×Dic312yes—
194C32×C8—32no—
195C32×C8—61no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—22no—
199C3×C3⋊C8—41no—
200C3×C3⋊C8C4×D312yes—
201C3×C3⋊C8C2×Dic312yes—
202#441—271noSylow
203C2×C33C4×D411yescharacteristic
204C2×C33#13711yescharacteristic
205C2×C33#13711yescharacteristic
206C2×C3×Q8—31no—
207C2×C3×Q8—31no—
208C2×C3×Q8—61no—
209C2×C3×Q8—91no—
210C2×Dic6—31no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×Dic6D3211yescharacteristic
214C3×Q16—94no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—61no—
221C3×C4⋊C4—91no—
222C3×C42—91no—
223#216—34no—
224C4⋊Dic3—31no—
225C4⋊Dic3—31no—
226C4⋊Dic3—61no—
227C4⋊Dic3—91no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—181no—
231C4⋊Dic3—181no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235C4⋊Dic3—181no—
236C4⋊Dic3—182no—
237C4⋊Dic3—361no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—181no—
241C4×Dic3—31no—
242C4×Dic3—31no—
243C4×Dic3—61no—
244C4×Dic3—181no—
245C4×Dic3—181no—
246C4×Dic3—181no—
247C4×Dic3—181no—
248C4×Dic3—181no—
249C4×Dic3—181no—
250C4×Dic3—182no—
251C2×C3⋊C8D3211yescharacteristic
252C2×C3⋊C8D3211yescharacteristic
253Dic12—34no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32C2×D1211yescharacteristic
259C22×C32D3×D411yescharacteristic
260C22×C32D3×D411yescharacteristic
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—62no—
270C32×C4C2×C4×D311yescharacteristic
271C32×C4C2×C4×D311yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23711yescharacteristic
274C32×C4#23711yescharacteristic
275C32×C4C22×Dic311yescharacteristic
276C3⋊Dic3—21no—
277C3⋊Dic3—21no—
278C3⋊Dic3—22no—
279C3⋊Dic3—41no—
280C3⋊Dic3—121no—
281C3⋊Dic3—122no—
282C3⋊Dic3—121no—
283C3⋊Dic3—121no—
284C3⋊Dic3—241no—
285C3⋊Dic3—241no—
286C3⋊Dic3—242no—
287C3×Dic3—24no—
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—62no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—122no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C3×Dic3—241no—
302C2×Q16—91no—
303C4×Q8—271no—
304C4×Q8—271no—
305#106—271no—
306#103—91no—
307#103—271no—
308#97—91no—
309C33#44111yescharacteristic, complemented, Sylow
310C3×Q8—32no—
311C3×Q8—32no—
312C3×Q8—62no—
313C3×Q8—62no—
314C3×Q8—61no—
315C3×Q8—61no—
316C3×Q8—92no—
317C3×Q8—181no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—62no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×Dic3—21no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—121no—
344C2×Dic3—121no—
345C2×Dic3—182no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—361no—
350C2×Dic3—361no—
351C2×Dic3—362no—
352Dic6—21no—
353Dic6—32no—
354Dic6—32no—
355Dic6—62no—
356Dic6—61no—
357Dic6—61no—
358Dic6—121no—
359Dic6D3×Dic312yes—
360C3×C8—32no—
361C3×C8—32no—
362C3×C8—62no—
363C3×C8—61no—
364C3×C8—61no—
365C3×C8—121no—
366C3⋊C8—21no—
367C3⋊C8D3×Dic312yes—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32C4×D1211yescharacteristic
381C2×C32#67⋊C411yescharacteristic
382C2×C32#90911yescharacteristic
383C2×C32#90911yescharacteristic
384C2×C32#91511yescharacteristic
385C2×C32#91811yescharacteristic
386C2×C32#91811yescharacteristic
387C2×C32#91811yescharacteristic
388C2×C32D4×Dic311yescharacteristic
389C2×Q8—31no—
390C2×Q8—91no—
391Q16—94no—
392C2×C8—31no—
393C2×C8—31no—
394C4⋊C4—91no—
395C4⋊C4—271no—
396C4⋊C4—541no—
397C4⋊C4—541no—
398C42—91no—
399C42—541no—
400C42—541no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3D3×D1211yescharacteristic
406C22×C3D3×D1211yescharacteristic
407C22×C3#381111yescharacteristic
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—64no—
417C3×C4—62no—
418C3×C4—62no—
419C3×C4—182no—
420C3×C4—181no—
421C3×C4—181no—
422C3×C4—361no—
423C3×C4#379511yescharacteristic
424C3×C4C4×D3211yescharacteristic
425C3×C4C2×D3×Dic311yescharacteristic
426C3×C4C2×D3×Dic311yescharacteristic
427C3×C4#380411yescharacteristic
428C3×C4#380511yescharacteristic
429Dic3—22no—
430Dic3—62no—
431Dic3—61no—
432Dic3—61no—
433Dic3—61no—
434Dic3—61no—
435Dic3—62no—
436Dic3—62no—
437Dic3—122no—
438Dic3—121no—
439Dic3—121no—
440Dic3—121no—
441Dic3—241no—
442Dic3—362no—
443Dic3—361no—
444Dic3—361no—
445Dic3—361no—
446Dic3—721no—
447Dic3—721no—
448Dic3—722no—
449C32—21no—
450C32—21no—
451C32—21no—
452C32—41no—
453C32#471611yescharacteristic, complemented
454C32#478911yescharacteristic, complemented
455C32#518411yescharacteristic, complemented
456Q8—32no—
457Q8—61no—
458Q8—92no—
459Q8—181no—
460C2×C4—61no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—541no—
466C2×C4D3311yescharacteristic
467C8—32no—
468C8—61no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3#6380011yescharacteristic
482C2×C3#6380111yescharacteristic
483C2×C3#6380111yescharacteristic
484C2×C3#6380111yescharacteristic
485C2×C3#6388011yescharacteristic
486C2×C3Dic3×D1211yescharacteristic
487C2×C3#6394511yescharacteristic
488C2×C3#6395111yescharacteristic
489C2×C3#67⋊Dic311yescharacteristic
490C22#8895511yescharacteristic
491C4—62no—
492C4—181no—
493C4—181no—
494C4—182no—
495C4—361no—
496C4—1081no—
497C4D32×Dic311yescharacteristic
498C4#8896111yescharacteristic
499C3—21no—
500C3—21no—
501C3—21no—
502C3—41no—
503C3#1059079411yescharacteristic, complemented
504C3#1059081211yescharacteristic, complemented
505C3#1059158211yescharacteristic, complemented
506C2#1172289611yescharacteristic
507C2#1172289711yescharacteristic
508C2#1172292811yescharacteristic
subgroupG/H
centerC22#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
22. C3×#3719C4semidirect
110. #3719Dic3semidirect
309. C33#441semidirect
453. C32#4716semidirect
454. C32#4789semidirect
455. C32#5184semidirect
503. C3#10590794semidirect
504. C3#10590812semidirect
505. C3#10591582semidirect

Maximal subgroups

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

subgroupindex
#117236682
Dic6×C3⋊Dic32
C2×C3×#37192
#117220522
#117220402
C3×#635662
C3×#635552
#105915823
#105908123
#105907943