Finite Group Databasebeta

← #13156358

Subgroups — #13156358

Order 1728.

498 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63869C211yesmaximal, characteristic
2C3×Dic3×Dic6C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722053C211yesmaximal, characteristic
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591547—31nomaximal
9#10590491—31nomaximal
10#10590286—31nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C3×#3724—21no—
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×Q8×Dic3—31no—
23C3×#890—31no—
24C3×#870—31no—
25C3×C4×Dic6—31no—
26#63869D311yescharacteristic
27Dic3×Dic6D311yescharacteristic
28#63662—31no—
29#63635—31no—
30C4⋊(C3⋊(C3⋊C8))—31no—
31C3×#835—31no—
32C3×#810—31no—
33#63576—31no—
34#63575—31no—
35#63575—31no—
36#63574—31no—
37#63574—31no—
38#63574—61no—
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63565—31no—
43#63565—31no—
44#63565—61no—
45#63565D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×Dic6—21no—
54C32×Dic6Q812yes—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C3×C3⋊(C3⋊C8)—21no—
57#5063—91no—
58#5022—91no—
59#4796—91no—
60C2×C32×Q8—31no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6C2×D311yescharacteristic
63C2×C3×Dic6C2×D311yescharacteristic
64C32×C4⋊C4—31no—
65C32×C42—31no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—61no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—61no—
74C2×C3⋊(C3⋊C8)—121no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×Dic3⋊C4—61no—
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×C4×Dic3—21no—
92C3×C4×Dic3—61no—
93C3×C4×Dic3—61no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99#3724—21no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—21no—
102Dic32—21no—
103C22×C33C2×D411yescharacteristic
104C33×C4C2×Q811yescharacteristic
105C33×C4#4111yescharacteristic
106C3×C3⋊Dic3—41no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—22no—
109C32×Dic3—21no—
110C32×Dic3—21no—
111C32×Dic3—41no—
112C3×#125—91no—
113C3×C4×Q8—91no—
114Q8×Dic3—31no—
115#890—31no—
116#870—31no—
117C4×Dic6—31no—
118C3×#103—91no—
119#835—31no—
120#835—91no—
121#835—91no—
122#835—181no—
123#818—91no—
124Dic3⋊C8—91no—
125#810—31no—
126#810—91no—
127#808—91no—
128#808—91no—
129#808—91no—
130#808—181no—
131#807—91no—
132#807—91no—
133C4⋊(C3⋊C8)—91no—
134C4⋊(C3⋊C8)—91no—
135C4⋊(C3⋊C8)—181no—
136C32×Q8—32no—
137C32×Q8—61no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—31no—
142C2×C32×C4—31no—
143C2×C32×C4—41no—
144C2×C32×C4—61no—
145C2×C32×C4—61no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—41no—
158C2×C3×Dic3—41no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—121no—
164C2×C3×Dic3—121no—
165C2×C3×Dic3#6711yescharacteristic
166C2×C3×Dic3#6711yescharacteristic
167C2×C3×Dic3#6711yescharacteristic
168C2×C3×Dic3#6711yescharacteristic
169C3×Dic6—21no—
170C3×Dic6—21no—
171C3×Dic6—22no—
172C3×Dic6—41no—
173C3×Dic6Dic612yes—
174C3×Dic6Dic612yes—
175C3⋊(C3⋊C8)—21no—
176C3⋊(C3⋊C8)—61no—
177C3⋊(C3⋊C8)—62no—
178C3⋊(C3⋊C8)—61no—
179C3⋊(C3⋊C8)—61no—
180C3⋊(C3⋊C8)—121no—
181C3⋊(C3⋊C8)—121no—
182C3⋊(C3⋊C8)—122no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185C3×C3⋊C8—121no—
186#479—271noSylow
187C2×C33C22⋊Q811yescharacteristic
188C2×C33#13511yescharacteristic
189C2×C33#13711yescharacteristic
190C2×C3×Q8—31no—
191C2×C3×Q8—31no—
192C2×C3×Q8—61no—
193C2×Dic6D3211yescharacteristic
194C2×C3×C8—91no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4—91no—
200C3×C4⋊C4—181no—
201C3×C4⋊C4—181no—
202C3×C4⋊C4—181no—
203C3×C42—31no—
204C3×C42—31no—
205C3×C42—61no—
206C3×C42—181no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—61no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4⋊Dic3D3211yescharacteristic
216Dic3⋊C4—61no—
217Dic3⋊C4—61no—
218Dic3⋊C4—61no—
219Dic3⋊C4—61no—
220C4×Dic3—61no—
221C4×Dic3—61no—
222C4×Dic3D3211yescharacteristic
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—91no—
227C2×C3⋊C8—91no—
228C2×C3⋊C8—91no—
229C2×C3⋊C8—181no—
230C2×C3⋊C8—181no—
231C2×C3⋊C8—181no—
232C2×C3⋊C8—361no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—41no—
237C22×C32D3×D411yescharacteristic
238C22×C32C2×#6711yescharacteristic
239C22×C32C2×#6711yescharacteristic
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4—61no—
249C32×C4—61no—
250C32×C4—62no—
251C32×C4—121no—
252C32×C4C2×Dic611yescharacteristic
253C32×C4C2×Dic611yescharacteristic
254C32×C4#23511yescharacteristic
255C32×C4#23711yescharacteristic
256C32×C4D3×Q811yescharacteristic
257C32×C4#23911yescharacteristic
258C3⋊Dic3—41no—
259C3⋊Dic3—41no—
260C3×Dic3—22no—
261C3×Dic3—22no—
262C3×Dic3—22no—
263C3×Dic3—21no—
264C3×Dic3—21no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—42no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C3×Dic3—41no—
271C3×Dic3—81no—
272C3×Dic3—121no—
273C3×Dic3—121no—
274C3×Dic3—121no—
275C3×Dic3—121no—
276C3×Dic3—241no—
277C3×Dic3—241no—
278#125—91no—
279C4×Q8—91no—
280#107—271no—
281#106—271no—
282C4⋊C8—271no—
283#103—91no—
284#103—271no—
285C33#47911yescharacteristic, complemented, Sylow
286C3×Q8—32no—
287C3×Q8—32no—
288C3×Q8—62no—
289C3×Q8—61no—
290C3×Q8—61no—
291C3×Q8—121no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—31no—
299C2×C3×C4—41no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—121no—
307C2×C3×C4—121no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—121no—
320C2×Dic3—121no—
321C2×Dic3#62511yescharacteristic
322C2×Dic3#62511yescharacteristic
323Dic6—21no—
324Dic6#62712yes—
325C3×C8—181no—
326C3⋊C8—61no—
327C3⋊C8—61no—
328C3⋊C8—121no—
329C3⋊C8—182no—
330C3⋊C8—181no—
331C3⋊C8—181no—
332C3⋊C8—181no—
333C3⋊C8—361no—
334C3⋊C8—361no—
335C3⋊C8—362no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32#87811yescharacteristic
349C2×C32#91511yescharacteristic
350C2×C32#91911yescharacteristic
351C2×C32#92411yescharacteristic
352C2×C32#94211yescharacteristic
353C2×C32#94611yescharacteristic
354C2×C32#95011yescharacteristic
355C2×C32#95011yescharacteristic
356C2×C32#95111yescharacteristic
357C2×Q8—31no—
358C2×C8—91no—
359C2×C8—271no—
360C4⋊C4—31no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—181no—
364C4⋊C4—181no—
365C4⋊C4—181no—
366C42—31no—
367C42—181no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—41no—
372C22×C3C2×#62511yescharacteristic
373C22×C3D3×#6711yescharacteristic
374C22×C3D3×#6711yescharacteristic
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—62no—
384C3×C4—62no—
385C3×C4—62no—
386C3×C4—61no—
387C3×C4—61no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—121no—
391C3×C4—122no—
392C3×C4—121no—
393C3×C4—121no—
394C3×C4—241no—
395C3×C4—361no—
396C3×C4—361no—
397C3×C4D3×Dic611yescharacteristic
398C3×C4D3×Dic611yescharacteristic
399C3×C4#379611yescharacteristic
400C3×C4#379911yescharacteristic
401C3×C4#380411yescharacteristic
402C3×C4C2×#62711yescharacteristic
403Dic3—21no—
404Dic3—21no—
405Dic3—22no—
406Dic3—41no—
407Dic3—121no—
408Dic3—121no—
409Dic3—121no—
410Dic3—121no—
411Dic3—241no—
412Dic3—241no—
413C32—21no—
414C32—21no—
415C32—21no—
416C32—41no—
417C32#479611yescharacteristic, complemented
418C32#502211yescharacteristic, complemented
419C32#506311yescharacteristic, complemented
420Q8—32no—
421Q8—61no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4D3311yescharacteristic
429C8—181no—
430C8—541no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3#6382211yescharacteristic
444C2×C3#6382711yescharacteristic
445C2×C3#6385611yescharacteristic
446C2×C3#6388711yescharacteristic
447C2×C3#6392911yescharacteristic
448C2×C3#6394611yescharacteristic
449C2×C3#6394911yescharacteristic
450C2×C3#6394911yescharacteristic
451C2×C3#6396411yescharacteristic
452C22D3×#62511yescharacteristic
453C4—61no—
454C4—61no—
455C4—62no—
456C4—121no—
457C4—361no—
458C4—361no—
459C4D3×#62711yescharacteristic
460C4#8896211yescharacteristic
461C3—21no—
462C3—21no—
463C3—21no—
464C3—41no—
465C3#1059028611yescharacteristic, complemented
466C3#1059049111yescharacteristic, complemented
467C3#1059154711yescharacteristic, complemented
468C2#1172285211yescharacteristic
469C2#1172288211yescharacteristic
470C2#1172296911yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#479
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
285. C33#479semidirect
417. C32#4796semidirect
418. C32#5022semidirect
419. C32#5063semidirect
465. C3#10590286semidirect
466. C3#10590491semidirect
467. C3#10591547semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
C3×Dic3×Dic62
#117220802
#117220532
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
C3×#635652
#105915473
#105904913
#105902863