Finite Group Databasebeta

← #13156331

Subgroups — #13156331

Order 1728.

554 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
1C3×#63894C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic
4#11722053C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic
7C3×#63576C211yesmaximal, characteristic, complemented
8#10590542—31nomaximal
9#10590468—31nomaximal
10#10590270—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#624—21no—
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×#3722—21no—
22C3×#893—31no—
23C3×#890—31no—
24C3×C4×D12—31no—
25C3×#870—31no—
26#63894D311yescharacteristic
27#63869D311yescharacteristic, complemented
28#63636—31no—
29#63635—31no—
30C4⋊(C3⋊(C3⋊C8))—31no—
31C3×#807—31no—
32C3×#807—31no—
33#63576D311yescharacteristic, complemented
34#63574—31no—
35#63574—31no—
36#63574—61no—
37Dic3⋊(C3⋊C8)—31no—
38Dic3⋊(C3⋊C8)—31no—
39Dic3⋊(C3⋊C8)—61no—
40#63564—31no—
41#63563—31no—
42#63562—31no—
43#63562—31no—
44#63562—31no—
45#63562—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)Q812yes—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C3×#624—42no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5066—91no—
59#5009—91no—
60#4891—91no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—61no—
66C2×C3×C4×D3—61no—
67C2×#624—21no—
68C32×C4⋊C4—31no—
69C32×C4⋊C4—31no—
70C32×C42—31no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—61no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—121no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×(C2×D3)⋊C4—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101#3724—21no—
102Dic3⋊Dic3—21no—
103#3722—21no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3—41no—
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3×C3⋊Dic3—41no—
111C32×Dic3—21no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C32×Dic3—41no—
115C3×#125—91no—
116C3×C4×D4—91no—
117#893—31no—
118#890—31no—
119C4×D12—31no—
120#870—31no—
121C3×#107—91no—
122#832—91no—
123#832—91no—
124#821—91no—
125Dic3⋊C8—91no—
126#809—91no—
127#809—91no—
128#809—91no—
129#809—181no—
130#808—91no—
131#808—91no—
132#808—91no—
133#808—181no—
134#807—31no—
135#807—31no—
136C4⋊(C3⋊C8)—91no—
137C4⋊(C3⋊C8)—91no—
138C4⋊(C3⋊C8)—181no—
139C22×C3⋊D3—21no—
140C22×C3×D3—61no—
141C22×C3×D3—61no—
142C22×C3×D3—121no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—31no—
147C2×C32×C4—31no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C4⋊(C3⋊D3)—21no—
156C4⋊(C3⋊D3)Dic612yes—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—41no—
164C2×C3×Dic3—41no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3#6711yescharacteristic
169C2×C3×Dic3#6711yescharacteristic
170C2×C3×Dic3#6711yescharacteristic
171C2×C3×Dic3#6711yescharacteristic
172C3×D12—32no—
173C3×D12—32no—
174C3×D12—62no—
175C3×D12—61no—
176C3×D12—61no—
177C3×D12—121no—
178C3×C4×D3—122no—
179C3×C4×D3—122no—
180#624—42no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—62no—
184C3⋊(C3⋊C8)—61no—
185C3⋊(C3⋊C8)—61no—
186C3⋊(C3⋊C8)—121no—
187C3⋊(C3⋊C8)—121no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—61no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—121no—
192#478—271noSylow
193C2×C33C22⋊Q811yescharacteristic
194C2×C33#13511yescharacteristic
195C2×C33AGL(1,ℤ/8)11yescharacteristic
196C3×C3⋊D3—42no—
197C2×C3×D4—91no—
198C22×C3×C4—181no—
199C2×D12—31no—
200C2×D12—31no—
201C2×D12—61no—
202C2×C4×D3—61no—
203C2×C4×D3—61no—
204C2×C3×C8—91no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—91no—
212C3×C4⋊C4—181no—
213C3×C4⋊C4—181no—
214C3×C22⋊C4—181no—
215C3×C42—31no—
216C3×C42—31no—
217C3×C42—61no—
218(C2×D3)⋊C4—61no—
219(C2×D3)⋊C4—61no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—61no—
224C4⋊Dic3D3211yescharacteristic
225C4⋊Dic3D3211yescharacteristic
226Dic3⋊C4—61no—
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229C4×Dic3D3211yescharacteristic
230C2×C3⋊C8—31no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—61no—
233C2×C3⋊C8—91no—
234C2×C3⋊C8—91no—
235C2×C3⋊C8—91no—
236C2×C3⋊C8—181no—
237C2×C3⋊C8—181no—
238C2×C3⋊C8—181no—
239C2×C3⋊C8—361no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32D3×D411yescharacteristic
245C22×C32C2×#6711yescharacteristic
246C22×C32C2×#6711yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—122no—
253C2×C3×D3—121no—
254C2×C3×D3—121no—
255C2×C3×D3—121no—
256C2×C3×D3—121no—
257C2×C3×D3—241no—
258C2×C3×D3—241no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—61no—
268C32×C4—61no—
269C32×C4—121no—
270C32×C4—121no—
271C32×C4C2×Dic611yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23511yescharacteristic
274C32×C4D3×Q811yescharacteristic
275C32×C4D3×Q811yescharacteristic
276C32×C4#23911yescharacteristic
277C3⋊Dic3—41no—
278C3×Dic3—22no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—41no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—41no—
288C3×Dic3—81no—
289C3×Dic3—81no—
290C3×Dic3—121no—
291C3×Dic3—121no—
292C3×Dic3—241no—
293#125—91no—
294C4×D4—91no—
295#107—91no—
296#106—271no—
297C4⋊C8—271no—
298#102—271no—
299#102—271no—
300C33#47811yescharacteristic, complemented, Sylow
301C23×C3—181no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—121no—
305C3×D4—92no—
306C3×D4—181no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—41no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—121no—
322C2×C3×C4—121no—
323C2×C3×C4—181no—
324C2×C3×C4—362no—
325C2×C3×C4C2×D3211yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—121no—
333C2×Dic3#62511yescharacteristic
334C2×Dic3#62511yescharacteristic
335D12—32no—
336D12—32no—
337D12—62no—
338D12—61no—
339D12—61no—
340D12—121no—
341C4×D3—122no—
342C4×D3—122no—
343C3×C8—181no—
344C3⋊C8—61no—
345C3⋊C8—61no—
346C3⋊C8—121no—
347C3⋊C8—182no—
348C3⋊C8—181no—
349C3⋊C8—181no—
350C3⋊C8—181no—
351C3⋊C8—361no—
352C3⋊C8—361no—
353C3⋊C8—362no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32#89611yescharacteristic
367C2×C32#91111yescharacteristic
368C2×C32#91911yescharacteristic
369C2×C32#92411yescharacteristic
370C2×C32#93211yescharacteristic
371C2×C32#94611yescharacteristic
372C2×C32#94911yescharacteristic
373C2×C32#95011yescharacteristic
374C2×C32#95011yescharacteristic
375C3⋊D3—42no—
376C3×D3—122no—
377C3×D3—122no—
378C3×D3—242no—
379C2×D4—91no—
380C22×C4—181no—
381C2×C8—91no—
382C2×C8—271no—
383C4⋊C4—31no—
384C4⋊C4—31no—
385C4⋊C4—91no—
386C4⋊C4—181no—
387C4⋊C4—181no—
388C22⋊C4—181no—
389C42—31no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—182no—
395C22×C3—361no—
396C22×C3—361no—
397C22×C3C2×#62511yescharacteristic
398C22×C3D3×#6711yescharacteristic
399C22×C3D3×#6711yescharacteristic
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—122no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—121no—
407C2×D3—241no—
408C2×D3—241no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—62no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—61no—
422C3×C4—121no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—241no—
428C3×C4—241no—
429C3×C4—361no—
430C3×C4D3×Dic611yescharacteristic
431C3×C4D3×Dic611yescharacteristic
432C3×C4#379711yescharacteristic
433C3×C4#379811yescharacteristic
434C3×C4#379911yescharacteristic
435C3×C4#379911yescharacteristic
436Dic3—21no—
437Dic3—21no—
438Dic3—41no—
439Dic3—41no—
440Dic3—121no—
441Dic3—121no—
442Dic3—241no—
443C32—21no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32#489111yescharacteristic, complemented
448C32#500911yescharacteristic, complemented
449C32#506611yescharacteristic, complemented
450C23—181no—
451D4—92no—
452D4—181no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—181no—
458C2×C4—362no—
459C2×C4D3311yescharacteristic
460C8—181no—
461C8—541no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—362no—
475C2×C3#6382211yescharacteristic
476C2×C3#6382311yescharacteristic
477C2×C3#6385111yescharacteristic
478C2×C3#6385611yescharacteristic
479C2×C3#6388811yescharacteristic
480C2×C3#6392511yescharacteristic
481C2×C3#6393211yescharacteristic
482C2×C3#6394911yescharacteristic
483C2×C3#6394911yescharacteristic
484D3—122no—
485D3—122no—
486D3—242no—
487C22—182no—
488C22—361no—
489C22—361no—
490C22D3×#62511yescharacteristic
491C4—61no—
492C4—61no—
493C4—121no—
494C4—121no—
495C4—361no—
496C4#8895711yescharacteristic
497C4#8896511yescharacteristic
498C3—21no—
499C3—21no—
500C3—21no—
501C3—41no—
502C3#1059027011yescharacteristic, complemented
503C3#1059046811yescharacteristic, complemented
504C3#1059054211yescharacteristic, complemented
505C2—362no—
506C2#1172285211yescharacteristic
507C2#1172286611yescharacteristic
508C2#1172297011yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#478
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
2. C3×#63869C2semidirect
4. #11722053C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
7. C3×#63576C2semidirect
27. #63869D3semidirect
33. #63576D3semidirect
300. C33#478semidirect
447. C32#4891semidirect
448. C32#5009semidirect
449. C32#5066semidirect
502. C3#10590270semidirect
503. C3#10590468semidirect
504. C3#10590542semidirect

Maximal subgroups

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

subgroupindex
C3×#638942
C3×#638692
#117220752
#117220532
Dic3⋊(C3⋊(C3⋊C8))2
#117220332
C3×#635762
#105905423
#105904683
#105902703