Finite Group Databasebeta

← #13156336

Subgroups — #13156336

Order 1728.

576 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723454C211yesmaximal, characteristic, complemented
2C3×Dic3×D12C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic, complemented
4#11722052C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8#10591409—31nomaximal
9#10590772—31nomaximal
10#10590271—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C3×D3×Dic3—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#88807—21no—
20#88806—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64079—31no—
23C3×D4×Dic3—31no—
24C3×C4×D12—31no—
25Dic3×D12D311yescharacteristic
26#63869—31no—
27#63869—31no—
28#63869—61no—
29#63659—31no—
30#63644—31no—
31C3×#832—31no—
32C3×#821—31no—
33C3×Dic3⋊C8—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63573—31no—
39#63573—31no—
40#63573—61no—
41Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
42#63563—31no—
43#63563—31no—
44#63563—61no—
45#63563D311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×D12—21no—
54C32×D12Q812yes—
55C3×D3×Dic3—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#5009—91no—
59#4847—91no—
60#4763—91no—
61C2×C32×D4—31no—
62C22×C3×Dic3—61no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×C4×D3—61no—
67C2×D3×Dic3—21no—
68C2×C32×C8—31no—
69C32×C42—31no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—121no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79#3750—61no—
80#3750—61no—
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#217—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4×Dic3—21no—
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—62no—
100#3724—61no—
101#3724—61no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105(C2×D3)⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C2×C32×D3—22no—
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3⋊(C3⋊Dic3)—41no—
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—21no—
117C3×C4×D4—91no—
118D4×Dic3—31no—
119#890—91no—
120C4×D12—31no—
121#870—91no—
122#870—91no—
123#870—181no—
124C3×C4⋊C8—91no—
125C3×#102—91no—
126#832—31no—
127#832—91no—
128#832—91no—
129#832—181no—
130#821—31no—
131#821—91no—
132#818—91no—
133#817—91no—
134#817—91no—
135#817—181no—
136Dic3⋊C8—31no—
137#808—91no—
138#807—91no—
139#807—91no—
140C4⋊(C3⋊C8)—31no—
141C23×C32—61no—
142C22×C3×D3—21no—
143C22×C3×D3—21no—
144C22×C3×D3—41no—
145C32×D4—32no—
146C32×D4—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—31no—
151C2×C32×C4—31no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—62no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—121no—
164C2×C3⋊Dic3—122no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—121no—
173C2×C3×Dic3—122no—
174C2×C3×Dic3D1211yescharacteristic
175C2×C3×Dic3D1211yescharacteristic
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3#6711yescharacteristic
178C3×D12—21no—
179C3×D12—21no—
180C3×D12—22no—
181C3×D12—41no—
182C3×D12Dic612yes—
183C3×D12Dic612yes—
184C3×C4×D3—122no—
185D3×Dic3—42no—
186C32×C8—61no—
187C3⋊(C3⋊C8)—21no—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—61no—
192C3×C3⋊C8—61no—
193C3×C3⋊C8—121no—
194#478—271noSylow
195C2×C33C22⋊Q811yescharacteristic
196C2×C33#13511yescharacteristic
197C2×C33AGL(1,ℤ/8)11yescharacteristic
198C32×D3—42no—
199C2×C3×D4—31no—
200C2×C3×D4—31no—
201C2×C3×D4—61no—
202C22×C3×C4—181no—
203C22×Dic3—61no—
204C2×D12D3211yescharacteristic
205C2×C4×D3—61no—
206C2×C3×C8—31no—
207C2×C3×C8—31no—
208C2×C3×C8—61no—
209C2×C3×C8—91no—
210C3×C4⋊C4—91no—
211C3×C4⋊C4—91no—
212C3×C22⋊C4—181no—
213C3×C42—31no—
214C3×C42—31no—
215C3×C42—61no—
216#217—61no—
217(C2×D3)⋊C4—61no—
218C4⋊Dic3—31no—
219C4⋊Dic3—31no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—61no—
224C4⋊Dic3—91no—
225C4⋊Dic3—91no—
226C4⋊Dic3—91no—
227C4⋊Dic3—181no—
228C4⋊Dic3—181no—
229C4⋊Dic3—181no—
230C4⋊Dic3—181no—
231C4⋊Dic3—361no—
232Dic3⋊C4—181no—
233Dic3⋊C4—181no—
234Dic3⋊C4—181no—
235Dic3⋊C4—181no—
236Dic3⋊C4—182no—
237Dic3⋊C4—182no—
238Dic3⋊C4—181no—
239C4×Dic3D3211yescharacteristic
240C2×C3⋊C8—31no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8D3211yescharacteristic
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32—62no—
249C22×C32—121no—
250C22×C32—121no—
251C22×C32C2×D1211yescharacteristic
252C22×C32D3×D411yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C2×C3×D3—22no—
255C2×C3×D3—22no—
256C2×C3×D3—42no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—41no—
261C2×C3×D3—81no—
262C2×C3×D3—81no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—61no—
272C32×C4—61no—
273C32×C4C2×Dic611yescharacteristic
274C32×C4C2×Dic611yescharacteristic
275C32×C4#23511yescharacteristic
276C32×C4#23711yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4D3×Q811yescharacteristic
279C3⋊Dic3—41no—
280C3⋊Dic3—41no—
281C3⋊Dic3—121no—
282C3⋊Dic3—122no—
283C3⋊Dic3—121no—
284C3⋊Dic3—121no—
285C3⋊Dic3—241no—
286C3⋊Dic3—241no—
287C3⋊Dic3—242no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—122no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—241no—
300#125—271no—
301C4×D4—91no—
302#107—271no—
303#106—271no—
304C4⋊C8—91no—
305#102—91no—
306#102—271no—
307C33#47811yescharacteristic, complemented, Sylow
308C23×C3—61no—
309C23×C3—61no—
310C23×C3—121no—
311C22×D3—21no—
312C3×D4—32no—
313C3×D4—32no—
314C3×D4—62no—
315C3×D4—61no—
316C3×D4—61no—
317C3×D4—121no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—41no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—362no—
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×Dic3—62no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—121no—
340C2×Dic3—122no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—361no—
346C2×Dic3—361no—
347C2×Dic3—362no—
348C2×Dic3#62611yescharacteristic
349C2×Dic3#62611yescharacteristic
350D12—21no—
351D12#62712yes—
352C4×D3—122no—
353C3×C8—62no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—181no—
357C3⋊C8—21no—
358C3⋊C8—61no—
359C3⋊C8—61no—
360C3⋊C8—121no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32—122no—
374C2×C32#87811yescharacteristic
375C2×C32#89711yescharacteristic
376C2×C32#90411yescharacteristic
377C2×C32#90811yescharacteristic
378C2×C32#91111yescharacteristic
379C2×C32#91611yescharacteristic
380C2×C32#92411yescharacteristic
381C2×C32#93211yescharacteristic
382C2×C32#95011yescharacteristic
383C3×D3—42no—
384C3×D3—42no—
385C3×D3—82no—
386C2×D4—31no—
387C22×C4—181no—
388C2×C8—31no—
389C2×C8—91no—
390C4⋊C4—91no—
391C4⋊C4—91no—
392C4⋊C4—271no—
393C4⋊C4—541no—
394C4⋊C4—541no—
395C22⋊C4—181no—
396C42—31no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—122no—
404C22×C3—121no—
405C22×C3—121no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3—241no—
409C22×C3—241no—
410C22×C3D3×D1211yescharacteristic
411C22×C3C2×#62611yescharacteristic
412C22×C3D3×#6711yescharacteristic
413C2×D3—22no—
414C2×D3—41no—
415C2×D3—41no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—61no—
428C3×C4—121no—
429C3×C4—121no—
430C3×C4—361no—
431C3×C4—361no—
432C3×C4D3×Dic611yescharacteristic
433C3×C4D3×Dic611yescharacteristic
434C3×C4#379511yescharacteristic
435C3×C4#380411yescharacteristic
436C3×C4#380511yescharacteristic
437C3×C4C2×#62711yescharacteristic
438Dic3—21no—
439Dic3—21no—
440Dic3—122no—
441Dic3—121no—
442Dic3—121no—
443Dic3—121no—
444Dic3—121no—
445Dic3—241no—
446Dic3—362no—
447Dic3—361no—
448Dic3—361no—
449Dic3—361no—
450Dic3—721no—
451Dic3—721no—
452Dic3—722no—
453C32—21no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32#476311yescharacteristic, complemented
458C32#484711yescharacteristic, complemented
459C32#500911yescharacteristic, complemented
460C23—61no—
461D4—32no—
462D4—61no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—362no—
468C2×C4—541no—
469C2×C4D3311yescharacteristic
470C8—61no—
471C8—181no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—122no—
485C2×C3—122no—
486C2×C3—242no—
487C2×C3#6379511yescharacteristic
488C2×C3#6380611yescharacteristic
489C2×C3#6382911yescharacteristic
490C2×C3#6383011yescharacteristic
491C2×C3#6385711yescharacteristic
492C2×C3#6389311yescharacteristic
493C2×C3#6393211yescharacteristic
494C2×C3#6393511yescharacteristic
495C2×C3#6396411yescharacteristic
496D3—42no—
497C22—62no—
498C22—121no—
499C22—121no—
500C22D3×#62611yescharacteristic
501C4—61no—
502C4—61no—
503C4—361no—
504C4—361no—
505C4—1081no—
506C4D3×#62711yescharacteristic
507C4#8896111yescharacteristic
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3#1059027111yescharacteristic, complemented
513C3#1059077211yescharacteristic, complemented
514C3#1059140911yescharacteristic, complemented
515C2—122no—
516C2#1172285311yescharacteristic
517C2#1172286911yescharacteristic
518C2#1172297111yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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
1. #11723454C2semidirect
3. #11722080C2semidirect
4. #11722052C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
41. Dic3⋊(C3⋊C8)D3semidirect
307. C33#478semidirect
457. C32#4763semidirect
458. C32#4847semidirect
459. C32#5009semidirect
512. C3#10590271semidirect
513. C3#10590772semidirect
514. C3#10591409semidirect

Maximal subgroups

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

subgroupindex
#117234542
C3×Dic3×D122
#117220802
#117220522
#117220322
C3×Dic3⋊(C3⋊C8)2
C3×#635632
#105914093
#105907723
#105902713