Finite Group Databasebeta

← #13156356

Subgroups — #13156356

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#11723691C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic, complemented
3#11722052C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590774—31nomaximal
9#10590772—31nomaximal
10#10590473—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×#6294—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88806—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64099—31no—
23#63898—31no—
24#63894—31no—
25Dic3×D12—31no—
26#63869—31no—
27#63869—31no—
28#63869—61no—
29#63648—31no—
30#63644—31no—
31C3×#821—31no—
32C3×#818—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×#809—31no—
36C3×#807—31no—
37#63575D311yescharacteristic, complemented
38#63573—31no—
39#63573—31no—
40#63573—61no—
41#63572D311yescharacteristic, complemented
42#63564D311yescharacteristic
43#63563—31no—
44#63563—31no—
45#63563—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)Q812yes—
54C2×C32×Dic3—21no—
55#6294—42no—
56C32×C3⋊C8—21no—
57C32×C3⋊C8—21no—
58#4891—91no—
59#4847—91no—
60#4763—91no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×#624—61no—
66C2×D3×Dic3—61no—
67C2×D3×Dic3—61no—
68C2×C32×C8—31no—
69C2×C32×C8—31no—
70C32×C4⋊C4—31no—
71C4⋊(C3⋊Dic3)—31no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—121no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80#3750—61no—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
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
97#3724—61no—
98#3724—61no—
99#3724—62no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—62no—
103#3722—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3⋊(C3⋊Dic3)—41no—
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—41no—
117D4×Dic3—91no—
118#893—91no—
119#890—91no—
120#890—91no—
121#890—181no—
122C4×D12—91no—
123#870—91no—
124C3×#107—91no—
125C3×C4⋊C8—91no—
126C3×#102—91no—
127#832—91no—
128#821—31no—
129#821—91no—
130#821—91no—
131#821—181no—
132#818—31no—
133#817—91no—
134#817—91no—
135#817—181no—
136Dic3⋊C8—31no—
137Dic3⋊C8—31no—
138#809—31no—
139#808—91no—
140#807—31no—
141C22×C3⋊D3—21no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—121no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—61no—
155C2×C3⋊Dic3—62no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—121no—
159C2×C3⋊Dic3—121no—
160C2×C3⋊Dic3—122no—
161C2×C3⋊Dic3D1211yescharacteristic
162C2×C3⋊Dic3D1211yescharacteristic
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)Dic612yes—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—121no—
177C3×D12—32no—
178C3×D12—32no—
179C3×D12—62no—
180C3×D12—61no—
181C3×D12—61no—
182C3×D12—121no—
183#624—122no—
184D3×Dic3—122no—
185D3×Dic3—122no—
186C32×C8—61no—
187C32×C8—61no—
188C3×C3⋊C8—22no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—21no—
193C3×C3⋊C8—41no—
194#478—271noSylow
195C2×C33C22⋊Q811yescharacteristic
196C2×C33#13511yescharacteristic
197C2×C33AGL(1,ℤ/8)11yescharacteristic
198C3×C3⋊D3—42no—
199C2×C3×D4—91no—
200C22×Dic3—181no—
201C2×D12—31no—
202C2×D12—31no—
203C2×D12—61no—
204C2×C4×D3—181no—
205C2×C4×D3—181no—
206C2×C3×C8—31no—
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—31no—
210C2×C3×C8—61no—
211C2×C3×C8—61no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—91no—
216C3×C42—91no—
217#217—181no—
218(C2×D3)⋊C4—181no—
219(C2×D3)⋊C4—181no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—91no—
224C4⋊Dic3—91no—
225C4⋊Dic3—91no—
226C4⋊Dic3—181no—
227C4⋊Dic3—181no—
228C4⋊Dic3—181no—
229C4⋊Dic3—181no—
230C4⋊Dic3—181no—
231C4⋊Dic3—182no—
232C4⋊Dic3—361no—
233C4⋊Dic3D3211yescharacteristic
234Dic3⋊C4—181no—
235Dic3⋊C4—181no—
236Dic3⋊C4—181no—
237Dic3⋊C4—181no—
238Dic3⋊C4—182no—
239C4×Dic3—31no—
240C4×Dic3—31no—
241C4×Dic3—61no—
242C2×C3⋊C8D3211yescharacteristic
243C2×C3⋊C8D3211yescharacteristic
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32C2×D1211yescharacteristic
249C22×C32D3×D411yescharacteristic
250C22×C32D3×D411yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—122no—
257C2×C3×D3—121no—
258C2×C3×D3—121no—
259C2×C3×D3—121no—
260C2×C3×D3—121no—
261C2×C3×D3—241no—
262C2×C3×D3—241no—
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—121no—
272C32×C4C2×Dic611yescharacteristic
273C32×C4#23511yescharacteristic
274C32×C4#23711yescharacteristic
275C32×C4#23711yescharacteristic
276C32×C4D3×Q811yescharacteristic
277C32×C4D3×Q811yescharacteristic
278C3⋊Dic3—21no—
279C3⋊Dic3—21no—
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—42no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—241no—
300#125—271no—
301C4×D4—271no—
302#107—91no—
303#106—271no—
304C4⋊C8—91no—
305#102—91no—
306#102—271no—
307C33#47811yescharacteristic, complemented, Sylow
308C23×C3—181no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—121no—
312C3×D4—92no—
313C3×D4—181no—
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—91no—
322C2×C3×C4—91no—
323C2×C3×C4—181no—
324C2×C3×C4C2×D3211yescharacteristic
325C2×C3×C4C2×D3211yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×Dic3—21no—
328C2×Dic3—31no—
329C2×Dic3—31no—
330C2×Dic3—31no—
331C2×Dic3—31no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—121no—
337C2×Dic3—182no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—361no—
342C2×Dic3—361no—
343C2×Dic3—362no—
344C2×Dic3—362no—
345D12—32no—
346D12—32no—
347D12—62no—
348D12—61no—
349D12—61no—
350D12—121no—
351C4×D3—362no—
352C4×D3—362no—
353C3×C8—62no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—61no—
357C3×C8—61no—
358C3×C8—121no—
359C3⋊C8—21no—
360C3⋊C8—21no—
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#87811yescharacteristic
374C2×C32#89611yescharacteristic
375C2×C32#89711yescharacteristic
376C2×C32#90411yescharacteristic
377C2×C32#90811yescharacteristic
378C2×C32#91111yescharacteristic
379C2×C32#91111yescharacteristic
380C2×C32#91611yescharacteristic
381C2×C32#91911yescharacteristic
382C3⋊D3—42no—
383C3×D3—122no—
384C3×D3—122no—
385C3×D3—242no—
386C2×D4—91no—
387C22×C4—541no—
388C2×C8—31no—
389C2×C8—31no—
390C4⋊C4—31no—
391C4⋊C4—91no—
392C4⋊C4—271no—
393C4⋊C4—541no—
394C4⋊C4—541no—
395C22⋊C4—541no—
396C42—91no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—182no—
402C22×C3—361no—
403C22×C3—361no—
404C22×C3D3×D1211yescharacteristic
405C22×C3D3×D1211yescharacteristic
406C22×C3#381111yescharacteristic
407C2×D3—62no—
408C2×D3—62no—
409C2×D3—122no—
410C2×D3—121no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—121no—
414C2×D3—241no—
415C2×D3—241no—
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—122no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—181no—
428C3×C4—181no—
429C3×C4—361no—
430C3×C4D3×Dic611yescharacteristic
431C3×C4D3×Dic611yescharacteristic
432C3×C4#379511yescharacteristic
433C3×C4#379711yescharacteristic
434C3×C4#380411yescharacteristic
435C3×C4#380511yescharacteristic
436Dic3—41no—
437Dic3—61no—
438Dic3—61no—
439Dic3—61no—
440Dic3—61no—
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#489111yescharacteristic, complemented
460C23—181no—
461D4—92no—
462D4—181no—
463C2×C4—61no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—181no—
467C2×C4—541no—
468C2×C4—1082no—
469C2×C4D3311yescharacteristic
470C8—61no—
471C8—61no—
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—362no—
485C2×C3#6379511yescharacteristic
486C2×C3#6379511yescharacteristic
487C2×C3#6380611yescharacteristic
488C2×C3#6380811yescharacteristic
489C2×C3#6389311yescharacteristic
490C2×C3#6389511yescharacteristic
491C2×C3#6389611yescharacteristic
492C2×C3#6392711yescharacteristic
493C2×C3#6395011yescharacteristic
494D3—122no—
495D3—122no—
496D3—242no—
497C22—182no—
498C22—361no—
499C22—361no—
500C22#8895511yescharacteristic
501C4—121no—
502C4—181no—
503C4—181no—
504C4—361no—
505C4—1081no—
506C4#8895711yescharacteristic
507C4#8896111yescharacteristic
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3#1059047311yescharacteristic, complemented
513C3#1059077211yescharacteristic, complemented
514C3#1059077411yescharacteristic, complemented
515C2—362no—
516C2#1172286511yescharacteristic
517C2#1172288011yescharacteristic
518C2#1172298411yescharacteristic
subgroupG/H
centerC22#88955
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. #11723447C2semidirect
3. #11722052C2semidirect
5. C3×#63575C2semidirect
6. C3×#63572C2semidirect
37. #63575D3semidirect
41. #63572D3semidirect
307. C33#478semidirect
457. C32#4763semidirect
458. C32#4847semidirect
459. C32#4891semidirect
512. C3#10590473semidirect
513. C3#10590772semidirect
514. C3#10590774semidirect

Maximal subgroups

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

subgroupindex
#117236912
#117234472
#117220522
#117220342
C3×#635752
C3×#635722
C3×#635642
#105907743
#105907723
#105904733