Finite Group Databasebeta

← #13156543

Subgroups — #13156543

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C211yesmaximal, characteristic, complemented
2C3×#63864C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic
4#11722037C211yesmaximal, characteristic, complemented
5#11722033C211yesmaximal, characteristic
6#11722019C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591624—31nomaximal
9#10590535—31nomaximal
10#10590280—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#627—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×#3722—21no—
21C3×#3722—21no—
22C3×#947—31no—
23C3×#888—31no—
24C3×#875—31no—
25C3×#869—31no—
26#63918D311yescharacteristic, complemented
27#63864D311yescharacteristic
28#63662—31no—
29#63636—31no—
30#63630—31no—
31C3×#835—31no—
32C3×#810—31no—
33#63565—31no—
34#63565—31no—
35#63565—61no—
36#63565D311yescharacteristic, complemented
37#63564—31no—
38#63563—31no—
39#63562—31no—
40#63562—31no—
41#63562—31no—
42#63562—61no—
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×#627—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5089—91no—
61#5055—91no—
62#4794—91no—
63C2×C32×Q8—31no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×D12—31no—
67C2×C3×D12—61no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—61no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C2×D311yescharacteristic
73C2×#627—21no—
74C32×C4⋊C4—31no—
75C32×C42—31no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—121no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×(C2×D3)⋊C4—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103Dic3⋊Dic3—21no—
104#3722—21no—
105#3722—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—41no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—41no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C3×C4⋊Q8—91no—
118C3×#124—91no—
119#947—31no—
120#888—31no—
121#875—31no—
122#869—31no—
123C3×#103—91no—
124#835—31no—
125#835—91no—
126#835—91no—
127#835—181no—
128#832—91no—
129#832—91no—
130#821—91no—
131#815—91no—
132#810—31no—
133#810—91no—
134#809—91no—
135#809—91no—
136#809—91no—
137#809—181no—
138#803—91no—
139#803—91no—
140#803—181no—
141C22×C3⋊D3—21no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—121no—
145C32×Q8—61no—
146C32×Q8—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×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—21no—
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—121no—
172C2×C3×Dic3#6711yescharacteristic
173C2×C3×Dic3#6711yescharacteristic
174C2×C3×Dic3#6711yescharacteristic
175C2×C3×Dic3#6711yescharacteristic
176C3×D12—62no—
177C3×D12—61no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—61no—
181C3×D12—121no—
182C3×Dic6—22no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C3×Dic6—21no—
187C3×Dic6—41no—
188C3×Dic6—122no—
189C3×Dic6—122no—
190#627—42no—
191C3⋊(C3⋊C8)—21no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—62no—
194C3⋊(C3⋊C8)—61no—
195C3⋊(C3⋊C8)—61no—
196C3⋊(C3⋊C8)—121no—
197C3⋊(C3⋊C8)—121no—
198C3⋊(C3⋊C8)—122no—
199C3×C3⋊C8—62no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—61no—
202#489—271noSylow
203C2×C33#12411yescharacteristic
204C2×C33AGL(1,ℤ/8)11yescharacteristic
205C2×C33#13711yescharacteristic
206C3×C3⋊D3—81no—
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C2×C3×Q8—181no—
211C2×C3×D4—91no—
212C2×D12—31no—
213C2×D12—31no—
214C2×D12—61no—
215C2×Dic6—61no—
216C2×Dic6—61no—
217C2×Dic6D3211yescharacteristic
218C2×C3×C8—91no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—181no—
224C3×C22⋊C4—181no—
225C3×C22⋊C4—181no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—61no—
229(C2×D3)⋊C4—61no—
230(C2×D3)⋊C4—61no—
231(C2×D3)⋊C4—61no—
232(C2×D3)⋊C4—61no—
233C4⋊Dic3—31no—
234C4⋊Dic3—31no—
235C4⋊Dic3—61no—
236C4⋊Dic3—61no—
237C4⋊Dic3D3211yescharacteristic
238Dic3⋊C4—61no—
239C4×Dic3D3211yescharacteristic
240C2×C3⋊C8—31no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8—181no—
248C2×C3⋊C8—181no—
249C2×C3⋊C8—361no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32D3×D411yescharacteristic
255C22×C32C2×#6711yescharacteristic
256C22×C32C2×#6711yescharacteristic
257C2×C3⋊D3—41no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C2×C3×D3—122no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—241no—
268C2×C3×D3—241no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—61no—
278C32×C4—61no—
279C32×C4—121no—
280C32×C4—121no—
281C32×C4#23511yescharacteristic
282C32×C4#23711yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23711yescharacteristic
285C32×C4#23911yescharacteristic
286C32×C4#23911yescharacteristic
287C3⋊Dic3—41no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—42no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—81no—
300C3×Dic3—121no—
301C3×Dic3—121no—
302C3×Dic3—241no—
303C4⋊Q8—91no—
304#124—91no—
305#103—91no—
306#103—271no—
307#102—271no—
308#102—271no—
309#97—271no—
310C33#48911yescharacteristic, complemented, Sylow
311C23×C3—181no—
312C22×D3—61no—
313C22×D3—61no—
314C22×D3—121no—
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—61no—
318C3×Q8—61no—
319C3×Q8—61no—
320C3×Q8—121no—
321C3×Q8—362no—
322C3×D4—181no—
323C3×D4—181no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—41no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4—181no—
341C2×C3×C4C2×D3211yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—121no—
349C2×Dic3#62511yescharacteristic
350C2×Dic3#62511yescharacteristic
351D12—62no—
352D12—61no—
353D12—61no—
354D12—61no—
355D12—61no—
356D12—121no—
357Dic6—21no—
358Dic6—21no—
359Dic6—122no—
360Dic6—122no—
361C3×C8—181no—
362C3⋊C8—62no—
363C3⋊C8—61no—
364C3⋊C8—61no—
365C3⋊C8—182no—
366C3⋊C8—181no—
367C3⋊C8—181no—
368C3⋊C8—181no—
369C3⋊C8—361no—
370C3⋊C8—361no—
371C3⋊C8—362no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32#88511yescharacteristic
385C2×C32#91411yescharacteristic
386C2×C32#91811yescharacteristic
387C2×C32#93211yescharacteristic
388C2×C32#93611yescharacteristic
389C2×C32#94211yescharacteristic
390C2×C32#94711yescharacteristic
391C2×C32#94911yescharacteristic
392C2×C32#95111yescharacteristic
393C3⋊D3—81no—
394C3×D3—242no—
395C3×D3—241no—
396C3×D3—241no—
397C2×Q8—31no—
398C2×Q8—181no—
399C2×D4—91no—
400C2×C8—91no—
401C2×C8—271no—
402C4⋊C4—31no—
403C4⋊C4—91no—
404C4⋊C4—181no—
405C22⋊C4—181no—
406C22⋊C4—181no—
407C42—31no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3C2×#62511yescharacteristic
416C22×C3D3×#6711yescharacteristic
417C22×C3D3×#6711yescharacteristic
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—241no—
426C2×D3—241no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—61no—
439C3×C4—121no—
440C3×C4—121no—
441C3×C4—122no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—121no—
445C3×C4—121no—
446C3×C4—241no—
447C3×C4—361no—
448C3×C4#379611yescharacteristic
449C3×C4#379611yescharacteristic
450C3×C4#379611yescharacteristic
451C3×C4#379911yescharacteristic
452C3×C4#380411yescharacteristic
453C3×C4#380511yescharacteristic
454Dic3—21no—
455Dic3—21no—
456Dic3—41no—
457Dic3—41no—
458Dic3—121no—
459Dic3—121no—
460Dic3—241no—
461C32—21no—
462C32—21no—
463C32—21no—
464C32—41no—
465C32#479411yescharacteristic, complemented
466C32#505511yescharacteristic, complemented
467C32#508911yescharacteristic, complemented
468C23—181no—
469Q8—61no—
470Q8—61no—
471Q8—362no—
472D4—181no—
473D4—181no—
474C2×C4—31no—
475C2×C4—31no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—181no—
479C2×C4D3311yescharacteristic
480C8—181no—
481C8—541no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—721no—
495C2×C3#6382311yescharacteristic
496C2×C3#6382711yescharacteristic
497C2×C3#6386311yescharacteristic
498C2×C3#6390711yescharacteristic
499C2×C3#6393411yescharacteristic
500C2×C3#6393911yescharacteristic
501C2×C3#6394311yescharacteristic
502C2×C3#6394811yescharacteristic
503C2×C3#6395911yescharacteristic
504D3—242no—
505D3—241no—
506D3—241no—
507C22—361no—
508C22—361no—
509C22—361no—
510C22D3×#62511yescharacteristic
511C4—61no—
512C4—61no—
513C4—121no—
514C4—121no—
515C4—361no—
516C4#8896211yescharacteristic
517C4#8896411yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059028011yescharacteristic, complemented
523C3#1059053511yescharacteristic, complemented
524C3#1059162411yescharacteristic, complemented
525C2—721no—
526C2#1172286711yescharacteristic
527C2#1172288311yescharacteristic
528C2#1172300111yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#489
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. C3×#63918C2semidirect
4. #11722037C2semidirect
6. #11722019C2semidirect
7. C3×#63565C2semidirect
26. #63918D3semidirect
36. #63565D3semidirect
310. C33#489semidirect
465. C32#4794semidirect
466. C32#5055semidirect
467. C32#5089semidirect
522. C3#10590280semidirect
523. C3#10590535semidirect
524. C3#10591624semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#638642
#117220752
#117220372
#117220332
#117220192
C3×#635652
#105916243
#105905353
#105902803