Finite Group Databasebeta

← #13155648

Subgroups — #13155648

Order 1728.

706 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723691C211yesmaximal, characteristic
2#11723406C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic
4#11722042C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic, complemented
8#10590758—31nomaximal
9#10590744—31nomaximal
10#10590460—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6294—21no—
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#88796C414yescomplemented
23C2×#3744—31no—
24#64096—31no—
25#63898—31no—
26#63894—31no—
27Dic3×D12—31no—
28#63844—31no—
29#63844—31no—
30#63844—61no—
31C2×#3716—31no—
32C2×#3716—31no—
33C2×#3716—61no—
34#63643—31no—
35C3×#821—31no—
36C3×#817—31no—
37C3×C8×Dic3—31no—
38C3×C8×Dic3—31no—
39C3×#809—31no—
40C3×#808—31no—
41#63573D311yescharacteristic, complemented
42#63567—31no—
43#63567—31no—
44#63567—61no—
45#63564D311yescharacteristic
46#63553D311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308C2×C412yescomplemented
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)C2×C412yes—
57C2×C32×Dic3—21no—
58#6294—42no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#4852—91no—
62#4758—91no—
63C4×#204—91no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×#634—31no—
66C2×#634—31no—
67C2×#634—31no—
68C2×#634—61no—
69C2×#634—61no—
70C2×#634—61no—
71C2×#634—121no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×#624—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82#3750—61no—
83C4×C3⋊Dic3—61no—
84C4×C3⋊Dic3C2×D311yescharacteristic
85#3744—34no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—61no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—61no—
102#3724—61no—
103#3724—62no—
104#3722—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107Dic32—61no—
108Dic32—61no—
109Dic32—62no—
110#3716—34no—
111#3716—34no—
112#3716—64no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3⋊(C3⋊Dic3)—22no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—41no—
122C3×C3⋊Dic3SD1612yescomplemented
123C32×Dic3—41no—
124D4×Dic3—91no—
125C2×#214—91no—
126C2×#204—91no—
127C2×#204—91no—
128C2×#204—181no—
129#893—91no—
130#887—91no—
131#887—91no—
132#887—181no—
133C4×D12—91no—
134C4×Dic6—91no—
135C3×#106—91no—
136C3×#102—91no—
137C3×C4×C8—91no—
138#821—31no—
139#817—31no—
140#816—91no—
141#816—91no—
142#816—181no—
143C8×Dic3—31no—
144C8×Dic3—31no—
145#810—91no—
146#809—31no—
147#808—31no—
148C22×C3⋊D3—21no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—121no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—41no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—62no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—121no—
166C2×C3⋊Dic3—121no—
167C2×C3⋊Dic3—122no—
168C2×C3⋊Dic3D1211yescharacteristic
169C2×C3⋊Dic3D1211yescharacteristic
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)C4×D312yes—
172#634—32no—
173#634—32no—
174#634—32no—
175#634—62no—
176#634—62no—
177#634—62no—
178#634—61no—
179#634—62no—
180#634—61no—
181#634—61no—
182#634—122no—
183#634—121no—
184#634—121no—
185#634—122no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C3×D12—32no—
199C3×D12—32no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—121no—
204#624—122no—
205D3×Dic3—122no—
206D3×Dic3—122no—
207C32×C8—32no—
208C32×C8—61no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—22no—
212C3×C3⋊C8—41no—
213C3×C3⋊C8C4×D312yescomplemented
214C3×C3⋊C8C4×D312yescomplemented
215C4×SD16—271noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33C2×SD1611yescharacteristic
218C2×C33#13511yescharacteristic
219C3×C3⋊D3—42no—
220C2×C3×D4—91no—
221C22×Dic3—181no—
222C2×D12—31no—
223C2×D12—31no—
224C2×D12—61no—
225C2×C4×D3—181no—
226C2×C4×D3—181no—
227C2×Dic6—91no—
228C2×Dic6—91no—
229C2×Dic6—91no—
230C2×Dic6—181no—
231C2×Dic6—181no—
232C2×Dic6—181no—
233C2×Dic6—361no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—61no—
239C2×C3×C8—61no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—91no—
244C3×C42—91no—
245#217—181no—
246#214—94no—
247(C2×D3)⋊C4—181no—
248(C2×D3)⋊C4—181no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—61no—
252C4⋊Dic3D3211yescharacteristic
253Dic3⋊C4—181no—
254Dic3⋊C4—181no—
255Dic3⋊C4—181no—
256Dic3⋊C4—182no—
257C4×Dic3—31no—
258C4×Dic3—31no—
259C4×Dic3—61no—
260C4×Dic3—181no—
261C4×Dic3—181no—
262C4×Dic3—181no—
263C4×Dic3—182no—
264C2×C3⋊C8D3211yescharacteristic
265C2×C3⋊C8D3211yescharacteristic
266#204—94no—
267#204—94no—
268#204—184no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32C2×D1211yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32D3×D411yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—122no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—241no—
287C2×C3×D3—241no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—121no—
297C32×C4C2×C4×D311yescharacteristic
298C32×C4C2×C4×D311yescharacteristic
299C32×C4C2×C4×D311yescharacteristic
300C32×C4#23511yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23911yescharacteristic
303C3⋊Dic3—21no—
304C3⋊Dic3—41no—
305C3⋊Dic3—62no—
306C3⋊Dic3—64no—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3⋊Dic3—122no—
310C3⋊Dic3—122no—
311C3⋊Dic3—124no—
312C3⋊Dic3#20412yescomplemented
313C3×Dic3—32no—
314C3×Dic3—32no—
315C3×Dic3—42no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—62no—
319C3×Dic3—61no—
320C3×Dic3—61no—
321C3×Dic3—121no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3—241no—
325C2×SD16—271no—
326C4×Q8—271no—
327C4×D4—271no—
328#106—91no—
329#103—271no—
330#102—91no—
331C4×C8—91no—
332C33C4×SD1611yescharacteristic, complemented, Sylow
333C23×C3—181no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—121no—
337C3×D4—92no—
338C3×D4—181no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—41no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—91no—
347C2×C3×C4—91no—
348C2×C3×C4—181no—
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—121no—
362C2×Dic3—182no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—361no—
367C2×Dic3—361no—
368C2×Dic3—362no—
369C2×Dic3—362no—
370D12—32no—
371D12—32no—
372D12—62no—
373D12—61no—
374D12—61no—
375D12—121no—
376C4×D3—362no—
377C4×D3—362no—
378Dic6—92no—
379Dic6—92no—
380Dic6—92no—
381Dic6—182no—
382Dic6—182no—
383Dic6—182no—
384Dic6—182no—
385Dic6—181no—
386Dic6—181no—
387Dic6—181no—
388Dic6—362no—
389Dic6—361no—
390Dic6—361no—
391Dic6—362no—
392C3×C8—32no—
393C3×C8—32no—
394C3×C8—62no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—121no—
398C3⋊C8—21no—
399C3⋊C8#62412yescomplemented
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32C4×D1211yescharacteristic
413C2×C32#67⋊C411yescharacteristic
414C2×C32#89311yescharacteristic
415C2×C32C2×#20411yescharacteristic
416C2×C32#90411yescharacteristic
417C2×C32#91211yescharacteristic
418C2×C32D3×SD1611yescharacteristic
419C2×C32D3×SD1611yescharacteristic
420C2×C32#91611yescharacteristic
421C3⋊D3—42no—
422C3×D3—122no—
423C3×D3—122no—
424C3×D3—242no—
425C2×Q8—271no—
426C2×D4—91no—
427C22×C4—541no—
428SD16—274no—
429C2×C8—31no—
430C2×C8—31no—
431C4⋊C4—31no—
432C4⋊C4—91no—
433C4⋊C4—541no—
434C22⋊C4—541no—
435C42—91no—
436C42—541no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—182no—
442C22×C3—361no—
443C22×C3—361no—
444C22×C3D3×D1211yescharacteristic
445C22×C3D3×D1211yescharacteristic
446C22×C3#381111yescharacteristic
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—122no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—241no—
455C2×D3—241no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—41no—
463C3×C4—41no—
464C3×C4—92no—
465C3×C4—122no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—181no—
469C3×C4—361no—
470C3×C4#379611yescharacteristic
471C3×C4#379911yescharacteristic
472C3×C4C4×D3211yescharacteristic
473C3×C4C4×D3211yescharacteristic
474C3×C4#380411yescharacteristic
475C3×C4C2×#62411yescharacteristic
476Dic3—32no—
477Dic3—32no—
478Dic3—41no—
479Dic3—62no—
480Dic3—61no—
481Dic3—61no—
482Dic3—121no—
483Dic3—121no—
484Dic3—121no—
485Dic3—184no—
486Dic3—182no—
487Dic3—182no—
488Dic3—182no—
489Dic3—241no—
490Dic3—362no—
491Dic3—362no—
492Dic3—364no—
493C32—21no—
494C32—21no—
495C32—21no—
496C32—41no—
497C32C4×#20411yescharacteristic, complemented
498C32#475811yescharacteristic, complemented
499C32#485211yescharacteristic, complemented
500C23—181no—
501Q8—272no—
502Q8—541no—
503D4—92no—
504D4—181no—
505C2×C4—61no—
506C2×C4—91no—
507C2×C4—91no—
508C2×C4—181no—
509C2×C4—541no—
510C2×C4—1082no—
511C2×C4D3311yescharacteristic
512C8—32no—
513C8—61no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—362no—
527C2×C3D3×#20411yescharacteristic
528C2×C3D3×#20411yescharacteristic
529C2×C3#6380611yescharacteristic
530C2×C3#6380711yescharacteristic
531C2×C3#6388011yescharacteristic
532C2×C3#6388111yescharacteristic
533C2×C3#6389411yescharacteristic
534C2×C3#6393611yescharacteristic
535C2×C3#6394011yescharacteristic
536D3—122no—
537D3—122no—
538D3—242no—
539C22—182no—
540C22—361no—
541C22—361no—
542C22#8895511yescharacteristic
543C4—92no—
544C4—121no—
545C4—181no—
546C4—361no—
547C4—542no—
548C4D3×#62411yescharacteristic
549C4#8896211yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059046011yescharacteristic, complemented
555C3#1059074411yescharacteristic, complemented
556C3#1059075811yescharacteristic, complemented
557C2—362no—
558C2#1172284311yescharacteristic
559C2#1172286411yescharacteristic
560C2#1172293711yescharacteristic
subgroupG/H
centerC22#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723406C2semidirect
4. #11722042C2semidirect
5. C3×#63573C2semidirect
7. C3×#63553C2semidirect
22. #88796C4semidirect
41. #63573D3semidirect
46. #63553D3semidirect
51. #6308C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
122. C3×C3⋊Dic3SD16semidirect
213. C3×C3⋊C8C4×D3semidirect
214. C3×C3⋊C8C4×D3semidirect
312. C3⋊Dic3#204semidirect
332. C33C4×SD16semidirect
399. C3⋊C8#624semidirect
497. C32C4×#204semidirect
498. C32#4758semidirect
499. C32#4852semidirect
554. C3#10590460semidirect
555. C3#10590744semidirect
556. C3#10590758semidirect

Maximal subgroups

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

subgroupindex
#117236912
#117234062
C2×#887962
#117220422
C3×#635732
C3×#635642
C3×#635532
#105907583
#105907443
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