Finite Group Databasebeta

← #13155999

Subgroups — #13155999

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic
2C2×#88793C211yesmaximal, characteristic
3C3×#63894C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591565—31nomaximal
9#10590559—31nomaximal
10#10590363—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#624—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19#88811—22no—
20#88793—22no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic3⋊Dic3—21no—
23C3×#3722—21no—
24C2×#3755—31no—
25C3×#944—31no—
26C3×#893—31no—
27C3×C4×D12—31no—
28C3×#869—31no—
29#63894D311yescharacteristic
30#63866D311yescharacteristic, complemented
31C2×#3717—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34C2×#3715—31no—
35C2×#3715—31no—
36C2×#3715—61no—
37#63637—31no—
38C4⋊(C3⋊(C3⋊C8))—31no—
39C3×#835—31no—
40C3×#810—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—61no—
44#63565—31no—
45#63565—31no—
46#63565—61no—
47#63565D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)D412yes—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×#624—42no—
62C3⋊(C3⋊(C3⋊C8))—21no—
63C3×C3⋊(C3⋊C8)—21no—
64#5083—91no—
65#5019—91no—
66#4818—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634C2×D311yescharacteristic
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—61no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C2×#624—21no—
82C32×C4⋊C4—31no—
83C32×C42—31no—
84#3755—62no—
85C2×C3⋊(C3⋊C8)—31no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—61no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—121no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×(C2×D3)⋊C4—61no—
94C3×(C2×D3)⋊C4—61no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×Dic3⋊C4—61no—
101C3×Dic3⋊C4—61no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C2×D311yescharacteristic
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108Dic3⋊Dic3—21no—
109Dic3⋊Dic3—21no—
110#3722—21no—
111#3717—62no—
112#3716—62no—
113#3715—62no—
114#3715—62no—
115#3715—62no—
116#3715—64no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C33×C4C2×D411yescharacteristic
122C33×C4#4111yescharacteristic
123C3×C3⋊Dic3—41no—
124C32×Dic3—21no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C32×Dic3SD1612yescomplemented
128C3×C4⋊Q8—91no—
129C3×C4×D4—91no—
130#944—31no—
131C2×#215—91no—
132C2×#215—91no—
133C2×#215—91no—
134C2×#215—181no—
135C2×#214—91no—
136C2×#214—91no—
137C2×#204—91no—
138#893—31no—
139C4×D12—31no—
140#869—31no—
141C3×#103—91no—
142#835—31no—
143#835—91no—
144Dic3⋊C8—91no—
145#810—31no—
146#810—91no—
147#810—91no—
148#810—181no—
149C4⋊(C3⋊C8)—91no—
150C4⋊(C3⋊C8)—91no—
151C4⋊(C3⋊C8)—181no—
152C22×C3⋊D3—21no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—121no—
156C32×Q8—61no—
157C32×Q8—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4—61no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—21no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)D1212yes—
172#634—21no—
173#634—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—41no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C2×C3×Dic3#6711yescharacteristic
186C2×C3×Dic3#6711yescharacteristic
187C2×C3×Dic3#6711yescharacteristic
188C2×C3×Dic3#6711yescharacteristic
189C3×D12—32no—
190C3×D12—32no—
191C3×D12—62no—
192C3×D12—61no—
193C3×D12—61no—
194C3×D12—121no—
195C3×C4×D3—122no—
196C3×C4×D3—122no—
197C3×Dic6—22no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—41no—
203C3×Dic6—62no—
204C3×Dic6—61no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—121no—
209#624—42no—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)—61no—
212C3⋊(C3⋊C8)—62no—
213C3⋊(C3⋊C8)—61no—
214C3⋊(C3⋊C8)—61no—
215C3⋊(C3⋊C8)—121no—
216C3⋊(C3⋊C8)—121no—
217C3⋊(C3⋊C8)—122no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—121no—
221#461—271noSylow
222C2×C33#12111yescharacteristic
223C2×C33C2×SD1611yescharacteristic
224C2×C33#13711yescharacteristic
225C3×C3⋊D3—42no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—61no—
229C2×C3×Q8—91no—
230C2×C3×D4—91no—
231C22×C3×C4—181no—
232C2×D12—31no—
233C2×D12—31no—
234C2×D12—61no—
235C2×C4×D3—61no—
236C2×C4×D3—61no—
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—61no—
240C2×Dic6D3211yescharacteristic
241C2×C3×C8—91no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—181no—
246C3×C4⋊C4—181no—
247C3×C22⋊C4—181no—
248C3×C42—31no—
249C3×C42—31no—
250C3×C42—61no—
251#215—182no—
252#215—182no—
253#215—182no—
254#215—184no—
255#214—182no—
256#214—182no—
257(C2×D3)⋊C4—61no—
258(C2×D3)⋊C4—61no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3D3211yescharacteristic
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264C4×Dic3D3211yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—91no—
269C2×C3⋊C8—91no—
270C2×C3⋊C8—91no—
271C2×C3⋊C8—181no—
272C2×C3⋊C8—181no—
273C2×C3⋊C8—181no—
274C2×C3⋊C8—361no—
275#204—182no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32D3×D411yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C2×C3⋊D3—22no—
284C2×C3⋊D3—41no—
285C2×C3⋊D3—41no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—241no—
294C2×C3×D3—241no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—32no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—61no—
305C32×C4—121no—
306C32×C4—121no—
307C32×C4C2×D1211yescharacteristic
308C32×C4#23511yescharacteristic
309C32×C4#23511yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4#23711yescharacteristic
313C3⋊Dic3—41no—
314C3×Dic3—21no—
315C3×Dic3—21no—
316C3×Dic3—22no—
317C3×Dic3—41no—
318C3×Dic3—42no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—81no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3#21412yescomplemented
328C3×Dic3#21512yescomplemented
329C2×SD16—271no—
330C2×SD16—271no—
331C4⋊Q8—91no—
332C4×D4—91no—
333C4⋊C8—271no—
334#103—91no—
335#103—271no—
336C33#46111yescharacteristic, complemented, Sylow
337C23×C3—181no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—121no—
341C3×Q8—62no—
342C3×Q8—61no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—121no—
347C3×Q8—181no—
348C3×Q8—181no—
349C3×D4—92no—
350C3×D4—181no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—41no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—121no—
366C2×C3×C4—121no—
367C2×C3×C4—181no—
368C2×C3×C4—362no—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—121no—
377C2×Dic3#62511yescharacteristic
378C2×Dic3#62511yescharacteristic
379D12—32no—
380D12—32no—
381D12—62no—
382D12—61no—
383D12—61no—
384D12—121no—
385C4×D3—122no—
386C4×D3—122no—
387Dic6—21no—
388Dic6—21no—
389Dic6—62no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—61no—
394Dic6—121no—
395C3×C8—181no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—121no—
399C3⋊C8—182no—
400C3⋊C8—181no—
401C3⋊C8—181no—
402C3⋊C8—181no—
403C3⋊C8—361no—
404C3⋊C8—361no—
405C3⋊C8—362no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32#88411yescharacteristic
419C2×C32D3×SD1611yescharacteristic
420C2×C32#91811yescharacteristic
421C2×C32#93011yescharacteristic
422C2×C32C2×#21411yescharacteristic
423C2×C32#93811yescharacteristic
424C2×C32C2×#21511yescharacteristic
425C2×C32#94211yescharacteristic
426C2×C32#95111yescharacteristic
427C3⋊D3—42no—
428C3×D3—122no—
429C3×D3—122no—
430C3×D3—242no—
431C2×Q8—31no—
432C2×Q8—91no—
433C2×D4—91no—
434C22×C4—181no—
435SD16—542no—
436SD16—542no—
437C2×C8—91no—
438C2×C8—271no—
439C4⋊C4—31no—
440C4⋊C4—181no—
441C4⋊C4—181no—
442C22⋊C4—181no—
443C42—31no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—41no—
448C22×C3—182no—
449C22×C3—361no—
450C22×C3—361no—
451C22×C3C2×#62511yescharacteristic
452C22×C3D3×#6711yescharacteristic
453C22×C3D3×#6711yescharacteristic
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—122no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—121no—
460C2×D3—121no—
461C2×D3—241no—
462C2×D3—241no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—32no—
470C3×C4—32no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—62no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—121no—
477C3×C4—122no—
478C3×C4—121no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—241no—
483C3×C4—361no—
484C3×C4#379511yescharacteristic
485C3×C4#379811yescharacteristic
486C3×C4D3×D1211yescharacteristic
487C3×C4D3×D1211yescharacteristic
488C3×C4#380411yescharacteristic
489C3×C4#381111yescharacteristic
490Dic3—21no—
491Dic3—41no—
492Dic3—41no—
493Dic3—122no—
494Dic3—121no—
495Dic3—121no—
496Dic3#371512yescomplemented
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32#481811yescharacteristic, complemented
502C32#501911yescharacteristic, complemented
503C32#508311yescharacteristic, complemented
504C23—181no—
505Q8—61no—
506Q8—61no—
507Q8—181no—
508Q8—181no—
509D4—92no—
510D4—181no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—181no—
516C2×C4—362no—
517C2×C4D3311yescharacteristic
518C8—181no—
519C8—541no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—362no—
533C2×C3C2×#371511yescharacteristic
534C2×C3#6382711yescharacteristic
535C2×C3#6390611yescharacteristic
536C2×C3#6390811yescharacteristic
537C2×C3D3×#21411yescharacteristic
538C2×C3D3×#21511yescharacteristic
539C2×C3#6394311yescharacteristic
540C2×C3#6394811yescharacteristic
541C2×C3#6395811yescharacteristic
542D3—122no—
543D3—122no—
544D3—242no—
545C22—182no—
546C22—361no—
547C22—361no—
548C22D3×#62511yescharacteristic
549C4—32no—
550C4—61no—
551C4—121no—
552C4—121no—
553C4—361no—
554C4#8895511yescharacteristic
555C4#8896011yescharacteristic
556C3—21no—
557C3—21no—
558C3—21no—
559C3—41no—
560C3#1059036311yescharacteristic, complemented
561C3#1059055911yescharacteristic, complemented
562C3#1059156511yescharacteristic, complemented
563C2—362no—
564C2D3×#371511yescharacteristic
565C2#1172288311yescharacteristic
566C2#1172295711yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#461
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
4. C3×#63866C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722039C2semidirect
7. C3×#63565C2semidirect
30. #63866D3semidirect
47. #63565D3semidirect
127. C32×Dic3SD16semidirect
327. C3×Dic3#214semidirect
328. C3×Dic3#215semidirect
336. C33#461semidirect
496. Dic3#3715semidirect
501. C32#4818semidirect
502. C32#5019semidirect
503. C32#5083semidirect
560. C3#10590363semidirect
561. C3#10590559semidirect
562. C3#10591565semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
C2×#887932
C3×#638942
C3×#638662
Dic3⋊(C3⋊(C3⋊C8))2
#117220392
C3×#635652
#105915653
#105905593
#105903633