Finite Group Databasebeta

← #13156022

Subgroups — #13156022

Order 1728.

660 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723399C211yesmaximal, characteristic
2C2×#88800C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic
5#11722076C211yesmaximal, characteristic
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic
8#10591575—31nomaximal
9#10590808—31nomaximal
10#10590388—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806—21no—
19Dic3×C3⋊Dic3—21no—
20#88800—22no—
21#88799—22no—
22C3×Dic3⋊Dic3—21no—
23C3×Dic3⋊Dic3—21no—
24C2×#3756—31no—
25C2×#3746—31no—
26C4×#634—31no—
27C3×#944—31no—
28C3×#869—31no—
29#63866D311yescharacteristic
30#63844—31no—
31#63844—31no—
32#63844—61no—
33C2×#3719—31no—
34C2×#3719—31no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3719—61no—
38C2×#3719—61no—
39C3×#835—31no—
40C3×#816—31no—
41C3×Dic3⋊C8—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic
44#63567—31no—
45#63566—31no—
46#63566D311yescharacteristic
47#63565—31no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63#5090—91no—
64#4849—91no—
65#4795—91no—
66C2×C32×Q8—31no—
67C2×#634—31no—
68C2×#634—31no—
69C2×#634—31no—
70C2×#634—61no—
71C2×#634—61no—
72C2×#634—61no—
73C2×#634—121no—
74C2×#634C2×D311yescharacteristic
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×C32×C8—31no—
82C32×C42—31no—
83#3756—62no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C4×C3⋊Dic3—61no—
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88#3746—62no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105#3724—62no—
106#3724—61no—
107#3724—61no—
108Dic3⋊Dic3—21no—
109Dic3⋊Dic3—21no—
110Dic32—62no—
111Dic32—61no—
112Dic32—61no—
113#3719—62no—
114#3719—62no—
115#3719—64no—
116#3719—62no—
117#3719—62no—
118#3719—64no—
119C22×C33C2×D411yescharacteristic
120C33×C4C2×D411yescharacteristic
121C33×C4#4111yescharacteristic
122C3⋊(C3⋊Dic3)—22no—
123C3×C3⋊Dic3—41no—
124C3×C3⋊Dic3—41no—
125C32×Dic3—21no—
126C32×Dic3—41no—
127C32×Dic3Q1612yescomplemented
128C3×C4⋊Q8—91no—
129#944—31no—
130C2×#216—91no—
131C2×#216—91no—
132C2×#216—91no—
133C2×#216—181no—
134C2×Dic12—91no—
135C2×Dic12—91no—
136C2×Dic12—91no—
137C2×Dic12—181no—
138#887—91no—
139#869—31no—
140C4×Dic6—91no—
141C4×Dic6—91no—
142C4×Dic6—181no—
143C3×C4⋊C8—91no—
144C3×#103—91no—
145#835—31no—
146#835—91no—
147#816—31no—
148#816—91no—
149Dic3⋊C8—31no—
150#810—91no—
151C4⋊(C3⋊C8)—31no—
152C32×Q8—61no—
153C32×Q8—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—31no—
158C2×C32×C4—31no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—122no—
173#634—21no—
174#634—21no—
175#634—32no—
176#634—32no—
177#634—32no—
178#634—62no—
179#634—62no—
180#634—62no—
181#634—61no—
182#634—62no—
183#634—61no—
184#634—61no—
185#634—122no—
186#634—121no—
187#634—121no—
188#634—122no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3D1211yescharacteristic
201C2×C3×Dic3D1211yescharacteristic
202C2×C3×Dic3#6711yescharacteristic
203C2×C3×Dic3#6711yescharacteristic
204C3×Dic6—22no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—41no—
210C3×Dic6—62no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—121no—
216C32×C8—61no—
217C3⋊(C3⋊C8)—21no—
218C3×C3⋊C8—22no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—62no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224#462—271noSylow
225C2×C33#12111yescharacteristic
226C2×C33C2×Q1611yescharacteristic
227C2×C33#13711yescharacteristic
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×Dic6—31no—
233C2×Dic6—31no—
234C2×Dic6—61no—
235C2×Dic6—91no—
236C2×Dic6—91no—
237C2×Dic6—91no—
238C2×Dic6—181no—
239C2×Dic6—181no—
240C2×Dic6—181no—
241C2×Dic6—361no—
242C2×Dic6D3211yescharacteristic
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—61no—
246C2×C3×C8—91no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—181no—
249C3×C4⋊C4—181no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253#216—182no—
254#216—182no—
255#216—182no—
256#216—184no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3—61no—
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264Dic3⋊C4—181no—
265Dic3⋊C4—181no—
266Dic3⋊C4—182no—
267Dic3⋊C4—181no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—182no—
271C4×Dic3—181no—
272C4×Dic3D3211yescharacteristic
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8D3211yescharacteristic
277Dic12—182no—
278Dic12—182no—
279Dic12—184no—
280Dic12—182no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32C2×D1211yescharacteristic
286C22×C32D3×D411yescharacteristic
287C22×C32C2×#6711yescharacteristic
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—32no—
295C32×C4—41no—
296C32×C4—41no—
297C32×C4—61no—
298C32×C4—121no—
299C32×C4#23511yescharacteristic
300C32×C4D3×D411yescharacteristic
301C32×C4D3×D411yescharacteristic
302C32×C4D3×D411yescharacteristic
303C32×C4#23711yescharacteristic
304C32×C4#23911yescharacteristic
305C3⋊Dic3—41no—
306C3⋊Dic3—41no—
307C3⋊Dic3—62no—
308C3⋊Dic3—64no—
309C3⋊Dic3—62no—
310C3⋊Dic3—62no—
311C3⋊Dic3—122no—
312C3⋊Dic3—122no—
313C3⋊Dic3—124no—
314C3×Dic3—21no—
315C3×Dic3—21no—
316C3×Dic3—22no—
317C3×Dic3—41no—
318C3×Dic3—42no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—122no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3Dic1212yescomplemented
328C3×Dic3#21612yescomplemented
329C2×Q16—271no—
330C2×Q16—271no—
331C4⋊Q8—91no—
332C4×Q8—271no—
333C4⋊C8—91no—
334#103—91no—
335#103—271no—
336C33#46211yescharacteristic, complemented, Sylow
337C3×Q8—62no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—121no—
343C3×Q8—181no—
344C3×Q8—181no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—41no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—121no—
358C2×C3×C4—181no—
359C2×C3×C4—181no—
360C2×C3×C4C2×D3211yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C2×D3211yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—62no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—121no—
370C2×Dic3—182no—
371C2×Dic3—181no—
372C2×Dic3—181no—
373C2×Dic3—181no—
374C2×Dic3—361no—
375C2×Dic3—361no—
376C2×Dic3—362no—
377C2×Dic3#62611yescharacteristic
378C2×Dic3#62611yescharacteristic
379Dic6—21no—
380Dic6—21no—
381Dic6—62no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386Dic6—92no—
387Dic6—92no—
388Dic6—92no—
389Dic6—121no—
390Dic6—182no—
391Dic6—182no—
392Dic6—182no—
393Dic6—182no—
394Dic6—181no—
395Dic6—181no—
396Dic6—181no—
397Dic6—362no—
398Dic6—361no—
399Dic6—361no—
400Dic6—362no—
401C3×C8—62no—
402C3×C8—61no—
403C3×C8—61no—
404C3×C8—181no—
405C3⋊C8—21no—
406C3⋊C8—62no—
407C3⋊C8—61no—
408C3⋊C8—61no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—41no—
421C2×C32#88411yescharacteristic
422C2×C32#89511yescharacteristic
423C2×C32C2×Dic1211yescharacteristic
424C2×C32#90911yescharacteristic
425C2×C32#91511yescharacteristic
426C2×C32D3×Q1611yescharacteristic
427C2×C32#93811yescharacteristic
428C2×C32#94211yescharacteristic
429C2×C32C2×#21611yescharacteristic
430C2×Q8—31no—
431C2×Q8—91no—
432C2×Q8—271no—
433Q16—542no—
434Q16—542no—
435C2×C8—31no—
436C2×C8—91no—
437C4⋊C4—91no—
438C4⋊C4—181no—
439C4⋊C4—181no—
440C4⋊C4—541no—
441C42—31no—
442C42—541no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—41no—
447C22×C3D3×D1211yescharacteristic
448C22×C3C2×#62611yescharacteristic
449C22×C3D3×#6711yescharacteristic
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—32no—
457C3×C4—32no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—62no—
461C3×C4—61no—
462C3×C4—61no—
463C3×C4—121no—
464C3×C4—122no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—361no—
468C3×C4—361no—
469C3×C4#379611yescharacteristic
470C3×C4#379911yescharacteristic
471C3×C4#380211yescharacteristic
472C3×C4#380411yescharacteristic
473C3×C4#381111yescharacteristic
474C3×C4#381111yescharacteristic
475Dic3—21no—
476Dic3—41no—
477Dic3—122no—
478Dic3—122no—
479Dic3—121no—
480Dic3—121no—
481Dic3—121no—
482Dic3—121no—
483Dic3—184no—
484Dic3—182no—
485Dic3—182no—
486Dic3—182no—
487Dic3—362no—
488Dic3—362no—
489Dic3—364no—
490Dic3#371912yescomplemented
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#479511yescharacteristic, complemented
496C32#484911yescharacteristic, complemented
497C32#509011yescharacteristic, complemented
498Q8—61no—
499Q8—61no—
500Q8—181no—
501Q8—181no—
502Q8—272no—
503Q8—541no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—61no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4—541no—
510C2×C4D3311yescharacteristic
511C8—61no—
512C8—181no—
513C2×C3—21no—
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—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3D3×Dic1211yescharacteristic
526C2×C3#6380011yescharacteristic
527C2×C3#6383311yescharacteristic
528C2×C3C2×#371911yescharacteristic
529C2×C3#6391011yescharacteristic
530C2×C3#6391611yescharacteristic
531C2×C3D3×#21611yescharacteristic
532C2×C3#6394611yescharacteristic
533C2×C3#6395811yescharacteristic
534C22D3×#62611yescharacteristic
535C4—32no—
536C4—61no—
537C4—121no—
538C4—361no—
539C4—361no—
540C4—542no—
541C4#8895411yescharacteristic
542C4#8896211yescharacteristic
543C3—21no—
544C3—21no—
545C3—21no—
546C3—41no—
547C3#1059038811yescharacteristic, complemented
548C3#1059080811yescharacteristic, complemented
549C3#1059157511yescharacteristic, complemented
550C2D3×#371911yescharacteristic
551C2#1172288411yescharacteristic
552C2#1172296311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#462
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
127. C32×Dic3Q16semidirect
327. C3×Dic3Dic12semidirect
328. C3×Dic3#216semidirect
336. C33#462semidirect
490. Dic3#3719semidirect
495. C32#4795semidirect
496. C32#4849semidirect
497. C32#5090semidirect
547. C3#10590388semidirect
548. C3#10590808semidirect
549. C3#10591575semidirect

Maximal subgroups

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

subgroupindex
#117233992
C2×#888002
C2×#887992
C3×#638662
#117220762
C3×Dic3⋊(C3⋊C8)2
C3×#635662
#105915753
#105908083
#105903883