Finite Group Databasebeta

← #13156020

Subgroups — #13156020

Order 1728.

640 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88812C211yesmaximal, characteristic
2#11723443C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic
4C3×#63844C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8#10590808—31nomaximal
9#10590561—31nomaximal
10#10590385—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812—22no—
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88800—22no—
22C3×#3724—21no—
23C3×Dic32—21no—
24C2×#3746—31no—
25#64078—31no—
26C3×#887—31no—
27C3×C4×Dic6—31no—
28#63866—31no—
29#63866—31no—
30#63866—61no—
31#63844D311yescharacteristic
32C2×#3719—31no—
33C2×#3719—31no—
34C2×#3719—31no—
35C2×#3719—31no—
36C2×#3719—61no—
37C2×#3718—31no—
38#63637—31no—
39C3×#816—31no—
40C3×Dic3⋊C8—31no—
41C3×#810—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic
44#63567—31no—
45#63567—31no—
46#63567—61no—
47#63567D311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634—21no—
57C3×#634D412yes—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D411yescharacteristic
60C2×C32×Dic3D411yescharacteristic
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63#5031—91no—
64#4849—91no—
65#4795—91no—
66C2×#634—31no—
67C2×#634—31no—
68C2×#634—31no—
69C2×#634—61no—
70C2×#634—61no—
71C2×#634—61no—
72C2×#634—121no—
73C2×#634C2×D311yescharacteristic
74C2×#634C2×D311yescharacteristic
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—61no—
81C2×C32×C8—31no—
82C32×C4⋊C4—31no—
83C32×C42—31no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87#3746—62no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104#3724—21no—
105Dic3⋊Dic3—62no—
106Dic3⋊Dic3—62no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—61no—
110Dic3⋊Dic3—61no—
111Dic32—21no—
112#3719—62no—
113#3719—62no—
114#3719—64no—
115#3719—62no—
116#3719—62no—
117#3718—62no—
118C22×C33C2×D411yescharacteristic
119C33×C4C2×D411yescharacteristic
120C33×C4#4111yescharacteristic
121C3⋊(C3⋊Dic3)—41no—
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—41no—
124C32×Dic3—21no—
125C32×Dic3—41no—
126C32×Dic3Q1612yescomplemented
127C3×C4×Q8—91no—
128#944—91no—
129C2×#216—91no—
130C2×#216—91no—
131C2×#216—91no—
132C2×Dic12—91no—
133C2×Dic12—91no—
134C2×Dic12—91no—
135C2×Dic12—181no—
136#887—31no—
137#869—91no—
138#869—91no—
139#869—181no—
140C4×Dic6—31no—
141C3×C4⋊C8—91no—
142C3×#103—91no—
143#816—31no—
144#816—91no—
145Dic3⋊C8—31no—
146#810—31no—
147#810—91no—
148#810—91no—
149#810—181no—
150C4⋊(C3⋊C8)—31no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—41no—
157C2×C32×C4—61no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—62no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—121no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—122no—
170#634—21no—
171#634—21no—
172#634—21no—
173#634—61no—
174#634—61no—
175#634—62no—
176#634—62no—
177#634—61no—
178#634—61no—
179#634—61no—
180#634—61no—
181#634—121no—
182#634—121no—
183#634—121no—
184#634—121no—
185#634—122no—
186#634—122no—
187#634D1212yes—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3D1211yescharacteristic
200C2×C3×Dic3D1211yescharacteristic
201C2×C3×Dic3#6711yescharacteristic
202C2×C3×Dic3#6711yescharacteristic
203C3×Dic6—32no—
204C3×Dic6—32no—
205C3×Dic6—62no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—121no—
213C3×Dic6—121no—
214C3×Dic6—121no—
215C32×C8—61no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—22no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#462—271noSylow
224C2×C33#12111yescharacteristic
225C2×C33C2×Q1611yescharacteristic
226C2×C33#13711yescharacteristic
227C2×C3×Q8—91no—
228C2×C3×Q8—91no—
229C2×Dic6—31no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—31no—
233C2×Dic6—61no—
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×C3×C8—31no—
243C2×C3×C8—31no—
244C2×C3×C8—61no—
245C2×C3×C8—91no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—181no—
250C3×C42—31no—
251C3×C42—31no—
252C3×C42—61no—
253C3×C42—181no—
254#216—182no—
255#216—182no—
256#216—182no—
257C4⋊Dic3—181no—
258C4⋊Dic3—181no—
259C4⋊Dic3—181no—
260C4⋊Dic3—181no—
261C4⋊Dic3—182no—
262C4⋊Dic3—182no—
263C4⋊Dic3D3211yescharacteristic
264Dic3⋊C4—61no—
265Dic3⋊C4—61no—
266Dic3⋊C4—181no—
267Dic3⋊C4—181no—
268C4×Dic3—61no—
269C4×Dic3—61no—
270C4×Dic3D3211yescharacteristic
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8D3211yescharacteristic
275Dic12—182no—
276Dic12—182no—
277Dic12—184no—
278Dic12—182no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32C2×D1211yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32C2×#6711yescharacteristic
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—32no—
293C32×C4—41no—
294C32×C4—41no—
295C32×C4—61no—
296C32×C4—121no—
297C32×C4C2×D1211yescharacteristic
298C32×C4#23511yescharacteristic
299C32×C4#23511yescharacteristic
300C32×C4D3×D411yescharacteristic
301C32×C4D3×D411yescharacteristic
302C32×C4#23911yescharacteristic
303C3⋊Dic3—22no—
304C3⋊Dic3—41no—
305C3⋊Dic3—122no—
306C3⋊Dic3—122no—
307C3⋊Dic3—122no—
308C3⋊Dic3—124no—
309C3⋊Dic3—121no—
310C3⋊Dic3—121no—
311C3⋊Dic3—121no—
312C3×Dic3—22no—
313C3×Dic3—21no—
314C3×Dic3—21no—
315C3×Dic3—22no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—81no—
321C3×Dic3—122no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C3×Dic3Dic1212yescomplemented
326C3×Dic3#21612yescomplemented
327C2×Q16—271no—
328C2×Q16—271no—
329C4⋊Q8—271no—
330C4×Q8—91no—
331C4⋊C8—91no—
332#103—91no—
333#103—271no—
334C33#46211yescharacteristic, complemented, Sylow
335C3×Q8—92no—
336C3×Q8—181no—
337C3×Q8—181no—
338C3×Q8—181no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—41no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—121no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—62no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—121no—
364C2×Dic3—182no—
365C2×Dic3—181no—
366C2×Dic3—181no—
367C2×Dic3—181no—
368C2×Dic3—361no—
369C2×Dic3—361no—
370C2×Dic3—362no—
371C2×Dic3#62611yescharacteristic
372C2×Dic3#62611yescharacteristic
373Dic6—32no—
374Dic6—32no—
375Dic6—62no—
376Dic6—61no—
377Dic6—61no—
378Dic6—61no—
379Dic6—61no—
380Dic6—61no—
381Dic6—61no—
382Dic6—121no—
383Dic6—121no—
384Dic6—121no—
385Dic6—182no—
386Dic6—182no—
387Dic6—181no—
388Dic6—181no—
389Dic6—181no—
390Dic6—181no—
391Dic6—181no—
392Dic6—181no—
393Dic6—361no—
394Dic6—361no—
395Dic6—361no—
396Dic6—361no—
397Dic6—362no—
398Dic6—362no—
399C3×C8—62no—
400C3×C8—61no—
401C3×C8—61no—
402C3×C8—181no—
403C3⋊C8—21no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—121no—
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—21no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32#88411yescharacteristic
420C2×C32#89511yescharacteristic
421C2×C32C2×Dic1211yescharacteristic
422C2×C32#90911yescharacteristic
423C2×C32#91511yescharacteristic
424C2×C32D3×Q1611yescharacteristic
425C2×C32#93011yescharacteristic
426C2×C32C2×#21611yescharacteristic
427C2×C32#95111yescharacteristic
428C2×Q8—91no—
429C2×Q8—91no—
430C2×Q8—271no—
431Q16—542no—
432Q16—542no—
433C2×C8—31no—
434C2×C8—91no—
435C4⋊C4—31no—
436C4⋊C4—181no—
437C4⋊C4—541no—
438C4⋊C4—541no—
439C42—31no—
440C42—181no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—41no—
445C22×C3D3×D1211yescharacteristic
446C22×C3C2×#62611yescharacteristic
447C22×C3D3×#6711yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—32no—
455C3×C4—32no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—61no—
461C3×C4—61no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—182no—
465C3×C4—241no—
466C3×C4—361no—
467C3×C4#379811yescharacteristic
468C3×C4#379911yescharacteristic
469C3×C4#379911yescharacteristic
470C3×C4D3×D1211yescharacteristic
471C3×C4D3×D1211yescharacteristic
472C3×C4#381111yescharacteristic
473Dic3—21no—
474Dic3—41no—
475Dic3—62no—
476Dic3—62no—
477Dic3—122no—
478Dic3—122no—
479Dic3—121no—
480Dic3—121no—
481Dic3—362no—
482Dic3—362no—
483Dic3—362no—
484Dic3—364no—
485Dic3—361no—
486Dic3—361no—
487Dic3—361no—
488Dic3#371912yescomplemented
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#479511yescharacteristic, complemented
494C32#484911yescharacteristic, complemented
495C32#503111yescharacteristic, complemented
496Q8—92no—
497Q8—181no—
498Q8—181no—
499Q8—181no—
500Q8—541no—
501Q8—541no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—61no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—541no—
508C2×C4D3311yescharacteristic
509C8—61no—
510C8—181no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3D3×Dic1211yescharacteristic
524C2×C3#6380011yescharacteristic
525C2×C3#6383111yescharacteristic
526C2×C3C2×#371911yescharacteristic
527C2×C3#6390611yescharacteristic
528C2×C3#6390911yescharacteristic
529C2×C3#6391011yescharacteristic
530C2×C3#6392911yescharacteristic
531C2×C3D3×#21611yescharacteristic
532C22D3×#62611yescharacteristic
533C4—32no—
534C4—61no—
535C4—121no—
536C4—182no—
537C4—361no—
538C4—1081no—
539C4#8895511yescharacteristic
540C4#8896511yescharacteristic
541C3—21no—
542C3—21no—
543C3—21no—
544C3—41no—
545C3#1059038511yescharacteristic, complemented
546C3#1059056111yescharacteristic, complemented
547C3#1059080811yescharacteristic, complemented
548C2D3×#371911yescharacteristic
549C2#1172288611yescharacteristic
550C2#1172296111yescharacteristic
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
126. C32×Dic3Q16semidirect
325. C3×Dic3Dic12semidirect
326. C3×Dic3#216semidirect
334. C33#462semidirect
488. Dic3#3719semidirect
493. C32#4795semidirect
494. C32#4849semidirect
495. C32#5031semidirect
545. C3#10590385semidirect
546. C3#10590561semidirect
547. C3#10590808semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117234432
C2×#888002
C3×#638442
#117220412
C3×Dic3⋊(C3⋊C8)2
C3×#635672
#105908083
#105905613
#105903853