Finite Group Databasebeta

← #13156023

Subgroups — #13156023

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
1#11723455C211yesmaximal, characteristic
2C2×#88799C211yesmaximal, characteristic
3C3×#63843C211yesmaximal, characteristic
4C2×C3×#3718C211yesmaximal, characteristic
5#11722078C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591625—31nomaximal
9#10591580—31nomaximal
10#10590561—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19#88807—21no—
20#88799—22no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3718—22no—
24#64153—31no—
25C2×#3756—31no—
26C3×Q8×Dic3—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C3×#887—31no—
30#63866—31no—
31#63866—31no—
32#63866—61no—
33#63843D311yescharacteristic
34C2×#3719—31no—
35C2×#3719—31no—
36C2×#3719—61no—
37C2×#3718D311yescharacteristic
38#63637—31no—
39C3×#835—31no—
40C3×#810—31no—
41#63572—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—31no—
44#63567—31no—
45#63567—31no—
46#63567—61no—
47#63565D311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50#6308—21no—
51#6308—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63#5182—91no—
64#5031—91no—
65#4795—91no—
66C2×C32×Q8—31no—
67C2×C32×Q8—31no—
68C2×#634—31no—
69C2×#634—31no—
70C2×#634—31no—
71C2×#634—61no—
72C2×#634—61no—
73C2×#634—61no—
74C2×#634—121no—
75C2×C3×Dic6—21no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C2×D311yescharacteristic
81C32×C4⋊C4—31no—
82#3756—62no—
83#3750—61no—
84#3750—61no—
85C4×C3⋊Dic3C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×#216—62no—
89C3×#216—62no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106Dic3⋊Dic3—21no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—61no—
110Dic3⋊Dic3—61no—
111Dic3⋊Dic3—62no—
112Dic3⋊Dic3—62no—
113Dic32—21no—
114#3719—64no—
115#3719—62no—
116#3719—62no—
117#3718—22no—
118C22×C33C2×D411yescharacteristic
119C33×C4C2×D411yescharacteristic
120C33×C4#4111yescharacteristic
121C3⋊(C3⋊Dic3)—41no—
122C3×C3⋊Dic3—21no—
123C3×C3⋊Dic3Q1612yescomplemented
124C32×Dic3—22no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×Q16—91no—
128C3×C4×Q8—91no—
129Q8×Dic3—31no—
130#944—91no—
131#944—91no—
132#944—181no—
133C2×#216—31no—
134C2×#216—31no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#216—181no—
138C2×Dic12—91no—
139#887—31no—
140#869—91no—
141C3×#103—91no—
142#835—31no—
143#816—91no—
144Dic3⋊C8—91no—
145Dic3⋊C8—91no—
146#810—31no—
147#810—91no—
148#810—91no—
149#810—181no—
150C4⋊(C3⋊C8)—91no—
151C32×Q8—32no—
152C32×Q8—61no—
153C32×Q8—61no—
154C32×Q8—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—62no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—122no—
172C2×C3⋊Dic3#6711yescharacteristic
173C2×C3⋊Dic3#6711yescharacteristic
174#634—61no—
175#634—61no—
176#634—62no—
177#634—62no—
178#634—61no—
179#634—61no—
180#634—61no—
181#634—61no—
182#634—121no—
183#634—121no—
184#634—121no—
185#634—121no—
186#634—122no—
187#634—122no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—41no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—22no—
210C3×Dic6—41no—
211C3×Dic6—41no—
212C3×Dic6—41no—
213C3×Dic6D1212yes—
214C3×Dic6#6712yes—
215C3⋊(C3⋊C8)—21no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—62no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
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—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—31no—
231C2×C3×Q8—61no—
232C2×C3×Q8—61no—
233C2×Dic6—91no—
234C2×Dic6—91no—
235C2×Dic6—91no—
236C2×Dic6—181no—
237C2×Dic6—181no—
238C2×Dic6—181no—
239C2×Dic6—361no—
240C2×Dic6D3211yescharacteristic
241C2×Dic6D3211yescharacteristic
242C3×Q16—182no—
243C2×C3×C8—91no—
244C2×C3×C8—91no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—181no—
249C3×C42—91no—
250C3×C42—181no—
251#216—62no—
252#216—62no—
253#216—182no—
254#216—182no—
255#216—184no—
256C4⋊Dic3—61no—
257C4⋊Dic3—181no—
258C4⋊Dic3—181no—
259C4⋊Dic3D3211yescharacteristic
260Dic3⋊C4—61no—
261Dic3⋊C4—181no—
262Dic3⋊C4—181no—
263Dic3⋊C4—181no—
264Dic3⋊C4—181no—
265Dic3⋊C4—182no—
266Dic3⋊C4—182no—
267C4×Dic3—31no—
268C4×Dic3—31no—
269C4×Dic3—61no—
270C4×Dic3—61no—
271C4×Dic3—61no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8—61no—
278Dic12—182no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32D3×D411yescharacteristic
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—41no—
293C32×C4—41no—
294C32×C4—62no—
295C32×C4—121no—
296C32×C4—121no—
297C32×C4C2×D1211yescharacteristic
298C32×C4#23511yescharacteristic
299C32×C4#23511yescharacteristic
300C32×C4D3×D411yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4C2×#6711yescharacteristic
303C3⋊Dic3—21no—
304C3⋊Dic3—122no—
305C3⋊Dic3—122no—
306C3⋊Dic3—122no—
307C3⋊Dic3—124no—
308C3⋊Dic3—121no—
309C3⋊Dic3—121no—
310C3⋊Dic3—121no—
311C3⋊Dic3#21612yescomplemented
312C3×Dic3—22no—
313C3×Dic3—22no—
314C3×Dic3—32no—
315C3×Dic3—32no—
316C3×Dic3—42no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—61no—
325C3×Dic3—61no—
326C3×Dic3—81no—
327C2×Q16—91no—
328C2×Q16—271no—
329C4⋊Q8—271no—
330C4×Q8—91no—
331C4⋊C8—271no—
332#103—91no—
333#103—271no—
334C33#46211yescharacteristic, complemented, Sylow
335C3×Q8—32no—
336C3×Q8—32no—
337C3×Q8—62no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—61no—
344C3×Q8—121no—
345C3×Q8—121no—
346C3×Q8—121no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—62no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—121no—
361C2×C3×C4—121no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—182no—
375C2×Dic3—181no—
376C2×Dic3—181no—
377C2×Dic3—181no—
378C2×Dic3—361no—
379C2×Dic3—361no—
380C2×Dic3—362no—
381Dic6—21no—
382Dic6—21no—
383Dic6—21no—
384Dic6—182no—
385Dic6—182no—
386Dic6—181no—
387Dic6—181no—
388Dic6—181no—
389Dic6—181no—
390Dic6—181no—
391Dic6—181no—
392Dic6—361no—
393Dic6—361no—
394Dic6—361no—
395Dic6—361no—
396Dic6—362no—
397Dic6—362no—
398Dic6#62612yes—
399C3×C8—181no—
400C3×C8—181no—
401C3⋊C8—62no—
402C3⋊C8—61no—
403C3⋊C8—61no—
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#91511yescharacteristic
421C2×C32D3×Q1611yescharacteristic
422C2×C32D3×Q1611yescharacteristic
423C2×C32#91811yescharacteristic
424C2×C32#93011yescharacteristic
425C2×C32#93911yescharacteristic
426C2×C32C2×#21611yescharacteristic
427C2×C32#95111yescharacteristic
428C2×Q8—31no—
429C2×Q8—31no—
430C2×Q8—271no—
431Q16—182no—
432Q16—542no—
433C2×C8—91no—
434C2×C8—91no—
435C4⋊C4—31no—
436C4⋊C4—181no—
437C4⋊C4—541no—
438C4⋊C4—541no—
439C42—91no—
440C42—181no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—41no—
445C22×C3D3×#6711yescharacteristic
446C22×C3D3×#6711yescharacteristic
447C22×C3#381111yescharacteristic
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—92no—
459C3×C4—122no—
460C3×C4—122no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—121no—
465C3×C4—181no—
466C3×C4—241no—
467C3×C4#379811yescharacteristic
468C3×C4D3×D1211yescharacteristic
469C3×C4#380411yescharacteristic
470C3×C4#380411yescharacteristic
471C3×C4C2×#62611yescharacteristic
472C3×C4D3×#6711yescharacteristic
473Dic3—22no—
474Dic3—32no—
475Dic3—32no—
476Dic3—41no—
477Dic3—41no—
478Dic3—62no—
479Dic3—62no—
480Dic3—61no—
481Dic3—61no—
482Dic3—362no—
483Dic3—362no—
484Dic3—362no—
485Dic3—364no—
486Dic3—361no—
487Dic3—361no—
488Dic3—361no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#479511yescharacteristic, complemented
494C32#503111yescharacteristic, complemented
495C32#518211yescharacteristic, complemented
496Q8—32no—
497Q8—61no—
498Q8—61no—
499Q8—61no—
500Q8—541no—
501Q8—541no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—91no—
506C2×C4—91no—
507C2×C4—541no—
508C2×C4D3311yescharacteristic
509C8—181no—
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×C3#6390611yescharacteristic
524C2×C3#6392911yescharacteristic
525C2×C3D3×#21611yescharacteristic
526C2×C3D3×#21611yescharacteristic
527C2×C3#6394311yescharacteristic
528C2×C3#6394411yescharacteristic
529C2×C3#6395111yescharacteristic
530C2×C3#6397111yescharacteristic
531C2×C3#6397811yescharacteristic
532C22#8895211yescharacteristic
533C4—62no—
534C4—92no—
535C4—121no—
536C4—121no—
537C4—181no—
538C4—1081no—
539C4D3×#62611yescharacteristic
540C4#8895911yescharacteristic
541C3—21no—
542C3—21no—
543C3—21no—
544C3—41no—
545C3#1059056111yescharacteristic, complemented
546C3#1059158011yescharacteristic, complemented
547C3#1059162511yescharacteristic, complemented
548C2#1172284611yescharacteristic
549C2#1172288811yescharacteristic
550C2#1172295811yescharacteristic
subgroupG/H
centerC22#88952
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
123. C3×C3⋊Dic3Q16semidirect
311. C3⋊Dic3#216semidirect
334. C33#462semidirect
493. C32#4795semidirect
494. C32#5031semidirect
495. C32#5182semidirect
545. C3#10590561semidirect
546. C3#10591580semidirect
547. C3#10591625semidirect

Maximal subgroups

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

subgroupindex
#117234552
C2×#887992
C3×#638432
C2×C3×#37182
#117220782
#117220412
C3×#635652
#105916253
#105915803
#105905613