Finite Group Databasebeta

← #13156017

Subgroups — #13156017

Order 1728.

638 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#11723451C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic
4C2×C3×#3718C211yesmaximal, characteristic
5#11722076C211yesmaximal, characteristic
6#11722050C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic
8#10591633—31nomaximal
9#10591580—31nomaximal
10#10590798—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806—21no—
19#88806—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22C3×#3719—22no—
23C3×#3718—22no—
24Q8×C3⋊Dic3—31no—
25#64153—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C2×C3×Dic12—31no—
30#63873—31no—
31#63873—31no—
32#63873—61no—
33#63843—31no—
34#63843—31no—
35#63843—61no—
36C2×#3719D311yescharacteristic
37C2×#3718D311yescharacteristic
38#63641—31no—
39C3×#835—31no—
40C3×#816—31no—
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63567—31no—
45#63566—31no—
46#63566D311yescharacteristic
47#63565—31no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—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—
62C32×C3⋊C8—21no—
63#5182—91no—
64#5182—91no—
65#4795—91no—
66C2×C32×Q8—31no—
67C2×C32×Q8—31no—
68C2×#634C2×D311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C2×D311yescharacteristic
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C32×C8—31no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—61no—
82#3750—61no—
83C4×C3⋊Dic3—31no—
84C4×C3⋊Dic3—31no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—61no—
87C4×C3⋊Dic3—61no—
88C4×C3⋊Dic3—61no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—121no—
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#216—62no—
93C3×#216—62no—
94C3×#216—62no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
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
104C3×Dic12—62no—
105#3724—61no—
106#3724—61no—
107#3724—61no—
108#3724—61no—
109#3724—62no—
110#3724—62no—
111Dic3⋊Dic3—61no—
112Dic3⋊Dic3—61no—
113Dic3⋊Dic3—62no—
114Dic32—61no—
115Dic32—61no—
116Dic32—62no—
117#3719—22no—
118#3718—22no—
119C22×C33C2×D411yescharacteristic
120C33×C4C2×D411yescharacteristic
121C33×C4#4111yescharacteristic
122C3⋊(C3⋊Dic3)—21no—
123C3⋊(C3⋊Dic3)Q1612yescomplemented
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—22no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×Q16—91no—
130Q8×Dic3—91no—
131Q8×Dic3—91no—
132Q8×Dic3—181no—
133#944—91no—
134#944—91no—
135#944—91no—
136#944—181no—
137C2×#216—31no—
138C2×#216—31no—
139C2×#216—31no—
140C2×Dic12—31no—
141#887—91no—
142C3×#103—91no—
143#835—31no—
144#835—91no—
145#816—31no—
146#816—91no—
147Dic3⋊C8—91no—
148Dic3⋊C8—91no—
149Dic3⋊C8—91no—
150Dic3⋊C8—181no—
151#810—91no—
152C32×Q8—32no—
153C32×Q8—61no—
154C32×Q8—61no—
155C32×Q8—61no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—121no—
180C2×C3⋊Dic3—121no—
181#634—21no—
182#634—21no—
183C2×C3×Dic3—22no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C3×Dic6—22no—
196C3×Dic6—21no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—22no—
203C3×Dic6—41no—
204C3×Dic6—41no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—121no—
210C3×Dic6—121no—
211C3×Dic6#6712yes—
212C3×Dic6#6712yes—
213C32×C8—61no—
214C3⋊(C3⋊C8)—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—41no—
218C3×C3⋊C8—61no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—121no—
221#462—271noSylow
222C2×C33#12111yescharacteristic
223C2×C33C2×Q1611yescharacteristic
224C2×C33#13711yescharacteristic
225C2×C3×Q8—31no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×Dic6—31no—
233C2×Dic6—31no—
234C2×Dic6—61no—
235C2×Dic6D3211yescharacteristic
236C2×Dic6D3211yescharacteristic
237C3×Q16—182no—
238C3×Q16—182no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—91no—
243C3×C4⋊C4—91no—
244#216—62no—
245#216—62no—
246#216—62no—
247C4⋊Dic3—31no—
248C4⋊Dic3—31no—
249C4⋊Dic3—61no—
250C4⋊Dic3—181no—
251C4⋊Dic3—181no—
252C4⋊Dic3—182no—
253Dic3⋊C4—181no—
254Dic3⋊C4—181no—
255Dic3⋊C4—181no—
256Dic3⋊C4—181no—
257Dic3⋊C4—181no—
258Dic3⋊C4—181no—
259Dic3⋊C4—181no—
260Dic3⋊C4—182no—
261Dic3⋊C4—182no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—181no—
266C4×Dic3—181no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—181no—
271C4×Dic3—182no—
272C4×Dic3—361no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8D3211yescharacteristic
277Dic12—62no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32D3×D411yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—41no—
292C32×C4—41no—
293C32×C4—62no—
294C32×C4—121no—
295C32×C4#23511yescharacteristic
296C32×C4D3×D411yescharacteristic
297C32×C4#23711yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4C2×#6711yescharacteristic
300C32×C4C2×#6711yescharacteristic
301C3⋊Dic3—32no—
302C3⋊Dic3—32no—
303C3⋊Dic3—32no—
304C3⋊Dic3—41no—
305C3⋊Dic3—41no—
306C3⋊Dic3—62no—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3⋊Dic3—61no—
310C3⋊Dic3—62no—
311C3⋊Dic3—61no—
312C3⋊Dic3—61no—
313C3⋊Dic3—122no—
314C3⋊Dic3—121no—
315C3⋊Dic3—121no—
316C3⋊Dic3—122no—
317C3×Dic3—24no—
318C3×Dic3—22no—
319C3×Dic3—22no—
320C3×Dic3—42no—
321C3×Dic3—41no—
322C3×Dic3—41no—
323C3×Dic3—122no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C2×Q16—91no—
330C2×Q16—91no—
331C4⋊Q8—271no—
332C4×Q8—271no—
333C4⋊C8—271no—
334#103—91no—
335#103—271no—
336C33#46211yescharacteristic, complemented, Sylow
337C3×Q8—32no—
338C3×Q8—32no—
339C3×Q8—62no—
340C3×Q8—62no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—121no—
348C3×Q8—121no—
349C3×Q8—181no—
350C3×Q8—181no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—181no—
362C2×C3×C4—181no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—62no—
369C2×Dic3—62no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—91no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C2×Dic3—181no—
384C2×Dic3—181no—
385C2×Dic3—181no—
386C2×Dic3—361no—
387C2×Dic3—361no—
388Dic6—21no—
389Dic6—21no—
390Dic6—21no—
391Dic6—61no—
392Dic6—61no—
393Dic6—61no—
394Dic6—61no—
395Dic6—121no—
396Dic6—121no—
397Dic6#62512yes—
398C3×C8—61no—
399C3×C8—61no—
400C3×C8—121no—
401C3×C8—181no—
402C3⋊C8—21no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—121no—
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×C32#91511yescharacteristic
420C2×C32D3×Q1611yescharacteristic
421C2×C32D3×Q1611yescharacteristic
422C2×C32D3×Q1611yescharacteristic
423C2×C32#91811yescharacteristic
424C2×C32#91811yescharacteristic
425C2×C32#93911yescharacteristic
426C2×C32#93911yescharacteristic
427C2×Q8—31no—
428C2×Q8—31no—
429C2×Q8—91no—
430Q16—182no—
431Q16—182no—
432C2×C8—31no—
433C2×C8—91no—
434C4⋊C4—91no—
435C4⋊C4—541no—
436C4⋊C4—541no—
437C4⋊C4—541no—
438C42—271no—
439C42—541no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3#380211yescharacteristic
445C22×C3#381111yescharacteristic
446C22×C3#381111yescharacteristic
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—64no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—361no—
462C3×C4—361no—
463C3×C4#379511yescharacteristic
464C3×C4#380411yescharacteristic
465C3×C4#380511yescharacteristic
466C3×C4C2×#62511yescharacteristic
467C3×C4D3×#6711yescharacteristic
468C3×C4D3×#6711yescharacteristic
469Dic3—22no—
470Dic3—41no—
471Dic3—92no—
472Dic3—92no—
473Dic3—92no—
474Dic3—122no—
475Dic3—122no—
476Dic3—121no—
477Dic3—121no—
478Dic3—121no—
479Dic3—121no—
480Dic3—182no—
481Dic3—182no—
482Dic3—182no—
483Dic3—182no—
484Dic3—181no—
485Dic3—181no—
486Dic3—181no—
487Dic3—362no—
488Dic3—361no—
489Dic3—361no—
490Dic3—362no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#479511yescharacteristic, complemented
496C32#518211yescharacteristic, complemented
497C32#518211yescharacteristic, complemented
498Q8—32no—
499Q8—61no—
500Q8—61no—
501Q8—61no—
502Q8—181no—
503Q8—181no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—271no—
509C2×C4—271no—
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×C3#6380211yescharacteristic
526C2×C3#6381111yescharacteristic
527C2×C3#6388511yescharacteristic
528C2×C3#6394411yescharacteristic
529C2×C3#6394411yescharacteristic
530C2×C3#6394511yescharacteristic
531C2×C3#6395111yescharacteristic
532C2×C3#6397111yescharacteristic
533C2×C3#6398011yescharacteristic
534C22#8895411yescharacteristic
535C4—62no—
536C4—121no—
537C4—272no—
538C4—361no—
539C4—361no—
540C4—541no—
541C4D3×#62511yescharacteristic
542C4#8896111yescharacteristic
543C3—21no—
544C3—21no—
545C3—21no—
546C3—41no—
547C3#1059079811yescharacteristic, complemented
548C3#1059158011yescharacteristic, complemented
549C3#1059163311yescharacteristic, complemented
550C2#1172284811yescharacteristic
551C2#1172289411yescharacteristic
552C2#1172295411yescharacteristic
subgroupG/H
centerC22#88954
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⋊Dic3)Q16semidirect
336. C33#462semidirect
495. C32#4795semidirect
496. C32#5182semidirect
497. C32#5182semidirect
547. C3#10590798semidirect
548. C3#10591580semidirect
549. C3#10591633semidirect

Maximal subgroups

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

subgroupindex
#117234512
#117233972
C2×C3×#37192
C2×C3×#37182
#117220762
#117220502
C3×#635662
#105916333
#105915803
#105907983