Finite Group Databasebeta

← #13156026

Subgroups — #13156026

Order 1728.

594 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2C2×#88798C211yesmaximal, characteristic
3C3×#63873C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic
5#11722051C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic
8#10591575—31nomaximal
9#10591574—31nomaximal
10#10590798—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88813—21no—
20#88798—22no—
21C3×#3724—21no—
22C3×#3724—21no—
23C3×#3719—22no—
24C2×#3756—31no—
25C3×#944—31no—
26C3×#944—31no—
27C2×C3×#216—31no—
28C2×C3×Dic12—31no—
29#63873D311yescharacteristic
30#63844—31no—
31#63843—31no—
32Dic3×Dic6—31no—
33C2×#3719D311yescharacteristic
34C2×#3718—31no—
35C2×#3718—31no—
36C2×#3718—61no—
37#63662—31no—
38#63641—31no—
39C3×#835—31no—
40C3×#816—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—61no—
44#63566D311yescharacteristic
45#63565—31no—
46#63565—31no—
47#63565—61no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×#634—21no—
54C3×#634D412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3⋊(C3⋊(C3⋊C8))—21no—
62C32×C3⋊C8—21no—
63#5182—91no—
64#5090—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—
79#3756—62no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4×C3⋊Dic3C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—121no—
89C3×#216—62no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C3×Dic12—62no—
104#3724—21no—
105#3724—21no—
106#3724—61no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109Dic32—61no—
110Dic32—61no—
111Dic32—61no—
112#3719—22no—
113#3718—62no—
114#3718—62no—
115#3718—64no—
116C22×C33C2×D411yescharacteristic
117C33×C4C2×D411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—41no—
122C3×C3⋊Dic3Q1612yescomplemented
123C32×Dic3—41no—
124C32×Dic3—41no—
125C2×C3×Q16—91no—
126C3×C4⋊Q8—91no—
127Q8×Dic3—91no—
128#944—31no—
129#944—31no—
130C2×#216—31no—
131C2×#216—91no—
132C2×#216—91no—
133C2×#216—91no—
134C2×#216—181no—
135C2×Dic12—31no—
136#887—91no—
137C4×Dic6—91no—
138C3×#103—91no—
139#835—31no—
140#835—91no—
141#835—91no—
142#835—181no—
143#816—31no—
144Dic3⋊C8—91no—
145Dic3⋊C8—91no—
146Dic3⋊C8—181no—
147#810—91no—
148C4⋊(C3⋊C8)—91no—
149C32×Q8—61no—
150C32×Q8—61no—
151C32×Q8—61no—
152C32×Q8—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3#6711yescharacteristic
165C2×C3⋊Dic3#6711yescharacteristic
166#634—21no—
167#634#6712yes—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—121no—
185C2×C3×Dic3—121no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—32no—
197C3×Dic6—32no—
198C3×Dic6—41no—
199C3×Dic6—41no—
200C3×Dic6—62no—
201C3×Dic6—61no—
202C3×Dic6—61no—
203C3×Dic6—121no—
204C32×C8—61no—
205C3⋊(C3⋊C8)—61no—
206C3⋊(C3⋊C8)—62no—
207C3⋊(C3⋊C8)—61no—
208C3⋊(C3⋊C8)—61no—
209C3⋊(C3⋊C8)—121no—
210C3⋊(C3⋊C8)—121no—
211C3⋊(C3⋊C8)—122no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—41no—
215#462—271noSylow
216C2×C33#12111yescharacteristic
217C2×C33C2×Q1611yescharacteristic
218C2×C33#13711yescharacteristic
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×C3×Q8—61no—
225C2×C3×Q8—91no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—61no—
229C2×Dic6D3211yescharacteristic
230C2×Dic6D3211yescharacteristic
231C3×Q16—182no—
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—61no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—181no—
237C3×C4⋊C4—181no—
238C3×C42—91no—
239#216—62no—
240#216—182no—
241#216—182no—
242#216—182no—
243#216—184no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247C4⋊Dic3—181no—
248Dic3⋊C4—61no—
249Dic3⋊C4—61no—
250Dic3⋊C4—61no—
251Dic3⋊C4—61no—
252Dic3⋊C4—181no—
253Dic3⋊C4—181no—
254C4×Dic3—31no—
255C4×Dic3—31no—
256C4×Dic3—61no—
257C4×Dic3—181no—
258C4×Dic3—181no—
259C4×Dic3—181no—
260C2×C3⋊C8—91no—
261C2×C3⋊C8—91no—
262C2×C3⋊C8—91no—
263C2×C3⋊C8—181no—
264C2×C3⋊C8—181no—
265C2×C3⋊C8—181no—
266C2×C3⋊C8—361no—
267C2×C3⋊C8D3211yescharacteristic
268Dic12—62no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32D3×D411yescharacteristic
274C22×C32D3×D411yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—121no—
285C32×C4—121no—
286C32×C4#23511yescharacteristic
287C32×C4D3×D411yescharacteristic
288C32×C4D3×D411yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23711yescharacteristic
291C32×C4C2×#6711yescharacteristic
292C3⋊Dic3—22no—
293C3⋊Dic3—21no—
294C3⋊Dic3—41no—
295C3⋊Dic3#21612yescomplemented
296C3×Dic3—32no—
297C3×Dic3—32no—
298C3×Dic3—42no—
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—62no—
305C3×Dic3—62no—
306C3×Dic3—61no—
307C3×Dic3—61no—
308C3×Dic3—62no—
309C3×Dic3—122no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C3×Dic3—241no—
314C2×Q16—91no—
315C2×Q16—271no—
316C4⋊Q8—91no—
317C4×Q8—271no—
318C4⋊C8—271no—
319#103—91no—
320#103—271no—
321C33#46211yescharacteristic, complemented, Sylow
322C3×Q8—62no—
323C3×Q8—62no—
324C3×Q8—61no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—61no—
330C3×Q8—61no—
331C3×Q8—61no—
332C3×Q8—92no—
333C3×Q8—121no—
334C3×Q8—121no—
335C3×Q8—181no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—41no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—121no—
366C2×Dic3—121no—
367Dic6—21no—
368Dic6—21no—
369Dic6—21no—
370Dic6—21no—
371Dic6—32no—
372Dic6—32no—
373Dic6—62no—
374Dic6—61no—
375Dic6—61no—
376Dic6—121no—
377C3×C8—61no—
378C3×C8—61no—
379C3×C8—121no—
380C3⋊C8—21no—
381C3⋊C8—182no—
382C3⋊C8—181no—
383C3⋊C8—181no—
384C3⋊C8—181no—
385C3⋊C8—361no—
386C3⋊C8—361no—
387C3⋊C8—362no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32#88411yescharacteristic
401C2×C32#91511yescharacteristic
402C2×C32D3×Q1611yescharacteristic
403C2×C32D3×Q1611yescharacteristic
404C2×C32#91811yescharacteristic
405C2×C32#93811yescharacteristic
406C2×C32#93911yescharacteristic
407C2×C32#94211yescharacteristic
408C2×C32C2×#21611yescharacteristic
409C2×Q8—31no—
410C2×Q8—31no—
411C2×Q8—91no—
412Q16—182no—
413Q16—542no—
414C2×C8—31no—
415C2×C8—271no—
416C4⋊C4—91no—
417C4⋊C4—181no—
418C4⋊C4—181no—
419C4⋊C4—541no—
420C42—91no—
421C42—541no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—41no—
426C22×C3#380211yescharacteristic
427C22×C3D3×#6711yescharacteristic
428C22×C3D3×#6711yescharacteristic
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—92no—
438C3×C4—122no—
439C3×C4—122no—
440C3×C4—121no—
441C3×C4—121no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—181no—
445C3×C4—182no—
446C3×C4—361no—
447C3×C4#379511yescharacteristic
448C3×C4#380411yescharacteristic
449C3×C4#380511yescharacteristic
450C3×C4D3×#6711yescharacteristic
451C3×C4D3×#6711yescharacteristic
452C3×C4#381111yescharacteristic
453Dic3—32no—
454Dic3—32no—
455Dic3—41no—
456Dic3—41no—
457Dic3—62no—
458Dic3—62no—
459Dic3—61no—
460Dic3—61no—
461Dic3—62no—
462Dic3—122no—
463Dic3—121no—
464Dic3—121no—
465Dic3—121no—
466Dic3—241no—
467C32—21no—
468C32—21no—
469C32—21no—
470C32—41no—
471C32#479511yescharacteristic, complemented
472C32#509011yescharacteristic, complemented
473C32#518211yescharacteristic, complemented
474Q8—61no—
475Q8—61no—
476Q8—61no—
477Q8—61no—
478Q8—92no—
479Q8—181no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4D3311yescharacteristic
487C8—61no—
488C8—541no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3#6380211yescharacteristic
502C2×C3#6381111yescharacteristic
503C2×C3#6388511yescharacteristic
504C2×C3D3×#21611yescharacteristic
505C2×C3D3×#21611yescharacteristic
506C2×C3#6394611yescharacteristic
507C2×C3#6394811yescharacteristic
508C2×C3#6395811yescharacteristic
509C2×C3#6397011yescharacteristic
510C22#8895311yescharacteristic
511C4—92no—
512C4—121no—
513C4—121no—
514C4—181no—
515C4—182no—
516C4—361no—
517C4#8895211yescharacteristic
518C4#8896111yescharacteristic
519C3—21no—
520C3—21no—
521C3—21no—
522C3—41no—
523C3#1059079811yescharacteristic, complemented
524C3#1059157411yescharacteristic, complemented
525C3#1059157511yescharacteristic, complemented
526C2#1172284711yescharacteristic
527C2#1172289211yescharacteristic
528C2#1172296511yescharacteristic
subgroupG/H
centerC22#88953
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
122. C3×C3⋊Dic3Q16semidirect
295. C3⋊Dic3#216semidirect
321. C33#462semidirect
471. C32#4795semidirect
472. C32#5090semidirect
473. C32#5182semidirect
523. C3#10590798semidirect
524. C3#10591574semidirect
525. C3#10591575semidirect

Maximal subgroups

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

subgroupindex
#117236682
C2×#887982
C3×#638732
C2×C3×#37192
#117220512
#117220372
C3×#635662
#105915753
#105915743
#105907983