Finite Group Databasebeta

← #13161803

Subgroups — #13161803

Order 1728.

994 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722072C214yesmaximal, complemented
5#10591073—31nomaximal
6#10591071—31nomaximal
7#10591070—31nomaximal
8C22×C33×C4C2211yescharacteristic
9C2×C3×#638C412yes—
10C3×#3828—22no—
11C3×#3828—22no—
12C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×#634C411yescharacteristic
15C3×#3748—22no—
16C2×C3×C3⋊(C3⋊C8)—22no—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18C2×C3×C3⋊(C3⋊C8)C2212yes—
19C2×#3828D311yescharacteristic
20C2×C3×#235—31no—
21C2×C3×#235—31no—
22C2×C3×#235—61no—
23C2×#3748D311yescharacteristic, complemented
24C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
25C2×C3×#208—31no—
26C2×C3×#208—31no—
27C2×C3×#208—61no—
28C22×C3×C3⋊C8—31no—
29C22×C3×C3⋊C8—31no—
30C22×C3×C3⋊C8—61no—
31#63557—34no—
32(C2×D3)⋊(C3⋊C8)—34no—
33(C2×D3)⋊(C3⋊C8)—34no—
34C23×C33C2×C411yescharacteristic
35C22×C3×C3⋊D3C2×C412yes—
36C2×C33×C4C2×C411yescharacteristic
37C2×C33×C4D412yes—
38C2×C33×C4D412yes—
39C2×C33×C4C2311yescharacteristic
40C3×#638—24no—
41C2×C3×C3⋊Dic3C2×C412yes—
42C3×C4⋊(C3⋊D3)—22no—
43C3×C4×C3⋊D3—22no—
44C3×C4×C3⋊D3—22no—
45C3×#634—22no—
46C3×C3⋊(C3⋊C8)—22no—
47C3×C3⋊(C3⋊C8)—22no—
48#5234—91no—
49#5129—91no—
50#5110—91no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4—41no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C2×D311yescharacteristic
58C2×#638Dic312yes—
59#3828—22no—
60#3828—22no—
61C2×C4⋊(C3⋊D3)Dic311yescharacteristic
62C2×C4×C3⋊D3C2×D312yes—
63C2×#634Dic311yescharacteristic
64C2×C3×#67—32no—
65C2×C3×#67—32no—
66C2×C3×#67—62no—
67C3×#235—64no—
68C3×#235—64no—
69C3×#235—62no—
70C3×#235—62no—
71C3×#235—62no—
72C3×#235—62no—
73C2×C3×D12—31no—
74C2×C3×D12—31no—
75C2×C3×D12—61no—
76C2×C3×C4×D3—32no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—62no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82#3748—22no—
83C2×C3⋊(C3⋊C8)—22no—
84C2×C3⋊(C3⋊C8)C2×D312yes—
85C2×C3⋊(C3⋊C8)C2×D312yes—
86C3×#208—64no—
87C3×#208—62no—
88C3×#208—62no—
89C2×C3×C3⋊C8—32no—
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—32no—
93C2×C3×C3⋊C8—62no—
94C2×C3×C3⋊C8—62no—
95C2×C3×C3⋊C8—64no—
96C2×C3×C3⋊C8—62no—
97C2×C3×C3⋊C8—62no—
98C22×C33—22no—
99C22×C33C22⋊C412yes—
100C22×C33C22×C411yescharacteristic
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—24no—
103C33×C4C22⋊C412yes—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—22no—
107C2×C3×#41—91no—
108C2×#235—31no—
109C2×#235—31no—
110C2×#235—61no—
111C2×C3×M16—91no—
112C22×C3×C8—91no—
113C2×#208—31no—
114C2×#208—31no—
115C2×#208—61no—
116C22×C3⋊C8—31no—
117C22×C3⋊C8—31no—
118C22×C3⋊C8—61no—
119#830—94no—
120(C2×D3)⋊C8—94no—
121(C2×D3)⋊C8—94no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—41no—
126C23×C32C4×D311yescharacteristic
127C23×C32C4×D311yescharacteristic
128C23×C32C2×Dic311yescharacteristic
129C22×C3⋊D3C2×Dic312yes—
130C22×C3×D3—32no—
131C22×C3×D3—32no—
132C22×C3×D3—62no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—41no—
146C2×C32×C4—42no—
147C2×C32×C4—42no—
148C2×C32×C4—41no—
149C2×C32×C4C4×D311yescharacteristic
150C2×C32×C4C4×D311yescharacteristic
151C2×C32×C4D1212yes—
152C2×C32×C4D1212yes—
153C2×C32×C4C2×Dic311yescharacteristic
154C2×C32×C4#6712yes—
155C2×C32×C4#6712yes—
156C2×C32×C4#6712yes—
157C2×C32×C4#6712yes—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161#638—24no—
162C2×C3⋊Dic3C2×Dic312yes—
163C4⋊(C3⋊D3)—22no—
164C4×C3⋊D3—22no—
165C4×C3⋊D3—22no—
166#634—22no—
167C3×#67—68no—
168C3×#67—64no—
169C3×#67—64no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—62no—
173C3×D12—64no—
174C3×D12—62no—
175C3×D12—62no—
176C3×C4×D3—64no—
177C3×C4×D3—64no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—62no—
181C3×C4×D3—62no—
182C3×Dic6—64no—
183C3×Dic6—62no—
184C3×Dic6—62no—
185C3⋊(C3⋊C8)—22no—
186C3⋊(C3⋊C8)—22no—
187C3×C3⋊C8—64no—
188C3×C3⋊C8—64no—
189C3×C3⋊C8—62no—
190C3×C3⋊C8—62no—
191C3×C3⋊C8—62no—
192C3×C3⋊C8—62no—
193#408—271noSylow
194C2×C33—22no—
195C2×C33C2×C22⋊C411yescharacteristic
196C2×C33#13112yes—
197C3×C3⋊D3—42no—
198C3×#41—182no—
199C3×#41—182no—
200C2×C3×Q8—91no—
201C2×C3×D4—92no—
202C2×C3×D4—91no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—41no—
207C22×C3×C4—92no—
208C22×C3×C4D3211yescharacteristic
209C22×C3×C4D3211yescharacteristic
210C22×C3×C4D3211yescharacteristic
211C2×#67—32no—
212C2×#67—32no—
213C2×#67—62no—
214#235—64no—
215#235—64no—
216#235—62no—
217#235—62no—
218#235—62no—
219#235—62no—
220C2×D12—31no—
221C2×D12—31no—
222C2×D12—61no—
223C2×C4×D3—32no—
224C2×C4×D3—32no—
225C2×C4×D3—62no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—61no—
229C3×M16—182no—
230C2×C3×C8—92no—
231C2×C3×C8—92no—
232C2×C3×C8—182no—
233#208—64no—
234#208—62no—
235#208—62no—
236C2×C3⋊C8—32no—
237C2×C3⋊C8—32no—
238C2×C3⋊C8—32no—
239C2×C3⋊C8—32no—
240C2×C3⋊C8—62no—
241C2×C3⋊C8—62no—
242C2×C3⋊C8—64no—
243C2×C3⋊C8—62no—
244C2×C3⋊C8—62no—
245C22×C32—22no—
246C22×C32—24no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—42no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—44no—
260C22×C32(C2×D3)⋊C412yes—
261C22×C32(C2×D3)⋊C412yes—
262C22×C32#21712yes—
263C22×C32C2×C4×D311yescharacteristic
264C22×C32C2×C4×D311yescharacteristic
265C22×C32C22×Dic311yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—24no—
268C2×C3×D3—64no—
269C2×C3×D3—64no—
270C2×C3×D3—64no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—124no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4(C2×D3)⋊C412yes—
287C32×C4(C2×D3)⋊C412yes—
288C32×C4#21712yes—
289C32×C4C2×D1211yescharacteristic
290C32×C4C2×D1211yescharacteristic
291C32×C4C2×#6711yescharacteristic
292C32×C4C2×#6711yescharacteristic
293C32×C4C2×#6711yescharacteristic
294C32×C4C2×#6711yescharacteristic
295C3⋊Dic3—22no—
296C3×Dic3—64no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C2×#41—91no—
300C2×M16—91no—
301C22×C8—91no—
302C22⋊C8—274no—
303C33#40811yescharacteristic, complemented, Sylow
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—92no—
309C23×C3D3×Dic311yescharacteristic
310C23×C3D3×Dic311yescharacteristic
311C23×C3#62411yescharacteristic
312C22×D3—32no—
313C22×D3—32no—
314C22×D3—62no—
315C3×Q8—182no—
316C3×D4—184no—
317C3×D4—182no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—92no—
335C2×C3×C4—182no—
336C2×C3×C4—182no—
337C2×C3×C4D3×Dic311yescharacteristic
338C2×C3×C4D3×Dic311yescharacteristic
339C2×C3×C4#62411yescharacteristic
340C2×C3×C4#62512yes—
341C2×C3×C4#62512yes—
342C2×C3×C4#62612yes—
343C2×C3×C4#62612yes—
344C2×C3×C4#62612yes—
345C2×C3×C4#62612yes—
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—68no—
350#67—64no—
351#67—64no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—62no—
355D12—64no—
356D12—62no—
357D12—62no—
358C4×D3—64no—
359C4×D3—64no—
360C4×D3—62no—
361C4×D3—62no—
362C4×D3—62no—
363C4×D3—62no—
364Dic6—64no—
365Dic6—62no—
366Dic6—62no—
367C3×C8—182no—
368C3×C8—182no—
369C3⋊C8—64no—
370C3⋊C8—64no—
371C3⋊C8—62no—
372C3⋊C8—62no—
373C3⋊C8—62no—
374C3⋊C8—62no—
375C2×C32—22no—
376C2×C32—24no—
377C2×C32—22no—
378C2×C32—22no—
379C2×C32—22no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32—42no—
389C2×C32—44no—
390C2×C32#90112yes—
391C2×C32#90712yes—
392C2×C32C2×(C2×D3)⋊C411yescharacteristic
393C2×C32C2×(C2×D3)⋊C411yescharacteristic
394C2×C32#94812yes—
395C2×C32C2×#21711yescharacteristic
396C3⋊D3—42no—
397C3×D3—124no—
398C3×D3—122no—
399C3×D3—122no—
400#41—182no—
401#41—182no—
402C2×Q8—91no—
403C2×D4—92no—
404C2×D4—91no—
405C22×C4—92no—
406C22×C4#111311yescharacteristic
407M16—182no—
408C2×C8—92no—
409C2×C8—92no—
410C2×C8—182no—
411C22×C3—24no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—44no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—184no—
427C22×C3—182no—
428C22×C3(C2×D3)⋊Dic312yes—
429C22×C3(C2×D3)⋊Dic312yes—
430C22×C3#372212yes—
431C22×C3C2×D3×Dic311yescharacteristic
432C22×C3C2×D3×Dic311yescharacteristic
433C22×C3C2×#62411yescharacteristic
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—124no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—42no—
452C3×C4—182no—
453C3×C4(C2×D3)⋊Dic312yes—
454C3×C4(C2×D3)⋊Dic312yes—
455C3×C4#372212yes—
456C3×C4C2×#62511yescharacteristic
457C3×C4C2×#62511yescharacteristic
458C3×C4C2×#62611yescharacteristic
459C3×C4C2×#62611yescharacteristic
460C3×C4C2×#62611yescharacteristic
461C3×C4C2×#62611yescharacteristic
462Dic3—64no—
463Dic3—62no—
464Dic3—62no—
465C32—21no—
466C32—21no—
467C32—21no—
468C32—41no—
469C32#511011yescharacteristic, complemented
470C32#512911yescharacteristic, complemented
471C32#523411yescharacteristic, complemented
472C23—92no—
473C23#629411yescharacteristic
474Q8—182no—
475D4—184no—
476D4—182no—
477C2×C4—92no—
478C2×C4—182no—
479C2×C4—182no—
480C2×C4#629411yescharacteristic
481C2×C4#629512yes—
482C2×C4#629512yes—
483C2×C4C2×#111311yescharacteristic
484C8—182no—
485C8—182no—
486C2×C3—24no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—42no—
497C2×C3—42no—
498C2×C3—44no—
499C2×C3—42no—
500C2×C3—41no—
501C2×C3—362no—
502C2×C3#6381412yes—
503C2×C3#6381512yes—
504C2×C3#6381712yes—
505C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
506C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
507C2×C3C2×#372211yescharacteristic
508D3—124no—
509D3—122no—
510D3—122no—
511C22—22no—
512C22—184no—
513C22—182no—
514C22#8881412yes—
515C22C2×#629411yescharacteristic
516C4—182no—
517C4#8881412yes—
518C4C2×#629511yescharacteristic
519C4C2×#629511yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059107011yescharacteristic, complemented
525C3#1059107111yescharacteristic, complemented
526C3#1059107311yescharacteristic, complemented
527C2—22no—
528C2—362no—
529C2#1172365712yes—
530C2C2×#8881411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C22
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. #11722072C2semidirect
23. C2×#3748D3semidirect
24. C22×C3⋊(C3⋊C8)D3semidirect
303. C33#408semidirect
469. C32#5110semidirect
470. C32#5129semidirect
471. C32#5234semidirect
524. C3#10591070semidirect
525. C3#10591071semidirect
526. C3#10591073semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C3×#37482
C22×C3×C3⋊(C3⋊C8)2
#117220722
#105910733
#105910713
#105910703