Finite Group Databasebeta

← #13161806

Subgroups — #13161806

Order 1728.

954 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C2×C3×#3748C212yesmaximal, complemented
3#11722074C212yesmaximal, complemented
4#11722073C212yesmaximal, complemented
5#10591078—31nomaximal
6#10591076—32nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C3×#638C412yes—
9C3×#3828—22no—
10C3×#3828—22no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C412yes—
13C2×C3×#634C2211yescharacteristic
14C3×#3748—22no—
15C3×#3748C2214yes—
16C2×C3×C3⋊(C3⋊C8)—22no—
17C2×#3828D311yescharacteristic
18C2×C3×#235—32no—
19C2×C3×#235—61no—
20C2×#3748D312yescomplemented
21C2×C3×#208—32no—
22C2×C3×#208—32no—
23C2×C3×#208—62no—
24#63561—32no—
25#63560—34no—
26#63559—32no—
27#63558—34no—
28C23×C33C2×C411yescharacteristic
29C22×C3×C3⋊D3C2×C412yes—
30C2×C33×C4C2×C411yescharacteristic
31C2×C33×C4D412yes—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34C3×#638—24no—
35C2×C3×C3⋊Dic3C2×C412yes—
36C3×C4⋊(C3⋊D3)—22no—
37C3×C4×C3⋊D3—22no—
38C3×C4×C3⋊D3—22no—
39C3×#634—22no—
40C3×C3⋊(C3⋊C8)—24no—
41#5237—91no—
42#5132—92no—
43C22×C32×C4—22no—
44C22×C32×C4—21no—
45C22×C32×C4—41no—
46C22×C32×C4C2×D312yes—
47C22×C32×C4C2×D311yescharacteristic
48C2×#638Dic312yes—
49#3828—22no—
50#3828—22no—
51C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C4×C3⋊D3Dic312yes—
53C2×#634C2×D311yescharacteristic
54C2×C3×#67—34no—
55C2×C3×#67—62no—
56C3×#235—64no—
57C3×#235—64no—
58C3×#235—62no—
59C3×#235—62no—
60C3×#235—62no—
61C3×#235—62no—
62C2×C3×D12—32no—
63C2×C3×D12—61no—
64C2×C3×C4×D3—34no—
65C2×C3×C4×D3—62no—
66C2×C3×Dic6—32no—
67C2×C3×Dic6—61no—
68#3748—22no—
69#3748C2×D314yes—
70C2×C3⋊(C3⋊C8)—22no—
71C3×#208—34no—
72C3×#208—34no—
73C3×#208—64no—
74C3×#208—64no—
75C3×#208—62no—
76C3×#208—62no—
77C2×C3×C3⋊C8—64no—
78C2×C3×C3⋊C8—62no—
79C2×C3×C3⋊C8—62no—
80C22×C33—41no—
81C22×C33C22⋊C412yes—
82C22×C33C22×C411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—42no—
85C33×C4C22⋊C411yescharacteristic
86C33×C4C22⋊C411yescharacteristic
87C33×C4C2×D412yes—
88C3×C3⋊Dic3—22no—
89C2×C3×#41—91no—
90C2×#235—32no—
91C2×#235—61no—
92C2×C3×M16—92no—
93C2×#208—32no—
94C2×#208—32no—
95C2×#208—62no—
96#836—92no—
97#833—92no—
98#824—94no—
99#823—94no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32C4×D312yes—
104C23×C32C2×Dic311yescharacteristic
105C22×C3⋊D3C2×Dic312yes—
106C22×C3×D3—34no—
107C22×C3×D3—62no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—41no—
122C2×C32×C4C4×D312yes—
123C2×C32×C4D1212yes—
124C2×C32×C4D1212yes—
125C2×C32×C4C2×Dic311yescharacteristic
126C2×C32×C4#6712yes—
127C2×C32×C4#6712yes—
128C2×C32×C4#6712yes—
129C2×C32×C4#6712yes—
130C2×C32×C4C22×D312yes—
131C2×C32×C4C22×D311yescharacteristic
132#638—24no—
133C2×C3⋊Dic3C2×Dic312yes—
134C4⋊(C3⋊D3)—22no—
135C4×C3⋊D3—22no—
136C4×C3⋊D3—22no—
137#634—22no—
138C3×#67—68no—
139C3×#67—64no—
140C3×#67—64no—
141C2×C3×Dic3—34no—
142C2×C3×Dic3—62no—
143C3×D12—64no—
144C3×D12—62no—
145C3×D12—62no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148C3×C4×D3—64no—
149C3×C4×D3—64no—
150C3×Dic6—64no—
151C3×Dic6—62no—
152C3×Dic6—62no—
153C3⋊(C3⋊C8)—24no—
154C3×C3⋊C8—68no—
155C3×C3⋊C8—64no—
156C3×C3⋊C8—64no—
157#413—271noSylow
158C2×C33—21no—
159C2×C33—22no—
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—42no—
162C3×#41—182no—
163C3×#41—182no—
164C2×C3×Q8—91no—
165C2×C3×D4—92no—
166C2×C3×D4—91no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—41no—
170C22×C3×C4—92no—
171C22×C3×C4D3212yes—
172C22×C3×C4D3211yescharacteristic
173C2×#67—34no—
174C2×#67—62no—
175#235—64no—
176#235—64no—
177#235—62no—
178#235—62no—
179#235—62no—
180#235—62no—
181C2×D12—32no—
182C2×D12—61no—
183C2×C4×D3—34no—
184C2×C4×D3—62no—
185C2×Dic6—32no—
186C2×Dic6—61no—
187C3×M16—94no—
188C3×M16—182no—
189C2×C3×C8—182no—
190#208—34no—
191#208—34no—
192#208—64no—
193#208—64no—
194#208—62no—
195#208—62no—
196C2×C3⋊C8—64no—
197C2×C3⋊C8—62no—
198C2×C3⋊C8—62no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—42no—
205C22×C32—41no—
206C22×C32—42no—
207C22×C32—41no—
208C22×C32—44no—
209C22×C32—42no—
210C22×C32—44no—
211C22×C32(C2×D3)⋊C412yes—
212C22×C32(C2×D3)⋊C412yes—
213C22×C32#21712yes—
214C22×C32C2×C4×D312yes—
215C22×C32C22×Dic311yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—42no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—22no—
227C32×C4—22no—
228C32×C4—22no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4—42no—
232C32×C4(C2×D3)⋊C412yes—
233C32×C4(C2×D3)⋊C412yes—
234C32×C4#21711yescharacteristic
235C32×C4#21711yescharacteristic
236C32×C4C2×D1212yes—
237C32×C4C2×#6712yes—
238C32×C4C2×#6712yes—
239C3⋊Dic3—22no—
240C3×Dic3—64no—
241C3×Dic3—64no—
242C2×#41—91no—
243C2×M16—92no—
244#101—272no—
245#100—272no—
246C33#41311yescharacteristic, complemented, Sylow
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—41no—
250C23×C3—92no—
251C23×C3D3×Dic312yes—
252C23×C3#62411yescharacteristic
253C22×D3—34no—
254C22×D3—62no—
255C3×Q8—182no—
256C3×D4—184no—
257C3×D4—182no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—92no—
273C2×C3×C4—182no—
274C2×C3×C4—182no—
275C2×C3×C4D3×Dic312yes—
276C2×C3×C4#62411yescharacteristic
277C2×C3×C4#62512yes—
278C2×C3×C4#62512yes—
279C2×C3×C4#62612yes—
280C2×C3×C4#62612yes—
281C2×C3×C4#62612yes—
282C2×C3×C4#62612yes—
283C2×C3×C4C2×D3212yes—
284C2×C3×C4C2×D3211yescharacteristic
285#67—68no—
286#67—64no—
287#67—64no—
288C2×Dic3—34no—
289C2×Dic3—62no—
290D12—64no—
291D12—62no—
292D12—62no—
293C4×D3—64no—
294C4×D3—64no—
295C4×D3—64no—
296C4×D3—64no—
297Dic6—64no—
298Dic6—62no—
299Dic6—62no—
300C3×C8—184no—
301C3⋊C8—68no—
302C3⋊C8—64no—
303C3⋊C8—64no—
304C2×C32—22no—
305C2×C32—21no—
306C2×C32—22no—
307C2×C32—24no—
308C2×C32—22no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—41no—
314C2×C32—42no—
315C2×C32—42no—
316C2×C32—42no—
317C2×C32—42no—
318C2×C32—44no—
319C2×C32C2×(C2×D3)⋊C412yes—
320C2×C32C2×#21711yescharacteristic
321C3⋊D3—42no—
322C3×D3—124no—
323C3×D3—124no—
324#41—182no—
325#41—182no—
326C2×Q8—91no—
327C2×D4—92no—
328C2×D4—91no—
329C22×C4—92no—
330C22×C4#111311yescharacteristic
331M16—94no—
332M16—182no—
333C2×C8—182no—
334C22×C3—21no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—42no—
340C22×C3—41no—
341C22×C3—42no—
342C22×C3—44no—
343C22×C3—44no—
344C22×C3—41no—
345C22×C3—42no—
346C22×C3—182no—
347C22×C3—362no—
348C22×C3(C2×D3)⋊Dic312yes—
349C22×C3(C2×D3)⋊Dic312yes—
350C22×C3#372212yes—
351C22×C3C2×D3×Dic312yes—
352C22×C3C2×#62411yescharacteristic
353C2×D3—64no—
354C2×D3—64no—
355C2×D3—124no—
356C2×D3—124no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—21no—
361C3×C4—22no—
362C3×C4—21no—
363C3×C4—22no—
364C3×C4—42no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—182no—
368C3×C4(C2×D3)⋊Dic312yes—
369C3×C4(C2×D3)⋊Dic312yes—
370C3×C4#372211yescharacteristic
371C3×C4#372211yescharacteristic
372C3×C4C2×#62512yes—
373C3×C4C2×#62612yes—
374C3×C4C2×#62612yes—
375Dic3—64no—
376Dic3—64no—
377C32—22no—
378C32—21no—
379C32—41no—
380C32#513212yescomplemented
381C32#523711yescharacteristic, complemented
382C23—92no—
383C23#629411yescharacteristic
384Q8—182no—
385D4—184no—
386D4—182no—
387C2×C4—92no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4#629411yescharacteristic
391C2×C4#629512yes—
392C2×C4#629512yes—
393C2×C4C2×#111311yescharacteristic
394C8—184no—
395C2×C3—22no—
396C2×C3—24no—
397C2×C3—22no—
398C2×C3—22no—
399C2×C3—22no—
400C2×C3—21no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—42no—
405C2×C3—42no—
406C2×C3—42no—
407C2×C3—42no—
408C2×C3—44no—
409C2×C3—41no—
410C2×C3—362no—
411C2×C3C2×(C2×D3)⋊Dic312yes—
412C2×C3C2×#372211yescharacteristic
413D3—124no—
414D3—124no—
415C22—41no—
416C22—182no—
417C22—362no—
418C22#8881412yes—
419C22C2×#629411yescharacteristic
420C4—182no—
421C4#8881411yescharacteristic
422C4#8881411yescharacteristic
423C4C2×#629512yes—
424C3—21no—
425C3—22no—
426C3—41no—
427C3#1059107612yescomplemented
428C3#1059107811yescharacteristic, complemented
429C2—21no—
430C2—22no—
431C2—362no—
432C2C2×#8881411yescharacteristic
subgroupG/H
centerC4#88814
commutatorC22×C33C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3828C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. #11722074C2semidirect
4. #11722073C2semidirect
20. C2×#3748D3semidirect
246. C33#413semidirect
380. C32#5132semidirect
381. C32#5237semidirect
427. C3#10591076semidirect
428. C3#10591078semidirect

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
#117220742
#117220732
#105910783
#105910763