Finite Group Databasebeta

← #13161485

Subgroups — #13161485

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64083C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic, complemented
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722075C214yesmaximal, complemented
5#10590281—31nomaximal
6#10590277—31nomaximal
7#10590273—31nomaximal
8C2×C3×C4⋊(C3⋊D3)—21no—
9C2×C3×C4⋊(C3⋊D3)—21no—
10C2×C3×C4⋊(C3⋊D3)C2212yes—
11C2×C3×#634—21no—
12C33×C42C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)—21no—
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C2×C3×C3⋊(C3⋊C8)C2212yes—
16#64083D311yescharacteristic
17#64078D311yescharacteristic, complemented
18C3×#874—31no—
19C3×#874—31no—
20C3×#874—61no—
21C3×#869—31no—
22C3×#869—31no—
23C3×#869—61no—
24C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
25C3×C4×C3⋊C8—31no—
26C3×C4×C3⋊C8—31no—
27C3×C4×C3⋊C8—61no—
28#63564—34no—
29#63563—34no—
30#63562—34no—
31C22×C3×C3⋊D3—22no—
32C2×C33×C4D411yescharacteristic
33C2×C33×C4D411yescharacteristic
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C3×C4⋊(C3⋊D3)—24no—
37C3×C4⋊(C3⋊D3)—42no—
38C3×C4⋊(C3⋊D3)—42no—
39C3×#634—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5088—91no—
42#5070—91no—
43#4694—91no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)C2×D312yes—
47C2×#634—21no—
48C2×C3×D12—32no—
49C2×C3×D12—32no—
50C2×C3×D12—62no—
51C2×C3×D12—62no—
52C2×C3×D12—62no—
53C2×C3×D12—61no—
54C2×C3×D12—61no—
55C2×C3×D12—61no—
56C2×C3×D12—61no—
57C2×C3×Dic6—62no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—61no—
60C32×C42—21no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42—41no—
64C32×C42C2×D311yescharacteristic
65C32×C42C2×D311yescharacteristic
66C32×C42C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)—21no—
68C4⋊(C3⋊Dic3)C2×D312yes—
69C2×C3⋊(C3⋊C8)C2×D312yes—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3—62no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—61no—
76C2×C3×C3⋊C8—32no—
77C2×C3×C3⋊C8—32no—
78C2×C3×C3⋊C8—62no—
79C22×C33C2×D411yescharacteristic
80C2×C3×C3⋊D3—42no—
81C2×C3×C3⋊D3—42no—
82C2×C3×C3⋊D3—42no—
83C33×C4D812yes—
84C33×C4SD1612yes—
85C33×C4#4112yes—
86C3×C3⋊Dic3—42no—
87C3×C4⋊Q8—91no—
88C3×#127—91no—
89#874—31no—
90#874—31no—
91#874—61no—
92#869—31no—
93#869—31no—
94#869—61no—
95C3×C4×C8—91no—
96#832—94no—
97#821—94no—
98#809—94no—
99C4×C3⋊C8—31no—
100C4×C3⋊C8—31no—
101C4×C3⋊C8—61no—
102C22×C3⋊D3—22no—
103C22×C3×D3—64no—
104C22×C3×D3—62no—
105C22×C3×D3—62no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4D1211yescharacteristic
119C2×C32×C4D1211yescharacteristic
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—22no—
128C4⋊(C3⋊D3)—24no—
129C4⋊(C3⋊D3)—42no—
130C4⋊(C3⋊D3)—42no—
131#634—42no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—62no—
135C3×D12—64no—
136C3×D12—64no—
137C3×D12—124no—
138C3×D12—124no—
139C3×D12—124no—
140C3×D12—122no—
141C3×D12—122no—
142C3×D12—122no—
143C3×D12—122no—
144C3×Dic6—124no—
145C3×Dic6—122no—
146C3×Dic6—122no—
147C3⋊(C3⋊C8)—22no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—62no—
151#486—271noSylow
152C2×C33#12411yescharacteristic
153C2×C33C2×D811yescharacteristic
154C2×C33C2×SD1611yescharacteristic
155C3×C3⋊D3—82no—
156C2×C3×Q8—181no—
157C2×C3×D4—92no—
158C2×C3×D4—181no—
159C2×C3×D4—181no—
160C2×D12—32no—
161C2×D12—32no—
162C2×D12—62no—
163C2×D12—62no—
164C2×D12—62no—
165C2×D12—61no—
166C2×D12—61no—
167C2×D12—61no—
168C2×D12—61no—
169C2×Dic6—62no—
170C2×Dic6—61no—
171C2×Dic6—61no—
172C2×C3×C8—92no—
173C3×C4⋊C4—92no—
174C3×C4⋊C4—181no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42—21no—
178C3×C42—41no—
179C3×C42D3211yescharacteristic
180C3×C42D3211yescharacteristic
181C3×C42D3211yescharacteristic
182C4⋊Dic3—32no—
183C4⋊Dic3—32no—
184C4⋊Dic3—62no—
185C4⋊Dic3—62no—
186C4⋊Dic3—61no—
187C4⋊Dic3—61no—
188C2×C3⋊C8—32no—
189C2×C3⋊C8—32no—
190C2×C3⋊C8—62no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32C2×D1211yescharacteristic
196C22×C32C2×#6711yescharacteristic
197C22×C32C2×#6711yescharacteristic
198C2×C3⋊D3—42no—
199C2×C3⋊D3—42no—
200C2×C3⋊D3—42no—
201C2×C3×D3—124no—
202C2×C3×D3—124no—
203C2×C3×D3—124no—
204C2×C3×D3—122no—
205C2×C3×D3—122no—
206C2×C3×D3—122no—
207C2×C3×D3—122no—
208C2×C3×D3—122no—
209C2×C3×D3—122no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—22no—
216C32×C4—22no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4—42no—
222C32×C4#20412yes—
223C32×C4D2412yes—
224C32×C4#21312yes—
225C32×C4#21312yes—
226C32×C4#21412yes—
227C32×C4#21512yes—
228C32×C4#23512yes—
229C32×C4#23712yes—
230C32×C4#23912yes—
231C3⋊Dic3—42no—
232C3×Dic3—124no—
233C3×Dic3—122no—
234C3×Dic3—122no—
235C4⋊Q8—91no—
236#127—91no—
237#102—274no—
238C4×C8—91no—
239C33#48611yescharacteristic, complemented, Sylow
240C23×C3—182no—
241C22×D3—64no—
242C22×D3—62no—
243C22×D3—62no—
244C3×Q8—362no—
245C3×D4—184no—
246C3×D4—362no—
247C3×D4—362no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—182no—
261C2×C3×C4#62511yescharacteristic
262C2×C3×C4#62511yescharacteristic
263C2×C3×C4#62611yescharacteristic
264C2×C3×C4#62611yescharacteristic
265C2×C3×C4#62611yescharacteristic
266C2×C3×C4#62611yescharacteristic
267C2×C3×C4C2×D3211yescharacteristic
268C2×C3×C4C2×D3211yescharacteristic
269C2×C3×C4C2×D3211yescharacteristic
270C2×Dic3—64no—
271C2×Dic3—62no—
272C2×Dic3—62no—
273D12—64no—
274D12—64no—
275D12—124no—
276D12—124no—
277D12—124no—
278D12—122no—
279D12—122no—
280D12—122no—
281D12—122no—
282Dic6—124no—
283Dic6—122no—
284Dic6—122no—
285C3×C8—182no—
286C3⋊C8—64no—
287C3⋊C8—62no—
288C3⋊C8—62no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—41no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32#87511yescharacteristic
302C2×C32C2×#20411yescharacteristic
303C2×C32C2×D2411yescharacteristic
304C2×C32C2×#21311yescharacteristic
305C2×C32C2×#21311yescharacteristic
306C2×C32C2×#21411yescharacteristic
307C2×C32#93611yescharacteristic
308C2×C32C2×#21511yescharacteristic
309C2×C32#94711yescharacteristic
310C3⋊D3—82no—
311C3×D3—244no—
312C3×D3—242no—
313C3×D3—242no—
314C2×Q8—181no—
315C2×D4—92no—
316C2×D4—181no—
317C2×D4—181no—
318C2×C8—92no—
319C4⋊C4—92no—
320C4⋊C4—181no—
321C42#111311yescharacteristic
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—362no—
327C22×C3—362no—
328C22×C3—362no—
329C22×C3C2×#62511yescharacteristic
330C22×C3C2×#62611yescharacteristic
331C22×C3C2×#62611yescharacteristic
332C2×D3—124no—
333C2×D3—124no—
334C2×D3—124no—
335C2×D3—122no—
336C2×D3—122no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—122no—
340C2×D3—122no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—42no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—362no—
354C3×C4#371312yes—
355C3×C4#371412yes—
356C3×C4#371412yes—
357C3×C4#371512yes—
358C3×C4#371612yes—
359C3×C4#371712yes—
360C3×C4#379512yes—
361C3×C4#379612yes—
362C3×C4#379912yes—
363Dic3—124no—
364Dic3—122no—
365Dic3—122no—
366C32—21no—
367C32—21no—
368C32—21no—
369C32—41no—
370C32#469411yescharacteristic, complemented
371C32#507011yescharacteristic, complemented
372C32#508811yescharacteristic, complemented
373C23—182no—
374Q8—362no—
375D4—184no—
376D4—362no—
377D4—362no—
378C2×C4—182no—
379C2×C4#629511yescharacteristic
380C2×C4#629511yescharacteristic
381C2×C4C2×#111311yescharacteristic
382C8—182no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—722no—
396C2×C3C2×#371311yescharacteristic
397C2×C3C2×#371411yescharacteristic
398C2×C3C2×#371411yescharacteristic
399C2×C3C2×#371511yescharacteristic
400C2×C3C2×#371611yescharacteristic
401C2×C3C2×#371711yescharacteristic
402C2×C3#6386011yescharacteristic
403C2×C3#6386311yescharacteristic
404C2×C3#6386411yescharacteristic
405D3—244no—
406D3—242no—
407D3—242no—
408C22—362no—
409C22—362no—
410C22—362no—
411C22C2×#629511yescharacteristic
412C4—362no—
413C4#8881012yes—
414C4#8881112yes—
415C4#8904112yes—
416C3—21no—
417C3—21no—
418C3—21no—
419C3—41no—
420C3#1059027311yescharacteristic, complemented
421C3#1059027711yescharacteristic, complemented
422C3#1059028111yescharacteristic, complemented
423C2—722no—
424C2C2×#8881011yescharacteristic
425C2C2×#8881111yescharacteristic
426C2#1172367711yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC3×#64083C2
hypercenterC42#1113
hypocenterC33#486
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64078C2semidirect
3. C3×C4×C3⋊(C3⋊C8)C2semidirect
4. #11722075C2semidirect
17. #64078D3semidirect
24. C4×C3⋊(C3⋊C8)D3semidirect
239. C33#486semidirect
370. C32#4694semidirect
371. C32#5070semidirect
372. C32#5088semidirect
420. C3#10590273semidirect
421. C3#10590277semidirect
422. C3#10590281semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C3×#640782
C3×C4×C3⋊(C3⋊C8)2
#117220752
#105902813
#105902773
#105902733