Finite Group Databasebeta

← #13175372

Subgroups — #13175372

Order 1728.

1612 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6308C211yesmaximal, characteristic, complemented
2C3×C42×C3⋊D3C211yesmaximal, characteristic
3C32×C4×Dic6C211yesmaximal, characteristic, complemented
4#11723535C211yesmaximal, characteristic, complemented
5C4×#6293C212yesmaximal, complemented
6#11723455C211yesmaximal, characteristic, complemented
7#11723445C212yesmaximal, complemented
8#11723437C212yesmaximal
9#11723435C211yesmaximal, characteristic
10#11723431C211yesmaximal, characteristic
11#11723418C212yesmaximal
12#10596049—31nomaximal
13#10593873—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C2×#6293C2212yes—
20C33×C42C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857C2212yes—
23C4×C3⋊(C3⋊Dic3)C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2212yes—
29C32×C4×Dic3C2212yes—
30#88807C2212yes—
31#88807C2212yes—
32#88807C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88803C2212yes—
36#88803C2212yes—
37C32×C4×Q8—31no—
38#64155—31no—
39#64153—31no—
40#64106—31no—
41#64105—32no—
42#64098—32no—
43#64097—31no—
44#64081—32no—
45C42×C3⋊D3D311yescharacteristic
46C4×#634—31no—
47C4×#634—32no—
48C4×#634—34no—
49C4×#634—64no—
50C3×C42×D3—34no—
51C3×C4×Dic6D314yescomplemented
52#63918—34no—
53C4×#627—38no—
54#63867—38no—
55#63866—34no—
56#63857—34no—
57#63856—34no—
58#63855—38no—
59#63851—38no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C2×C33×C4C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4×C3⋊D3—22no—
71C3×C4×C3⋊D3—22no—
72C3×C4×C3⋊D3Q814yescomplemented
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C32×Dic6—22no—
76#6293—24no—
77#5578—94no—
78#5470—91no—
79C2×C32×Q8—31no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×#634—31no—
84C2×#634—34no—
85C2×#634—32no—
86C2×#634—64no—
87C2×C3×C4×D3—34no—
88C2×C3×C4×D3—34no—
89C2×C3×C4×D3—34no—
90C2×C3×Dic6C2×D314yes—
91C2×#627—38no—
92C32×C4⋊C4—31no—
93C32×C4⋊C4—32no—
94C32×C42—24no—
95C32×C42—32no—
96C32×C42C2×D314yes—
97C32×C42C2×D311yescharacteristic
98#3752—32no—
99#3752—32no—
100C4⋊(C3⋊Dic3)—31no—
101C4⋊(C3⋊Dic3)—34no—
102C4⋊(C3⋊Dic3)—32no—
103C4⋊(C3⋊Dic3)—64no—
104#3750—32no—
105#3750—38no—
106#3750—32no—
107#3750—32no—
108#3750—32no—
109#3750—32no—
110#3750—68no—
111C4×C3⋊Dic3—32no—
112C4×C3⋊Dic3—38no—
113C4×C3⋊Dic3—68no—
114C4×C3⋊Dic3C2×D311yescharacteristic
115C4×C3⋊Dic3C2×D311yescharacteristic
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×C4⋊Dic3C2×D314yes—
118C3×Dic3⋊C4C2×D318yes—
119C3×C4×Dic3—34no—
120C3×C4×Dic3—34no—
121C3×C4×Dic3—34no—
122C3×C4×Dic3C2×D318yes—
123#3724—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130C22×C33C2411yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3C2×Q812yes—
134C33×C4—21no—
135C33×C4C2×Q812yes—
136C33×C4#4112yes—
137C3⋊(C3⋊Dic3)—22no—
138C3⋊(C3⋊Dic3)—22no—
139C3×C3⋊Dic3—21no—
140C3×C3⋊Dic3C2×Q812yes—
141C3×C3⋊Dic3#4112yes—
142C32×Dic3—22no—
143C32×Dic3—22no—
144C3×C4×Q8—34no—
145C2×C3×C42—91no—
146#946—94no—
147#944—94no—
148#926—92no—
149#924—91no—
150#924—91no—
151#924—92no—
152#897—94no—
153#896—98no—
154#889—98no—
155#888—94no—
156#872—98no—
157C42×D3—34no—
158#870—92no—
159#869—91no—
160#869—91no—
161C4×Dic6—98no—
162C4×Dic6—94no—
163C4×Dic6—91no—
164C4×Dic6—92no—
165C4×Dic6—184no—
166C4×Dic6C3⋊D311yescharacteristic, complemented
167C22×C3⋊D3C22×D311yescharacteristic
168C22×C3×D3—34no—
169C32×Q8—62no—
170C2×C32×C4—24no—
171C2×C32×C4—24no—
172C2×C32×C4—24no—
173C2×C32×C4—32no—
174C2×C32×C4—32no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D314yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—32no—
183C2×C3⋊Dic3—38no—
184C2×C3⋊Dic3—38no—
185C2×C3⋊Dic3—68no—
186C2×C3⋊Dic3—68no—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D311yescharacteristic
190C4×C3⋊D3—22no—
191C4×C3⋊D3—22no—
192C4×C3⋊D3Dic614yescomplemented
193#634—616no—
194#634—68no—
195#634—62no—
196#634—64no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3C22×D318yes—
201C2×C3×Dic3C22×D318yes—
202C3×C4×D3—316no—
203C3×C4×D3—68no—
204C3×C4×D3—68no—
205C3×Dic6—28no—
206#627—616no—
207#533—271noSylow
208C2×C33C22×Q811yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C33C2×#4111yescharacteristic
211C3×C3⋊D3—22no—
212C2×C3×Q8—34no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C2×C4×D3—34no—
217C2×C4×D3—34no—
218C2×C4×D3—34no—
219C2×Dic6—98no—
220C2×Dic6—94no—
221C2×Dic6—91no—
222C2×Dic6—92no—
223C2×Dic6—184no—
224C2×Dic6C2×C3⋊D311yescharacteristic
225C3×C4⋊C4—34no—
226C3×C4⋊C4—38no—
227C3×C42—24no—
228C3×C42—38no—
229C3×C42—91no—
230C3×C42—91no—
231C3×C42—91no—
232C3×C42D3214yes—
233C3×C42C2×C3⋊D311yescharacteristic
234#217—92no—
235#217—92no—
236(C2×D3)⋊C4—98no—
237(C2×D3)⋊C4—98no—
238C4⋊Dic3—98no—
239C4⋊Dic3—94no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—184no—
245C4⋊Dic3C2×C3⋊D311yescharacteristic
246Dic3⋊C4—98no—
247Dic3⋊C4—98no—
248Dic3⋊C4—98no—
249Dic3⋊C4—98no—
250Dic3⋊C4—98no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—188no—
255Dic3⋊C4C2×C3⋊D312yes—
256C4×Dic3—34no—
257C4×Dic3—34no—
258C4×Dic3—34no—
259C4×Dic3—98no—
260C4×Dic3—92no—
261C4×Dic3—188no—
262C4×Dic3C2×C3⋊D312yes—
263C22×C32—24no—
264C22×C32C23×D314yes—
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—21no—
267C2×C3⋊D3—21no—
268C2×C3⋊D3C2×Dic612yes—
269C2×C3×D3—38no—
270C2×C3×D3—64no—
271C2×C3×D3—64no—
272C32×C4—21no—
273C32×C4—28no—
274C32×C4—28no—
275C32×C4—24no—
276C32×C4—44no—
277C32×C4—62no—
278C32×C4—62no—
279C32×C4C2×Dic612yes—
280C32×C4#23518yes—
281C32×C4#23512yes—
282C32×C4D3×Q818yes—
283C3⋊Dic3—21no—
284C3⋊Dic3—616no—
285C3⋊Dic3—68no—
286C3⋊Dic3—62no—
287C3⋊Dic3—62no—
288C3⋊Dic3—68no—
289C3⋊Dic3—128no—
290C3⋊Dic3C2×Dic612yes—
291C3⋊Dic3#23512yes—
292C3×Dic3—28no—
293C3×Dic3—28no—
294C3×Dic3—38no—
295C3×Dic3—38no—
296C3×Dic3—64no—
297C4⋊Q8—271no—
298C4⋊Q8—271no—
299#125—272no—
300C22⋊Q8—272no—
301C22⋊Q8—271no—
302C22⋊Q8—271no—
303C4×Q8—31no—
304C4×Q8—272no—
305C4×Q8—271no—
306#117—272no—
307C2×C42—91no—
308C33#53311yescharacteristic, complemented, Sylow
309C23×C3—91no—
310C22×D3—34no—
311C3×Q8—68no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—38no—
316C2×C3×C4—38no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—94no—
320C2×C3×C4—91no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4C2×D3214yes—
324C2×C3×C4C2×D3214yes—
325C2×C3×C4C2×D3214yes—
326C2×C3×C4C22×C3⋊D311yescharacteristic
327C2×C3×C4C22×C3⋊D311yescharacteristic
328C2×C3×C4C22×C3⋊D311yescharacteristic
329C2×Dic3—34no—
330C2×Dic3—34no—
331C2×Dic3—34no—
332C2×Dic3—98no—
333C2×Dic3—98no—
334C2×Dic3—92no—
335C2×Dic3—92no—
336C2×Dic3—188no—
337C2×Dic3—188no—
338C2×Dic3C22×C3⋊D312yes—
339C2×Dic3C22×C3⋊D312yes—
340C4×D3—316no—
341C4×D3—68no—
342C4×D3—68no—
343Dic6—22no—
344Dic6—184no—
345Dic6—1816no—
346Dic6—1816no—
347Dic6—188no—
348Dic6—182no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32C22×Dic611yescharacteristic
353C2×C32C2×#23514yes—
354C2×C32C2×#23511yescharacteristic
355C2×C32C2×#23511yescharacteristic
356C2×C32C2×D3×Q814yes—
357C2×C32D3×#4114yes—
358C3⋊D3—22no—
359C3×D3—68no—
360C2×Q8—31no—
361C2×Q8—272no—
362C2×Q8—271no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C4⋊C4—32no—
367C4⋊C4—31no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C4⋊C4—271no—
375C22⋊C4—272no—
376C22⋊C4—272no—
377C42—32no—
378C42—91no—
379C42—91no—
380C42—91no—
381C42—272no—
382C42D3×C3⋊D311yescharacteristic
383C22×C3—24no—
384C22×C3—92no—
385C22×C3—181no—
386C22×C3—181no—
387C22×C3C22×D3214yes—
388C22×C3C23×C3⋊D311yescharacteristic
389C2×D3—38no—
390C2×D3—64no—
391C2×D3—64no—
392C3×C4—24no—
393C3×C4—21no—
394C3×C4—28no—
395C3×C4—28no—
396C3×C4—44no—
397C3×C4—68no—
398C3×C4—68no—
399C3×C4—92no—
400C3×C4—92no—
401C3×C4—181no—
402C3×C4D3×Dic618yes—
403C3×C4#379818yes—
404C3×C4#382812yes—
405C3×C4Q8×C3⋊D312yes—
406Dic3—22no—
407Dic3—22no—
408Dic3—38no—
409Dic3—38no—
410Dic3—64no—
411Dic3—1816no—
412Dic3—188no—
413Dic3—188no—
414Dic3—182no—
415Dic3—182no—
416Dic3—368no—
417C32—24no—
418C32#547011yescharacteristic, complemented
419C32#557814yescomplemented
420C23—91no—
421Q8—62no—
422Q8—544no—
423Q8—542no—
424C2×C4—32no—
425C2×C4—32no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—94no—
429C2×C4—91no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—272no—
433C2×C4—272no—
434C2×C4C2×D3×C3⋊D311yescharacteristic
435C2×C4C2×D3×C3⋊D311yescharacteristic
436C2×C4C2×D3×C3⋊D311yescharacteristic
437C2×C3—24no—
438C2×C3—24no—
439C2×C3—24no—
440C2×C3—182no—
441C2×C3C2×D3×Dic614yes—
442C2×C3C2×#379814yes—
443C2×C3D3×#23514yes—
444C2×C3C2×#382811yescharacteristic
445C2×C3C2×Q8×C3⋊D311yescharacteristic
446C2×C3#41×C3⋊D311yescharacteristic
447D3—68no—
448C22—92no—
449C22—181no—
450C22—181no—
451C22C22×D3×C3⋊D311yescharacteristic
452C4—21no—
453C4—62no—
454C4—62no—
455C4—92no—
456C4—92no—
457C4—181no—
458C4—542no—
459C4—542no—
460C4C3⋊D3×Dic612yes—
461C4#8902012yes—
462C3—24no—
463C3#1059387314yescomplemented
464C3#1059604911yescharacteristic, complemented
465C2—182no—
466C2C2×C3⋊D3×Dic611yescharacteristic
467C2C2×#8902011yescharacteristic
468C2C3⋊D3×#23511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC42D3×C3⋊D3
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6308C2semidirect
3. C32×C4×Dic6C2semidirect
4. #11723535C2semidirect
5. C4×#6293C2semidirect
6. #11723455C2semidirect
7. #11723445C2semidirect
51. C3×C4×Dic6D3semidirect
72. C3×C4×C3⋊D3Q8semidirect
166. C4×Dic6C3⋊D3semidirect
192. C4×C3⋊D3Dic6semidirect
308. C33#533semidirect
418. C32#5470semidirect
419. C32#5578semidirect
463. C3#10593873semidirect
464. C3#10596049semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C3×C42×C3⋊D32
C32×C4×Dic62
#117235352
C4×#62932
#117234552
#117234452
#117234372
#117234352
#117234312
#117234182
#105960493
#105938733