Finite Group Databasebeta

← #13176501

Subgroups — #13176501

Order 1728.

1592 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723931C211yesmaximal, characteristic, complemented
2C3×#64099C211yesmaximal, characteristic, complemented
3C32×#893C211yesmaximal, characteristic
4#11723534C211yesmaximal, characteristic, complemented
5C4×#6293C211yesmaximal, characteristic, complemented
6#11723500C212yesmaximal, complemented
7#11723499C211yesmaximal, characteristic
8#11723491C212yesmaximal, complemented
9D12×C3⋊Dic3C211yesmaximal, characteristic
10D3×C4⋊(C3⋊Dic3)C212yesmaximal, complemented
11#11723415C212yesmaximal, complemented
12#10596014—31nomaximal
13#10594452—34nomaximal
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3C2212yes—
16C2×#6293C2211yescharacteristic
17C2×#6293C2212yes—
18C2×D3×C3⋊Dic3C2212yes—
19C2×D3×C3⋊Dic3C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4⋊(C3⋊(C3⋊Dic3))C2212yes—
23#88857C2212yes—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4⋊(C3⋊Dic3)C2212yes—
27C3×#3750C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×(C2×D3)⋊C4C2212yes—
30C32×C4×Dic3C2211yescharacteristic
31#88807C2211yescharacteristic
32#88806C2211yescharacteristic
33Dic3⋊(C3⋊Dic3)C2211yescharacteristic
34Dic3⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38D4×C3⋊Dic3—31no—
39D4×C3⋊Dic3—31no—
40C2×C4⋊(C3⋊Dic3)—32no—
41#64133—32no—
42#64099—31no—
43#64099—34no—
44#64099—64no—
45#64099D311yescharacteristic, complemented
46#64087—32no—
47#64087—32no—
48#64078—31no—
49C4×#634—31no—
50C3×#893D314yes—
51C3×#890—34no—
52#63918—34no—
53C4×#627—34no—
54#63899—38no—
55#63898—34no—
56#63893—38no—
57Dic3×D12—34no—
58D3×C4⋊Dic3—38no—
59#63851—38no—
60C22×C32×D3C2312yes—
61C2×C33×C4C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2311yescharacteristic
68C2×C3×C3⋊Dic3C2312yes—
69C2×C32×Dic3C2311yescharacteristic
70C2×C32×Dic3C2311yescharacteristic
71C32×D12Q814yescomplemented
72C32×C4×D3—24no—
73#6293—22no—
74#6293—42no—
75D3×C3⋊Dic3—42no—
76D3×C3⋊Dic3—42no—
77#5690—91no—
78#5543—94no—
79C2×C32×D4—31no—
80C22×C32×C4—32no—
81C22×C3⋊Dic3—32no—
82C22×C3⋊Dic3—32no—
83C2×#634—31no—
84C2×#634—32no—
85C2×C3×D12C2×D314yes—
86C2×C3×C4×D3C2×D318yes—
87C2×#627—34no—
88C2×#627—38no—
89C2×D3×Dic3—38no—
90C2×D3×Dic3—38no—
91C32×C4⋊C4—24no—
92C32×C4⋊C4C2×D314yes—
93C32×C4⋊C4C2×D311yescharacteristic
94C32×C22⋊C4—32no—
95C32×C42—31no—
96#3757—32no—
97#3757—32no—
98C4⋊(C3⋊Dic3)—31no—
99C4⋊(C3⋊Dic3)—32no—
100C4⋊(C3⋊Dic3)—38no—
101C4⋊(C3⋊Dic3)—34no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—68no—
105C4⋊(C3⋊Dic3)—64no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C4⋊(C3⋊Dic3)C2×D312yes—
108#3750—32no—
109#3750—38no—
110#3750—31no—
111#3750—31no—
112#3750—68no—
113#3750C2×D312yes—
114C4×C3⋊Dic3—31no—
115C4×C3⋊Dic3—34no—
116C4×C3⋊Dic3—64no—
117C4×C3⋊Dic3C2×D311yescharacteristic
118C3×(C2×D3)⋊C4C2×D318yes—
119C3×C4⋊Dic3—38no—
120C3×C4⋊Dic3—34no—
121C3×Dic3⋊C4—38no—
122C3×C4×Dic3—34no—
123C3×C4×Dic3C2×D314yes—
124#3724—34no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—34no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130C22×C33C2411yescharacteristic
131C2×C32×D3—22no—
132C2×C32×D3—22no—
133C2×C32×D3C2×Q814yes—
134C33×C4—22no—
135C33×C4C2×Q812yes—
136C3⋊(C3⋊Dic3)—21no—
137C3⋊(C3⋊Dic3)—21no—
138C3⋊(C3⋊Dic3)—22no—
139C3×C3⋊Dic3—21no—
140C3×C3⋊Dic3—21no—
141C3×C3⋊Dic3—22no—
142C32×Dic3—21no—
143C32×Dic3#4112yes—
144C3×#125—91no—
145C3×C4×D4—34no—
146D4×Dic3—94no—
147D4×Dic3—94no—
148C2×C4⋊Dic3—98no—
149#924—98no—
150#897—92no—
151#896—92no—
152#896—92no—
153#893—91no—
154#893C3⋊D311yescharacteristic
155D3×C4⋊C4—92no—
156#890—34no—
157#890—94no—
158#890—91no—
159#890—184no—
160#888—91no—
161#878—98no—
162#878—98no—
163C4×D12—91no—
164#869—94no—
165C4×Dic6—94no—
166C4×Dic6—91no—
167C23×C32—32no—
168C22×C3×D3C22×D318yes—
169C32×D4—34no—
170C2×C32×C4—24no—
171C2×C32×C4—28no—
172C2×C32×C4—31no—
173C2×C32×C4—31no—
174C2×C32×C4—64no—
175C2×C32×C4C22×D318yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—38no—
184C2×C3⋊Dic3—34no—
185C2×C3⋊Dic3—64no—
186C2×C3⋊Dic3—64no—
187C2×C3⋊Dic3—68no—
188C2×C3⋊Dic3—122no—
189C2×C3⋊Dic3—122no—
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193#634—62no—
194#634—122no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—38no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C22×D314yes—
200C3×D12Dic6116yescomplemented
201C3×C4×D3—216no—
202#627—68no—
203#627—128no—
204D3×Dic3—128no—
205D3×Dic3—128no—
206#554—271noSylow
207C2×C33C22×Q811yescharacteristic
208C2×C33C2×#4111yescharacteristic
209C2×C33H(5,𝔽2)11yescharacteristic
210C32×D3—24no—
211C2×C3×D4—34no—
212C22×C3×C4—38no—
213C22×Dic3—98no—
214C22×Dic3—98no—
215C2×D12C2×C3⋊D311yescharacteristic
216C2×C4×D3—92no—
217C2×C4×D3—92no—
218C2×C4×D3C2×C3⋊D312yes—
219C2×Dic6—94no—
220C2×Dic6—91no—
221C2×Dic6—98no—
222C2×Dic6—92no—
223C3×C4⋊C4—24no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C4⋊C4D3214yes—
228C3×C4⋊C4C2×C3⋊D311yescharacteristic
229C3×C22⋊C4—38no—
230C3×C42—34no—
231C3×C42—91no—
232#217—98no—
233#217—98no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4C2×C3⋊D312yes—
237C4⋊Dic3—34no—
238C4⋊Dic3—38no—
239C4⋊Dic3—94no—
240C4⋊Dic3—98no—
241C4⋊Dic3—94no—
242C4⋊Dic3—98no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—91no—
246C4⋊Dic3—188no—
247C4⋊Dic3—184no—
248Dic3⋊C4—38no—
249Dic3⋊C4—94no—
250Dic3⋊C4—94no—
251Dic3⋊C4—98no—
252Dic3⋊C4—92no—
253Dic3⋊C4—91no—
254Dic3⋊C4—91no—
255Dic3⋊C4—92no—
256Dic3⋊C4—188no—
257C4×Dic3—34no—
258C4×Dic3—94no—
259C4×Dic3—91no—
260C4×Dic3—184no—
261C4×Dic3C2×C3⋊D311yescharacteristic
262C22×C32—24no—
263C22×C32—34no—
264C22×C32—62no—
265C22×C32—62no—
266C22×C32C23×D314yes—
267C22×C32C23×D311yescharacteristic
268C2×C3×D3—28no—
269C2×C3×D3—28no—
270C2×C3×D3C2×Dic6116yes—
271C32×C4—22no—
272C32×C4—28no—
273C32×C4—28no—
274C32×C4—32no—
275C32×C4—48no—
276C32×C4—61no—
277C32×C4C2×Dic618yes—
278C32×C4D3×Q812yes—
279C3⋊Dic3—21no—
280C3⋊Dic3—22no—
281C3⋊Dic3—21no—
282C3⋊Dic3—61no—
283C3⋊Dic3—62no—
284C3⋊Dic3—61no—
285C3⋊Dic3—64no—
286C3⋊Dic3—64no—
287C3⋊Dic3—68no—
288C3⋊Dic3—124no—
289C3⋊Dic3—128no—
290C3⋊Dic3—124no—
291C3×Dic3—24no—
292C3×Dic3—64no—
293C3×Dic3—64no—
294C3×Dic3—68no—
295C3×Dic3#23718yes—
296C4⋊Q8—271no—
297#125—91no—
298#125—271no—
299C22⋊Q8—272no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C4×Q8—271no—
303C4×D4—31no—
304C4×D4—271no—
305C4×D4—271no—
306C2×C4⋊C4—272no—
307C33#55411yescharacteristic, complemented, Sylow
308C23×C3—38no—
309C22×D3C22×C3⋊D312yes—
310C3×D4—316no—
311C2×C3×C4—24no—
312C2×C3×C4—28no—
313C2×C3×C4—34no—
314C2×C3×C4—34no—
315C2×C3×C4—616no—
316C2×C3×C4—91no—
317C2×C3×C4—92no—
318C2×C3×C4—91no—
319C2×C3×C4C2×D3218yes—
320C2×C3×C4C2×D3214yes—
321C2×C3×C4C22×C3⋊D312yes—
322C2×C3×C4C22×C3⋊D311yescharacteristic
323C2×Dic3—34no—
324C2×Dic3—34no—
325C2×Dic3—38no—
326C2×Dic3—94no—
327C2×Dic3—94no—
328C2×Dic3—98no—
329C2×Dic3—91no—
330C2×Dic3—91no—
331C2×Dic3—92no—
332C2×Dic3—184no—
333C2×Dic3—184no—
334C2×Dic3—188no—
335C2×Dic3—368no—
336C2×Dic3—368no—
337C2×Dic3C22×C3⋊D311yescharacteristic
338C2×Dic3C22×C3⋊D311yescharacteristic
339D12#63414yescomplemented
340C4×D3—24no—
341C4×D3—362no—
342C4×D3—362no—
343Dic6—188no—
344Dic6—182no—
345Dic6—362no—
346Dic6—368no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—64no—
351C2×C32C22×Dic614yes—
352C2×C32C2×#23714yes—
353C2×C32C2×D3×Q811yescharacteristic
354C2×C32D3×#4111yescharacteristic
355C2×C32#100914yes—
356C2×C32#100911yescharacteristic
357C3×D3—216no—
358C2×Q8—271no—
359C2×Q8—272no—
360C2×D4—31no—
361C22×C4—32no—
362C22×C4—272no—
363C22×C4—272no—
364C4⋊C4—92no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—271no—
368C4⋊C4—272no—
369C4⋊C4—271no—
370C4⋊C4—271no—
371C4⋊C4—272no—
372C4⋊C4—271no—
373C4⋊C4—272no—
374C4⋊C4D3×C3⋊D311yescharacteristic
375C22⋊C4—32no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—31no—
379C42—91no—
380C42—271no—
381C22×C3—24no—
382C22×C3—316no—
383C22×C3—68no—
384C22×C3—68no—
385C22×C3C22×D3214yes—
386C22×C3C23×C3⋊D311yescharacteristic
387C2×D3—22no—
388C2×D3—22no—
389C2×D3C2×#63414yes—
390C3×C4—28no—
391C3×C4—22no—
392C3×C4—28no—
393C3×C4—38no—
394C3×C4—48no—
395C3×C4—64no—
396C3×C4—181no—
397C3×C4—182no—
398C3×C4—181no—
399C3×C4D3×Dic618yes—
400C3×C4C2×#63412yes—
401Dic3—21no—
402Dic3—64no—
403Dic3—68no—
404Dic3—64no—
405Dic3—184no—
406Dic3—188no—
407Dic3—184no—
408Dic3—181no—
409Dic3—182no—
410Dic3—181no—
411Dic3—364no—
412Dic3—364no—
413Dic3—368no—
414Dic3#383012yes—
415C32—24no—
416C32#554314yescomplemented
417C32#569011yescharacteristic, complemented
418C23—32no—
419Q8—542no—
420Q8—1082no—
421D4—34no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—64no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—92no—
428C2×C4—271no—
429C2×C4—271no—
430C2×C4—272no—
431C2×C4—1082no—
432C2×C4—1082no—
433C2×C4C2×D3×C3⋊D312yes—
434C2×C4C2×D3×C3⋊D311yescharacteristic
435C2×C3—24no—
436C2×C3—24no—
437C2×C3—24no—
438C2×C3—616no—
439C2×C3C2×D3×Dic614yes—
440C2×C3D3×#23714yes—
441C2×C3#6432014yes—
442C2×C3C22×#63411yescharacteristic
443C2×C3C2×#383011yescharacteristic
444C2×C3#6436611yescharacteristic
445D3—24no—
446C22—34no—
447C22—62no—
448C22—62no—
449C22C22×D3×C3⋊D311yescharacteristic
450C4—22no—
451C4—32no—
452C4—61no—
453C4—181no—
454C4—181no—
455C4—182no—
456C4—541no—
457C4—541no—
458C4—542no—
459C4D3×#63412yes—
460C3—24no—
461C3#1059445214yescomplemented
462C3#1059601411yescharacteristic, complemented
463C2—64no—
464C2C2×D3×#63411yescharacteristic
465C2D3×#383011yescharacteristic
466C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C32×D12C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723931C2semidirect
2. C3×#64099C2semidirect
4. #11723534C2semidirect
5. C4×#6293C2semidirect
6. #11723500C2semidirect
8. #11723491C2semidirect
10. D3×C4⋊(C3⋊Dic3)C2semidirect
11. #11723415C2semidirect
45. #64099D3semidirect
71. C32×D12Q8semidirect
200. C3×D12Dic6semidirect
307. C33#554semidirect
339. D12#634semidirect
416. C32#5543semidirect
417. C32#5690semidirect
461. C3#10594452semidirect
462. C3#10596014semidirect

Maximal subgroups

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

subgroupindex
#117239312
C3×#640992
C32×#8932
#117235342
C4×#62932
#117235002
#117234992
#117234912
D12×C3⋊Dic32
D3×C4⋊(C3⋊Dic3)2
#117234152
#105960143
#105944523