Finite Group Databasebeta

← #13177276

Subgroups — #13177276

Order 1728.

1628 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723965C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4#11723640C212yesmaximal
5#11723594C212yesmaximal
6#11723534C211yesmaximal, characteristic, complemented
7#11723444C212yesmaximal, complemented
8#11723443C211yesmaximal, characteristic, complemented
9#11723399C211yesmaximal, characteristic, complemented
10Dic3×#634C211yesmaximal, characteristic, complemented
11#11723389C212yesmaximal, complemented
12#10595941—31nomaximal
13#10594808—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×#6308C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C2×#6293C2212yes—
20#88864C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857C2212yes—
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3750C2212yes—
26C32×C4×Dic3C2211yescharacteristic
27C32×C4×Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29C32×C4×Dic3C2211yescharacteristic
30#88807C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33Dic3⋊(C3⋊Dic3)C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C2×C32×C42—31no—
38#64135—32no—
39#64133—31no—
40#64133—32no—
41#64133—34no—
42#64133—64no—
43#64133D311yescharacteristic
44#64079—32no—
45#64078—31no—
46#64078—31no—
47C4×#634—31no—
48C4×#634—31no—
49C4×#634—32no—
50C3×#924—34no—
51C2×C3×C4×Dic3D314yes—
52#63964—38no—
53#63955—38no—
54#63918—34no—
55#63867—38no—
56#63866—34no—
57#63844—34no—
58Dic3×Dic6—34no—
59#63837—38no—
60C23×C33C2311yescharacteristic
61C2×C33×C4—22no—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2312yes—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66#6308—22no—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×#634—22no—
70C2×C32×Dic3—22no—
71C2×C32×Dic3Q814yescomplemented
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2311yescharacteristic
76#6293—24no—
77#5622—91no—
78#5470—94no—
79C22×C32×C4—24no—
80C22×C32×C4—31no—
81C22×C32×C4—31no—
82C22×C32×C4C2×D314yes—
83C22×C32×C4C2×D311yescharacteristic
84C2×#634—31no—
85C2×#634—34no—
86C2×#634—32no—
87C2×#634—64no—
88C2×#634C2×D311yescharacteristic
89C22×C3×Dic3C2×D314yes—
90C22×C3×Dic3C2×D314yes—
91C2×C3×Dic6—34no—
92C2×#627—38no—
93C32×C42—31no—
94C32×C42—31no—
95C32×C42—31no—
96C32×C42—31no—
97#3757—32no—
98#3757—38no—
99#3757—68no—
100#3757C2×D312yes—
101C4⋊(C3⋊Dic3)—31no—
102C4⋊(C3⋊Dic3)—34no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—32no—
105C4⋊(C3⋊Dic3)—32no—
106C4⋊(C3⋊Dic3)—64no—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108#3750—32no—
109#3750—38no—
110#3750—32no—
111#3750—32no—
112#3750—68no—
113#3750C2×D312yes—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—32no—
116C3×#217—38no—
117C3×C4⋊Dic3—34no—
118C3×Dic3⋊C4—38no—
119C3×C4×Dic3C2×D314yes—
120C3×C4×Dic3C2×D314yes—
121C3×C4×Dic3C2×D314yes—
122C3×C4×Dic3C2×D314yes—
123#3724—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—38no—
128Dic32—38no—
129Dic32—38no—
130C22×C33—21no—
131C22×C33—21no—
132C22×C33C2×Q812yes—
133C22×C33C2411yescharacteristic
134C33×C4—21no—
135C33×C4#4112yes—
136C3⋊(C3⋊Dic3)—22no—
137C3⋊(C3⋊Dic3)—22no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—22no—
140C32×Dic3—21no—
141C32×Dic3C2×Q812yes—
142C32×Dic3C2×Q812yes—
143C32×Dic3#4112yes—
144C3×C22⋊Q8—91no—
145C2×C3×C42—34no—
146Q8×Dic3—91no—
147#944—91no—
148#926—98no—
149#924—34no—
150#924—98no—
151#924—94no—
152#924—91no—
153#924—184no—
154C2×C4×Dic3C3⋊D311yescharacteristic
155#889—92no—
156#888—91no—
157#887—92no—
158#887—91no—
159#878—92no—
160#877—92no—
161#870—98no—
162#869—94no—
163#869—94no—
164C4×Dic6—98no—
165C4×Dic6—94no—
166C4×Dic6—94no—
167C23×C32—24no—
168C23×C32C22×D314yes—
169C23×C32C22×D311yescharacteristic
170C2×C32×C4—22no—
171C2×C32×C4—24no—
172C2×C32×C4—24no—
173C2×C32×C4—28no—
174C2×C32×C4—34no—
175C2×C32×C4—31no—
176C2×C32×C4—31no—
177C2×C32×C4—31no—
178C2×C32×C4—31no—
179C2×C32×C4—48no—
180C2×C32×C4—62no—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—38no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D312yes—
193#634—22no—
194#634—64no—
195#634—62no—
196#634—68no—
197#634—128no—
198C2×C3×Dic3—28no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3Dic6116yescomplemented
202C2×C3×Dic3C22×D314yes—
203C2×C3×Dic3C22×D314yes—
204C2×C3×Dic3C22×D314yes—
205C2×C3×Dic3C22×D314yes—
206C3×Dic6—68no—
207#627—616no—
208#533—271noSylow
209C2×C33—22no—
210C2×C33C22×Q811yescharacteristic
211C2×C33C2×#4111yescharacteristic
212C2×C33C2×#4111yescharacteristic
213C2×C3×Q8—91no—
214C22×C3×C4—24no—
215C22×C3×C4—34no—
216C22×C3×C4—34no—
217C22×C3×C4D3214yes—
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C22×Dic3C2×C3⋊D311yescharacteristic
220C22×Dic3C2×C3⋊D311yescharacteristic
221C2×Dic6—34no—
222C2×Dic6—98no—
223C2×Dic6—94no—
224C2×Dic6—91no—
225C2×Dic6—92no—
226C2×Dic6—184no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—92no—
229C3×C22⋊C4—92no—
230C3×C42—34no—
231C3×C42—34no—
232C3×C42—34no—
233C3×C42—34no—
234#217—38no—
235#217—98no—
236#217—92no—
237#217—188no—
238C4⋊Dic3—34no—
239C4⋊Dic3—98no—
240C4⋊Dic3—98no—
241C4⋊Dic3—98no—
242C4⋊Dic3—94no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—184no—
246Dic3⋊C4—38no—
247Dic3⋊C4—98no—
248Dic3⋊C4—98no—
249Dic3⋊C4—98no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—188no—
256C4×Dic3—98no—
257C4×Dic3—98no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3C2×C3⋊D311yescharacteristic
261C4×Dic3C2×C3⋊D311yescharacteristic
262C4×Dic3C2×C3⋊D311yescharacteristic
263C4×Dic3C2×C3⋊D311yescharacteristic
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—24no—
267C22×C32—28no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—44no—
271C22×C32—44no—
272C22×C32C2×Dic618yes—
273C22×C32D3×Q812yes—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C32×C4—21no—
277C32×C4—28no—
278C32×C4—24no—
279C32×C4—32no—
280C32×C4—32no—
281C32×C4—32no—
282C32×C4—44no—
283C32×C4—61no—
284C32×C4#23518yes—
285C32×C4#23712yes—
286C3⋊Dic3—22no—
287C3⋊Dic3—22no—
288C3⋊Dic3—62no—
289C3⋊Dic3—62no—
290C3⋊Dic3—68no—
291C3⋊Dic3—68no—
292C3⋊Dic3—128no—
293C3⋊Dic3—128no—
294C3×Dic3—24no—
295C3×Dic3—68no—
296C3×Dic3—68no—
297C3×Dic3C2×Dic618yes—
298C3×Dic3C2×Dic618yes—
299C3×Dic3#23518yes—
300C4⋊Q8—271no—
301C4⋊Q8—271no—
302#125—272no—
303C22⋊Q8—91no—
304C22⋊Q8—272no—
305C22⋊Q8—271no—
306C4×Q8—272no—
307C4×Q8—271no—
308C4×Q8—271no—
309#117—272no—
310C2×C42—31no—
311C33#53311yescharacteristic, complemented, Sylow
312C23×C3—24no—
313C23×C3C2×D3214yes—
314C23×C3C22×C3⋊D311yescharacteristic
315C3×Q8—182no—
316C2×C3×C4—28no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—34no—
321C2×C3×C4—34no—
322C2×C3×C4—34no—
323C2×C3×C4—316no—
324C2×C3×C4—34no—
325C2×C3×C4—48no—
326C2×C3×C4—68no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4C2×D3214yes—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C22×C3⋊D311yescharacteristic
332C2×C3×C4C22×C3⋊D311yescharacteristic
333C2×Dic3—22no—
334C2×Dic3—38no—
335C2×Dic3—38no—
336C2×Dic3—98no—
337C2×Dic3—98no—
338C2×Dic3—92no—
339C2×Dic3—92no—
340C2×Dic3—188no—
341C2×Dic3—188no—
342C2×Dic3#63414yescomplemented
343C2×Dic3C22×C3⋊D311yescharacteristic
344C2×Dic3C22×C3⋊D311yescharacteristic
345C2×Dic3C22×C3⋊D311yescharacteristic
346C2×Dic3C22×C3⋊D311yescharacteristic
347Dic6—68no—
348Dic6—1816no—
349Dic6—188no—
350Dic6—182no—
351Dic6—184no—
352Dic6—368no—
353C2×C32—22no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—28no—
358C2×C32—48no—
359C2×C32C22×Dic614yes—
360C2×C32C2×#23514yes—
361C2×C32C2×#23514yes—
362C2×C32C2×#23711yescharacteristic
363C2×C32C2×D3×Q811yescharacteristic
364C2×C32D3×#4111yescharacteristic
365C2×Q8—91no—
366C2×Q8—272no—
367C2×Q8—271no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4D3×C3⋊D311yescharacteristic
371C4⋊C4—92no—
372C4⋊C4—91no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C22⋊C4—92no—
381C22⋊C4—272no—
382C42—31no—
383C42—31no—
384C42—31no—
385C42—31no—
386C42—272no—
387C42—272no—
388C22×C3—24no—
389C22×C3—24no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3D3×Dic618yes—
397C22×C3C2×#63412yes—
398C22×C3C22×D3214yes—
399C22×C3C23×C3⋊D311yescharacteristic
400C3×C4—24no—
401C3×C4—21no—
402C3×C4—28no—
403C3×C4—38no—
404C3×C4—38no—
405C3×C4—38no—
406C3×C4—44no—
407C3×C4—64no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4#380418yes—
411C3×C4#382812yes—
412Dic3—21no—
413Dic3—68no—
414Dic3—68no—
415Dic3—188no—
416Dic3—188no—
417Dic3—182no—
418Dic3—182no—
419Dic3—368no—
420Dic3—368no—
421Dic3C2×#63412yes—
422Dic3C2×#63412yes—
423Dic3#382812yes—
424C32—24no—
425C32#547014yescomplemented
426C32#562211yescharacteristic, complemented
427C23C2×D3×C3⋊D311yescharacteristic
428Q8—182no—
429Q8—544no—
430Q8—542no—
431C2×C4—22no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—34no—
436C2×C4—31no—
437C2×C4—62no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—272no—
441C2×C4—272no—
442C2×C4C2×D3×C3⋊D311yescharacteristic
443C2×C4C2×D3×C3⋊D311yescharacteristic
444C2×C3—22no—
445C2×C3—28no—
446C2×C3—24no—
447C2×C3—24no—
448C2×C3—24no—
449C2×C3—48no—
450C2×C3C2×D3×Dic614yes—
451C2×C3D3×#23514yes—
452C2×C3C2×#380414yes—
453C2×C3C22×#63411yescharacteristic
454C2×C3C2×#382811yescharacteristic
455C2×C3C2×#382811yescharacteristic
456C22—21no—
457C22—21no—
458C22D3×#63412yes—
459C22C22×D3×C3⋊D311yescharacteristic
460C4—21no—
461C4—32no—
462C4—32no—
463C4—32no—
464C4—61no—
465C4—182no—
466C4—182no—
467C4—542no—
468C4—542no—
469C4#8903112yes—
470C3—24no—
471C3#1059480814yescomplemented
472C3#1059594111yescharacteristic, complemented
473C2—22no—
474C2C2×D3×#63411yescharacteristic
475C2D3×#382811yescharacteristic
476C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11723534C2semidirect
7. #11723444C2semidirect
8. #11723443C2semidirect
9. #11723399C2semidirect
10. Dic3×#634C2semidirect
11. #11723389C2semidirect
71. C2×C32×Dic3Q8semidirect
201. C2×C3×Dic3Dic6semidirect
311. C33#533semidirect
342. C2×Dic3#634semidirect
425. C32#5470semidirect
426. C32#5622semidirect
471. C3#10594808semidirect
472. C3#10595941semidirect

Maximal subgroups

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

subgroupindex
#117239652
C3×#641332
C2×C32×C4×Dic32
#117236402
#117235942
#117235342
#117234442
#117234432
#117233992
Dic3×#6342
#117233892
#105959413
#105948083