Finite Group Databasebeta

← #13177281

Subgroups — #13177281

Order 1728.

1764 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C32×#924C211yesmaximal, characteristic
4#11723642C212yesmaximal
5#11723598C212yesmaximal
6#11723536C211yesmaximal, characteristic, complemented
7#11723451C211yesmaximal, characteristic, complemented
8#11723448C212yesmaximal, complemented
9#11723410C211yesmaximal, characteristic, complemented
10#11723397C211yesmaximal, characteristic, complemented
11#11723391C212yesmaximal, complemented
12#10596093—31nomaximal
13#10594810—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C2×#6293C2212yes—
20C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3757C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×#3750C2212yes—
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38Q8×C3⋊Dic3—31no—
39#64153—31no—
40#64133D311yescharacteristic
41C2×C4×C3⋊Dic3—31no—
42C2×C4×C3⋊Dic3—34no—
43C2×C4×C3⋊Dic3—64no—
44#64098—32no—
45#64097—31no—
46#64096—31no—
47#64096—32no—
48#64087—32no—
49#64086—32no—
50C3×#924—34no—
51C3×#924D314yes—
52#63982—38no—
53#63957—38no—
54#63919—34no—
55#63873—34no—
56#63870—38no—
57#63843—34no—
58#63843—34no—
59#63838—38no—
60C23×C33C2311yescharacteristic
61C2×C33×C4—22no—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)—22no—
65C2×C3⋊(C3⋊Dic3)Q814yescomplemented
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C3×#634—22no—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C32×Dic6—22no—
76#6293—24no—
77#5622—94no—
78#5622—91no—
79C2×C32×Q8—31no—
80C22×C32×C4—24no—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C2×D311yescharacteristic
83C22×C3⋊Dic3—31no—
84C22×C3⋊Dic3—34no—
85C22×C3⋊Dic3—31no—
86C22×C3⋊Dic3—34no—
87C22×C3⋊Dic3—64no—
88C22×C3⋊Dic3—64no—
89C2×#634—32no—
90C2×#634C2×D311yescharacteristic
91C2×C3×Dic6—34no—
92C2×C3×Dic6C2×D314yes—
93C2×#627—38no—
94C32×C4⋊C4—31no—
95C32×C4⋊C4—32no—
96C32×C22⋊C4—32no—
97#3757C2×D312yes—
98C4⋊(C3⋊Dic3)—32no—
99C4⋊(C3⋊Dic3)C2×D311yescharacteristic
100#3750—32no—
101#3750—32no—
102#3750—32no—
103#3750—32no—
104#3750C2×D312yes—
105C4×C3⋊Dic3—31no—
106C4×C3⋊Dic3—31no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—34no—
110C4×C3⋊Dic3—34no—
111C4×C3⋊Dic3—34no—
112C4×C3⋊Dic3—34no—
113C4×C3⋊Dic3—32no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—64no—
116C4×C3⋊Dic3—64no—
117C4×C3⋊Dic3—64no—
118C4×C3⋊Dic3—64no—
119C3×#217—38no—
120C3×#217C2×D318yes—
121C3×C4⋊Dic3—34no—
122C3×C4⋊Dic3C2×D314yes—
123C3×Dic3⋊C4—38no—
124C3×Dic3⋊C4C2×D318yes—
125#3724—38no—
126#3724—38no—
127#3724—38no—
128Dic3⋊Dic3—38no—
129Dic3⋊Dic3—38no—
130Dic32—38no—
131Dic32—38no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33C2×Q812yes—
135C22×C33C2411yescharacteristic
136C33×C4—21no—
137C33×C4#4112yes—
138C3⋊(C3⋊Dic3)—21no—
139C3⋊(C3⋊Dic3)C2×Q812yes—
140C3⋊(C3⋊Dic3)C2×Q812yes—
141C3⋊(C3⋊Dic3)#4112yes—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—22no—
144C32×Dic3—22no—
145C32×Dic3—22no—
146C3×C22⋊Q8—34no—
147C3×C22⋊Q8—91no—
148Q8×Dic3—94no—
149Q8×Dic3—91no—
150#944—94no—
151#944—91no—
152#924—34no—
153#924C3⋊D311yescharacteristic
154C2×C4×Dic3—94no—
155C2×C4×Dic3—91no—
156C2×C4×Dic3—184no—
157#889—98no—
158#889—92no—
159#888—94no—
160#888—91no—
161#887—98no—
162#887—94no—
163#887—91no—
164#887—92no—
165#878—98no—
166#878—92no—
167#877—98no—
168#877—92no—
169C23×C32—24no—
170C23×C32C22×D314yes—
171C23×C32C22×D311yescharacteristic
172C32×Q8—62no—
173C2×C32×C4—216no—
174C2×C32×C4—28no—
175C2×C32×C4—22no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—32no—
179C2×C32×C4—32no—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—34no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—316no—
190C2×C3⋊Dic3—34no—
191C2×C3⋊Dic3—34no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—34no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3—616no—
199C2×C3⋊Dic3—62no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3—616no—
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204#634—22no—
205#634—64no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3C22×D318yes—
209C2×C3×Dic3C22×D318yes—
210C3×Dic6—28no—
211C3×Dic6—68no—
212#627—616no—
213#533—271noSylow
214C2×C33—22no—
215C2×C33C22×Q811yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C2×C33C2×#4111yescharacteristic
218C2×C3×Q8—34no—
219C2×C3×Q8—91no—
220C22×C3×C4—24no—
221C22×C3×C4D3214yes—
222C22×C3×C4C2×C3⋊D311yescharacteristic
223C22×Dic3—94no—
224C22×Dic3—91no—
225C22×Dic3—94no—
226C22×Dic3—91no—
227C22×Dic3—184no—
228C22×Dic3—184no—
229C2×Dic6—34no—
230C2×Dic6—98no—
231C2×Dic6—92no—
232C2×Dic6C2×C3⋊D311yescharacteristic
233C3×C4⋊C4—34no—
234C3×C4⋊C4—38no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C22⋊C4—38no—
238C3×C22⋊C4—92no—
239#217—38no—
240#217C2×C3⋊D312yes—
241C4⋊Dic3—34no—
242C4⋊Dic3—98no—
243C4⋊Dic3—92no—
244C4⋊Dic3C2×C3⋊D311yescharacteristic
245Dic3⋊C4—38no—
246Dic3⋊C4—98no—
247Dic3⋊C4—98no—
248Dic3⋊C4—98no—
249Dic3⋊C4—98no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×C3⋊D312yes—
255C4×Dic3—98no—
256C4×Dic3—98no—
257C4×Dic3—94no—
258C4×Dic3—94no—
259C4×Dic3—94no—
260C4×Dic3—94no—
261C4×Dic3—91no—
262C4×Dic3—91no—
263C4×Dic3—91no—
264C4×Dic3—91no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3—184no—
268C4×Dic3—184no—
269C4×Dic3—184no—
270C4×Dic3—184no—
271C22×C32—28no—
272C22×C32—28no—
273C22×C32—24no—
274C22×C32—21no—
275C22×C32—24no—
276C22×C32—21no—
277C22×C32—24no—
278C22×C32—28no—
279C22×C32D3×Q818yes—
280C22×C32D3×Q812yes—
281C22×C32C23×D314yes—
282C22×C32C23×D311yescharacteristic
283C32×C4—28no—
284C32×C4—24no—
285C32×C4—21no—
286C32×C4—28no—
287C32×C4—62no—
288C32×C4—62no—
289C32×C4#23718yes—
290C32×C4#23712yes—
291C3⋊Dic3—22no—
292C3⋊Dic3—22no—
293C3⋊Dic3—32no—
294C3⋊Dic3—32no—
295C3⋊Dic3—32no—
296C3⋊Dic3—38no—
297C3⋊Dic3—38no—
298C3⋊Dic3—38no—
299C3⋊Dic3—68no—
300C3⋊Dic3—68no—
301C3⋊Dic3—68no—
302C3⋊Dic3—68no—
303C3⋊Dic3—64no—
304C3⋊Dic3—61no—
305C3×Dic3—28no—
306C3×Dic3—28no—
307C3×Dic3—68no—
308C3×Dic3—68no—
309C4⋊Q8—271no—
310C4⋊Q8—271no—
311#125—272no—
312C22⋊Q8—31no—
313C22⋊Q8—91no—
314C22⋊Q8—272no—
315C4×Q8—272no—
316C4×Q8—271no—
317C4×Q8—271no—
318#117—272no—
319C2×C42—271no—
320C33#53311yescharacteristic, complemented, Sylow
321C23×C3—24no—
322C23×C3C2×D3214yes—
323C23×C3C22×C3⋊D311yescharacteristic
324C3×Q8—68no—
325C3×Q8—182no—
326C2×C3×C4—216no—
327C2×C3×C4—28no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—38no—
332C2×C3×C4—38no—
333C2×C3×C4—92no—
334C2×C3×C4—92no—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C2×D3214yes—
337C2×C3×C4C22×C3⋊D311yescharacteristic
338C2×C3×C4C22×C3⋊D311yescharacteristic
339C2×Dic3—38no—
340C2×Dic3—38no—
341C2×Dic3—94no—
342C2×Dic3—94no—
343C2×Dic3—94no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—916no—
348C2×Dic3—94no—
349C2×Dic3—94no—
350C2×Dic3—91no—
351C2×Dic3—184no—
352C2×Dic3—184no—
353C2×Dic3—184no—
354C2×Dic3—184no—
355C2×Dic3—1816no—
356C2×Dic3—188no—
357C2×Dic3—182no—
358C2×Dic3—1816no—
359C2×Dic3C22×C3⋊D312yes—
360C2×Dic3C22×C3⋊D312yes—
361Dic6—22no—
362Dic6—68no—
363Dic6—184no—
364Dic6—1816no—
365C2×C32—216no—
366C2×C32—28no—
367C2×C32—22no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32C2×#23714yes—
372C2×C32C2×#23711yescharacteristic
373C2×C32C2×D3×Q814yes—
374C2×C32C2×D3×Q811yescharacteristic
375C2×C32D3×#4114yes—
376C2×C32D3×#4111yescharacteristic
377C2×Q8—31no—
378C2×Q8—91no—
379C2×Q8—272no—
380C22×C4—271no—
381C22×C4—271no—
382C22×C4D3×C3⋊D311yescharacteristic
383C4⋊C4—32no—
384C4⋊C4—31no—
385C4⋊C4—92no—
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—32no—
393C22⋊C4—92no—
394C42—272no—
395C42—272no—
396C42—271no—
397C42—271no—
398C42—271no—
399C42—271no—
400C22×C3—28no—
401C22×C3—28no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—24no—
407C22×C3—28no—
408C22×C3#379718yes—
409C22×C3Q8×C3⋊D312yes—
410C22×C3C22×D3214yes—
411C22×C3C23×C3⋊D311yescharacteristic
412C3×C4—28no—
413C3×C4—24no—
414C3×C4—21no—
415C3×C4—28no—
416C3×C4—68no—
417C3×C4—68no—
418C3×C4—182no—
419C3×C4—182no—
420C3×C4#380518yes—
421C3×C4#383012yes—
422Dic3—22no—
423Dic3—22no—
424Dic3—68no—
425Dic3—68no—
426Dic3—98no—
427Dic3—98no—
428Dic3—98no—
429Dic3—92no—
430Dic3—92no—
431Dic3—92no—
432Dic3—188no—
433Dic3—188no—
434Dic3—188no—
435Dic3—188no—
436Dic3—184no—
437Dic3—181no—
438C32—24no—
439C32#562214yescomplemented
440C32#562211yescharacteristic, complemented
441C23C2×D3×C3⋊D311yescharacteristic
442Q8—62no—
443Q8—182no—
444Q8—544no—
445C2×C4—22no—
446C2×C4—32no—
447C2×C4—32no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—271no—
451C2×C4—271no—
452C2×C4—271no—
453C2×C4—274no—
454C2×C4—271no—
455C2×C4—542no—
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C4C2×D3×C3⋊D311yescharacteristic
458C2×C3—216no—
459C2×C3—28no—
460C2×C3—22no—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3C2×#379714yes—
465C2×C3#6430614yes—
466C2×C3C2×#380514yes—
467C2×C3C2×#383011yescharacteristic
468C2×C3C2×Q8×C3⋊D311yescharacteristic
469C2×C3#41×C3⋊D311yescharacteristic
470C22—21no—
471C22—21no—
472C22#8901912yes—
473C22C22×D3×C3⋊D311yescharacteristic
474C4—21no—
475C4—62no—
476C4—62no—
477C4—182no—
478C4—182no—
479C4—272no—
480C4—272no—
481C4—272no—
482C4—541no—
483C4#8902912yes—
484C3—24no—
485C3#1059481014yescomplemented
486C3#1059609311yescharacteristic, complemented
487C2—22no—
488C2C2×#8901911yescharacteristic
489C2#1172457811yescharacteristic
490C2C2×#8902911yescharacteristic
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. #11723536C2semidirect
7. #11723451C2semidirect
8. #11723448C2semidirect
9. #11723410C2semidirect
10. #11723397C2semidirect
11. #11723391C2semidirect
65. C2×C3⋊(C3⋊Dic3)Q8semidirect
320. C33#533semidirect
439. C32#5622semidirect
440. C32#5622semidirect
485. C3#10594810semidirect
486. C3#10596093semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊(C3⋊Dic3)2
C3×#641332
C32×#9242
#117236422
#117235982
#117235362
#117234512
#117234482
#117234102
#117233972
#117233912
#105960933
#105948103