Finite Group Databasebeta

← #13177433

Subgroups — #13177433

Order 1728.

2516 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
1C2×#89078C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic, complemented
3C3×#64145C211yesmaximal, characteristic, complemented
4C32×#940C211yesmaximal, characteristic, complemented
5#67⋊(C3⋊Dic3)C214yesmaximal, complemented
6#11723599C214yesmaximal, complemented
7C4×#6289C211yesmaximal, characteristic, complemented
8#11723443C211yesmaximal, characteristic, complemented
9#11723429C211yesmaximal, characteristic, complemented
10#10596187—31nomaximal
11#10595167—34nomaximal
12C2×C33×D4C2211yescharacteristic
13C2×#6312C2212yes—
14C22×C3⋊(C3⋊Dic3)C2212yes—
15#89078—24no—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×#6308C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C2×C32×#67C2214yes—
20C2×C32×D12C2211yescharacteristic
21C2×#6291C2212yes—
22C2×#6289C2212yes—
23C2×D3×C3⋊Dic3C2212yes—
24#89015—24no—
25C3×#3757C2214yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×C4×Dic3C2211yescharacteristic
28Dic3⋊(C3⋊Dic3)C2214yes—
29(C2×D3)⋊(C3⋊Dic3)C2214yes—
30Dic3×C3⋊Dic3C2212yes—
31C2×#3830—31no—
32C2×#3830—31no—
33C2×#3830—34no—
34C2×#3830—64no—
35C2×#3796—34no—
36C32×#127—31no—
37#64147—34no—
38#64145—31no—
39#64145D311yescharacteristic, complemented
40D4×C3⋊Dic3—34no—
41C42×C3⋊D3—31no—
42#64078—31no—
43C3×#940D314yescomplemented
44C3×#936—34no—
45#67⋊Dic3—316no—
46#63958—316no—
47C4×#624—34no—
48#63866—34no—
49#63860—34no—
50C23×C33C2312yes—
51C22×C3⋊(C3⋊D3)C2311yescharacteristic
52C22×C32×D3C2312yes—
53C33×D4—22no—
54C2×C33×C4C2311yescharacteristic
55#6312—24no—
56C2×C3⋊(C3⋊Dic3)—24no—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
59C4×C3⋊(C3⋊D3)D414yescomplemented
60#6308—22no—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C3×#634—22no—
64C32×#67—28no—
65C2×C32×Dic3C2312yes—
66C32×D12—22no—
67#6291—24no—
68#6289—24no—
69D3×C3⋊Dic3—24no—
70#5716—94no—
71#5668—91no—
72C2×C32×D4—24no—
73C2×C32×D4—34no—
74C2×C32×D4—31no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C2×D311yescharacteristic
77C2×#638—32no—
78C2×#638—38no—
79C2×#638—32no—
80C2×#638—68no—
81C22×C3⋊Dic3—32no—
82C22×C3⋊Dic3—38no—
83C22×C3⋊Dic3—32no—
84C22×C3⋊Dic3—68no—
85#3830—64no—
86#3830—64no—
87#3830—616no—
88#3830—632no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—34no—
91C2×C4×C3⋊D3—32no—
92C2×C4×C3⋊D3—64no—
93C2×#634—31no—
94C2×#634—34no—
95C2×#634—64no—
96C2×#634C2×D311yescharacteristic
97C2×C3×#67C2×D3116yes—
98C2×C3×D12C2×D314yes—
99C2×C3×Dic6—34no—
100C2×#626—38no—
101C2×#624—38no—
102C2×D3×Dic3—38no—
103#3796—616no—
104C32×C42—31no—
105#3757—34no—
106#3757C2×D314yes—
107C4⋊(C3⋊Dic3)—34no—
108C4×C3⋊Dic3—32no—
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×#217—316no—
111C3×C4×Dic3—34no—
112C3×C4×Dic3C2×D314yes—
113Dic3⋊Dic3—316no—
114(C2×D3)⋊Dic3—316no—
115Dic32—38no—
116C22×C33—22no—
117C22×C33—24no—
118C22×C33C2411yescharacteristic
119C2×C3⋊(C3⋊D3)—22no—
120C2×C3⋊(C3⋊D3)C2×D412yes—
121C2×C32×D3—22no—
122C2×C32×D3—24no—
123C33×C4C2×D412yes—
124C3⋊(C3⋊Dic3)—22no—
125C3⋊(C3⋊Dic3)C2×D412yes—
126C3×C3⋊Dic3—22no—
127C3×C3⋊Dic3—22no—
128C32×Dic3#4114yescomplemented
129C2×#239—91no—
130C2×#237—94no—
131C2×#237—94no—
132C2×#237—91no—
133C2×#237—184no—
134C3×#127—34no—
135C3×#124—91no—
136#944—91no—
137#940C3⋊D311yescharacteristic, complemented
138#938—916no—
139#936—34no—
140#936—94no—
141D4×Dic3—916no—
142#884—94no—
143#67⋊C4—94no—
144#875—91no—
145C42×D3—94no—
146C42×D3—91no—
147#869—94no—
148C23×C32—28no—
149C23×C32—32no—
150C23×C32C22×D318yes—
151C23×C32C22×D312yes—
152C22×C3⋊D3—31no—
153C22×C3⋊D3—34no—
154C22×C3⋊D3—64no—
155C22×C3×D3C22×D318yes—
156C32×D4—216no—
157C32×D4—28no—
158C32×D4—22no—
159C32×D4—68no—
160C32×D4—62no—
161C2×C32×C4—24no—
162C2×C32×C4—32no—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D311yescharacteristic
165#638—64no—
166#638—64no—
167#638—616no—
168#638—632no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—34no—
172C2×C3⋊Dic3—38no—
173C2×C3⋊Dic3—68no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—64no—
177C2×C3⋊Dic3—616no—
178C2×C3⋊Dic3—632no—
179C2×C3⋊Dic3C22×D312yes—
180C2×C3⋊Dic3C22×D312yes—
181C4×C3⋊D3—34no—
182C4×C3⋊D3—316no—
183C4×C3⋊D3—616no—
184C4×C3⋊D3—64no—
185#634—22no—
186#634—616no—
187#634—68no—
188#634—62no—
189C3×#67—232no—
190C2×C3×Dic3—38no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3C22×D318yes—
193C3×D12—28no—
194C3×Dic6—68no—
195#626—616no—
196#624—616no—
197D3×Dic3—616no—
198#532—271noSylow
199C2×C33—42no—
200C2×C33C22×D411yescharacteristic
201C2×C33C2×#4112yes—
202C3⋊(C3⋊D3)—22no—
203C32×D3—42no—
204C2×C3×Q8—91no—
205C2×C3×D4—24no—
206C2×C3×D4—316no—
207C2×C3×D4—34no—
208C2×C3×D4D3214yes—
209C2×C3×D4C2×C3⋊D311yescharacteristic
210C2×#67—98no—
211C2×#67—98no—
212C2×#67—92no—
213C2×#67—188no—
214C2×#67C2×C3⋊D314yes—
215C22×Dic3—98no—
216C22×Dic3—98no—
217C22×Dic3—92no—
218C22×Dic3—188no—
219#239—184no—
220#237—1816no—
221#237—1816no—
222#237—184no—
223#237—1832no—
224C2×D12—92no—
225C2×D12C2×C3⋊D311yescharacteristic
226C2×C4×D3—94no—
227C2×C4×D3—91no—
228C2×C4×D3—98no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—184no—
232C2×Dic6—34no—
233C2×Dic6—94no—
234C2×Dic6—91no—
235C2×Dic6—184no—
236C3×C22⋊C4—94no—
237C3×C42—34no—
238C3×C42—91no—
239#217—316no—
240#217—916no—
241(C2×D3)⋊C4—94no—
242C4⋊Dic3—916no—
243Dic3⋊C4—94no—
244C4×Dic3—34no—
245C4×Dic3—98no—
246C4×Dic3—92no—
247C4×Dic3C2×C3⋊D311yescharacteristic
248C22×C32—216no—
249C22×C32—28no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—24no—
253C22×C32—216no—
254C22×C32—416no—
255C22×C32—62no—
256C22×C32—64no—
257C22×C32C23×D314yes—
258C22×C32C23×D311yescharacteristic
259C2×C3⋊D3—32no—
260C2×C3⋊D3—38no—
261C2×C3⋊D3—68no—
262C2×C3⋊D3—62no—
263C2×C3⋊D3—68no—
264C2×C3⋊D3—128no—
265C2×C3×D3—28no—
266C2×C3×D3—216no—
267C32×C4—28no—
268C32×C4—34no—
269C32×C4D3×D418yes—
270C32×C4D3×D412yes—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3⋊Dic3—32no—
274C3⋊Dic3—38no—
275C3⋊Dic3—68no—
276C3⋊Dic3—616no—
277C3⋊Dic3—68no—
278C3⋊Dic3—62no—
279C3×Dic3—68no—
280C3×Dic3—68no—
281C3×Dic3#237116yescomplemented
282C2×#41—271no—
283C2×#41—271no—
284C4⋊Q8—271no—
285#127—31no—
286#124—91no—
287#124—271no—
288#121—274no—
289C4×D4—274no—
290C2×C42—271no—
291C33#53211yescharacteristic, complemented, Sylow
292C23×C3—28no—
293C23×C3—38no—
294C23×C3C2×D3218yes—
295C23×C3C22×C3⋊D312yes—
296C22×D3—94no—
297C22×D3—91no—
298C22×D3—184no—
299C22×D3C22×C3⋊D312yes—
300C3×Q8—182no—
301C3×D4—216no—
302C3×D4—28no—
303C3×D4—22no—
304C3×D4—632no—
305C3×D4—68no—
306C2×C3×C4—24no—
307C2×C3×C4—38no—
308C2×C3×C4—92no—
309C2×C3×C4—92no—
310C2×C3×C4C2×D3214yes—
311C2×C3×C4C22×C3⋊D311yescharacteristic
312#67—28no—
313#67—1816no—
314#67—1816no—
315#67—184no—
316#67—1832no—
317C2×Dic3—38no—
318C2×Dic3—38no—
319C2×Dic3—94no—
320C2×Dic3—91no—
321C2×Dic3—98no—
322C2×Dic3—92no—
323C2×Dic3—188no—
324C2×Dic3—184no—
325C2×Dic3—1816no—
326C2×Dic3—1816no—
327C2×Dic3—184no—
328C2×Dic3—1832no—
329C2×Dic3C22×C3⋊D312yes—
330D12—22no—
331D12—184no—
332C4×D3—916no—
333C4×D3—94no—
334C4×D3—1816no—
335C4×D3—1816no—
336C4×D3—184no—
337C4×D3—184no—
338Dic6—68no—
339Dic6—1816no—
340Dic6—188no—
341Dic6—182no—
342C2×C32—28no—
343C2×C32—24no—
344C2×C32—416no—
345C2×C32—42no—
346C2×C32—48no—
347C2×C32—122no—
348C2×C32C2×D3×D414yes—
349C2×C32C2×D3×D411yescharacteristic
350C2×C32C2×#23718yes—
351C2×C32D3×#4112yes—
352C3⋊D3—62no—
353C3⋊D3—68no—
354C3⋊D3—128no—
355C3×D3—48no—
356#41—544no—
357#41—544no—
358C2×Q8—91no—
359C2×Q8—271no—
360C2×D4—34no—
361C2×D4—31no—
362C2×D4—272no—
363C2×D4—272no—
364C2×D4D3×C3⋊D311yescharacteristic
365C22×C4—272no—
366C22×C4—272no—
367C22×C4—271no—
368C22×C4—272no—
369C4⋊C4—274no—
370C22⋊C4—94no—
371C22⋊C4—274no—
372C42—31no—
373C42—91no—
374C42—272no—
375C22×C3—216no—
376C22×C3—216no—
377C22×C3—28no—
378C22×C3—22no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—416no—
382C22×C3—616no—
383C22×C3—68no—
384C22×C3C22×D3214yes—
385C22×C3C23×C3⋊D311yescharacteristic
386C2×D3—22no—
387C2×D3—24no—
388C2×D3—98no—
389C2×D3—92no—
390C2×D3—188no—
391C2×D3—188no—
392C2×D3—182no—
393C2×D3—368no—
394C3×C4—28no—
395C3×C4—316no—
396C3×C4—182no—
397C3×C4—182no—
398C3×C4#381118yes—
399C3×C4D4×C3⋊D312yes—
400Dic3—68no—
401Dic3—68no—
402Dic3—98no—
403Dic3—92no—
404Dic3—188no—
405Dic3—1816no—
406Dic3—188no—
407Dic3—182no—
408Dic3#383014yescomplemented
409C32—24no—
410C32#566811yescharacteristic, complemented
411C32#571614yescomplemented
412C23—32no—
413C23—271no—
414C23C2×D3×C3⋊D312yes—
415Q8—182no—
416Q8—542no—
417D4—22no—
418D4—68no—
419D4—62no—
420D4—544no—
421D4—544no—
422C2×C4—32no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—271no—
426C2×C4—274no—
427C2×C4—272no—
428C2×C4—544no—
429C2×C4—544no—
430C2×C4—544no—
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C2×C3—28no—
433C2×C3—24no—
434C2×C3—416no—
435C2×C3—42no—
436C2×C3—48no—
437C2×C3—128no—
438C2×C3D3×#23718yes—
439C2×C3C2×#381114yes—
440C2×C3C2×D4×C3⋊D311yescharacteristic
441C2×C3C2×#383012yes—
442D3—42no—
443D3—182no—
444D3—188no—
445D3—368no—
446C22—24no—
447C22—22no—
448C22—64no—
449C22—62no—
450C22—272no—
451C22—542no—
452C22C22×D3×C3⋊D311yescharacteristic
453C4—34no—
454C4—182no—
455C4—182no—
456C4—272no—
457C4—542no—
458C4#8903912yes—
459C3—24no—
460C3#1059516714yescomplemented
461C3#1059618711yescharacteristic, complemented
462C2—42no—
463C2—122no—
464C2—542no—
465C2D3×#383012yes—
466C2C2×#8903911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C2×#89015C2semidirect
3. C3×#64145C2semidirect
4. C32×#940C2semidirect
5. #67⋊(C3⋊Dic3)C2semidirect
6. #11723599C2semidirect
7. C4×#6289C2semidirect
8. #11723443C2semidirect
9. #11723429C2semidirect
39. #64145D3semidirect
43. C3×#940D3semidirect
59. C4×C3⋊(C3⋊D3)D4semidirect
128. C32×Dic3#41semidirect
137. #940C3⋊D3semidirect
281. C3×Dic3#237semidirect
291. C33#532semidirect
408. Dic3#3830semidirect
410. C32#5668semidirect
411. C32#5716semidirect
460. C3#10595167semidirect
461. C3#10596187semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×#890152
C3×#641452
C32×#9402
#67⋊(C3⋊Dic3)2
#117235992
C4×#62892
#117234432
#117234292
#105961873
#105951673