Finite Group Databasebeta

← #13177480

Subgroups — #13177480

Order 1728.

2008 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#11723978C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×#935C211yesmaximal, characteristic, complemented
4#11723628C212yesmaximal, complemented
5#11723627C212yesmaximal, complemented
6#11723618C212yesmaximal, complemented
7#11723597C212yesmaximal, complemented
8#11723489C212yesmaximal, complemented
9#11723469C212yesmaximal, complemented
10#10596130—31nomaximal
11#10595208—34nomaximal
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2212yes—
15C2×C3×#634C2211yescharacteristic, complemented
16C22×C32×Dic3C2211yescharacteristic, complemented
17C2×#6289C2212yes—
18C2×#6289C2212yes—
19#88864C2211yescharacteristic
20#88859C2212yes—
21C3×#3757C2212yes—
22C3×#3757C2212yescomplemented
23C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
24C32×#217C2212yes—
25C32×#217C2211yescharacteristic, complemented
26C32×Dic3⋊C4C2212yescomplemented
27#88807C2212yescomplemented
28#88806C2212yescomplemented
29Dic3⋊(C3⋊Dic3)C2212yescomplemented
30#88804C2212yes—
31#88804C2212yes—
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×#123—31no—
34#64146—31no—
35#64146—34no—
36#64146—64no—
37#64145D311yescharacteristic, complemented
38C4×#638—32no—
39#64105—32no—
40#64101—32no—
41#64092—32no—
42C22⋊C4×C3⋊D3—32no—
43#64088—32no—
44C3×#936—34no—
45C3×#935D314yescomplemented
46#63975—38no—
47#63974—38no—
48#63956—38no—
49#63952—38no—
50#63888—38no—
51#63836—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3⋊(C3⋊D3)—24no—
55C22×C3⋊(C3⋊D3)—41no—
56C22×C3⋊(C3⋊D3)C2312yes—
57C33×D4—41no—
58C2×C33×C4C2311yescharacteristic
59#6312—24no—
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C3×#634—41no—
64C2×C32×Dic3—41no—
65C2×C32×Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67#6289—24no—
68#6289—42no—
69#5679—91no—
70#5659—94no—
71C23×C3⋊D3—31no—
72C23×C3⋊D3—34no—
73C23×C3⋊D3—64no—
74C2×C32×D4—24no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C2×D311yescharacteristic
77C22×C32×C4—31no—
78C2×#638—32no—
79C2×#638—38no—
80C2×#638—68no—
81C2×C4×C3⋊D3—32no—
82C2×C4×C3⋊D3—32no—
83C2×#634C2×D311yescharacteristic, complemented
84C22×C3×Dic3C2×D314yescomplemented
85C2×C3×Dic6—34no—
86C2×#624—38no—
87C2×#624—38no—
88C32×C4⋊C4—32no—
89C32×C22⋊C4—32no—
90C32×C22⋊C4—31no—
91#3757—31no—
92#3757—34no—
93#3757—64no—
94#3757C2×D312yes—
95#3757C2×D312yescomplemented
96#3752—32no—
97#3752—38no—
98#3752—32no—
99#3752—32no—
100#3752—68no—
101C4⋊(C3⋊Dic3)—32no—
102#3750—32no—
103#3750—32no—
104C4×C3⋊Dic3—32no—
105C4×C3⋊Dic3C2×D311yescharacteristic, complemented
106C3×#217—38no—
107C3×#217—38no—
108C3×#217C2×D318yes—
109C3×#217C2×D314yescomplemented
110C3×Dic3⋊C4C2×D318yescomplemented
111C3×C4×Dic3—34no—
112#3724—38no—
113Dic3⋊Dic3—38no—
114Dic3⋊Dic3—38no—
115#3722—38no—
116#3722—38no—
117Dic32—38no—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—21no—
121C22×C33—21no—
122C22×C33C2411yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)—44no—
126C2×C3⋊(C3⋊D3)#4114yes—
127C33×C4—21no—
128C3⋊(C3⋊Dic3)—22no—
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3—22no—
131C32×Dic3—22no—
132C32×Dic3—22no—
133C3×#124—91no—
134C3×#123—34no—
135#946—92no—
136#937—94no—
137#937—91no—
138#937—184no—
139#936—34no—
140#935C3⋊D311yescharacteristic, complemented
141C4×#67—98no—
142#896—98no—
143#892—98no—
144#883—98no—
145#883—92no—
146#67⋊C4—92no—
147D3×C22⋊C4—98no—
148D3×C22⋊C4—92no—
149#879—98no—
150#879—92no—
151#872—92no—
152C23×C32—24no—
153C23×C32—24no—
154C23×C32C22×D314yes—
155C23×C32C22×D314yes—
156C23×C32C22×D311yescharacteristic
157C23×C32C22×D311yescharacteristic
158C22×C3⋊D3—32no—
159C22×C3⋊D3—38no—
160C22×C3⋊D3—68no—
161C22×C3⋊D3—64no—
162C22×C3⋊D3—616no—
163C22×C3⋊D3—1216no—
164C22×C3⋊D3—121no—
165C22×C3⋊D3—124no—
166C22×C3⋊D3—128no—
167C32×D4—41no—
168C32×D4—44no—
169C32×D4—48no—
170C2×C32×C4—24no—
171C2×C32×C4—32no—
172C2×C32×C4—32no—
173C2×C32×C4—121no—
174C2×C32×C4C22×D314yes—
175C2×C32×C4C22×D311yescharacteristic
176#638—64no—
177#638—616no—
178#638—1216no—
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—38no—
181C2×C3⋊Dic3—68no—
182C2×C3⋊Dic3C22×D312yes—
183C2×C3⋊Dic3C22×D312yes—
184C4×C3⋊D3—64no—
185C4×C3⋊D3—122no—
186#634—41no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—38no—
189C2×C3×Dic3—44no—
190C2×C3×Dic3C22×D318yes—
191C2×C3×Dic3C22×D318yes—
192C3×Dic6—124no—
193#624—616no—
194#624—128no—
195#551—271noSylow
196C2×C33—41no—
197C2×C33—41no—
198C2×C33C2×#4112yes—
199C2×C33H(5,𝔽2)11yescharacteristic
200C3⋊(C3⋊D3)—24no—
201C23×D3—94no—
202C23×D3—91no—
203C23×D3—184no—
204C2×C3×Q8—91no—
205C2×C3×D4—24no—
206C2×C3×D4D3214yes—
207C2×C3×D4C2×C3⋊D311yescharacteristic
208C22×C3×C4—34no—
209C2×#67—98no—
210C2×#67—92no—
211C2×#67—188no—
212C22×Dic3C2×C3⋊D311yescharacteristic, complemented
213C2×C4×D3—98no—
214C2×C4×D3—92no—
215C2×C4×D3—98no—
216C2×C4×D3—92no—
217C2×Dic6—34no—
218C3×C4⋊C4—38no—
219C3×C22⋊C4—38no—
220C3×C22⋊C4—34no—
221C3×C22⋊C4—92no—
222C3×C22⋊C4—92no—
223C3×C42—91no—
224#217—38no—
225#217—38no—
226#217—94no—
227#217—91no—
228#217—184no—
229#217C2×C3⋊D312yes—
230#217C2×C3⋊D311yescharacteristic, complemented
231(C2×D3)⋊C4—98no—
232(C2×D3)⋊C4—98no—
233(C2×D3)⋊C4—98no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—188no—
238C4⋊Dic3—98no—
239C4⋊Dic3—92no—
240Dic3⋊C4—98no—
241Dic3⋊C4—98no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4C2×C3⋊D312yescomplemented
245C4×Dic3—34no—
246C4×Dic3—98no—
247C4×Dic3—92no—
248C22×C32—28no—
249C22×C32—21no—
250C22×C32—24no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—24no—
255C22×C32—24no—
256C22×C32—28no—
257C22×C32—28no—
258C22×C32—48no—
259C22×C32—48no—
260C22×C32—44no—
261C22×C32C23×D314yes—
262C22×C32C23×D311yescharacteristic
263C2×C3⋊D3—34no—
264C2×C3⋊D3—316no—
265C2×C3⋊D3—616no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3⋊D3—68no—
269C2×C3⋊D3—68no—
270C2×C3⋊D3—128no—
271C2×C3⋊D3—128no—
272C2×C3⋊D3—124no—
273C2×C3⋊D3—1216no—
274C2×C3⋊D3—1232no—
275C32×C4—21no—
276C32×C4—24no—
277C32×C4—44no—
278C32×C4—62no—
279C32×C4—62no—
280C3⋊Dic3—22no—
281C3⋊Dic3—22no—
282C3⋊Dic3—62no—
283C3⋊Dic3—68no—
284C3⋊Dic3—128no—
285C3×Dic3—28no—
286C3×Dic3—28no—
287C3×Dic3—68no—
288C3×Dic3—68no—
289#126—272no—
290#124—91no—
291#123—31no—
292#123—272no—
293C22⋊Q8—272no—
294#120—271no—
295C4×D4—272no—
296#117—272no—
297C2×C22⋊C4—272no—
298C33#55111yescharacteristic, complemented, Sylow
299C23×C3—24no—
300C23×C3—24no—
301C23×C3C2×D3214yes—
302C23×C3C2×D3214yes—
303C23×C3C22×C3⋊D311yescharacteristic
304C23×C3C22×C3⋊D311yescharacteristic
305C22×D3—98no—
306C22×D3—92no—
307C22×D3—188no—
308C22×D3—184no—
309C22×D3—1816no—
310C22×D3—3616no—
311C22×D3—364no—
312C22×D3—361no—
313C22×D3—368no—
314C3×Q8—361no—
315C3×D4—44no—
316C3×D4—41no—
317C3×D4—48no—
318C2×C3×C4—24no—
319C2×C3×C4—38no—
320C2×C3×C4—38no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4—124no—
324C2×C3×C4C2×D3214yes—
325C2×C3×C4C22×C3⋊D311yescharacteristic
326#67—184no—
327#67—1816no—
328#67—3616no—
329C2×Dic3—38no—
330C2×Dic3—38no—
331C2×Dic3—41no—
332C2×Dic3—98no—
333C2×Dic3—92no—
334C2×Dic3—188no—
335C2×Dic3C22×C3⋊D312yes—
336C2×Dic3C22×C3⋊D312yes—
337C4×D3—184no—
338C4×D3—1816no—
339C4×D3—362no—
340C4×D3—368no—
341Dic6—124no—
342C2×C32—28no—
343C2×C32—24no—
344C2×C32—48no—
345C2×C32—44no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32—44no—
349C2×C32—84no—
350C2×C32#100414yes—
351C2×C32#100411yescharacteristic
352C2×C32D3×#4118yes—
353C2×C32D3×#4112yes—
354C3⋊D3—64no—
355C3⋊D3—616no—
356C3⋊D3—1216no—
357C24—271no—
358C2×Q8—91no—
359C2×D4—272no—
360C2×D4D3×C3⋊D311yescharacteristic
361C22×C4—31no—
362C22×C4—272no—
363C22×C4—272no—
364C4⋊C4—32no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4—272no—
368C22⋊C4—32no—
369C22⋊C4—31no—
370C22⋊C4—92no—
371C22⋊C4—92no—
372C22⋊C4—272no—
373C22⋊C4—272no—
374C22⋊C4—271no—
375C22⋊C4—272no—
376C42—91no—
377C42—272no—
378C22×C3—28no—
379C22×C3—28no—
380C22×C3—28no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—48no—
389C22×C3—48no—
390C22×C3—44no—
391C22×C3C22×D3214yes—
392C22×C3C23×C3⋊D311yescharacteristic
393C2×D3—916no—
394C2×D3—94no—
395C2×D3—1816no—
396C2×D3—188no—
397C2×D3—188no—
398C2×D3—182no—
399C2×D3—182no—
400C2×D3—368no—
401C2×D3—368no—
402C2×D3—364no—
403C2×D3—3616no—
404C2×D3—3632no—
405C3×C4—24no—
406C3×C4—21no—
407C3×C4—44no—
408C3×C4—68no—
409C3×C4—68no—
410C3×C4—182no—
411C3×C4—182no—
412Dic3—22no—
413Dic3—22no—
414Dic3—68no—
415Dic3—68no—
416Dic3—188no—
417Dic3—182no—
418Dic3—368no—
419C32—24no—
420C32#565914yescomplemented
421C32#567911yescharacteristic, complemented
422C23—272no—
423C23—544no—
424C23—1081no—
425C23C2×D3×C3⋊D311yescharacteristic
426C23C2×D3×C3⋊D311yescharacteristic
427Q8—361no—
428D4—41no—
429D4—544no—
430C2×C4—32no—
431C2×C4—32no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—121no—
435C2×C4—272no—
436C2×C4—544no—
437C2×C4—1082no—
438C2×C4C2×D3×C3⋊D311yescharacteristic
439C2×C3—28no—
440C2×C3—24no—
441C2×C3—48no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—44no—
445C2×C3—44no—
446C2×C3—84no—
447C2×C3#6431218yes—
448C2×C3#6433014yes—
449C2×C3#6436111yescharacteristic
450C2×C3#41×C3⋊D312yes—
451D3—184no—
452D3—1816no—
453D3—3616no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—274no—
459C22—542no—
460C22—542no—
461C22—1084no—
462C22C22×D3×C3⋊D311yescharacteristic
463C4—21no—
464C4—62no—
465C4—62no—
466C4—182no—
467C4—182no—
468C4—542no—
469C3—24no—
470C3#1059520814yescomplemented
471C3#1059613011yescharacteristic, complemented
472C2—41no—
473C2—41no—
474C2—544no—
475C2#1172458612yes—
476C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radical#11723978C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#551
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. C32×#935C2semidirect
4. #11723628C2semidirect
5. #11723627C2semidirect
6. #11723618C2semidirect
7. #11723597C2semidirect
8. #11723489C2semidirect
9. #11723469C2semidirect
15. C2×C3×#634C22semidirect
16. C22×C32×Dic3C22semidirect
22. C3×#3757C22semidirect
23. C3×C4×C3⋊Dic3C22semidirect
25. C32×#217C22semidirect
26. C32×Dic3⋊C4C22semidirect
27. #88807C22semidirect
28. #88806C22semidirect
29. Dic3⋊(C3⋊Dic3)C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
37. #64145D3semidirect
45. C3×#935D3semidirect
83. C2×#634C2×D3semidirect
84. C22×C3×Dic3C2×D3semidirect
95. #3757C2×D3semidirect
105. C4×C3⋊Dic3C2×D3semidirect
109. C3×#217C2×D3semidirect
110. C3×Dic3⋊C4C2×D3semidirect
140. #935C3⋊D3semidirect
212. C22×Dic3C2×C3⋊D3semidirect
230. #217C2×C3⋊D3semidirect
244. Dic3⋊C4C2×C3⋊D3semidirect
298. C33#551semidirect
420. C32#5659semidirect
421. C32#5679semidirect
470. C3#10595208semidirect
471. C3#10596130semidirect

Maximal subgroups

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

subgroupindex
#117239782
C3×#641452
C32×#9352
#117236282
#117236272
#117236182
#117235972
#117234892
#117234692
#105961303
#105952083