Finite Group Databasebeta

← #13177547

Subgroups — #13177547

Order 1728.

2292 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723981C211yesmaximal, characteristic, complemented
4C32×#936C211yesmaximal, characteristic, complemented
5#11723624C214yesmaximal, complemented
6#11723622C214yesmaximal, complemented
7C4×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
8#11723451C211yesmaximal, characteristic, complemented
9#11723441C211yesmaximal, characteristic, complemented
10#10596139—31nomaximal
11#10595153—34nomaximal
12C2×C33×D4C2211yescharacteristic
13C2×#6312C2214yes—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C3×#638C2212yes—
16C22×C3×C3⋊Dic3C2212yes—
17C3×#3830—24no—
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C3×#634C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic
21C2×#6292C2212yes—
22C2×#6289C2212yes—
23C2×Dic3×C3⋊D3C2212yes—
24#89018—24no—
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C32×#217C2214yes—
27C32×C4×Dic3C2211yescharacteristic
28#88806C2214yes—
29#88804C2214yes—
30Dic3×C3⋊Dic3C2212yes—
31C2×#3832—31no—
32C2×#3830D311yescharacteristic, complemented
33C2×C3×#237—34no—
34C2×#3799—34no—
35C32×#124—31no—
36#64153—31no—
37#64149—31no—
38#64149—34no—
39#64149—64no—
40#64093—34no—
41#64090—34no—
42#64084—31no—
43C42×C3⋊D3—31no—
44C3×#936D314yescomplemented
45#63971—316no—
46#63952—316no—
47C4×D3×Dic3—34no—
48#63873—34no—
49#63864—34no—
50C23×C33C2312yes—
51C22×C3⋊(C3⋊D3)C2312yes—
52C22×C3×C3⋊D3C2311yescharacteristic
53C33×D4—22no—
54C2×C33×C4C2311yescharacteristic
55#6312—28no—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C4⋊(C3⋊(C3⋊D3))—22no—
58C3×#638—24no—
59C2×C3×C3⋊Dic3—24no—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2311yescharacteristic
62C3×C4×C3⋊D3D414yescomplemented
63C3×#634—22no—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2312yes—
66C32×Dic6—22no—
67#6292—24no—
68#6289—24no—
69Dic3×C3⋊D3—24no—
70#5832—91no—
71#5668—94no—
72C2×C32×Q8—31no—
73C2×C32×D4—24no—
74C2×C32×D4C2×D314yes—
75C2×C32×D4C2×D311yescharacteristic
76C2×#638—34no—
77C2×#638—316no—
78C2×#638—616no—
79C2×#638C2×D312yes—
80C22×C3⋊Dic3C2×D312yes—
81#3832—64no—
82#3830—24no—
83C2×C4⋊(C3⋊D3)—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C4⋊(C3⋊D3)—34no—
86C2×C4⋊(C3⋊D3)—64no—
87C2×C4×C3⋊D3—32no—
88C2×C4×C3⋊D3—32no—
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C2×#634C2×D311yescharacteristic
91C2×C3×#67—38no—
92C22×C3×Dic3—38no—
93C3×#237—616no—
94C2×C3×C4×D3—34no—
95C2×C3×Dic6—34no—
96C2×C3×Dic6C2×D314yes—
97C2×#626—38no—
98C2×#624—38no—
99C2×D3×Dic3—38no—
100#3799—616no—
101C32×C22⋊C4—34no—
102C32×C42—31no—
103#3752—34no—
104#3750—34no—
105C4×C3⋊Dic3—31no—
106C4×C3⋊Dic3—34no—
107C4×C3⋊Dic3—32no—
108C4×C3⋊Dic3—64no—
109C3×#217C2×D3116yes—
110C3×C4×Dic3C2×D314yes—
111#3724—316no—
112#3722—316no—
113Dic32—38no—
114C22×C33—22no—
115C22×C33—24no—
116C22×C33C2411yescharacteristic
117C2×C3⋊(C3⋊D3)—22no—
118C2×C3⋊(C3⋊D3)—24no—
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3C2×D412yes—
121C33×C4C2×D412yes—
122C3⋊(C3⋊Dic3)#4114yescomplemented
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3C2×D412yes—
125C32×Dic3—22no—
126C32×Dic3—22no—
127C2×C3×#41—91no—
128C2×#239—94no—
129C2×#237—34no—
130C2×#235—91no—
131C3×#124—34no—
132#947—91no—
133#944—94no—
134#944—91no—
135#940—94no—
136#940—91no—
137#940—184no—
138#939—94no—
139#936C3⋊D311yescharacteristic, complemented
140C4×#67—94no—
141C2×C4×Dic3—91no—
142#884—916no—
143#67⋊C4—916no—
144#875—94no—
145C42×D3—94no—
146C23×C32—28no—
147C23×C32C22×D318yes—
148C23×C32C22×D312yes—
149C22×C3⋊D3—32no—
150C22×C3⋊D3—38no—
151C22×C3⋊D3—68no—
152C22×C3⋊D3C22×D311yescharacteristic
153C22×C3×D3—34no—
154C32×Q8—62no—
155C32×D4—22no—
156C32×D4—28no—
157C32×D4—48no—
158C2×C32×C4—24no—
159C2×C32×C4—32no—
160C2×C32×C4—32no—
161C2×C32×C4C22×D314yes—
162C2×C32×C4C22×D311yescharacteristic
163#638—24no—
164#638—68no—
165#638—632no—
166#638—1232no—
167C2×C3⋊Dic3—24no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—38no—
170C2×C3⋊Dic3—68no—
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D311yescharacteristic
173C4⋊(C3⋊D3)—64no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—68no—
176C4⋊(C3⋊D3)—128no—
177C4×C3⋊D3—64no—
178C4×C3⋊D3—64no—
179C4×C3⋊D3#6714yescomplemented
180#634—22no—
181C3×#67—616no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3—616no—
185C2×C3×Dic3C22×D318yes—
186C2×C3×Dic3C22×D318yes—
187C3×C4×D3—316no—
188C3×Dic6—28no—
189C3×Dic6—68no—
190#626—616no—
191#624—616no—
192D3×Dic3—616no—
193#532—271noSylow
194C2×C33—42no—
195C2×C33C22×D411yescharacteristic
196C2×C33C2×#4112yes—
197C3⋊(C3⋊D3)—42no—
198C3×C3⋊D3—22no—
199C3×#41—184no—
200C2×C3×Q8—34no—
201C2×C3×Q8—91no—
202C2×C3×D4—24no—
203C2×C3×D4—92no—
204C2×C3×D4D3214yes—
205C2×C3×D4C2×C3⋊D311yescharacteristic
206C22×C3×C4—91no—
207C22×C3×C4—92no—
208C2×#67—38no—
209C2×#67—916no—
210C2×#67—94no—
211C2×#67—92no—
212C2×#67—1816no—
213C22×Dic3—38no—
214C22×Dic3—92no—
215#239—1816no—
216#237—616no—
217#235—184no—
218C2×D12—98no—
219C2×D12—94no—
220C2×D12—91no—
221C2×D12—184no—
222C2×C4×D3—34no—
223C2×C4×D3—98no—
224C2×C4×D3—98no—
225C2×C4×D3—92no—
226C2×Dic6—34no—
227C2×Dic6C2×C3⋊D311yescharacteristic
228C3×C22⋊C4—316no—
229C3×C42—34no—
230#217C2×C3⋊D314yes—
231(C2×D3)⋊C4—916no—
232(C2×D3)⋊C4—94no—
233Dic3⋊C4—916no—
234Dic3⋊C4—94no—
235C4×Dic3—98no—
236C4×Dic3—94no—
237C4×Dic3—91no—
238C4×Dic3—92no—
239C4×Dic3—184no—
240C4×Dic3C2×C3⋊D311yescharacteristic
241C22×C32—24no—
242C22×C32—22no—
243C22×C32—24no—
244C22×C32—28no—
245C22×C32—216no—
246C22×C32—416no—
247C22×C32—48no—
248C22×C32C23×D314yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3—64no—
252C2×C3⋊D3—62no—
253C2×C3⋊D3—68no—
254C2×C3⋊D3—616no—
255C2×C3⋊D3—128no—
256C2×C3⋊D3—1216no—
257C2×C3⋊D3C2×#6712yes—
258C2×C3×D3—38no—
259C2×C3×D3—68no—
260C32×C4—28no—
261C32×C4—62no—
262C32×C4—62no—
263C32×C4D3×D418yes—
264C32×C4C2×#6712yes—
265C3⋊Dic3—22no—
266C3⋊Dic3—34no—
267C3⋊Dic3—316no—
268C3⋊Dic3—616no—
269C3⋊Dic3C2×#6712yes—
270C3×Dic3—28no—
271C3×Dic3—28no—
272C3×Dic3—38no—
273C3×Dic3—68no—
274C2×#41—91no—
275C2×#41—271no—
276C4⋊Q8—271no—
277#127—271no—
278#124—31no—
279#124—271no—
280#121—274no—
281C4×D4—274no—
282C2×C42—271no—
283C33#53211yescharacteristic, complemented, Sylow
284C23×C3—28no—
285C23×C3—91no—
286C23×C3C2×D3218yes—
287C23×C3C22×C3⋊D312yes—
288C22×D3—34no—
289C22×D3—98no—
290C22×D3—92no—
291C22×D3—188no—
292C3×Q8—68no—
293C3×Q8—182no—
294C3×D4—28no—
295C3×D4—22no—
296C3×D4—48no—
297C3×D4—184no—
298C2×C3×C4—24no—
299C2×C3×C4—38no—
300C2×C3×C4—38no—
301C2×C3×C4—91no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—184no—
305C2×C3×C4C2×D3214yes—
306C2×C3×C4C22×C3⋊D311yescharacteristic
307#67—616no—
308#67—184no—
309#67—188no—
310#67—1832no—
311#67—3632no—
312C2×Dic3—34no—
313C2×Dic3—38no—
314C2×Dic3—616no—
315C2×Dic3—98no—
316C2×Dic3—92no—
317C2×Dic3—188no—
318C2×Dic3—184no—
319C2×Dic3C22×C3⋊D312yes—
320C2×Dic3C22×C3⋊D312yes—
321D12—1816no—
322D12—188no—
323D12—182no—
324D12—368no—
325C4×D3—316no—
326C4×D3—184no—
327C4×D3—1816no—
328C4×D3—1816no—
329Dic6—22no—
330Dic6—68no—
331C2×C32—28no—
332C2×C32—24no—
333C2×C32—42no—
334C2×C32—48no—
335C2×C32—88no—
336C2×C32C2×D3×D414yes—
337C2×C32D3×#4118yes—
338C2×C32D3×#4112yes—
339C2×C32C22×#6711yescharacteristic
340C3⋊D3—22no—
341C3⋊D3—122no—
342C3⋊D3—128no—
343C3⋊D3—248no—
344C3×D3—68no—
345#41—184no—
346#41—544no—
347C2×Q8—31no—
348C2×Q8—91no—
349C2×D4—92no—
350C2×D4—272no—
351C2×D4—274no—
352C2×D4—271no—
353C2×D4D3×C3⋊D311yescharacteristic
354C22×C4—91no—
355C22×C4—92no—
356C22×C4—272no—
357C22×C4—272no—
358C4⋊C4—274no—
359C22⋊C4—34no—
360C22⋊C4—274no—
361C42—31no—
362C42—272no—
363C42—271no—
364C22×C3—216no—
365C22×C3—28no—
366C22×C3—22no—
367C22×C3—24no—
368C22×C3—24no—
369C22×C3—416no—
370C22×C3—48no—
371C22×C3—92no—
372C22×C3—182no—
373C22×C3C22×D3214yes—
374C22×C3C23×C3⋊D311yescharacteristic
375C2×D3—38no—
376C2×D3—68no—
377C2×D3—1816no—
378C2×D3—188no—
379C2×D3—182no—
380C2×D3—184no—
381C2×D3—368no—
382C2×D3—3616no—
383C3×C4—28no—
384C3×C4—68no—
385C3×C4—68no—
386C3×C4—92no—
387C3×C4—182no—
388C3×C4D3×#6718yes—
389C3×C4D4×C3⋊D312yes—
390Dic3—22no—
391Dic3—22no—
392Dic3—38no—
393Dic3—68no—
394Dic3—916no—
395Dic3—94no—
396Dic3—1816no—
397C32—24no—
398C32#566814yescomplemented
399C32#583211yescharacteristic, complemented
400C23—91no—
401C23—272no—
402C23C2×D3×C3⋊D312yes—
403Q8—62no—
404Q8—182no—
405D4—22no—
406D4—184no—
407D4—544no—
408D4—548no—
409D4—542no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—91no—
413C2×C4—94no—
414C2×C4—92no—
415C2×C4—184no—
416C2×C4—272no—
417C2×C4—544no—
418C2×C4—544no—
419C2×C4C2×D3×C3⋊D311yescharacteristic
420C2×C3—28no—
421C2×C3—24no—
422C2×C3—42no—
423C2×C3—48no—
424C2×C3—88no—
425C2×C3—182no—
426C2×C3#6431218yes—
427C2×C3C2×D3×#6714yes—
428C2×C3C2×D4×C3⋊D311yescharacteristic
429C2×C3#41×C3⋊D312yes—
430D3—68no—
431D3—362no—
432D3—368no—
433D3—728no—
434C22—24no—
435C22—22no—
436C22—92no—
437C22—182no—
438C22—544no—
439C22—542no—
440C22C22×D3×C3⋊D311yescharacteristic
441C4—62no—
442C4—62no—
443C4—92no—
444C4—182no—
445C4—274no—
446C4C3⋊D3×#6712yes—
447C3—24no—
448C3#1059515314yescomplemented
449C3#1059613911yescharacteristic, complemented
450C2—42no—
451C2—182no—
452C2—1082no—
453C2#1172458612yes—
454C2C2×C3⋊D3×#6711yescharacteristic
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×C3×#3830C2semidirect
2. C2×#89018C2semidirect
3. #11723981C2semidirect
4. C32×#936C2semidirect
5. #11723624C2semidirect
6. #11723622C2semidirect
7. C4×Dic3×C3⋊D3C2semidirect
8. #11723451C2semidirect
9. #11723441C2semidirect
32. C2×#3830D3semidirect
44. C3×#936D3semidirect
62. C3×C4×C3⋊D3D4semidirect
122. C3⋊(C3⋊Dic3)#41semidirect
139. #936C3⋊D3semidirect
179. C4×C3⋊D3#67semidirect
283. C33#532semidirect
398. C32#5668semidirect
399. C32#5832semidirect
448. C3#10595153semidirect
449. C3#10596139semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×#890182
#117239812
C32×#9362
#117236242
#117236222
C4×Dic3×C3⋊D32
#117234512
#117234412
#105961393
#105951533