Finite Group Databasebeta

← #13175195

Subgroups — #13175195

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×#89017C211yesmaximal, characteristic, complemented
2C2×#89015C211yesmaximal, characteristic, complemented
3C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C3×#64084C211yesmaximal, characteristic, complemented
5C32×#875C211yesmaximal, characteristic, complemented
6#11723529C211yesmaximal, characteristic, complemented
7#11723527C214yesmaximal, complemented
8#11723508C214yesmaximal, complemented
9#11723451C211yesmaximal, characteristic, complemented
10#10596139—31nomaximal
11#10593971—34nomaximal
12C2×C4×C3⋊(C3⋊D3)C2212yes—
13C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6292C2212yes—
19C2×#6291C2212yes—
20C2×#6290C2214yes—
21C2×Dic3×C3⋊D3C2212yes—
22C2×D3×C3⋊Dic3C2212yes—
23#89017—24no—
24#89015—24no—
25C33×C42C2211yescharacteristic
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2212yes—
28C3×#3752C2214yes—
29C32×(C2×D3)⋊C4C2214yes—
30#88806C2214yes—
31C2×#3832—31no—
32C2×#3830—31no—
33C2×#3796—34no—
34C2×#3796—34no—
35C32×#124—31no—
36#64153—31no—
37#64149—31no—
38#64093—34no—
39#64090—34no—
40#64084D311yescharacteristic, complemented
41C42×C3⋊D3—31no—
42C42×C3⋊D3—34no—
43C42×C3⋊D3—64no—
44C3×#875—34no—
45C3×#875D314yescomplemented
46#63914—34no—
47#63912—316no—
48#63902—316no—
49#63873—34no—
50C22×C3⋊(C3⋊D3)C2311yescharacteristic
51C22×C3×C3⋊D3C2312yes—
52C22×C32×D3C2312yes—
53C2×C33×C4C2312yes—
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C4×C3⋊(C3⋊D3)—24no—
58C4×C3⋊(C3⋊D3)D414yescomplemented
59C2×C3×C3⋊Dic3C2312yes—
60C3×C4⋊(C3⋊D3)—22no—
61C3×#634—22no—
62C2×C32×Dic3C2312yes—
63C32×D12—22no—
64C32×Dic6—22no—
65#6292—24no—
66#6291—24no—
67#6290—28no—
68Dic3×C3⋊D3—24no—
69D3×C3⋊Dic3—24no—
70#5668—94no—
71#5668—91no—
72C2×C32×Q8—31no—
73C2×C32×D4—31no—
74C2×#638—34no—
75C2×#638—32no—
76C22×C3⋊Dic3—32no—
77#3832—64no—
78#3830—64no—
79C2×C4⋊(C3⋊D3)—32no—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3—31no—
82C2×C4×C3⋊D3—34no—
83C2×C4×C3⋊D3—32no—
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—38no—
86C2×C4×C3⋊D3—64no—
87C2×C4×C3⋊D3—68no—
88C2×#634C2×D311yescharacteristic
89C2×C3×D12—34no—
90C2×C3×D12C2×D314yes—
91C2×C3×Dic6—34no—
92C2×C3×Dic6C2×D314yes—
93C2×#626—38no—
94C2×#626—38no—
95C2×#625—316no—
96C2×D3×Dic3—38no—
97C2×D3×Dic3—38no—
98#3796—616no—
99#3796—616no—
100C32×C22⋊C4—34no—
101C32×C42—24no—
102C32×C42C2×D314yes—
103C32×C42C2×D311yescharacteristic
104#3752C2×D314yes—
105#3750—34no—
106C4×C3⋊Dic3—32no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—34no—
109C4×C3⋊Dic3—38no—
110C4×C3⋊Dic3—68no—
111C4×C3⋊Dic3—64no—
112C3×(C2×D3)⋊C4—316no—
113C3×(C2×D3)⋊C4C2×D3116yes—
114#3724—316no—
115C22×C33C2411yescharacteristic
116C2×C3⋊(C3⋊D3)—22no—
117C2×C3⋊(C3⋊D3)C2×D412yes—
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—24no—
120C2×C32×D3—22no—
121C2×C32×D3—24no—
122C33×C4—22no—
123C33×C4C2×D412yes—
124C3⋊(C3⋊Dic3)C2×D412yes—
125C3⋊(C3⋊Dic3)#4114yescomplemented
126C3×C3⋊Dic3—22no—
127C32×Dic3—22no—
128C2×#239—94no—
129C2×#239—91no—
130C2×#237—94no—
131C2×#237—91no—
132C3×#124—34no—
133C3×#124—91no—
134#944—94no—
135#944—91no—
136#940—94no—
137#940—91no—
138#884—916no—
139#884—94no—
140#67⋊C4—916no—
141#67⋊C4—94no—
142#875—34no—
143#875C3⋊D311yescharacteristic, complemented
144C42×D3—94no—
145C42×D3—91no—
146C42×D3—184no—
147C23×C32—32no—
148C22×C3⋊D3—31no—
149C22×C3⋊D3—34no—
150C22×C3⋊D3—64no—
151C22×C3⋊D3C22×D312yes—
152C22×C3×D3—38no—
153C22×C3×D3C22×D318yes—
154C32×Q8—62no—
155C32×D4—62no—
156C2×C32×C4—28no—
157C2×C32×C4—24no—
158C2×C32×C4—32no—
159C2×C32×C4C22×D318yes—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D311yescharacteristic
163#638—64no—
164#638—68no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—31no—
167C2×C3⋊Dic3—38no—
168C2×C3⋊Dic3—34no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3C22×D312yes—
173C4⋊(C3⋊D3)—22no—
174C4⋊(C3⋊D3)—64no—
175C4×C3⋊D3—34no—
176C4×C3⋊D3—316no—
177C4×C3⋊D3—616no—
178C4×C3⋊D3—64no—
179C4×C3⋊D3—616no—
180C4×C3⋊D3—64no—
181C4×C3⋊D3—632no—
182#634—22no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3C22×D318yes—
185C3×D12—28no—
186C3×D12—68no—
187C3×Dic6—28no—
188C3×Dic6—68no—
189#626—616no—
190#626—616no—
191#625—632no—
192D3×Dic3—616no—
193D3×Dic3—616no—
194#532—271noSylow
195C2×C33C22×D411yescharacteristic
196C2×C33C2×#4112yes—
197C3⋊(C3⋊D3)—22no—
198C3×C3⋊D3—42no—
199C32×D3—42no—
200C2×C3×Q8—34no—
201C2×C3×Q8—91no—
202C2×C3×D4—34no—
203C2×C3×D4—91no—
204C2×#67—98no—
205C2×#67—916no—
206C2×#67—94no—
207C2×#67—92no—
208C22×Dic3—98no—
209C22×Dic3—92no—
210#239—184no—
211#239—1816no—
212#237—184no—
213#237—1816no—
214C2×D12—34no—
215C2×D12—98no—
216C2×D12—92no—
217C2×D12C2×C3⋊D311yescharacteristic
218C2×C4×D3—94no—
219C2×C4×D3—91no—
220C2×C4×D3—98no—
221C2×C4×D3—92no—
222C2×C4×D3—98no—
223C2×C4×D3—92no—
224C2×C4×D3—184no—
225C2×C4×D3—188no—
226C2×Dic6—34no—
227C2×Dic6C2×C3⋊D311yescharacteristic
228C3×C22⋊C4—316no—
229C3×C22⋊C4—94no—
230C3×C42—24no—
231C3×C42D3214yes—
232C3×C42C2×C3⋊D311yescharacteristic
233(C2×D3)⋊C4—316no—
234(C2×D3)⋊C4C2×C3⋊D314yes—
235Dic3⋊C4—916no—
236Dic3⋊C4—94no—
237C4×Dic3—98no—
238C4×Dic3—94no—
239C4×Dic3—92no—
240C4×Dic3—91no—
241C4×Dic3—188no—
242C4×Dic3—184no—
243C22×C32—24no—
244C22×C32—62no—
245C22×C32—64no—
246C22×C32C23×D314yes—
247C22×C32C23×D311yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—24no—
250C2×C3⋊D3—32no—
251C2×C3⋊D3—38no—
252C2×C3⋊D3—68no—
253C2×C3⋊D3—616no—
254C2×C3⋊D3—68no—
255C2×C3⋊D3—62no—
256C2×C3×D3—28no—
257C2×C3×D3—216no—
258C2×C3×D3—68no—
259C2×C3×D3—616no—
260C32×C4—216no—
261C32×C4—28no—
262C32×C4—22no—
263C32×C4—28no—
264C32×C4—62no—
265C32×C4D3×D418yes—
266C32×C4D3×D412yes—
267C3⋊Dic3—22no—
268C3⋊Dic3—32no—
269C3⋊Dic3—34no—
270C3⋊Dic3—38no—
271C3⋊Dic3—316no—
272C3⋊Dic3—68no—
273C3⋊Dic3—616no—
274C3×Dic3—28no—
275C3×Dic3—68no—
276C2×#41—271no—
277C2×#41—271no—
278C4⋊Q8—271no—
279#127—271no—
280#124—31no—
281#124—91no—
282#121—274no—
283C4×D4—274no—
284C2×C42—271no—
285C33#53211yescharacteristic, complemented, Sylow
286C23×C3—38no—
287C23×C3—92no—
288C22×D3—38no—
289C22×D3—94no—
290C22×D3—91no—
291C22×D3—184no—
292C22×D3C22×C3⋊D312yes—
293C3×Q8—68no—
294C3×Q8—182no—
295C3×D4—68no—
296C3×D4—182no—
297C2×C3×C4—28no—
298C2×C3×C4—24no—
299C2×C3×C4—38no—
300C2×C3×C4—92no—
301C2×C3×C4C2×D3218yes—
302C2×C3×C4C2×D3214yes—
303C2×C3×C4C22×C3⋊D312yes—
304C2×C3×C4C22×C3⋊D311yescharacteristic
305#67—184no—
306#67—1816no—
307#67—188no—
308#67—1832no—
309C2×Dic3—38no—
310C2×Dic3—98no—
311C2×Dic3—94no—
312C2×Dic3—92no—
313C2×Dic3—91no—
314C2×Dic3—188no—
315C2×Dic3—184no—
316C2×Dic3—184no—
317C2×Dic3—1816no—
318C2×Dic3C22×C3⋊D312yes—
319D12—22no—
320D12—68no—
321D12—1816no—
322D12—184no—
323C4×D3—916no—
324C4×D3—94no—
325C4×D3—1816no—
326C4×D3—1816no—
327C4×D3—184no—
328C4×D3—1816no—
329C4×D3—184no—
330C4×D3—1832no—
331Dic6—22no—
332Dic6—68no—
333C2×C32—24no—
334C2×C32—28no—
335C2×C32—122no—
336C2×C32C2×D3×D414yes—
337C2×C32C2×D3×D411yescharacteristic
338C2×C32D3×#4118yes—
339C2×C32D3×#4112yes—
340C3⋊D3—42no—
341C3⋊D3—616no—
342C3⋊D3—68no—
343C3⋊D3—62no—
344C3×D3—48no—
345C3×D3—128no—
346#41—544no—
347#41—544no—
348C2×Q8—31no—
349C2×Q8—91no—
350C2×D4—31no—
351C2×D4—91no—
352C2×D4—272no—
353C2×D4—272no—
354C2×D4—274no—
355C22×C4—272no—
356C22×C4—272no—
357C22×C4—271no—
358C22×C4—272no—
359C4⋊C4—274no—
360C22⋊C4—34no—
361C22⋊C4—94no—
362C42—272no—
363C42—271no—
364C42D3×C3⋊D311yescharacteristic
365C22×C3—24no—
366C22×C3—68no—
367C22×C3—616no—
368C22×C3—182no—
369C22×C3—184no—
370C22×C3C22×D3214yes—
371C22×C3C23×C3⋊D311yescharacteristic
372C2×D3—22no—
373C2×D3—24no—
374C2×D3—68no—
375C2×D3—616no—
376C2×D3—98no—
377C2×D3—92no—
378C2×D3—188no—
379C2×D3—1816no—
380C2×D3—188no—
381C2×D3—182no—
382C3×C4—216no—
383C3×C4—28no—
384C3×C4—22no—
385C3×C4—28no—
386C3×C4—68no—
387C3×C4—182no—
388C3×C4#380218yes—
389C3×C4D4×C3⋊D312yes—
390Dic3—22no—
391Dic3—68no—
392Dic3—98no—
393Dic3—916no—
394Dic3—92no—
395Dic3—94no—
396Dic3—188no—
397Dic3—1816no—
398C32—24no—
399C32#566814yescomplemented
400C32#566811yescharacteristic, complemented
401C23—32no—
402C23—92no—
403C23—271no—
404Q8—62no—
405Q8—182no—
406D4—62no—
407D4—182no—
408D4—544no—
409D4—544no—
410D4—548no—
411C2×C4—32no—
412C2×C4—92no—
413C2×C4—272no—
414C2×C4—271no—
415C2×C4—274no—
416C2×C4—544no—
417C2×C4—544no—
418C2×C4—544no—
419C2×C4C2×D3×C3⋊D312yes—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C2×C3—24no—
422C2×C3—28no—
423C2×C3—128no—
424C2×C3—362no—
425C2×C3C2×#380214yes—
426C2×C3#6430618yes—
427C2×C3C2×D4×C3⋊D311yescharacteristic
428C2×C3#41×C3⋊D312yes—
429D3—42no—
430D3—128no—
431D3—1816no—
432D3—182no—
433D3—188no—
434C22—62no—
435C22—64no—
436C22—182no—
437C22—184no—
438C22—272no—
439C22—542no—
440C22C22×D3×C3⋊D311yescharacteristic
441C4—22no—
442C4—62no—
443C4—182no—
444C4—272no—
445C4—274no—
446C4#8902612yes—
447C3—24no—
448C3#1059397114yescomplemented
449C3#1059613911yescharacteristic, complemented
450C2—122no—
451C2—362no—
452C2—542no—
453C2C2×#8902611yescharacteristic
454C2#1172457812yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89017C2semidirect
2. C2×#89015C2semidirect
3. C42×C3⋊(C3⋊D3)C2semidirect
4. C3×#64084C2semidirect
5. C32×#875C2semidirect
6. #11723529C2semidirect
7. #11723527C2semidirect
8. #11723508C2semidirect
9. #11723451C2semidirect
40. #64084D3semidirect
45. C3×#875D3semidirect
58. C4×C3⋊(C3⋊D3)D4semidirect
125. C3⋊(C3⋊Dic3)#41semidirect
143. #875C3⋊D3semidirect
285. C33#532semidirect
399. C32#5668semidirect
400. C32#5668semidirect
448. C3#10593971semidirect
449. C3#10596139semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
C2×#890152
C42×C3⋊(C3⋊D3)2
C3×#640842
C32×#8752
#117235292
#117235272
#117235082
#117234512
#105961393
#105939713