Finite Group Databasebeta

← #13178741

Subgroups — #13178741

Order 1728.

1168 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C212yesmaximal, complemented
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C4⋊#6296C211yesmaximal, characteristic, complemented
4C4×#6295C211yesmaximal, characteristic
5#11723692C214yesmaximal, complemented
6#11723689C212yesmaximal, complemented
7#11723685C212yesmaximal
8#11723679C212yesmaximal, complemented
9#10595407—31nomaximal
10#10595371—32nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×#6296C2212yes—
14C2×#6295C2211yescharacteristic
15C2×#6294C2214yes—
16C3×#3752C2214yes—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2212yes—
19C3×C4⋊(C3⋊Dic3)C2212yes—
20C3×#3750C2214yes—
21C3×C4×C3⋊Dic3C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23#88815C2211yescharacteristic
24#88815C2212yes—
25#88815C2214yes—
26#88814C2212yes—
27#64155D312yescomplemented
28Q8×C3⋊Dic3D311yescharacteristic, complemented
29C3×#946—32no—
30C3×#946—32no—
31C3×#946—62no—
32C3×Q8×Dic3—32no—
33C3×Q8×Dic3—61no—
34#63919—31no—
35#63918—32no—
36C4×#626—32no—
37#63902—32no—
38C4×#625—31no—
39#63900—32no—
40#63899—34no—
41#63895—34no—
42#63893—34no—
43D3×C4⋊Dic3—32no—
44D3×C4⋊Dic3—32no—
45#63880—34no—
46#63870—32no—
47#63867—34no—
48C22×C3×C3⋊D3C2312yes—
49C33×Q8—22no—
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2314yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4×C3⋊D3—24no—
58#6296—42no—
59#6295Q814yescomplemented
60#6294—44no—
61#5635—92no—
62#5579—91no—
63C2×C32×Q8—21no—
64C2×C32×Q8—22no—
65C2×C32×Q8—41no—
66C2×C32×Q8C2×D312yes—
67C2×C32×Q8C2×D311yescharacteristic
68C2×C4×C3⋊D3C2×D312yes—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—62no—
72C2×#627—32no—
73C2×#627—34no—
74C2×#626—32no—
75C2×#625—31no—
76C2×#624—34no—
77C2×D3×Dic3—34no—
78C2×D3×Dic3—34no—
79#3752C2×D314yes—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D312yes—
82C4⋊(C3⋊Dic3)C2×D312yes—
83#3750C2×D314yes—
84C4×C3⋊Dic3C2×D312yes—
85C4×C3⋊Dic3C2×D311yescharacteristic
86C3×(C2×D3)⋊C4—34no—
87C3×(C2×D3)⋊C4—34no—
88C3×(C2×D3)⋊C4—64no—
89C3×C4⋊Dic3—32no—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—62no—
95C3×C4⋊Dic3—62no—
96C3×Dic3⋊C4—34no—
97C3×Dic3⋊C4—34no—
98C3×Dic3⋊C4—64no—
99C3×C4×Dic3—34no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—62no—
103#3724—34no—
104#3724—31no—
105#3724—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—34no—
108Dic3⋊Dic3—34no—
109Dic3⋊Dic3—34no—
110#3722—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113C22×C33C2411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3C2×Q814yes—
117C33×C4—22no—
118C33×C4#4112yes—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3—24no—
122C3×C3⋊Dic3—21no—
123C3×C3⋊Dic3C2×Q812yes—
124C3×C22⋊Q8—92no—
125C3×C4×Q8—91no—
126#946—32no—
127#946—32no—
128#946—62no—
129Q8×Dic3—32no—
130Q8×Dic3—61no—
131C4×#67—92no—
132C2×C4⋊Dic3—92no—
133#897—94no—
134#896—94no—
135D3×C4⋊C4—92no—
136D3×C4⋊C4—92no—
137#889—94no—
138#888—92no—
139#67⋊C4—94no—
140#878—94no—
141C4×D12—92no—
142C4×D12—91no—
143#870—92no—
144#869—91no—
145C22×C3⋊D3C22×D312yes—
146C22×C3×D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3—62no—
149C32×Q8—22no—
150C32×Q8—24no—
151C32×Q8—24no—
152C32×Q8—44no—
153C32×Q8—44no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—24no—
158C2×C32×C4—41no—
159C2×C32×C4—42no—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D312yes—
164C2×C3⋊Dic3C22×D311yescharacteristic
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D314yes—
168C2×C3⋊Dic3C22×D311yescharacteristic
169C4×C3⋊D3—24no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—34no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—64no—
182C3×C4×D3—64no—
183C3×C4×D3—64no—
184C3×C4×D3—124no—
185#627—122no—
186#627—124no—
187#626—38no—
188#625—34no—
189#624—124no—
190D3×Dic3—124no—
191D3×Dic3—124no—
192#554—271noSylow
193C2×C33C22×Q811yescharacteristic
194C2×C33C2×#4111yescharacteristic
195C2×C33H(5,𝔽2)11yescharacteristic
196C3×C3⋊D3—24no—
197C2×C3×Q8—21no—
198C2×C3×Q8—22no—
199C2×C3×Q8—41no—
200C2×C3×Q8D3212yes—
201C2×C3×Q8D3211yescharacteristic
202C22×C3×C4—92no—
203C2×#67—92no—
204C22×Dic3—94no—
205C2×D12—91no—
206C2×C4×D3—32no—
207C2×C4×D3—32no—
208C2×C4×D3—62no—
209C2×C4×D3—94no—
210C2×C4×D3—94no—
211C2×Dic6—92no—
212C2×Dic6—94no—
213C3×C4⋊C4—92no—
214C3×C4⋊C4—94no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—92no—
217C3×C22⋊C4—94no—
218C3×C42—91no—
219C3×C42—92no—
220#217—92no—
221(C2×D3)⋊C4—34no—
222(C2×D3)⋊C4—34no—
223(C2×D3)⋊C4—64no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226C4⋊Dic3—32no—
227C4⋊Dic3—34no—
228C4⋊Dic3—32no—
229C4⋊Dic3—32no—
230C4⋊Dic3—61no—
231C4⋊Dic3—62no—
232C4⋊Dic3—62no—
233C4⋊Dic3—92no—
234C4⋊Dic3—91no—
235C4⋊Dic3—94no—
236Dic3⋊C4—34no—
237Dic3⋊C4—34no—
238Dic3⋊C4—64no—
239Dic3⋊C4—94no—
240Dic3⋊C4—94no—
241Dic3⋊C4—94no—
242Dic3⋊C4—92no—
243C4×Dic3—34no—
244C4×Dic3—32no—
245C4×Dic3—62no—
246C4×Dic3—61no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—41no—
250C22×C32C23×D311yescharacteristic
251C22×C32C23×D312yes—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—22no—
254C2×C3⋊D3D3×Q814yes—
255C2×C3×D3—34no—
256C2×C3×D3—34no—
257C2×C3×D3—64no—
258C2×C3×D3—62no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264C32×C4—22no—
265C32×C4—24no—
266C32×C4—24no—
267C32×C4—24no—
268C32×C4—22no—
269C32×C4—42no—
270C32×C4—44no—
271C32×C4—44no—
272C32×C4#23512yes—
273C32×C4#23714yes—
274C3⋊Dic3—22no—
275C3⋊Dic3—22no—
276C3⋊Dic3—24no—
277C3⋊Dic3—21no—
278C3⋊Dic3D3×Q812yes—
279C3×Dic3—34no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—64no—
284C3×Dic3—64no—
285C3×Dic3—64no—
286C3×Dic3—62no—
287C3×Dic3—122no—
288C3×Dic3—121no—
289C3×Dic3—124no—
290C3×Dic3—122no—
291C4⋊Q8—271no—
292#125—272no—
293C22⋊Q8—92no—
294C22⋊Q8—274no—
295C4×Q8—91no—
296C4×D4—272no—
297C4×D4—271no—
298C2×C4⋊C4—272no—
299C33#55411yescharacteristic, complemented, Sylow
300C23×C3—92no—
301C22×D3—32no—
302C22×D3—32no—
303C22×D3—62no—
304C3×Q8—24no—
305C3×Q8—24no—
306C3×Q8—22no—
307C3×Q8—44no—
308C3×Q8—44no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4—91no—
316C2×C3×C4—92no—
317C2×C3×C4—94no—
318C2×C3×C4—92no—
319C2×C3×C4—91no—
320C2×C3×C4—184no—
321C2×C3×C4C2×D3214yes—
322C2×C3×C4C2×D3212yes—
323C2×C3×C4C2×D3212yes—
324C2×C3×C4C2×D3211yescharacteristic
325#67—98no—
326C2×Dic3—32no—
327C2×Dic3—34no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—34no—
332C2×Dic3—34no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—64no—
338C2×Dic3—364no—
339D12—94no—
340C4×D3—64no—
341C4×D3—64no—
342C4×D3—124no—
343C4×D3—364no—
344C4×D3—364no—
345Dic6—364no—
346Dic6—362no—
347C2×C32—21no—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—21no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32C2×#23511yescharacteristic
357C2×C32C2×#23712yes—
358C2×C32C2×D3×Q812yes—
359C2×C32C2×D3×Q811yescharacteristic
360C2×C32#100912yes—
361C2×C32#100911yescharacteristic
362C3⋊D3—24no—
363C3×D3—64no—
364C3×D3—64no—
365C3×D3—124no—
366C2×Q8—272no—
367C2×Q8#111311yescharacteristic
368C2×D4—271no—
369C22×C4—92no—
370C22×C4—274no—
371C4⋊C4—91no—
372C4⋊C4—92no—
373C4⋊C4—92no—
374C4⋊C4—94no—
375C4⋊C4—274no—
376C4⋊C4—271no—
377C4⋊C4—272no—
378C22⋊C4—94no—
379C22⋊C4—272no—
380C42—92no—
381C42—91no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—41no—
385C22×C3—94no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3C22×D3212yes—
389C22×C3C22×D3211yescharacteristic
390C2×D3—34no—
391C2×D3—34no—
392C2×D3—64no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—122no—
398C2×D3—122no—
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—22no—
402C3×C4—24no—
403C3×C4—22no—
404C3×C4—44no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—92no—
408C3×C4—181no—
409C3×C4—184no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4#380414yes—
413C3×C4#380512yes—
414Dic3—34no—
415Dic3—62no—
416Dic3—62no—
417Dic3—64no—
418Dic3—64no—
419Dic3—64no—
420Dic3—62no—
421Dic3—62no—
422Dic3—122no—
423Dic3—122no—
424Dic3—124no—
425Dic3—121no—
426C32—22no—
427C32—21no—
428C32—41no—
429C32#557911yescharacteristic, complemented
430C32#563512yescomplemented
431C23—92no—
432Q8—22no—
433Q8—1082no—
434D4—274no—
435C2×C4—91no—
436C2×C4—91no—
437C2×C4—92no—
438C2×C4—94no—
439C2×C4—92no—
440C2×C4—184no—
441C2×C4—1084no—
442C2×C4C2×#111312yes—
443C2×C4C2×#111311yescharacteristic
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—22no—
448C2×C3—22no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—184no—
454C2×C3D32×Q812yes—
455C2×C3D32×Q811yescharacteristic
456C2×C3#6432012yes—
457C2×C3#6432011yescharacteristic
458C2×C3C2×#380412yes—
459C2×C3C2×#380511yescharacteristic
460D3—64no—
461D3—64no—
462D3—124no—
463C22—94no—
464C22—182no—
465C22—182no—
466C22C22×#111311yescharacteristic
467C4—22no—
468C4—92no—
469C4—182no—
470C4—184no—
471C4—181no—
472C4—182no—
473C4#8904612yes—
474C3—22no—
475C3—21no—
476C3—41no—
477C3#1059537112yescomplemented
478C3#1059540711yescharacteristic, complemented
479C2—184no—
480C2Q8×#111311yescharacteristic
481C2#1172462711yescharacteristic
482C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×#6295C22
hypercenterC2×Q8#1113
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64155C2semidirect
2. C3×Q8×C3⋊Dic3C2semidirect
3. C4⋊#6296C2semidirect
5. #11723692C2semidirect
6. #11723689C2semidirect
8. #11723679C2semidirect
27. #64155D3semidirect
28. Q8×C3⋊Dic3D3semidirect
59. #6295Q8semidirect
299. C33#554semidirect
429. C32#5579semidirect
430. C32#5635semidirect
477. C3#10595371semidirect
478. C3#10595407semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C3×Q8×C3⋊Dic32
C4⋊#62962
C4×#62952
#117236922
#117236892
#117236852
#117236792
#105954073
#105953713