Finite Group Databasebeta

← #13177250

Subgroups — #13177250

Order 1728.

2420 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic, complemented
2C2×C3×#3828C211yesmaximal, characteristic, complemented
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11723632C214yesmaximal, complemented
5#11723532C212yesmaximal, complemented
6#11723526C212yesmaximal, complemented
7#11723449C212yesmaximal, complemented
8#11723431C212yesmaximal, complemented
9#10595981—31nomaximal
10#10595043—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×#6312C2212yes—
13#89076—24no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2212yes—
16C2×#6308C2211yescharacteristic
17C2×C3×#638C2212yes—
18C3×#3828—24no—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2212yes—
21C2×C3×#634C2211yescharacteristic
22C22×C32×Dic3C2212yes—
23C2×#6292C2214yes—
24C32×C4⋊Dic3C2212yes—
25C32×C4⋊Dic3C2212yes—
26Dic3⋊(C3⋊Dic3)C2214yes—
27#88804C2214yes—
28#88803C2214yes—
29C2×#3828—31no—
30C2×#3828—34no—
31C2×#3828—64no—
32C2×#3828D311yescharacteristic, complemented
33C2×C3×#235—34no—
34C2×C32×C4⋊C4—31no—
35#64139—34no—
36#64106—32no—
37#64106—32no—
38#64104—32no—
39#64104—32no—
40C2×C3×C4⋊Dic3D314yescomplemented
41#63978—316no—
42#63916—38no—
43#63909—38no—
44#63871—38no—
45#63857—38no—
46C23×C33C2311yescharacteristic
47C22×C3⋊(C3⋊D3)C2312yes—
48C22×C3×C3⋊D3C2312yes—
49C2×C33×C4D414yescomplemented
50C2×C33×C4C2311yescharacteristic
51C2×C33×C4C2311yescharacteristic
52#6312—24no—
53C2×C3⋊(C3⋊Dic3)C2312yes—
54C4⋊(C3⋊(C3⋊D3))—22no—
55C4×C3⋊(C3⋊D3)—24no—
56#6308—22no—
57C3×#638—24no—
58C2×C3×C3⋊Dic3C2312yes—
59C3×C4⋊(C3⋊D3)—22no—
60C3×C4×C3⋊D3—24no—
61C3×#634—22no—
62C2×C32×Dic3—42no—
63C2×C32×Dic3C2314yes—
64#6292—44no—
65#5833—91no—
66#5510—94no—
67C22×C32×C4—24no—
68C22×C32×C4—32no—
69C22×C32×C4C2×D314yes—
70C22×C32×C4C2×D311yescharacteristic
71C2×#638—32no—
72C2×#638—38no—
73C2×#638—68no—
74C2×#638C2×D312yes—
75#3828—24no—
76#3828—632no—
77#3828—616no—
78#3828—64no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—64no—
83C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
84C2×C4×C3⋊D3—32no—
85C2×C4×C3⋊D3—38no—
86C2×C4×C3⋊D3—68no—
87C2×C4×C3⋊D3C2×D312yes—
88C2×#634—31no—
89C2×#634—34no—
90C2×#634—64no—
91C2×#634C2×D311yescharacteristic
92C2×C3×#67—38no—
93C22×C3×Dic3C2×D318yes—
94C3×#235—616no—
95C2×C3×D12—34no—
96C2×C3×C4×D3—38no—
97C2×C3×Dic6—34no—
98C2×#626—316no—
99C32×C4⋊C4—32no—
100C32×C4⋊C4—32no—
101#3752—34no—
102#3752—34no—
103C4⋊(C3⋊Dic3)—34no—
104C3×C4⋊Dic3C2×D318yes—
105C3×C4⋊Dic3C2×D318yes—
106Dic3⋊Dic3—316no—
107#3722—316no—
108(C2×D3)⋊Dic3—316no—
109C22×C33—21no—
110C22×C33—21no—
111C22×C33C2×D412yes—
112C22×C33C2411yescharacteristic
113C2×C3⋊(C3⋊D3)—22no—
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)—22no—
116C2×C3×C3⋊D3—22no—
117C2×C3×C3⋊D3—22no—
118C2×C3×C3⋊D3—22no—
119C33×C4C2×D412yes—
120C33×C4C2×D412yes—
121C3⋊(C3⋊Dic3)—22no—
122C3×C3⋊Dic3—22no—
123C32×Dic3—24no—
124C2×C3×#41—91no—
125C2×#235—34no—
126C2×#235—94no—
127C2×#235—91no—
128C2×#235—184no—
129C2×C3×C4⋊C4—34no—
130#946—92no—
131#939—94no—
132#938—92no—
133#930—916no—
134#930—92no—
135C2×C4⋊Dic3C3⋊D311yescharacteristic, complemented
136#924—92no—
137#897—98no—
138#897—98no—
139#895—98no—
140#895—98no—
141C23×C32—24no—
142C23×C32C22×D314yes—
143C23×C32C22×D311yescharacteristic
144C22×C3⋊D3—32no—
145C22×C3⋊D3—38no—
146C22×C3⋊D3—68no—
147C22×C3⋊D3C22×D312yes—
148C22×C3×D3—38no—
149C2×C32×C4—216no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—34no—
153C2×C32×C4—122no—
154C2×C32×C4D12116yescomplemented
155C2×C32×C4#6714yescomplemented
156C2×C32×C4C22×D314yes—
157C2×C32×C4C22×D314yes—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160#638—24no—
161#638—632no—
162#638—616no—
163#638—64no—
164C2×C3⋊Dic3—32no—
165C2×C3⋊Dic3—38no—
166C2×C3⋊Dic3—68no—
167C2×C3⋊Dic3C22×D312yes—
168C4⋊(C3⋊D3)—22no—
169C4⋊(C3⋊D3)—616no—
170C4⋊(C3⋊D3)—68no—
171C4⋊(C3⋊D3)—62no—
172C4⋊(C3⋊D3)—124no—
173C4×C3⋊D3—24no—
174C4×C3⋊D3—632no—
175C4×C3⋊D3—616no—
176C4×C3⋊D3—64no—
177#634—22no—
178#634—616no—
179#634—68no—
180#634—62no—
181C3×#67—616no—
182C2×C3×Dic3—38no—
183C2×C3×Dic3—48no—
184C2×C3×Dic3C22×D3116yes—
185C3×D12—68no—
186C3×C4×D3—616no—
187C3×Dic6—68no—
188#626—1216no—
189#537—271noSylow
190C2×C33—22no—
191C2×C33C22×D411yescharacteristic
192C2×C33H(5,𝔽2)11yescharacteristic
193C2×C332−1+411yescharacteristic
194C3⋊(C3⋊D3)—42no—
195C3×C3⋊D3—42no—
196C3×#41—184no—
197C2×C3×Q8—91no—
198C2×C3×D4—92no—
199C2×C3×D4—91no—
200C22×C3×C4—24no—
201C22×C3×C4—38no—
202C22×C3×C4—92no—
203C22×C3×C4D3214yes—
204C22×C3×C4C2×C3⋊D311yescharacteristic
205C2×#67—38no—
206C2×#67—98no—
207C2×#67—92no—
208C2×#67—94no—
209C2×#67—188no—
210C22×Dic3C2×C3⋊D312yes—
211#235—616no—
212#235—1832no—
213#235—1816no—
214#235—184no—
215C2×D12—34no—
216C2×D12—94no—
217C2×D12—91no—
218C2×D12—916no—
219C2×D12—184no—
220C2×C4×D3—38no—
221C2×C4×D3—98no—
222C2×C4×D3—92no—
223C2×C4×D3—188no—
224C2×Dic6—34no—
225C2×Dic6—94no—
226C2×Dic6—91no—
227C2×Dic6—184no—
228C3×C4⋊C4—38no—
229C3×C4⋊C4—38no—
230#217—94no—
231(C2×D3)⋊C4—916no—
232(C2×D3)⋊C4—916no—
233(C2×D3)⋊C4—94no—
234C4⋊Dic3—916no—
235C4⋊Dic3C2×C3⋊D312yes—
236C4⋊Dic3C2×C3⋊D312yes—
237Dic3⋊C4—94no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—24no—
241C22×C32—28no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32—44no—
245C22×C32—44no—
246C22×C32C2×D1218yes—
247C22×C32C2×#6712yes—
248C22×C32C23×D314yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—616no—
254C2×C3⋊D3—68no—
255C2×C3⋊D3—62no—
256C2×C3⋊D3—62no—
257C2×C3⋊D3—62no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—68no—
260C2×C3⋊D3—128no—
261C2×C3⋊D3—128no—
262C2×C3×D3—68no—
263C2×C3×D3—68no—
264C2×C3×D3—68no—
265C32×C4—28no—
266C32×C4—28no—
267C32×C4—64no—
268C32×C4C2×D1218yes—
269C32×C4C2×D1218yes—
270C32×C4C2×#6712yes—
271C32×C4C2×#6712yes—
272C3⋊Dic3—22no—
273C3⋊Dic3—616no—
274C3⋊Dic3—68no—
275C3⋊Dic3—62no—
276C3×Dic3—216no—
277C3×Dic3—68no—
278C2×#41—91no—
279C2×#41—271no—
280C22⋊Q8—272no—
281C22⋊Q8—272no—
282#121—272no—
283#121—272no—
284#121—274no—
285C2×C4⋊C4—31no—
286C33#53711yescharacteristic, complemented, Sylow
287C23×C3—24no—
288C23×C3—92no—
289C23×C3C2×D3214yes—
290C23×C3C22×C3⋊D311yescharacteristic
291C22×D3—38no—
292C22×D3—98no—
293C22×D3—92no—
294C22×D3—188no—
295C3×Q8—182no—
296C3×D4—184no—
297C3×D4—182no—
298C2×C3×C4—216no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—316no—
302C2×C3×C4—92no—
303C2×C3×C4—128no—
304C2×C3×C4—184no—
305C2×C3×C4#626116yescomplemented
306C2×C3×C4C4⋊(C3⋊D3)14yescomplemented
307C2×C3×C4C2×D3214yes—
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C22×C3⋊D311yescharacteristic
310C2×C3×C4C22×C3⋊D311yescharacteristic
311#67—616no—
312#67—1832no—
313#67—1816no—
314#67—184no—
315#67—364no—
316C2×Dic3—38no—
317C2×Dic3—42no—
318C2×Dic3—98no—
319C2×Dic3—92no—
320C2×Dic3—188no—
321C2×Dic3C22×C3⋊D314yes—
322D12—68no—
323D12—1816no—
324D12—188no—
325D12—182no—
326D12—3616no—
327C4×D3—616no—
328C4×D3—1832no—
329C4×D3—1816no—
330C4×D3—184no—
331Dic6—68no—
332Dic6—1816no—
333Dic6—188no—
334Dic6—182no—
335C2×C32—22no—
336C2×C32—24no—
337C2×C32—24no—
338C2×C32—24no—
339C2×C32—28no—
340C2×C32—48no—
341C2×C32C22×D1214yes—
342C2×C32#100414yes—
343C2×C32#100714yes—
344C2×C32#100911yescharacteristic
345C2×C32#101011yescharacteristic
346C2×C32C22×#6711yescharacteristic
347C3⋊D3—42no—
348C3⋊D3—1216no—
349C3⋊D3—128no—
350C3⋊D3—122no—
351C3×D3—128no—
352#41—184no—
353#41—544no—
354C2×Q8—91no—
355C2×Q8—271no—
356C2×D4—92no—
357C2×D4—91no—
358C2×D4—274no—
359C2×D4—272no—
360C2×D4—271no—
361C22×C4—32no—
362C22×C4—92no—
363C22×C4—272no—
364C22×C4D3×C3⋊D311yescharacteristic
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—274no—
368C22⋊C4—274no—
369C22⋊C4—274no—
370C22×C3—24no—
371C22×C3—24no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—24no—
375C22×C3—28no—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—182no—
381C22×C3C2×#62618yes—
382C22×C3C2×C4⋊(C3⋊D3)12yes—
383C22×C3C22×D3214yes—
384C22×C3C23×C3⋊D311yescharacteristic
385C2×D3—68no—
386C2×D3—68no—
387C2×D3—68no—
388C2×D3—1816no—
389C2×D3—188no—
390C2×D3—188no—
391C2×D3—188no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—368no—
396C2×D3—368no—
397C3×C4—28no—
398C3×C4—28no—
399C3×C4—616no—
400C3×C4—182no—
401C3×C4C2×#62618yes—
402C3×C4C2×#62618yes—
403C3×C4C2×C4⋊(C3⋊D3)12yes—
404C3×C4C2×C4⋊(C3⋊D3)12yes—
405Dic3—24no—
406Dic3—68no—
407Dic3—1816no—
408Dic3—188no—
409Dic3—182no—
410C32—24no—
411C32#551014yescomplemented
412C32#583311yescharacteristic, complemented
413C23—92no—
414C23—272no—
415C23C2×D3×C3⋊D311yescharacteristic
416Q8—182no—
417Q8—542no—
418D4—184no—
419D4—182no—
420D4—544no—
421D4—542no—
422D4—1084no—
423C2×C4—34no—
424C2×C4—92no—
425C2×C4—122no—
426C2×C4—184no—
427C2×C4—272no—
428C2×C4—544no—
429C2×C4#629214yescomplemented
430C2×C4C2×D3×C3⋊D311yescharacteristic
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C2×C3—22no—
433C2×C3—28no—
434C2×C3—24no—
435C2×C3—24no—
436C2×C3—24no—
437C2×C3—48no—
438C2×C3—362no—
439C2×C3#6429714yes—
440C2×C3#6430714yes—
441C2×C3C22×#62614yes—
442C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
443C2×C3#6436111yescharacteristic
444C2×C3#6436411yescharacteristic
445D3—128no—
446D3—3616no—
447D3—362no—
448D3—368no—
449C22—21no—
450C22—21no—
451C22—182no—
452C22—182no—
453C22—182no—
454C22—542no—
455C22—542no—
456C22—542no—
457C22C2×#629212yes—
458C22C22×D3×C3⋊D311yescharacteristic
459C4—64no—
460C4—182no—
461C4—542no—
462C4C2×#629212yes—
463C4C2×#629212yes—
464C3—24no—
465C3#1059504314yescomplemented
466C3#1059598111yescharacteristic, complemented
467C2—22no—
468C2—362no—
469C2—1082no—
470C2#1172456611yescharacteristic
471C2#1172457911yescharacteristic
472C2C22×#629211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#537
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89076C2semidirect
2. C2×C3×#3828C2semidirect
3. C2×C32×C4⋊Dic3C2semidirect
4. #11723632C2semidirect
5. #11723532C2semidirect
6. #11723526C2semidirect
7. #11723449C2semidirect
8. #11723431C2semidirect
32. C2×#3828D3semidirect
40. C2×C3×C4⋊Dic3D3semidirect
49. C2×C33×C4D4semidirect
135. C2×C4⋊Dic3C3⋊D3semidirect
154. C2×C32×C4D12semidirect
155. C2×C32×C4#67semidirect
286. C33#537semidirect
305. C2×C3×C4#626semidirect
306. C2×C3×C4C4⋊(C3⋊D3)semidirect
411. C32#5510semidirect
412. C32#5833semidirect
429. C2×C4#6292semidirect
465. C3#10595043semidirect
466. C3#10595981semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C3×#38282
C2×C32×C4⋊Dic32
#117236322
#117235322
#117235262
#117234492
#117234312
#105959813
#105950433