Finite Group Databasebeta

← #13177248

Subgroups — #13177248

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×C32×#235C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#11723635C214yesmaximal, complemented
5#11723531C212yesmaximal, complemented
6#11723516C212yesmaximal, complemented
7#11723456C212yesmaximal, complemented
8#11723426C212yesmaximal, complemented
9#10596304—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
17C22×C3×C3⋊Dic3C2212yes—
18C2×C32×#67C2212yes—
19C32×#235—24no—
20C2×C32×D12C2211yescharacteristic
21C2×C32×C4×D3C2212yes—
22C2×C32×Dic6C2211yescharacteristic
23C2×#6291C2214yes—
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×C4⋊(C3⋊Dic3)C2212yes—
26#88807C2214yes—
27#88804C2214yes—
28(C2×D3)⋊(C3⋊Dic3)C2214yes—
29C2×C32×#41—31no—
30C2×#3828—31no—
31C2×#3828—34no—
32C2×#3828—64no—
33C2×C3×#235D314yescomplemented
34#64155—32no—
35#64148—34no—
36#64147—32no—
37#64139—32no—
38C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
39#64133—32no—
40C2×C3×C4⋊Dic3—34no—
41#63978—316no—
42#63916—38no—
43#63909—38no—
44#63871—38no—
45#63857—38no—
46C23×C33C2311yescharacteristic
47C22×C3⋊(C3⋊D3)C2312yes—
48C22×C32×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—
57C2×C3×C3⋊Dic3—42no—
58C2×C3×C3⋊Dic3C2314yes—
59C32×#67—24no—
60C2×C32×Dic3C2312yes—
61C32×D12—22no—
62C32×C4×D3—24no—
63C32×Dic6—22no—
64#6291—44no—
65#5833—94no—
66#5510—91no—
67C32×#41—64no—
68C2×C32×Q8—31no—
69C2×C32×D4—32no—
70C2×C32×D4—31no—
71C22×C32×C4—24no—
72C22×C32×C4—32no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D314yes—
75C2×#638—32no—
76C2×#638—38no—
77C2×#638—34no—
78C2×#638—68no—
79C22×C3⋊Dic3C2×D312yes—
80#3828—632no—
81#3828—64no—
82#3828—616no—
83C2×C4⋊(C3⋊D3)—31no—
84C2×C4⋊(C3⋊D3)—34no—
85C2×C4⋊(C3⋊D3)—64no—
86C2×C4×C3⋊D3—32no—
87C2×C4×C3⋊D3—38no—
88C2×C4×C3⋊D3—68no—
89C2×#634—31no—
90C2×#634—34no—
91C2×#634—64no—
92C2×C3×#67C2×D318yes—
93C22×C3×Dic3—38no—
94C3×#235—216no—
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3C2×D318yes—
97C2×C3×Dic6C2×D314yes—
98C2×#626—316no—
99#3757—34no—
100#3752—34no—
101C4⋊(C3⋊Dic3)C2×D312yes—
102C4⋊(C3⋊Dic3)C2×D312yes—
103#3750—34no—
104C3×C4⋊Dic3—38no—
105C3×C4⋊Dic3—38no—
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×C32×D3—22no—
117C2×C32×D3—22no—
118C2×C32×D3—22no—
119C33×C4C2×D412yes—
120C33×C4C2×D412yes—
121C3⋊(C3⋊Dic3)—22no—
122C3×C3⋊Dic3—24no—
123C32×Dic3—22no—
124C2×C3×#41—34no—
125C2×#235—94no—
126C2×#235—91no—
127C2×#235—184no—
128C2×#235C3⋊D311yescharacteristic, complemented
129C2×C3×C4⋊C4—91no—
130#946—98no—
131#939—916no—
132#938—98no—
133#930—98no—
134#930—94no—
135C2×C4⋊Dic3—34no—
136#924—98no—
137#897—92no—
138#897—92no—
139#895—92no—
140#895—92no—
141C23×C32—24no—
142C23×C32—32no—
143C23×C32C22×D314yes—
144C23×C32C22×D311yescharacteristic
145C22×C3⋊D3—32no—
146C22×C3⋊D3—38no—
147C22×C3⋊D3—68no—
148C22×C3×D3C22×D318yes—
149C32×Q8—62no—
150C32×D4—64no—
151C32×D4—62no—
152C2×C32×C4—216no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—32no—
156C2×C32×C4—64no—
157C2×C32×C4D1214yescomplemented
158C2×C32×C4#67116yescomplemented
159C2×C32×C4C22×D314yes—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163#638—632no—
164#638—64no—
165#638—616no—
166#638—124no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—38no—
169C2×C3⋊Dic3—42no—
170C2×C3⋊Dic3—68no—
171C2×C3⋊Dic3C22×D314yes—
172C4⋊(C3⋊D3)—616no—
173C4⋊(C3⋊D3)—62no—
174C4⋊(C3⋊D3)—68no—
175C4×C3⋊D3—632no—
176C4×C3⋊D3—64no—
177C4×C3⋊D3—616no—
178#634—616no—
179#634—62no—
180#634—68no—
181C3×#67—216no—
182C2×C3×Dic3—316no—
183C2×C3×Dic3—128no—
184C2×C3×Dic3C22×D318yes—
185C3×D12—28no—
186C3×C4×D3—216no—
187C3×Dic6—28no—
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—
195C32×D3—42no—
196C3×#41—616no—
197C2×C3×Q8—34no—
198C2×C3×D4—38no—
199C2×C3×D4—34no—
200C22×C3×C4—24no—
201C22×C3×C4—38no—
202C22×C3×C4—92no—
203C22×C3×C4D3214yes—
204C22×C3×C4C2×C3⋊D311yescharacteristic
205C2×#67—916no—
206C2×#67—98no—
207C2×#67—92no—
208C2×#67—188no—
209C2×#67C2×C3⋊D312yes—
210C22×Dic3—38no—
211#235—24no—
212#235—1832no—
213#235—1816no—
214#235—184no—
215C2×D12—94no—
216C2×D12—91no—
217C2×D12—94no—
218C2×D12—184no—
219C2×D12C2×C3⋊D311yescharacteristic
220C2×C4×D3—98no—
221C2×C4×D3—92no—
222C2×C4×D3—188no—
223C2×C4×D3C2×C3⋊D312yes—
224C2×Dic6—94no—
225C2×Dic6—91no—
226C2×Dic6—184no—
227C2×Dic6C2×C3⋊D311yescharacteristic
228C3×C4⋊C4—92no—
229C3×C4⋊C4—92no—
230#217—916no—
231(C2×D3)⋊C4—916no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234C4⋊Dic3—38no—
235C4⋊Dic3—38no—
236C4⋊Dic3—94no—
237Dic3⋊C4—916no—
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×C32—62no—
247C22×C32—62no—
248C22×C32—62no—
249C22×C32C2×D1212yes—
250C22×C32C2×#6718yes—
251C22×C32C23×D314yes—
252C22×C32C23×D311yescharacteristic
253C2×C3⋊D3—616no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—62no—
256C2×C3⋊D3—62no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—68no—
260C2×C3⋊D3—128no—
261C2×C3⋊D3—128no—
262C2×C3×D3—28no—
263C2×C3×D3—28no—
264C2×C3×D3—28no—
265C32×C4—28no—
266C32×C4—28no—
267C32×C4—62no—
268C32×C4C2×D1212yes—
269C32×C4C2×D1212yes—
270C32×C4C2×#6718yes—
271C32×C4C2×#6718yes—
272C3⋊Dic3—24no—
273C3⋊Dic3—616no—
274C3⋊Dic3—62no—
275C3⋊Dic3—68no—
276C3×Dic3—28no—
277C3×Dic3—616no—
278C2×#41—31no—
279C2×#41—271no—
280C22⋊Q8—272no—
281C22⋊Q8—272no—
282#121—272no—
283#121—272no—
284#121—274no—
285C2×C4⋊C4—91no—
286C33#53711yescharacteristic, complemented, Sylow
287C23×C3—24no—
288C23×C3—38no—
289C23×C3C2×D3214yes—
290C23×C3C22×C3⋊D311yescharacteristic
291C22×D3—98no—
292C22×D3—92no—
293C22×D3—188no—
294C22×D3C22×C3⋊D312yes—
295C3×Q8—68no—
296C3×D4—616no—
297C3×D4—68no—
298C2×C3×C4—216no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—38no—
302C2×C3×C4—616no—
303C2×C3×C4—94no—
304C2×C3×C4—362no—
305C2×C3×C4#626116yescomplemented
306C2×C3×C4#63814yescomplemented
307C2×C3×C4C2×D3214yes—
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C22×C3⋊D311yescharacteristic
310C2×C3×C4C22×C3⋊D311yescharacteristic
311#67—24no—
312#67—1832no—
313#67—1816no—
314#67—184no—
315#67—3616no—
316C2×Dic3—316no—
317C2×Dic3—98no—
318C2×Dic3—92no—
319C2×Dic3—128no—
320C2×Dic3—188no—
321C2×Dic3C22×C3⋊D312yes—
322D12—22no—
323D12—1816no—
324D12—188no—
325D12—182no—
326D12—364no—
327C4×D3—24no—
328C4×D3—1832no—
329C4×D3—1816no—
330C4×D3—184no—
331Dic6—22no—
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×C32—122no—
342C2×C32C22×D1211yescharacteristic
343C2×C32#100411yescharacteristic
344C2×C32#100711yescharacteristic
345C2×C32#100914yes—
346C2×C32#101014yes—
347C2×C32C22×#6714yes—
348C3⋊D3—1216no—
349C3⋊D3—122no—
350C3⋊D3—128no—
351C3×D3—48no—
352#41—64no—
353#41—544no—
354C2×Q8—31no—
355C2×Q8—271no—
356C2×D4—32no—
357C2×D4—31no—
358C2×D4—274no—
359C2×D4—272no—
360C2×D4—271no—
361C22×C4—32no—
362C22×C4—92no—
363C22×C4—272no—
364C22×C4D3×C3⋊D311yescharacteristic
365C4⋊C4—92no—
366C4⋊C4—92no—
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—68no—
379C22×C3—68no—
380C22×C3—68no—
381C22×C3C2×#62618yes—
382C22×C3C2×#63812yes—
383C22×C3C22×D3214yes—
384C22×C3C23×C3⋊D311yescharacteristic
385C2×D3—22no—
386C2×D3—22no—
387C2×D3—22no—
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—68no—
400C3×C4—184no—
401C3×C4C2×#62618yes—
402C3×C4C2×#62618yes—
403C3×C4C2×#63812yes—
404C3×C4C2×#63812yes—
405Dic3—22no—
406Dic3—616no—
407Dic3—1816no—
408Dic3—188no—
409Dic3—182no—
410C32—24no—
411C32#551011yescharacteristic, complemented
412C32#583314yescomplemented
413C23—32no—
414C23—272no—
415C23C2×D3×C3⋊D311yescharacteristic
416Q8—62no—
417Q8—542no—
418D4—64no—
419D4—62no—
420D4—544no—
421D4—542no—
422D4—1084no—
423C2×C4—32no—
424C2×C4—64no—
425C2×C4—94no—
426C2×C4—272no—
427C2×C4—362no—
428C2×C4—544no—
429C2×C4#629114yescomplemented
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—128no—
439C2×C3#6429714yes—
440C2×C3#6430714yes—
441C2×C3C22×#62614yes—
442C2×C3#6436611yescharacteristic
443C2×C3#6436711yescharacteristic
444C2×C3C22×#63811yescharacteristic
445D3—42no—
446D3—3616no—
447D3—362no—
448D3—368no—
449C22—21no—
450C22—21no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—542no—
455C22—542no—
456C22—542no—
457C22C2×#629112yes—
458C22C22×D3×C3⋊D311yescharacteristic
459C4—62no—
460C4—184no—
461C4—542no—
462C4C2×#629112yes—
463C4C2×#629112yes—
464C3—24no—
465C3#1059504314yescomplemented
466C3#1059630411yescharacteristic, complemented
467C2—22no—
468C2—122no—
469C2—1082no—
470C2#1172456211yescharacteristic
471C2#1172458011yescharacteristic
472C2C22×#629111yescharacteristic
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×C32×#235C2semidirect
3. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
4. #11723635C2semidirect
5. #11723531C2semidirect
6. #11723516C2semidirect
7. #11723456C2semidirect
8. #11723426C2semidirect
33. C2×C3×#235D3semidirect
38. C2×C4⋊(C3⋊Dic3)D3semidirect
49. C2×C33×C4D4semidirect
128. C2×#235C3⋊D3semidirect
157. C2×C32×C4D12semidirect
158. C2×C32×C4#67semidirect
286. C33#537semidirect
305. C2×C3×C4#626semidirect
306. C2×C3×C4#638semidirect
411. C32#5510semidirect
412. C32#5833semidirect
429. C2×C4#6291semidirect
465. C3#10595043semidirect
466. C3#10596304semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C32×#2352
C2×C3×C4⋊(C3⋊Dic3)2
#117236352
#117235312
#117235162
#117234562
#117234262
#105963043
#105950433