Finite Group Databasebeta

← #13177219

Subgroups — #13177219

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×#3828C211yesmaximal, characteristic, complemented
2C2×C32×#235C211yesmaximal, characteristic, complemented
3C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#11723637C214yesmaximal, complemented
5#11723529C211yesmaximal, characteristic, complemented
6C4×#6290C214yesmaximal, complemented
7#11723451C211yesmaximal, characteristic, complemented
8#11723436C211yesmaximal, characteristic, complemented
9#11723434C211yesmaximal, characteristic, complemented
10#10596303—31nomaximal
11#10595032—34nomaximal
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2212yes—
14C2×C3×#638C2212yes—
15C3×#3828—24no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yes—
18C2×C3×#634C2211yescharacteristic
19C2×C32×#67C2212yes—
20C32×#235—24no—
21C2×C32×D12C2211yescharacteristic
22C2×C32×C4×D3C2212yes—
23C2×C32×Dic6C2211yescharacteristic
24C2×#6290C2214yes—
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C4×C3⋊(C3⋊Dic3)C2212yes—
28#88806C2214yes—
29#88803C2214yes—
30(C2×D3)⋊(C3⋊Dic3)C2214yes—
31C2×C32×#41—31no—
32C2×#3828D311yescharacteristic, complemented
33C2×C3×#235—34no—
34C2×C3×#235D314yescomplemented
35#64156—31no—
36#64153—31no—
37#64149—31no—
38#64148—34no—
39#64145—31no—
40C4×#638—34no—
41C2×C4×C3⋊Dic3—31no—
42C2×C4×C3⋊Dic3—34no—
43C2×C4×C3⋊Dic3—64no—
44#63980—316no—
45#63914—34no—
46C4×#625—316no—
47#63873—34no—
48#63863—34no—
49#63863—34no—
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3C2312yes—
52C22×C32×D3C2312yes—
53C2×C33×C4D414yescomplemented
54C2×C33×C4C2311yescharacteristic
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)—24no—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C3×#638—24no—
60C2×C3×C3⋊Dic3C2312yes—
61C3×C4⋊(C3⋊D3)—22no—
62C3×C4×C3⋊D3—24no—
63C3×#634—22no—
64C32×#67—24no—
65C2×C32×Dic3C2312yes—
66C32×D12—22no—
67C32×C4×D3—24no—
68C32×Dic6—22no—
69#6290—28no—
70#5832—94no—
71#5832—91no—
72C32×#41—64no—
73C2×C32×Q8—31no—
74C2×C32×D4—32no—
75C2×C32×D4—31no—
76C22×C32×C4—24no—
77C22×C32×C4—32no—
78C22×C32×C4C2×D314yes—
79C22×C32×C4C2×D311yescharacteristic
80C2×#638—34no—
81C2×#638C2×D312yes—
82C22×C3⋊Dic3—32no—
83C22×C3⋊Dic3—38no—
84C22×C3⋊Dic3—68no—
85#3828—24no—
86C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
87C2×C4×C3⋊D3C2×D312yes—
88C2×#634C2×D311yescharacteristic
89C2×C3×#67—38no—
90C2×C3×#67C2×D318yes—
91C3×#235—216no—
92C3×#235—616no—
93C2×C3×D12—34no—
94C2×C3×D12C2×D314yes—
95C2×C3×C4×D3—38no—
96C2×C3×C4×D3C2×D318yes—
97C2×C3×Dic6—34no—
98C2×C3×Dic6C2×D314yes—
99C2×#625—316no—
100#3757—34no—
101#3752—34no—
102#3750—34no—
103C4×C3⋊Dic3—32no—
104C4×C3⋊Dic3—31no—
105C4×C3⋊Dic3—31no—
106C4×C3⋊Dic3—34no—
107C4×C3⋊Dic3—34no—
108C4×C3⋊Dic3—38no—
109C4×C3⋊Dic3—68no—
110C4×C3⋊Dic3—64no—
111C4×C3⋊Dic3—64no—
112#3724—316no—
113(C2×D3)⋊Dic3—316no—
114(C2×D3)⋊Dic3—316no—
115C22×C33—22no—
116C22×C33C2×D412yes—
117C22×C33C2411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—24no—
120C2×C32×D3—22no—
121C2×C32×D3—24no—
122C33×C4C2×D412yes—
123C33×C4C2×D412yes—
124C3⋊(C3⋊Dic3)—22no—
125C3⋊(C3⋊Dic3)#4114yescomplemented
126C3×C3⋊Dic3—22no—
127C32×Dic3—22no—
128C2×C3×#41—34no—
129C2×C3×#41—91no—
130C2×#235—34no—
131C2×#235C3⋊D311yescharacteristic, complemented
132#947—94no—
133#947—91no—
134#944—94no—
135#944—91no—
136#940—94no—
137#940—91no—
138#939—916no—
139#939—94no—
140#936—94no—
141#936—91no—
142C4×#67—916no—
143C4×#67—94no—
144C2×C4×Dic3—94no—
145C2×C4×Dic3—91no—
146C2×C4×Dic3—184no—
147C23×C32—24no—
148C23×C32—32no—
149C23×C32C22×D314yes—
150C23×C32C22×D311yescharacteristic
151C22×C3⋊D3C22×D312yes—
152C22×C3×D3—38no—
153C22×C3×D3C22×D318yes—
154C32×Q8—62no—
155C32×D4—64no—
156C32×D4—62no—
157C2×C32×C4—216no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—32no—
161C2×C32×C4—64no—
162C2×C32×C4#67116yescomplemented
163C2×C32×C4#6714yescomplemented
164C2×C32×C4C22×D314yes—
165C2×C32×C4C22×D314yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168#638—24no—
169#638—68no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—38no—
173C2×C3⋊Dic3—38no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—68no—
176C2×C3⋊Dic3—64no—
177C2×C3⋊Dic3—616no—
178C2×C3⋊Dic3—632no—
179C2×C3⋊Dic3C22×D312yes—
180C4⋊(C3⋊D3)—22no—
181C4×C3⋊D3—24no—
182#634—22no—
183C3×#67—216no—
184C3×#67—616no—
185C2×C3×Dic3—38no—
186C2×C3×Dic3C22×D318yes—
187C3×D12—28no—
188C3×D12—68no—
189C3×C4×D3—216no—
190C3×C4×D3—616no—
191C3×Dic6—28no—
192C3×Dic6—68no—
193#625—632no—
194#532—271noSylow
195C2×C33—22no—
196C2×C33C22×D411yescharacteristic
197C2×C33C2×#4112yes—
198C3×C3⋊D3—42no—
199C32×D3—42no—
200C3×#41—616no—
201C3×#41—184no—
202C2×C3×Q8—34no—
203C2×C3×Q8—91no—
204C2×C3×D4—38no—
205C2×C3×D4—34no—
206C2×C3×D4—92no—
207C2×C3×D4—91no—
208C22×C3×C4—24no—
209C22×C3×C4—38no—
210C22×C3×C4—92no—
211C22×C3×C4D3214yes—
212C22×C3×C4C2×C3⋊D311yescharacteristic
213C2×#67—38no—
214C2×#67—916no—
215C2×#67—94no—
216C2×#67C2×C3⋊D312yes—
217C22×Dic3—98no—
218C22×Dic3—92no—
219C22×Dic3—188no—
220#235—24no—
221#235—616no—
222C2×D12—34no—
223C2×D12C2×C3⋊D311yescharacteristic
224C2×C4×D3—38no—
225C2×C4×D3C2×C3⋊D312yes—
226C2×Dic6—34no—
227C2×Dic6C2×C3⋊D311yescharacteristic
228#217—916no—
229#217—94no—
230(C2×D3)⋊C4—916no—
231(C2×D3)⋊C4—94no—
232Dic3⋊C4—916no—
233Dic3⋊C4—94no—
234C4×Dic3—98no—
235C4×Dic3—94no—
236C4×Dic3—94no—
237C4×Dic3—92no—
238C4×Dic3—91no—
239C4×Dic3—91no—
240C4×Dic3—184no—
241C4×Dic3—184no—
242C4×Dic3—188no—
243C22×C32—216no—
244C22×C32—28no—
245C22×C32—22no—
246C22×C32—24no—
247C22×C32—28no—
248C22×C32—64no—
249C22×C32—62no—
250C22×C32C2×#6718yes—
251C22×C32C2×#6712yes—
252C22×C32C23×D314yes—
253C22×C32C23×D311yescharacteristic
254C2×C3⋊D3—24no—
255C2×C3⋊D3—22no—
256C2×C3×D3—216no—
257C2×C3×D3—28no—
258C2×C3×D3—616no—
259C2×C3×D3—68no—
260C32×C4—28no—
261C32×C4—28no—
262C32×C4—62no—
263C32×C4C2×#6718yes—
264C32×C4C2×#6718yes—
265C32×C4C2×#6712yes—
266C32×C4C2×#6712yes—
267C3⋊Dic3—22no—
268C3⋊Dic3—34no—
269C3⋊Dic3—316no—
270C3⋊Dic3—616no—
271C3⋊Dic3—616no—
272C3⋊Dic3—68no—
273C3⋊Dic3—62no—
274C3×Dic3—28no—
275C3×Dic3—68no—
276C2×#41—31no—
277C2×#41—91no—
278C4⋊Q8—271no—
279#127—271no—
280#124—271no—
281#124—271no—
282#121—274no—
283C4×D4—274no—
284C2×C42—271no—
285C33#53211yescharacteristic, complemented, Sylow
286C23×C3—24no—
287C23×C3—38no—
288C23×C3—92no—
289C23×C3C2×D3214yes—
290C23×C3C22×C3⋊D311yescharacteristic
291C22×D3—38no—
292C22×D3C22×C3⋊D312yes—
293C3×Q8—68no—
294C3×Q8—182no—
295C3×D4—616no—
296C3×D4—68no—
297C3×D4—184no—
298C3×D4—182no—
299C2×C3×C4—216no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—38no—
303C2×C3×C4—616no—
304C2×C3×C4—92no—
305C2×C3×C4—184no—
306C2×C3×C4#625116yescomplemented
307C2×C3×C4#63814yescomplemented
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C2×D3214yes—
310C2×C3×C4C22×C3⋊D311yescharacteristic
311C2×C3×C4C22×C3⋊D311yescharacteristic
312#67—24no—
313#67—616no—
314#67—188no—
315#67—1832no—
316C2×Dic3—38no—
317C2×Dic3—98no—
318C2×Dic3—92no—
319C2×Dic3—98no—
320C2×Dic3—92no—
321C2×Dic3—188no—
322C2×Dic3—188no—
323C2×Dic3—1816no—
324C2×Dic3—184no—
325C2×Dic3—1832no—
326C2×Dic3C22×C3⋊D312yes—
327D12—22no—
328D12—68no—
329C4×D3—24no—
330C4×D3—616no—
331Dic6—22no—
332Dic6—68no—
333C2×C32—216no—
334C2×C32—28no—
335C2×C32—22no—
336C2×C32—28no—
337C2×C32—24no—
338C2×C32—122no—
339C2×C32D3×#4118yes—
340C2×C32D3×#4112yes—
341C2×C32C22×#6714yes—
342C2×C32C22×#6711yescharacteristic
343C3⋊D3—42no—
344C3×D3—48no—
345C3×D3—128no—
346#41—64no—
347#41—184no—
348C2×Q8—31no—
349C2×Q8—91no—
350C2×D4—32no—
351C2×D4—31no—
352C2×D4—92no—
353C2×D4—91no—
354C2×D4—274no—
355C22×C4—32no—
356C22×C4—92no—
357C22×C4—272no—
358C22×C4D3×C3⋊D311yescharacteristic
359C4⋊C4—274no—
360C22⋊C4—274no—
361C22⋊C4—274no—
362C42—271no—
363C42—271no—
364C42—272no—
365C22×C3—216no—
366C22×C3—28no—
367C22×C3—22no—
368C22×C3—24no—
369C22×C3—28no—
370C22×C3—616no—
371C22×C3—68no—
372C22×C3—184no—
373C22×C3—182no—
374C22×C3C2×#62518yes—
375C22×C3C2×#63812yes—
376C22×C3C22×D3214yes—
377C22×C3C23×C3⋊D311yescharacteristic
378C2×D3—24no—
379C2×D3—22no—
380C2×D3—616no—
381C2×D3—68no—
382C3×C4—28no—
383C3×C4—28no—
384C3×C4—68no—
385C3×C4—182no—
386C3×C4C2×#62518yes—
387C3×C4C2×#62518yes—
388C3×C4C2×#63812yes—
389C3×C4C2×#63812yes—
390Dic3—22no—
391Dic3—68no—
392Dic3—916no—
393Dic3—94no—
394Dic3—1816no—
395Dic3—1816no—
396Dic3—188no—
397Dic3—182no—
398C32—24no—
399C32#583214yescomplemented
400C32#583211yescharacteristic, complemented
401C23—32no—
402C23—92no—
403C23C2×D3×C3⋊D311yescharacteristic
404Q8—62no—
405Q8—182no—
406D4—64no—
407D4—62no—
408D4—184no—
409D4—182no—
410D4—548no—
411C2×C4—32no—
412C2×C4—64no—
413C2×C4—92no—
414C2×C4—184no—
415C2×C4—272no—
416C2×C4—272no—
417C2×C4—544no—
418C2×C4#629014yescomplemented
419C2×C4C2×D3×C3⋊D311yescharacteristic
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C2×C3—216no—
422C2×C3—22no—
423C2×C3—28no—
424C2×C3—28no—
425C2×C3—24no—
426C2×C3—128no—
427C2×C3—362no—
428C2×C3#6430618yes—
429C2×C3C22×#62514yes—
430C2×C3#41×C3⋊D312yes—
431C2×C3C22×#63811yescharacteristic
432D3—42no—
433D3—128no—
434C22—22no—
435C22—64no—
436C22—62no—
437C22—184no—
438C22—182no—
439C22C2×#629012yes—
440C22C22×D3×C3⋊D311yescharacteristic
441C4—62no—
442C4—182no—
443C4—274no—
444C4—542no—
445C4C2×#629012yes—
446C4C2×#629012yes—
447C3—24no—
448C3#1059503214yescomplemented
449C3#1059630311yescharacteristic, complemented
450C2—22no—
451C2—122no—
452C2—362no—
453C2#1172457812yes—
454C2C22×#629011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C2×C32×#235C2semidirect
3. C2×C4×C3⋊(C3⋊Dic3)C2semidirect
4. #11723637C2semidirect
5. #11723529C2semidirect
6. C4×#6290C2semidirect
7. #11723451C2semidirect
8. #11723436C2semidirect
9. #11723434C2semidirect
32. C2×#3828D3semidirect
34. C2×C3×#235D3semidirect
53. C2×C33×C4D4semidirect
125. C3⋊(C3⋊Dic3)#41semidirect
131. C2×#235C3⋊D3semidirect
162. C2×C32×C4#67semidirect
163. C2×C32×C4#67semidirect
285. C33#532semidirect
306. C2×C3×C4#625semidirect
307. C2×C3×C4#638semidirect
399. C32#5832semidirect
400. C32#5832semidirect
418. C2×C4#6290semidirect
448. C3#10595032semidirect
449. C3#10596303semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C2×C32×#2352
C2×C4×C3⋊(C3⋊Dic3)2
#117236372
#117235292
C4×#62902
#117234512
#117234362
#117234342
#105963033
#105950323