Finite Group Databasebeta

← #13150516

Subgroups — #13150516

Order 1728.

1046 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×#88736C211yesmaximal, characteristic, complemented
2C2×C9×#235C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
4#11721288C214yesmaximal, complemented
5#11721045C212yesmaximal, complemented
6#11721038C212yesmaximal, complemented
7#11720961C212yesmaximal, complemented
8#11720929C212yesmaximal, complemented
9#10595043—31nomaximal
10#10589935—31nomaximal
11C22×C3×C4×C9C2211yescharacteristic
12C2×#6233C2212yes—
13#88736—24no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×#6226C2211yescharacteristic
17C2×C9×#67C2212yes—
18C22×C3×Dic9C2212yes—
19C9×#235—24no—
20C2×C9×D12C2211yescharacteristic
21C2×C4×D3×C9C2212yes—
22C2×C9×Dic6C2211yescharacteristic
23C2×#6195C2214yes—
24C3×C4⋊Dic9C2212yes—
25C3×C4⋊Dic9C2212yes—
26(C2×D3)⋊Dic9C2214yes—
27#88445C2214yes—
28Dic9⋊Dic3C2214yes—
29C2×#3828—31no—
30C2×C3×#235D311yescharacteristic
31C2×C3×C4⋊Dic3—31no—
32#63978—34no—
33#63916—32no—
34#63909—32no—
35#63871—32no—
36#63857—32no—
37C2×C9×#41—31no—
38C2×#3697—31no—
39C2×#3697—61no—
40#63508—32no—
41#63501—34no—
42#63500—32no—
43#63492—32no—
44C2×C4⋊Dic9D311yescharacteristic, complemented
45#63486—32no—
46C23×C3×C9C2311yescharacteristic
47C22×C3⋊D9C2312yes—
48C22×D3×C9C2312yes—
49C2×C3×C4×C9D414yescomplemented
50C2×C3×C4×C9C2311yescharacteristic
51C2×C3×C4×C9C2311yescharacteristic
52#6233—24no—
53C2×C3⋊Dic9C2312yes—
54C4⋊(C3⋊D9)—22no—
55C4×C3⋊D9—24no—
56#6226—22no—
57C9×#67—24no—
58C2×C9×Dic3C2312yes—
59C2×C3×Dic9—42no—
60C2×C3×Dic9C2314yes—
61C9×D12—22no—
62C4×D3×C9—24no—
63C9×Dic6—22no—
64#6195—44no—
65#5833—91no—
66#5510—91no—
67C22×C32×C4C2×D311yescharacteristic
68C2×#638—32no—
69#3828—64no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4×C3⋊D3—32no—
72C2×#634—31no—
73C2×C3×#67C2×D312yes—
74C22×C3×Dic3—32no—
75C3×#235—24no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×C4×D3C2×D312yes—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×#626—34no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3—32no—
82Dic3⋊Dic3—34no—
83#3722—34no—
84(C2×D3)⋊Dic3—34no—
85C9×#41—64no—
86C2×Q8×C9—31no—
87C2×D4×C9—32no—
88C2×D4×C9—31no—
89C22×C4×C9—21no—
90C22×C4×C9—32no—
91C22×C4×C9C2×D311yescharacteristic
92C2×#611—32no—
93C2×#611—34no—
94C2×#611—62no—
95C22×Dic9C2×D312yes—
96#3697—68no—
97#3697—64no—
98C2×D36—31no—
99C2×D36—61no—
100C2×C4×D9—32no—
101C2×C4×D9—62no—
102C2×Dic18—31no—
103C2×Dic18—61no—
104#3676—34no—
105(C2×D9)⋊C4—34no—
106C4⋊Dic9C2×D312yes—
107C4⋊Dic9C2×D312yes—
108Dic9⋊C4—34no—
109C22×C3×C9—21no—
110C22×C3×C9—21no—
111C22×C3×C9C2×D412yes—
112C22×C3×C9C2411yescharacteristic
113C2×C3⋊D9—22no—
114C2×C3⋊D9—22no—
115C2×C3⋊D9—22no—
116C2×D3×C9—22no—
117C2×D3×C9—22no—
118C2×D3×C9—22no—
119C3×C4×C9C2×D412yes—
120C3×C4×C9C2×D412yes—
121C3⋊Dic9—22no—
122C9×Dic3—22no—
123C3×Dic9—24no—
124C2×C3×#41—31no—
125C2×#235—91no—
126C2×#235—91no—
127C2×#235—181no—
128C2×#235D911yescharacteristic, complemented
129C2×C3×C4⋊C4—91no—
130#946—92no—
131#939—94no—
132#938—92no—
133#930—92no—
134#930—94no—
135C2×C4⋊Dic3—31no—
136#924—92no—
137#897—92no—
138#897—92no—
139#895—92no—
140#895—92no—
141C23×C32C22×D311yescharacteristic
142C22×C3⋊D3—32no—
143C22×C3×D3C22×D312yes—
144C2×C32×C4#6714yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147#638—64no—
148C2×C3⋊Dic3—32no—
149C4⋊(C3⋊D3)—62no—
150C4×C3⋊D3—64no—
151#634—62no—
152C3×#67—24no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—122no—
155C2×C3×Dic3C22×D312yes—
156C3×D12—22no—
157C3×C4×D3—24no—
158C3×Dic6—22no—
159#626—124no—
160C23×C9—21no—
161C23×C9—32no—
162C23×C9C22×D311yescharacteristic
163C22×D9—32no—
164C22×D9—62no—
165Q8×C9—62no—
166D4×C9—64no—
167D4×C9—62no—
168C2×C4×C9—24no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—32no—
172C2×C4×C9—64no—
173C2×C4×C9D1214yescomplemented
174C2×C4×C9C22×D311yescharacteristic
175C2×C4×C9C22×D311yescharacteristic
176#611—68no—
177#611—64no—
178#611—124no—
179C2×Dic9—32no—
180C2×Dic9—42no—
181C2×Dic9—62no—
182C2×Dic9C22×D314yes—
183D36—64no—
184D36—62no—
185C4×D9—68no—
186C4×D9—64no—
187Dic18—64no—
188Dic18—62no—
189#537—271noSylow
190C2×C3×C9—22no—
191C2×C3×C9C22×D411yescharacteristic
192C2×C3×C9H(5,𝔽2)11yescharacteristic
193C2×C3×C92−1+411yescharacteristic
194C3⋊D9—42no—
195D3×C9—42no—
196C3×#41—64no—
197C2×C3×Q8—31no—
198C2×C3×D4—32no—
199C2×C3×D4—31no—
200C22×C3×C4—21no—
201C22×C3×C4—32no—
202C22×C3×C4—92no—
203C22×C3×C4C2×D911yescharacteristic
204C22×C3×C4D3211yescharacteristic
205C2×#67—94no—
206C2×#67—92no—
207C2×#67—92no—
208C2×#67—182no—
209C2×#67C2×D912yes—
210C22×Dic3—32no—
211#235—24no—
212#235—188no—
213#235—184no—
214#235—184no—
215C2×D12—91no—
216C2×D12—91no—
217C2×D12—94no—
218C2×D12—181no—
219C2×D12C2×D911yescharacteristic
220C2×C4×D3—92no—
221C2×C4×D3—92no—
222C2×C4×D3—182no—
223C2×C4×D3C2×D912yes—
224C2×Dic6—91no—
225C2×Dic6—91no—
226C2×Dic6—181no—
227C2×Dic6C2×D911yescharacteristic
228C3×C4⋊C4—92no—
229C3×C4⋊C4—92no—
230#217—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234C4⋊Dic3—32no—
235C4⋊Dic3—32no—
236C4⋊Dic3—94no—
237Dic3⋊C4—94no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32C2×#6712yes—
241C22×C32C23×D311yescharacteristic
242C2×C3⋊D3—62no—
243C2×C3⋊D3—62no—
244C2×C3⋊D3—62no—
245C2×C3×D3—22no—
246C2×C3×D3—22no—
247C2×C3×D3—22no—
248C32×C4C2×#6712yes—
249C32×C4C2×#6712yes—
250C3⋊Dic3—62no—
251C3×Dic3—22no—
252C3×Dic3—64no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—22no—
257C22×C9—41no—
258C22×C9—41no—
259C22×C9—62no—
260C22×C9—62no—
261C22×C9—62no—
262C22×C9C2×D1212yes—
263C22×C9C23×D311yescharacteristic
264C2×D9—64no—
265C2×D9—62no—
266C2×D9—62no—
267C2×D9—62no—
268C2×D9—122no—
269C2×D9—122no—
270C4×C9—22no—
271C4×C9—22no—
272C4×C9—62no—
273C4×C9C2×D1212yes—
274C4×C9C2×D1212yes—
275Dic9—24no—
276Dic9—64no—
277Dic9—62no—
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—
286C3×C9#53711yescharacteristic, complemented, Sylow
287C23×C3—21no—
288C23×C3—32no—
289C23×C3C22×D911yescharacteristic
290C23×C3C2×D3211yescharacteristic
291C22×D3—92no—
292C22×D3—92no—
293C22×D3—182no—
294C22×D3C22×D912yes—
295C3×Q8—62no—
296C3×D4—64no—
297C3×D4—62no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—32no—
302C2×C3×C4—64no—
303C2×C3×C4—94no—
304C2×C3×C4—362no—
305C2×C3×C4#61114yescomplemented
306C2×C3×C4C22×D911yescharacteristic
307C2×C3×C4C22×D911yescharacteristic
308C2×C3×C4#62614yes—
309C2×C3×C4C2×D3211yescharacteristic
310C2×C3×C4C2×D3211yescharacteristic
311#67—24no—
312#67—188no—
313#67—184no—
314#67—184no—
315#67—364no—
316C2×Dic3—34no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—122no—
320C2×Dic3—182no—
321C2×Dic3C22×D912yes—
322D12—22no—
323D12—184no—
324D12—182no—
325D12—182no—
326D12—364no—
327C4×D3—24no—
328C4×D3—188no—
329C4×D3—184no—
330C4×D3—184no—
331Dic6—22no—
332Dic6—184no—
333Dic6—182no—
334Dic6—182no—
335C2×C32—22no—
336C2×C32#100911yescharacteristic
337C2×C32#101011yescharacteristic
338C2×C32C22×#6711yescharacteristic
339C3⋊D3—122no—
340C3×D3—42no—
341C2×C9—22no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9—42no—
346C2×C9—122no—
347C2×C9C22×D1211yescharacteristic
348C2×C9#100411yescharacteristic
349C2×C9#100711yescharacteristic
350D9—124no—
351D9—122no—
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×D911yescharacteristic
365C4⋊C4—92no—
366C4⋊C4—92no—
367C4⋊C4—274no—
368C22⋊C4—274no—
369C22⋊C4—274no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—22no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—62no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3C2×#61112yes—
382C22×C3C23×D911yescharacteristic
383C22×C3C2×#62612yes—
384C22×C3C22×D3211yescharacteristic
385C2×D3—22no—
386C2×D3—22no—
387C2×D3—22no—
388C2×D3—184no—
389C2×D3—182no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—182no—
394C2×D3—182no—
395C2×D3—362no—
396C2×D3—362no—
397C3×C4—22no—
398C3×C4—22no—
399C3×C4—62no—
400C3×C4—184no—
401C3×C4C2×#61112yes—
402C3×C4C2×#61112yes—
403C3×C4C2×#62612yes—
404C3×C4C2×#62612yes—
405Dic3—22no—
406Dic3—64no—
407Dic3—184no—
408Dic3—182no—
409Dic3—182no—
410C32#583311yescharacteristic
411C9—21no—
412C9#551011yescharacteristic, complemented
413C23—32no—
414C23—272no—
415C23C2×D3×D911yescharacteristic
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#619514yescomplemented
430C2×C4C2×D3×D911yescharacteristic
431C2×C4C2×D3×D911yescharacteristic
432C2×C3—22no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—42no—
438C2×C3—122no—
439C2×C3#6371511yescharacteristic
440C2×C3#6371611yescharacteristic
441C2×C3C22×#61111yescharacteristic
442C2×C3#6429711yescharacteristic
443C2×C3#6430711yescharacteristic
444C2×C3C22×#62611yescharacteristic
445D3—42no—
446D3—364no—
447D3—362no—
448D3—362no—
449C22—21no—
450C22—21no—
451C22—62no—
452C22—62no—
453C22—62no—
454C22—542no—
455C22—542no—
456C22—542no—
457C22C2×#619512yes—
458C22C22×D3×D911yescharacteristic
459C4—62no—
460C4—184no—
461C4—542no—
462C4C2×#619512yes—
463C4C2×#619512yes—
464C3—21no—
465C3#1058993511yescharacteristic, complemented
466C3#1059504311yescharacteristic
467C2—22no—
468C2—122no—
469C2—1082no—
470C2#1172246411yescharacteristic
471C2#1172247711yescharacteristic
472C2C22×#619511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#537
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88736C2semidirect
2. C2×C9×#235C2semidirect
3. C2×C3×C4⋊Dic9C2semidirect
4. #11721288C2semidirect
5. #11721045C2semidirect
6. #11721038C2semidirect
7. #11720961C2semidirect
8. #11720929C2semidirect
44. C2×C4⋊Dic9D3semidirect
49. C2×C3×C4×C9D4semidirect
128. C2×#235D9semidirect
173. C2×C4×C9D12semidirect
286. C3×C9#537semidirect
305. C2×C3×C4#611semidirect
412. C9#5510semidirect
429. C2×C4#6195semidirect
465. C3#10589935semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C2×C9×#2352
C2×C3×C4⋊Dic92
#117212882
#117210452
#117210382
#117209612
#117209292
#105950433
#105899353