Finite Group Databasebeta

← #13147687

Subgroups — #13147687

Order 1728.

814 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×#63434C211yesmaximal, characteristic, complemented
4#11721041C212yesmaximal, complemented
5#11721040C211yesmaximal, characteristic, complemented
6#11721036C211yesmaximal, characteristic, complemented
7#11721034C212yesmaximal, complemented
8#11721004C211yesmaximal, characteristic, complemented
9#11720957C212yesmaximal, complemented
10#11720954C212yesmaximal, complemented
11#11720935C211yesmaximal, characteristic, complemented
12#10593986—31nomaximal
13#10589664—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2212yes—
18C2×C3×C4×D9C2211yescharacteristic
19C2×#6195C2212yes—
20C2×#6195C2212yes—
21C2×#6194C2212yes—
22C2×#6192C2212yes—
23C3×C42×C9C2211yescharacteristic
24#88536C2212yes—
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C9×(C2×D3)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2211yescharacteristic
28C3×(C2×D9)⋊C4C2211yescharacteristic
29C9×C4⋊Dic3C2211yescharacteristic
30C3×Dic9⋊C4C2211yescharacteristic
31C3×Dic9⋊C4C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37C4×C4⋊(C3⋊D3)—31no—
38C3×C4×D12D311yescharacteristic
39C3×#872—31no—
40#63912—32no—
41#63911—31no—
42#63909—31no—
43#63908—31no—
44#63906—32no—
45#63869—31no—
46#63853—32no—
47#63850—32no—
48C4×D4×C9—31no—
49#63502—31no—
50#63500—31no—
51#63492—32no—
52#63491—32no—
53#63457—31no—
54#63452—31no—
55#63446—32no—
56#63445—32no—
57C4×D36—31no—
58C4×D36—61no—
59#63434D311yescharacteristic, complemented
60C22×C3⋊D9C2312yes—
61C22×D3×C9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C4⋊(C3⋊D9)—22no—
68C4×C3⋊D9—42no—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×D12—22no—
74C4×D3×C9—42no—
75C3×C4×D9—22no—
76#6195—42no—
77#6195—42no—
78#6194—42no—
79#6192—42no—
80#5562—91no—
81#5535—91no—
82C2×C4⋊(C3⋊D3)—31no—
83C2×C4×C3⋊D3—32no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3C2×D312yes—
87C2×#626—32no—
88C2×#626—32no—
89C2×#626—32no—
90C2×#625—32no—
91C32×C42C2×D311yescharacteristic
92#3752—32no—
93C4⋊(C3⋊Dic3)—31no—
94C3×(C2×D3)⋊C4—31no—
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4C2×D312yes—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4—31no—
100C3×C4×Dic3—31no—
101#3724—32no—
102Dic3⋊Dic3—32no—
103#3722—32no—
104(C2×D3)⋊Dic3—32no—
105C2×D4×C9—31no—
106C22×C4×C9—32no—
107C2×#611—32no—
108C2×#611—32no—
109C2×#611—32no—
110C2×D36—31no—
111C2×D36—32no—
112C2×D36—61no—
113C2×C4×D9—32no—
114C2×C4×D9—62no—
115C2×C4×D9C2×D311yescharacteristic
116C9×C4⋊C4—31no—
117C9×C22⋊C4—32no—
118C42×C9—21no—
119C42×C9C2×D311yescharacteristic
120#3676—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—62no—
124(C2×D9)⋊C4C2×D311yescharacteristic
125(C2×D9)⋊C4C2×D311yescharacteristic
126C4⋊Dic9—31no—
127C4⋊Dic9—32no—
128C4⋊Dic9—61no—
129Dic9⋊C4—32no—
130Dic9⋊C4C2×D311yescharacteristic
131Dic9⋊C4C2×D311yescharacteristic
132C4×Dic9C2×D311yescharacteristic
133C22×C3×C9C2411yescharacteristic
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×C3⋊D9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C2×D3×C9—22no—
140C2×C3×D9—21no—
141C2×C3×D9—21no—
142C2×C3×D9—21no—
143C3×C4×C9—21no—
144C3×C4×C9—21no—
145C3×C4×C9#4112yes—
146C3⋊Dic9—22no—
147C9×Dic3—22no—
148C3×Dic9—21no—
149C3×Dic9—21no—
150C3×Dic9—21no—
151C3×C4×D4—31no—
152C3×#117—91no—
153#940—91no—
154#938—91no—
155#930—92no—
156#930—91no—
157#930—91no—
158#929—92no—
159#895—91no—
160#894—92no—
161#894—92no—
162#890—91no—
163#884—92no—
164#884—92no—
165#884—92no—
166#883—92no—
167#874—91no—
168C4×D12—91no—
169C4×D12—91no—
170C4×D12—181no—
171C4×D12D911yescharacteristic, complemented
172#872—31no—
173#870—91no—
174C22×C3⋊D3—32no—
175C22×C3×D3—31no—
176C22×C3×D3C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—32no—
181C4⋊(C3⋊D3)—62no—
182C4×C3⋊D3—122no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3C22×D312yes—
187C3×D12—22no—
188C3×C4×D3—42no—
189C3×C4×D3—62no—
190#626—122no—
191#626—122no—
192#626—122no—
193#625—122no—
194C23×C9—32no—
195C22×D9—32no—
196C22×D9—62no—
197C22×D9C22×D311yescharacteristic
198D4×C9—62no—
199C2×C4×C9—21no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—32no—
203C2×C4×C9—122no—
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207#611—122no—
208#611—122no—
209#611—122no—
210C2×Dic9—32no—
211C2×Dic9—62no—
212C2×Dic9C22×D311yescharacteristic
213C2×Dic9C22×D311yescharacteristic
214C2×Dic9C22×D311yescharacteristic
215D36—64no—
216D36—62no—
217D36—122no—
218C4×D9—22no—
219C4×D9—122no—
220C4×D9—124no—
221#540—271noSylow
222C2×C3×C9C2×#4111yescharacteristic
223C2×C3×C9H(5,𝔽2)11yescharacteristic
224C2×C3×C9H(5,𝔽2)11yescharacteristic
225C3⋊D9—42no—
226D3×C9—42no—
227C3×D9—41no—
228C2×C3×D4—31no—
229C22×C3×C4—32no—
230C22×C3×C4—91no—
231C2×#67—92no—
232C2×#67—92no—
233C2×#67—92no—
234C2×#67—92no—
235C2×#67—92no—
236C2×D12—91no—
237C2×D12—91no—
238C2×D12—92no—
239C2×D12—92no—
240C2×D12—92no—
241C2×D12—181no—
242C2×D12C2×D911yescharacteristic
243C2×C4×D3—31no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3—182no—
247C2×C4×D3C2×D912yes—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—91no—
251C3×C22⋊C4—32no—
252C3×C22⋊C4—91no—
253C3×C22⋊C4—91no—
254C3×C42—21no—
255C3×C42—91no—
256C3×C42C2×D911yescharacteristic
257C3×C42D3211yescharacteristic
258#217—92no—
259(C2×D3)⋊C4—31no—
260(C2×D3)⋊C4—31no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—182no—
267(C2×D3)⋊C4C2×D912yes—
268C4⋊Dic3—92no—
269C4⋊Dic3—91no—
270C4⋊Dic3—91no—
271C4⋊Dic3—181no—
272C4⋊Dic3C2×D911yescharacteristic
273Dic3⋊C4—31no—
274Dic3⋊C4—31no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278C4×Dic3—31no—
279C22×C32C23×D311yescharacteristic
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3⋊D3—62no—
283C2×C3×D3—22no—
284C2×C3×D3—22no—
285C2×C3×D3—22no—
286C2×C3×D3—61no—
287C2×C3×D3—61no—
288C2×C3×D3—61no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4#23512yes—
292C3⋊Dic3—62no—
293C3×Dic3—22no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C22×C9—21no—
298C22×C9—62no—
299C22×C9—62no—
300C22×C9—62no—
301C22×C9C23×D311yescharacteristic
302C2×D9—21no—
303C2×D9—21no—
304C2×D9—21no—
305C2×D9—64no—
306C2×D9—62no—
307C2×D9—62no—
308C2×D9—62no—
309C2×D9—122no—
310C2×D9—122no—
311C4×C9—22no—
312C4×C9—21no—
313C4×C9—21no—
314C4×C9—22no—
315C4×C9—41no—
316C4×C9—62no—
317C4×C9#23512yes—
318Dic9—21no—
319Dic9—21no—
320Dic9—21no—
321Dic9—62no—
322Dic9—122no—
323#127—271no—
324#125—271no—
325#123—272no—
326#123—272no—
327#121—272no—
328#121—272no—
329#121—271no—
330#121—271no—
331C4×D4—31no—
332C4×D4—271no—
333#117—91no—
334C3×C9#54011yescharacteristic, complemented, Sylow
335C23×C3—32no—
336C23×C3—91no—
337C22×D3—31no—
338C22×D3—92no—
339C22×D3—92no—
340C22×D3—182no—
341C22×D3C22×D912yes—
342C3×D4—62no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—32no—
347C2×C3×C4—91no—
348C2×C3×C4—91no—
349C2×C3×C4—91no—
350C2×C3×C4—122no—
351C2×C3×C4—182no—
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C22×D911yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358#67—362no—
359#67—362no—
360#67—362no—
361#67—362no—
362#67—362no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—92no—
367C2×Dic3—92no—
368C2×Dic3—182no—
369C2×Dic3C22×D912yes—
370D12—22no—
371D12—184no—
372D12—182no—
373D12—182no—
374D12—362no—
375D12—362no—
376D12—362no—
377C4×D3—42no—
378C4×D3—62no—
379C4×D3—362no—
380C4×D3—362no—
381C4×D3—364no—
382C2×C32C2×#23511yescharacteristic
383C2×C32#100411yescharacteristic
384C2×C32#100911yescharacteristic
385C3⋊D3—122no—
386C3×D3—42no—
387C3×D3—121no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9—21no—
391C2×C9—122no—
392C2×C9C2×#23511yescharacteristic
393C2×C9#100911yescharacteristic
394C2×C9#100911yescharacteristic
395D9—41no—
396D9—124no—
397D9—122no—
398C2×D4—31no—
399C2×D4—272no—
400C2×D4—272no—
401C2×D4—272no—
402C2×D4—272no—
403C2×D4—271no—
404C22×C4—32no—
405C22×C4—91no—
406C22×C4—272no—
407C4⋊C4—31no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—32no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—272no—
417C22⋊C4—272no—
418C22⋊C4—272no—
419C42—91no—
420C42D3×D911yescharacteristic
421C22×C3—21no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3—181no—
428C22×C3C23×D911yescharacteristic
429C22×C3C22×D3211yescharacteristic
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—22no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—184no—
437C2×D3—182no—
438C2×D3—182no—
439C2×D3—182no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—362no—
444C2×D3—362no—
445C3×C4—22no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—62no—
453C3×C4—181no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4#369712yes—
457C3×C4#379812yes—
458Dic3—22no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—182no—
463Dic3—182no—
464Dic3—362no—
465C32#556211yescharacteristic
466C9—21no—
467C9#553511yescharacteristic, complemented
468C23—32no—
469C23—91no—
470C23—272no—
471D4—62no—
472D4—542no—
473D4—1082no—
474D4—1082no—
475D4—1082no—
476D4—1082no—
477C2×C4—32no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—122no—
482C2×C4—182no—
483C2×C4—272no—
484C2×C4—1082no—
485C2×C4C2×D3×D911yescharacteristic
486C2×C4C2×D3×D911yescharacteristic
487C2×C4C2×D3×D911yescharacteristic
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—122no—
492C2×C3—361no—
493C2×C3C2×#369711yescharacteristic
494C2×C3#6371011yescharacteristic
495C2×C3#6371511yescharacteristic
496C2×C3C2×#379811yescharacteristic
497C2×C3#6430711yescharacteristic
498C2×C3#6430811yescharacteristic
499D3—42no—
500D3—121no—
501D3—364no—
502D3—362no—
503D3—362no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—181no—
508C22—181no—
509C22—181no—
510C22—542no—
511C22—542no—
512C22—542no—
513C22C22×D3×D911yescharacteristic
514C4—21no—
515C4—21no—
516C4—62no—
517C4—181no—
518C4—181no—
519C4—181no—
520C4—542no—
521C4#8864012yes—
522C3—21no—
523C3#1058966411yescharacteristic, complemented
524C3#1059398611yescharacteristic
525C2—122no—
526C2—361no—
527C2—1082no—
528C2C2×#8864011yescharacteristic
529C2#1172247611yescharacteristic
530C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#540
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D9)C2semidirect
2. C4×C9×D12C2semidirect
3. C3×#63434C2semidirect
4. #11721041C2semidirect
5. #11721040C2semidirect
6. #11721036C2semidirect
7. #11721034C2semidirect
8. #11721004C2semidirect
9. #11720957C2semidirect
10. #11720954C2semidirect
11. #11720935C2semidirect
59. #63434D3semidirect
171. C4×D12D9semidirect
334. C3×C9#540semidirect
467. C9#5535semidirect
523. C3#10589664semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D9)2
C4×C9×D122
C3×#634342
#117210412
#117210402
#117210362
#117210342
#117210042
#117209572
#117209542
#117209352
#105939863
#105896643