Finite Group Databasebeta

← #13147681

Subgroups — #13147681

Order 1728.

808 conjugacy classes of subgroups.

The subgroup lattice has 546 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×#63432C211yesmaximal, characteristic, complemented
4#11721045C211yesmaximal, characteristic, complemented
5#11721045C211yesmaximal, characteristic, complemented
6#11721039C212yesmaximal, complemented
7#11721030C212yesmaximal, complemented
8#11721000C211yesmaximal, characteristic, complemented
9#11720965C212yesmaximal, complemented
10#11720954C212yesmaximal, complemented
11#11720922C211yesmaximal, characteristic, complemented
12#10593982—31nomaximal
13#10589660—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic
18C2×C4×D3×C9C2212yes—
19C2×#6195C2212yes—
20C2×#6195C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C3×C42×C9C2211yescharacteristic
23#88536C2211yescharacteristic
24#88536C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C9×(C2×D3)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37C4×C4⋊(C3⋊D3)—31no—
38C3×C4×D12D311yescharacteristic
39C3×#870—31no—
40#63916—31no—
41#63916—31no—
42#63910—32no—
43#63904—32no—
44#63900—31no—
45#63861—32no—
46#63850—32no—
47#63849—31no—
48C4×D4×C9—31no—
49#63500—31no—
50#63500—31no—
51#63491—32no—
52C4×#611—32no—
53#63454—31no—
54C4⋊C4×D9—31no—
55#63446—32no—
56#63441—32no—
57C4×D36—31no—
58C4×D36—61no—
59#63432D311yescharacteristic, complemented
60C22×C3⋊D9C2311yescharacteristic
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4⋊(C3⋊D9)—22no—
69C4×C3⋊D9—22no—
70C4×C3⋊D9—22no—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2312yes—
74C9×D12—22no—
75C4×D3×C9—42no—
76#6195—24no—
77#6195—42no—
78(C3⋊D9)⋊C4—42no—
79#5695—91no—
80#5558—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—31no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×C4×D3C2×D312yes—
86C2×#626—32no—
87C2×#626—32no—
88C2×#624—32no—
89C32×C42C2×D311yescharacteristic
90#3752—31no—
91#3752—31no—
92C4⋊(C3⋊Dic3)—31no—
93C3×(C2×D3)⋊C4C2×D312yes—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99#3724—32no—
100#3722—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—31no—
105C22×C4×C9—32no—
106C2×#611—32no—
107C2×#611—32no—
108C2×D36—31no—
109C2×D36—61no—
110C2×C4×D9—31no—
111C2×C4×D9—31no—
112C2×C4×D9—32no—
113C2×C4×D9—61no—
114C2×C4×D9—61no—
115C9×C4⋊C4—31no—
116C9×C22⋊C4—32no—
117C42×C9—21no—
118C42×C9C2×D311yescharacteristic
119#3676—32no—
120#3676—32no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—31no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—61no—
125(C2×D9)⋊C4—61no—
126C4⋊Dic9—31no—
127C4⋊Dic9—61no—
128C4⋊Dic9C2×D311yescharacteristic
129C4⋊Dic9C2×D311yescharacteristic
130Dic9⋊C4—32no—
131Dic9⋊C4C2×D312yes—
132Dic9⋊C4C2×D312yes—
133C4×Dic9—32no—
134C22×C3×C9C2411yescharacteristic
135C2×C3⋊D9—21no—
136C2×C3⋊D9—21no—
137C2×C3⋊D9—21no—
138C2×C3⋊D9—21no—
139C2×C3⋊D9—21no—
140C2×C3⋊D9#4112yes—
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C2×D3×C9—22no—
144C3×C4×C9—21no—
145C3×C4×C9—21no—
146C3×C4×C9#4112yes—
147C3⋊Dic9—21no—
148C3⋊Dic9—21no—
149C9×Dic3—22no—
150C3×Dic9—22no—
151C3×Dic9—22no—
152C3×#125—91no—
153C3×C4×D4—31no—
154#938—91no—
155#938—91no—
156#929—92no—
157C4×#67—92no—
158#898—92no—
159#895—91no—
160#895—91no—
161#895—92no—
162#894—92no—
163#893—92no—
164#892—91no—
165#892—91no—
166D3×C4⋊C4—91no—
167D3×C4⋊C4—91no—
168#884—92no—
169#879—92no—
170C4×D12—91no—
171C4×D12—91no—
172C4×D12—181no—
173C4×D12D911yescharacteristic, complemented
174#870—31no—
175C22×C3⋊D3—31no—
176C22×C3⋊D3—31no—
177C22×C3×D3C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—31no—
183C4⋊(C3⋊D3)—62no—
184C4×C3⋊D3—62no—
185C4×C3⋊D3—62no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3C22×D312yes—
189C3×D12—22no—
190C3×C4×D3—42no—
191#626—64no—
192#626—122no—
193#624—122no—
194C23×C9—32no—
195C22×D9—31no—
196C22×D9—31no—
197C22×D9—61no—
198C22×D9—61no—
199D4×C9—62no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—21no—
203C2×C4×C9—32no—
204C2×C4×C9—122no—
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D311yescharacteristic
208#611—64no—
209#611—122no—
210C2×Dic9—31no—
211C2×Dic9—31no—
212C2×Dic9—61no—
213C2×Dic9—61no—
214C2×Dic9C22×D312yes—
215C2×Dic9C22×D312yes—
216D36—62no—
217D36—64no—
218C4×D9—62no—
219C4×D9—62no—
220C4×D9—122no—
221C4×D9—122no—
222C4×D9—122no—
223#553—271noSylow
224C2×C3×C9C2×#4111yescharacteristic
225C2×C3×C9C2×#4111yescharacteristic
226C2×C3×C9H(5,𝔽2)11yescharacteristic
227C3⋊D9—22no—
228C3⋊D9—41no—
229D3×C9—42no—
230C2×C3×D4—31no—
231C22×C3×C4—32no—
232C2×#67—92no—
233C2×#67—92no—
234C2×D12—92no—
235C2×D12—91no—
236C2×D12—91no—
237C2×D12—92no—
238C2×D12—181no—
239C2×D12C2×D911yescharacteristic
240C2×C4×D3—91no—
241C2×C4×D3—91no—
242C2×C4×D3—91no—
243C2×C4×D3—91no—
244C2×C4×D3—92no—
245C2×C4×D3—92no—
246C2×C4×D3—181no—
247C2×C4×D3—181no—
248C2×C4×D3C2×D912yes—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4—92no—
253C3×C4⋊C4—92no—
254C3×C22⋊C4—32no—
255C3×C42—21no—
256C3×C42C2×D911yescharacteristic
257C3×C42D3211yescharacteristic
258#217—92no—
259#217—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266(C2×D3)⋊C4—91no—
267(C2×D3)⋊C4—92no—
268(C2×D3)⋊C4—181no—
269(C2×D3)⋊C4—181no—
270(C2×D3)⋊C4C2×D912yes—
271C4⋊Dic3—31no—
272C4⋊Dic3—31no—
273C4⋊Dic3—91no—
274C4⋊Dic3—91no—
275C4⋊Dic3—181no—
276C4⋊Dic3C2×D911yescharacteristic
277Dic3⋊C4—32no—
278Dic3⋊C4—32no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281C4×Dic3—92no—
282C4×Dic3—92no—
283C22×C32C23×D311yescharacteristic
284C2×C3⋊D3—32no—
285C2×C3⋊D3—61no—
286C2×C3⋊D3—61no—
287C2×C3⋊D3—61no—
288C2×C3⋊D3—61no—
289C2×C3⋊D3—61no—
290C2×C3×D3—22no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4#23512yes—
296C3⋊Dic3—61no—
297C3⋊Dic3—61no—
298C3×Dic3—22no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C22×C9—21no—
302C22×C9—62no—
303C22×C9—62no—
304C22×C9—62no—
305C22×C9C23×D311yescharacteristic
306C2×D9—32no—
307C2×D9—62no—
308C2×D9—61no—
309C2×D9—61no—
310C2×D9—62no—
311C2×D9—61no—
312C2×D9—61no—
313C2×D9—61no—
314C2×D9—121no—
315C2×D9—121no—
316C2×D9—121no—
317C2×D9—121no—
318C4×C9—22no—
319C4×C9—21no—
320C4×C9—21no—
321C4×C9—22no—
322C4×C9—41no—
323C4×C9—62no—
324C4×C9#23912yes—
325Dic9—22no—
326Dic9—22no—
327Dic9—61no—
328Dic9—61no—
329Dic9—121no—
330Dic9—121no—
331#126—272no—
332#125—91no—
333#123—272no—
334#121—272no—
335#121—271no—
336#121—271no—
337C4×D4—31no—
338C4×D4—272no—
339C4×D4—271no—
340#117—271no—
341C2×C4⋊C4—271no—
342C3×C9#55311yescharacteristic, complemented, Sylow
343C23×C3—32no—
344C22×D3—91no—
345C22×D3—91no—
346C22×D3—91no—
347C22×D3—91no—
348C22×D3—181no—
349C22×D3—181no—
350C22×D3C22×D912yes—
351C3×D4—62no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—32no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—122no—
359C2×C3×C4C22×D911yescharacteristic
360C2×C3×C4C22×D911yescharacteristic
361C2×C3×C4C22×D911yescharacteristic
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365#67—184no—
366#67—362no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—91no—
370C2×Dic3—91no—
371C2×Dic3—91no—
372C2×Dic3—91no—
373C2×Dic3—181no—
374C2×Dic3—181no—
375C2×Dic3C22×D912yes—
376D12—22no—
377D12—182no—
378D12—182no—
379D12—184no—
380D12—184no—
381D12—362no—
382C4×D3—42no—
383C4×D3—182no—
384C4×D3—182no—
385C4×D3—182no—
386C4×D3—182no—
387C4×D3—362no—
388C4×D3—362no—
389C4×D3—362no—
390C4×D3—362no—
391C2×C32C2×#23511yescharacteristic
392C2×C32#100411yescharacteristic
393C2×C32D3×#4111yescharacteristic
394C3⋊D3—62no—
395C3⋊D3—121no—
396C3×D3—42no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—21no—
400C2×C9—122no—
401C2×C9C2×#23911yescharacteristic
402C2×C9D3×#4111yescharacteristic
403C2×C9#100911yescharacteristic
404D9—62no—
405D9—122no—
406D9—122no—
407D9—121no—
408C2×D4—31no—
409C2×D4—272no—
410C2×D4—272no—
411C2×D4—271no—
412C22×C4—32no—
413C22×C4—272no—
414C22×C4—271no—
415C22×C4—271no—
416C4⋊C4—31no—
417C4⋊C4—91no—
418C4⋊C4—91no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—272no—
422C4⋊C4—271no—
423C22⋊C4—32no—
424C22⋊C4—272no—
425C22⋊C4—272no—
426C22⋊C4—272no—
427C22⋊C4—271no—
428C22⋊C4—271no—
429C42—272no—
430C42D3×D911yescharacteristic
431C22×C3—21no—
432C22×C3—62no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3C23×D911yescharacteristic
436C22×C3C22×D3211yescharacteristic
437C2×D3—22no—
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—92no—
441C2×D3—92no—
442C2×D3—182no—
443C2×D3—181no—
444C2×D3—181no—
445C2×D3—181no—
446C2×D3—181no—
447C2×D3—182no—
448C2×D3—181no—
449C2×D3—181no—
450C2×D3—181no—
451C2×D3—181no—
452C2×D3—181no—
453C2×D3—181no—
454C2×D3—361no—
455C2×D3—361no—
456C2×D3—361no—
457C2×D3—361no—
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—22no—
464C3×C4—41no—
465C3×C4—62no—
466C3×C4—182no—
467C3×C4—182no—
468C3×C4#369712yes—
469C3×C4#379912yes—
470Dic3—22no—
471Dic3—62no—
472Dic3—62no—
473Dic3—181no—
474Dic3—181no—
475Dic3—181no—
476Dic3—181no—
477Dic3—361no—
478Dic3—361no—
479C32#555811yescharacteristic
480C9—21no—
481C9#569511yescharacteristic, complemented
482C23—32no—
483C23—271no—
484C23—271no—
485D4—62no—
486D4—542no—
487D4—544no—
488D4—1082no—
489C2×C4—32no—
490C2×C4—92no—
491C2×C4—92no—
492C2×C4—122no—
493C2×C4—271no—
494C2×C4—271no—
495C2×C4—542no—
496C2×C4—542no—
497C2×C4—1082no—
498C2×C4C2×D3×D911yescharacteristic
499C2×C4C2×D3×D911yescharacteristic
500C2×C4C2×D3×D911yescharacteristic
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—122no—
505C2×C3C2×#369711yescharacteristic
506C2×C3#6371011yescharacteristic
507C2×C3#41×D911yescharacteristic
508C2×C3C2×#379911yescharacteristic
509C2×C3#6430611yescharacteristic
510C2×C3#6430711yescharacteristic
511D3—42no—
512D3—182no—
513D3—182no—
514D3—362no—
515D3—362no—
516D3—361no—
517D3—361no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—272no—
522C22—541no—
523C22—541no—
524C22—541no—
525C22—541no—
526C22—541no—
527C22C22×D3×D911yescharacteristic
528C4—21no—
529C4—21no—
530C4—62no—
531C4—182no—
532C4—182no—
533C4—541no—
534C4—541no—
535C4#8864212yes—
536C3—21no—
537C3#1058966011yescharacteristic, complemented
538C3#1059398211yescharacteristic
539C2—122no—
540C2—542no—
541C2—1081no—
542C2C2×#8864211yescharacteristic
543C2#1172247511yescharacteristic
544C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#553
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×#63432C2semidirect
4. #11721045C2semidirect
5. #11721045C2semidirect
6. #11721039C2semidirect
7. #11721030C2semidirect
8. #11721000C2semidirect
9. #11720965C2semidirect
10. #11720954C2semidirect
11. #11720922C2semidirect
59. #63432D3semidirect
173. C4×D12D9semidirect
342. C3×C9#553semidirect
481. C9#5695semidirect
537. C3#10589660semidirect

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×#634322
#117210452
#117210392
#117210302
#117210002
#117209652
#117209542
#117209222
#105939823
#105896603