Finite Group Databasebeta

← #13149015

Subgroups — #13149015

Order 1728.

808 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721751C211yesmaximal, characteristic, complemented
2C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
3C9×#890C211yesmaximal, characteristic, complemented
4#11721004C212yesmaximal, complemented
5#11721004C211yesmaximal, characteristic, complemented
6#11721002C212yesmaximal, complemented
7C4×#6192C211yesmaximal, characteristic, complemented
8#11720997C212yesmaximal, complemented
9#11720993C211yesmaximal, characteristic, complemented
10#11720922C211yesmaximal, characteristic, complemented
11#11720920C212yesmaximal, complemented
12#10594562—31nomaximal
13#10589797—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×#6192C2211yescharacteristic
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C3×C9×C4⋊C4C2211yescharacteristic
23#88536C2212yes—
24C9×C4⋊Dic3C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30#88445C2212yes—
31#88445C2211yescharacteristic
32#88445C2212yes—
33#88445C2212yes—
34(C2×D9)⋊Dic3C2211yescharacteristic
35Dic3⋊Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64104—31no—
38C3×D3×C4⋊C4—31no—
39C3×#890D311yescharacteristic
40#63909—32no—
41#63909—31no—
42#63904—32no—
43C4×#626—31no—
44#63897—32no—
45#63894—31no—
46#63849—31no—
47#63848—32no—
48C9×#125—31no—
49#63460—32no—
50#63457—32no—
51#63457—31no—
52#63457—31no—
53#63457—61no—
54#63456—32no—
55#63455—32no—
56#63454—31no—
57C4⋊C4×D9D311yescharacteristic, complemented
58C4×D36—31no—
59C4×D36—31no—
60C22×C3⋊D9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C4⋊(C3⋊D9)—22no—
67C4⋊(C3⋊D9)—42no—
68C4×C3⋊D9—22no—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9—22no—
75C3×C4×D9—42no—
76#6192—22no—
77#6192—24no—
78(C3⋊D9)⋊C4—42no—
79#5695—91no—
80#5514—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4⋊(C3⋊D3)—32no—
83C2×C4×C3⋊D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—32no—
86C2×#626—31no—
87C2×#626—32no—
88C2×#624—32no—
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—32no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3C2×D312yes—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4C2×D312yes—
96C3×C4×Dic3C2×D311yescharacteristic
97Dic3⋊Dic3—31no—
98#3722—32no—
99#3722—32no—
100#3722—32no—
101#3722—31no—
102(C2×D3)⋊Dic3—31no—
103Dic32—32no—
104C2×D36—31no—
105C2×D36—32no—
106C2×D36—31no—
107C2×D36—32no—
108C2×D36—61no—
109C2×D36—62no—
110C2×C4×D9—31no—
111C2×C4×D9—32no—
112C2×C4×D9—61no—
113C2×C4×D9C2×D311yescharacteristic
114C2×C4×D9C2×D312yes—
115C9×C4⋊C4—21no—
116C9×C4⋊C4—32no—
117C9×C4⋊C4—31no—
118C9×C4⋊C4—32no—
119C9×C4⋊C4C2×D311yescharacteristic
120C42×C9—31no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—31no—
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4—62no—
128C4⋊Dic9—31no—
129C4⋊Dic9C2×D311yescharacteristic
130Dic9⋊C4C2×D312yes—
131C4×Dic9—32no—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—21no—
136C2×C3⋊D9—22no—
137C2×C3⋊D9—21no—
138C2×C3⋊D9—22no—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9#4112yes—
142C3×C4×C9—22no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C9×Dic3—22no—
146C9×Dic3—21no—
147C9×Dic3—21no—
148C3×Dic9—21no—
149C3×Dic9—22no—
150C3×#125—31no—
151C2×C3×C4⋊C4—91no—
152#930—91no—
153#930—92no—
154C4×#67—92no—
155C4×#67—91no—
156#898—92no—
157#898—92no—
158#895—92no—
159#895—91no—
160#895—91no—
161#895—91no—
162#895—181no—
163#894—92no—
164#894—92no—
165#893—92no—
166#893—91no—
167#892—91no—
168#892—91no—
169D3×C4⋊C4—31no—
170#890D911yescharacteristic, complemented
171C4×D12—91no—
172C4×D12—91no—
173C22×C3⋊D3—31no—
174C22×C3⋊D3—32no—
175C22×C3×D3—31no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C4⋊(C3⋊D3)—62no—
180C4⋊(C3⋊D3)—122no—
181C4×C3⋊D3—62no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3C22×D312yes—
185C2×C3×Dic3C22×D311yescharacteristic
186C2×C3×Dic3C22×D311yescharacteristic
187C3×C4×D3—62no—
188C3×C4×D3—122no—
189#626—62no—
190#626—64no—
191#624—122no—
192C22×D9—31no—
193C22×D9—32no—
194C22×D9—61no—
195C22×D9—62no—
196C22×D9C22×D311yescharacteristic
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—32no—
200C2×C4×C9—31no—
201C2×C4×C9—31no—
202C2×C4×C9C22×D312yes—
203C2×C4×C9C22×D311yescharacteristic
204C2×Dic9—31no—
205C2×Dic9—61no—
206C2×Dic9C22×D312yes—
207C2×Dic9C22×D311yescharacteristic
208D36—62no—
209D36—64no—
210D36—62no—
211D36—122no—
212D36—122no—
213D36—124no—
214C4×D9—22no—
215C4×D9—42no—
216C4×D9—62no—
217C4×D9—122no—
218C4×D9—122no—
219#553—271noSylow
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223C3⋊D9—41no—
224C3⋊D9—42no—
225C3×D9—22no—
226C22×C3×C4—91no—
227C22×C3×C4—92no—
228C2×#67—92no—
229C2×#67—91no—
230C2×D12—91no—
231C2×D12—92no—
232C2×D12—91no—
233C2×D12—91no—
234C2×D12—92no—
235C2×D12—92no—
236C2×D12—181no—
237C2×D12—182no—
238C2×C4×D3—31no—
239C2×C4×D3—32no—
240C2×C4×D3—91no—
241C2×C4×D3—91no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3—181no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—32no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4C2×D911yescharacteristic
252C3×C4⋊C4D3211yescharacteristic
253C3×C42—31no—
254#217—91no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—182no—
267C4⋊Dic3—31no—
268C4⋊Dic3—91no—
269C4⋊Dic3C2×D911yescharacteristic
270C4⋊Dic3C2×D912yes—
271Dic3⋊C4—32no—
272Dic3⋊C4—91no—
273Dic3⋊C4C2×D912yes—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3C2×D911yescharacteristic
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—61no—
279C2×C3⋊D3—61no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—62no—
284C2×C3×D3—32no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C32×C4—22no—
288C32×C4#23912yes—
289C3⋊Dic3—61no—
290C3×Dic3—22no—
291C3×Dic3—21no—
292C3×Dic3—21no—
293C3×Dic3—61no—
294C3×Dic3—62no—
295C22×C9—21no—
296C22×C9C23×D311yescharacteristic
297C2×D9—21no—
298C2×D9—21no—
299C2×D9—62no—
300C2×D9—61no—
301C2×D9—62no—
302C2×D9—62no—
303C2×D9—61no—
304C2×D9—61no—
305C2×D9—62no—
306C2×D9—121no—
307C2×D9—121no—
308C2×D9—122no—
309C2×D9—122no—
310C2×D9—122no—
311C2×D9#23912yes—
312C4×C9—22no—
313C4×C9—22no—
314C4×C9—42no—
315C4×C9—61no—
316C4×C9—62no—
317C4×C9—61no—
318C4×C9#23512yes—
319Dic9—21no—
320Dic9—22no—
321Dic9—61no—
322Dic9—121no—
323#126—272no—
324#125—31no—
325#123—272no—
326#121—272no—
327#121—271no—
328#121—271no—
329C4×D4—271no—
330C4×D4—271no—
331C4×D4—272no—
332#117—271no—
333C2×C4⋊C4—91no—
334C3×C9#55311yescharacteristic, complemented, Sylow
335C23×C3—91no—
336C22×D3—31no—
337C22×D3—91no—
338C22×D3—92no—
339C22×D3—91no—
340C22×D3—92no—
341C22×D3—181no—
342C22×D3—182no—
343C2×C3×C4—21no—
344C2×C3×C4—22no—
345C2×C3×C4—32no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—92no—
349C2×C3×C4—91no—
350C2×C3×C4—182no—
351C2×C3×C4—362no—
352C2×C3×C4C22×D912yes—
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C2×D3212yes—
355C2×C3×C4C2×D3211yescharacteristic
356#67—184no—
357#67—182no—
358C2×Dic3—32no—
359C2×Dic3—31no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—181no—
363C2×Dic3C22×D912yes—
364C2×Dic3C22×D911yescharacteristic
365C2×Dic3C22×D911yescharacteristic
366D12—182no—
367D12—184no—
368D12—182no—
369D12—182no—
370D12—362no—
371D12—362no—
372D12—362no—
373D12—364no—
374C4×D3—62no—
375C4×D3—122no—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3—362no—
379C4×D3—362no—
380C4×D3—362no—
381C2×C32C2×#23911yescharacteristic
382C2×C32D3×#4111yescharacteristic
383C2×C32#100911yescharacteristic
384C3⋊D3—121no—
385C3⋊D3—122no—
386C3×D3—62no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9C2×#23511yescharacteristic
391C2×C9#100411yescharacteristic
392C2×C9C2×#23911yescharacteristic
393D9—22no—
394D9—122no—
395D9—121no—
396D9—122no—
397D9—242no—
398C2×D4—271no—
399C2×D4—272no—
400C2×D4—271no—
401C2×D4—272no—
402C22×C4—91no—
403C22×C4—92no—
404C22×C4—272no—
405C22×C4—271no—
406C4⋊C4—31no—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—91no—
410C4⋊C4—92no—
411C4⋊C4—271no—
412C4⋊C4D3×D911yescharacteristic
413C22⋊C4—272no—
414C22⋊C4—272no—
415C22⋊C4—271no—
416C22⋊C4—272no—
417C22⋊C4—271no—
418C22⋊C4—272no—
419C42—31no—
420C42—272no—
421C22×C3—21no—
422C22×C3—92no—
423C22×C3—181no—
424C22×C3—181no—
425C22×C3C23×D911yescharacteristic
426C22×C3C22×D3211yescharacteristic
427C2×D3—32no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—182no—
431C2×D3—181no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—181no—
435C2×D3—181no—
436C2×D3—182no—
437C2×D3—181no—
438C2×D3—182no—
439C2×D3—181no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—181no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—42no—
452C3×C4—61no—
453C3×C4—62no—
454C3×C4—61no—
455C3×C4—181no—
456C3×C4—182no—
457C3×C4#370112yes—
458C3×C4#379912yes—
459Dic3—22no—
460Dic3—21no—
461Dic3—21no—
462Dic3—61no—
463Dic3—62no—
464Dic3—181no—
465Dic3—181no—
466Dic3—361no—
467C32#569511yescharacteristic
468C9—21no—
469C9#551411yescharacteristic, complemented
470C23—91no—
471C23—272no—
472C23—271no—
473D4—542no—
474D4—544no—
475D4—542no—
476D4—1082no—
477C2×C4—31no—
478C2×C4—31no—
479C2×C4—32no—
480C2×C4—92no—
481C2×C4—91no—
482C2×C4—182no—
483C2×C4—271no—
484C2×C4—362no—
485C2×C4—542no—
486C2×C4—1082no—
487C2×C4C2×D3×D912yes—
488C2×C4C2×D3×D911yescharacteristic
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—182no—
493C2×C3C2×#370111yescharacteristic
494C2×C3#41×D911yescharacteristic
495C2×C3#6371511yescharacteristic
496C2×C3C2×#379911yescharacteristic
497C2×C3#6431411yescharacteristic
498C2×C3D3×#23911yescharacteristic
499D3—62no—
500D3—362no—
501D3—361no—
502D3—362no—
503D3—361no—
504D3—362no—
505D3—722no—
506C22—92no—
507C22—181no—
508C22—181no—
509C22—542no—
510C22—541no—
511C22—541no—
512C22—542no—
513C22—541no—
514C22—542no—
515C22C22×D3×D911yescharacteristic
516C4—22no—
517C4—61no—
518C4—61no—
519C4—62no—
520C4—182no—
521C4—181no—
522C4—541no—
523C4#8863512yes—
524C3—21no—
525C3#1058979711yescharacteristic, complemented
526C3#1059456211yescharacteristic
527C2—182no—
528C2—1081no—
529C2—1082no—
530C2C2×#8863511yescharacteristic
531C2#1172250111yescharacteristic
532C2D9×#23911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#553
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721751C2semidirect
2. C3×C4⋊C4×D9C2semidirect
3. C9×#890C2semidirect
4. #11721004C2semidirect
5. #11721004C2semidirect
6. #11721002C2semidirect
7. C4×#6192C2semidirect
8. #11720997C2semidirect
9. #11720993C2semidirect
10. #11720922C2semidirect
11. #11720920C2semidirect
57. C4⋊C4×D9D3semidirect
170. #890D9semidirect
334. C3×C9#553semidirect
469. C9#5514semidirect
525. C3#10589797semidirect

Maximal subgroups

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

subgroupindex
#117217512
C3×C4⋊C4×D92
C9×#8902
#117210042
#117210022
C4×#61922
#117209972
#117209932
#117209222
#117209202
#105945623
#105897973