Finite Group Databasebeta

← #13141038

Subgroups — #13141038

Order 1728.

572 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×#1009C211yesmaximal, characteristic, complemented
2#11721867C211yesmaximal, characteristic, complemented
3#11721040C211yesmaximal, characteristic, complemented
4#11720885C211yesmaximal, characteristic, complemented
5C3×#63396C211yesmaximal, characteristic, complemented
6#11720301C211yesmaximal, characteristic, complemented
7#11720296C211yesmaximal, characteristic, complemented
8#10591666—31nomaximal
9#10588922—31nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
12C9×#239—21no—
13D3×D4×C9—22no—
14D3×D4×C9—21no—
15C9×#235—21no—
16C9×#235C2211yescharacteristic
17C2×C9×D12—21no—
18C2×C9×D12C2211yescharacteristic
19C2×#6195—22no—
20#88548—21no—
21#88544—21no—
22#88532C2211yescharacteristic, complemented
23C3×C4×Dic9C2211yescharacteristic, complemented
24C3×#3667C2211yescharacteristic, complemented
25#88426—21no—
26#88422—21no—
27C3×#1009D311yescharacteristic
28#64158—31no—
29#63911—31no—
30#63825—31no—
31C9×H(5,𝔽2)—31no—
32C3×#837—31no—
33#63569—31no—
34#63558—31no—
35#63512—31no—
36#63512—31no—
37#63512—61no—
38#63502—31no—
39#63396—31no—
40#63396D311yescharacteristic, complemented
41#63391—31no—
42C22×C3⋊D9—21no—
43C22×D3×C9—22no—
44C22×D3×C9D412yes—
45C3×Q8×C9D411yescharacteristic, complemented
46C3×D4×C9—21no—
47C3×D4×C9D411yescharacteristic, complemented
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)—21no—
52C9×#67—22no—
53C9×#67—21no—
54C2×C3×Dic9—21no—
55C9×D12—22no—
56C9×D12—21no—
57C9×D12—21no—
58C9×D12D411yescharacteristic, complemented
59C4×D3×C9—22no—
60C4×D3×C9—21no—
61C9×Dic6D411yescharacteristic, complemented
62#6195—44no—
63C3⋊(C9⋊C8)—21no—
64C3×C9⋊C8—21no—
65#5241—91no—
66#4822—91no—
67C32×#41C2×D311yescharacteristic
68C2×C4⋊(C3⋊D3)—31no—
69C3×#239—21no—
70C3×D3×D4—22no—
71C3×D3×D4—21no—
72C3×#235—21no—
73C3×#235C2×D311yescharacteristic
74C2×C3×D12—21no—
75C2×C3×D12C2×D311yescharacteristic
76C2×#626—62no—
77#3755—61no—
78#3753—61no—
79#3748—31no—
80C3×C4×Dic3—31no—
81C3×#208—31no—
82#3717—61no—
83#3714—61no—
84C9×#41—21no—
85C9×#41—31no—
86C9×#41—61no—
87C9×#41—61no—
88C9×#41C2×D311yescharacteristic
89C2×D4×C9—31no—
90C2×D4×C9—61no—
91C2×D4×C9—62no—
92C2×D4×C9—61no—
93C2×#611—62no—
94C2×D36—31no—
95C2×D36—61no—
96#3675—61no—
97#3675—62no—
98#3675—61no—
99#3673—61no—
100#3673—62no—
101#3673—61no—
102C4×Dic9C2×D311yescharacteristic, complemented
103#3667—31no—
104#3667—61no—
105#3667C2×D311yescharacteristic, complemented
106C22×C3×C9—21no—
107C22×C3×C9C2×D411yescharacteristic
108C2×C3⋊D9—42no—
109C2×C3⋊D9—41no—
110C2×D3×C9—22no—
111C2×D3×C9—21no—
112C2×D3×C9—42no—
113C2×D3×C9—82no—
114C3×C4×C9—21no—
115C3×C4×C9C2×D411yescharacteristic
116C3×C4×C9C2×D411yescharacteristic
117C9×Dic3—21no—
118C3×Dic9—22no—
119C3×H(5,𝔽2)—31no—
120#1009D911yescharacteristic, complemented
121#949—91no—
122#949—91no—
123#949—91no—
124#949—181no—
125#940—91no—
126#908—91no—
127#874—91no—
128C3×#104—91no—
129#837—31no—
130#837—91no—
131#833—91no—
132#823—91no—
133#805—91no—
134C22×C3⋊D3—61no—
135C22×C3×D3—22no—
136C22×C3×D3#6712yes—
137C32×Q8#6711yescharacteristic
138C32×D4—21no—
139C32×D4#6711yescharacteristic
140C2×C32×C4—21no—
141C2×C32×C4C22×D311yescharacteristic
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—61no—
144C3×#67—22no—
145C3×#67—21no—
146C2×C3×Dic3—61no—
147C3×D12—22no—
148C3×D12—21no—
149C3×D12—21no—
150C3×D12#6711yescharacteristic
151C3×C4×D3—22no—
152C3×C4×D3—21no—
153C3×Dic6#6711yescharacteristic
154#626—124no—
155C23×C9—32no—
156C23×C9—62no—
157C22×D9—61no—
158C22×D9—121no—
159C3⋊(C3⋊C8)—61no—
160C3×C3⋊C8—61no—
161Q8×C9—21no—
162Q8×C9—31no—
163Q8×C9D1211yescharacteristic, complemented
164D4×C9—21no—
165D4×C9—21no—
166D4×C9—31no—
167D4×C9—41no—
168D4×C9—62no—
169D4×C9—62no—
170D4×C9—61no—
171D4×C9—61no—
172D4×C9—61no—
173D4×C9D1211yescharacteristic, complemented
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—41no—
177C2×C4×C9—62no—
178C2×C4×C9—61no—
179C2×C4×C9C22×D311yescharacteristic
180#611—124no—
181C2×Dic9—21no—
182D36—62no—
183D36—61no—
184D36—61no—
185D36—121no—
186C9⋊C8—21no—
187C9⋊C8—62no—
188C9⋊C8—61no—
189#453—271noSylow
190C2×C3×C9—21no—
191C2×C3×C9—41no—
192C2×C3×C9#12011yescharacteristic
193C3⋊D9—81no—
194D3×C9—42no—
195D3×C9—41no—
196C3×#41—21no—
197C3×#41—31no—
198C3×#41—61no—
199C3×#41—61no—
200C3×#41C2×D911yescharacteristic
201C3×#41D3211yescharacteristic
202C2×C3×D4—31no—
203C2×C3×D4—61no—
204C2×C3×D4—62no—
205C2×C3×D4—61no—
206C2×#67—182no—
207#239—21no—
208D3×D4—22no—
209D3×D4—21no—
210#235—21no—
211#235C2×D911yescharacteristic
212C2×D12—21no—
213C2×D12—91no—
214C2×D12—91no—
215C2×D12—181no—
216C2×D12—182no—
217C2×D12C2×D911yescharacteristic
218C3×M16—91no—
219C3×C42—91no—
220#215—181no—
221#215—182no—
222#215—181no—
223#215—181no—
224#213—181no—
225#213—182no—
226#213—181no—
227#213—181no—
228C4×Dic3—31no—
229#208—31no—
230#208—91no—
231#208—91no—
232#208—181no—
233D24—181no—
234#204—181no—
235C22×C32—21no—
236C22×C32C2×#6711yescharacteristic
237C2×C3⋊D3—121no—
238C2×C3⋊D3—122no—
239C2×C3×D3—22no—
240C2×C3×D3—21no—
241C2×C3×D3—42no—
242C2×C3×D3—82no—
243C32×C4—21no—
244C32×C4C2×#6711yescharacteristic
245C32×C4C2×#6711yescharacteristic
246C3×Dic3—21no—
247C3×Dic3—62no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—41no—
251C22×C9—61no—
252C22×C9—62no—
253C22×C9—122no—
254C22×C9—242no—
255C22×C9D3×D411yescharacteristic
256C2×D9—122no—
257C2×D9—122no—
258C2×D9—121no—
259C2×D9—242no—
260C4×C9—21no—
261C4×C9—21no—
262C4×C9—21no—
263C4×C9—41no—
264C4×C9—61no—
265C4×C9C2×D1211yescharacteristic
266C4×C9D3×D411yescharacteristic
267Dic9—22no—
268H(5,𝔽2)—31no—
269AGL(1,ℤ/8)—271no—
270AGL(1,ℤ/8)—271no—
271#127—271no—
272#104—91no—
273#104—271no—
274#100—271no—
275C3×C9#45311yescharacteristic, complemented, Sylow
276C23×C3—32no—
277C23×C3—62no—
278C22×D3—22no—
279C22×D3—181no—
280C22×D3—181no—
281C22×D3—361no—
282C22×D3#61112yes—
283C3×Q8—21no—
284C3×Q8—31no—
285C3×Q8#61111yescharacteristic, complemented
286C3×Q8#62611yescharacteristic
287C3×D4—21no—
288C3×D4—21no—
289C3×D4—21no—
290C3×D4—31no—
291C3×D4—41no—
292C3×D4—62no—
293C3×D4—62no—
294C3×D4—61no—
295C3×D4—61no—
296C3×D4—61no—
297C3×D4#61111yescharacteristic, complemented
298C3×D4#62611yescharacteristic
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—62no—
304C2×C3×C4—61no—
305C2×C3×C4—181no—
306C2×C3×C4C22×D911yescharacteristic
307C2×C3×C4C2×D3211yescharacteristic
308#67—22no—
309#67—21no—
310#67—364no—
311C2×Dic3—61no—
312D12—22no—
313D12—21no—
314D12—21no—
315D12—182no—
316D12—181no—
317D12—181no—
318D12—181no—
319D12—181no—
320D12—361no—
321D12—364no—
322D12#61111yescharacteristic, complemented
323C4×D3—22no—
324C4×D3—21no—
325Dic6#61111yescharacteristic, complemented
326C3×C8—181no—
327C3⋊C8—61no—
328C3⋊C8—182no—
329C3⋊C8—181no—
330C3⋊C8—181no—
331C2×C32—21no—
332C2×C32—41no—
333C2×C32#95311yescharacteristic
334C3⋊D3—241no—
335C3×D3—42no—
336C3×D3—41no—
337C2×C9—21no—
338C2×C9—21no—
339C2×C9—41no—
340C2×C9—41no—
341C2×C9—81no—
342C2×C9—122no—
343C2×C9—121no—
344C2×C9#88211yescharacteristic
345D9—242no—
346D9—241no—
347#41—31no—
348#41—61no—
349#41—61no—
350#41D3×D911yescharacteristic
351C2×D4—31no—
352C2×D4—61no—
353C2×D4—62no—
354C2×D4—61no—
355C2×D4—271no—
356C2×D4—542no—
357SD16—541no—
358SD16—541no—
359D8—541no—
360D8—541no—
361M16—91no—
362M16—271no—
363C42—91no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—61no—
369C22×C3—62no—
370C22×C3—122no—
371C22×C3—242no—
372C22×C3C2×#61111yescharacteristic
373C22×C3D3×#6711yescharacteristic
374C2×D3—22no—
375C2×D3—21no—
376C2×D3—42no—
377C2×D3—82no—
378C2×D3—362no—
379C2×D3—362no—
380C2×D3—362no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—722no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—61no—
390C3×C4—182no—
391C3×C4C2×#61111yescharacteristic
392C3×C4C2×#61111yescharacteristic
393C3×C4C2×#62611yescharacteristic
394C3×C4D3×#6711yescharacteristic
395Dic3—21no—
396Dic3—62no—
397C32#524111yescharacteristic
398C9—21no—
399C9#482211yescharacteristic, complemented
400C23—32no—
401C23—62no—
402C23—541no—
403Q8—31no—
404Q8#619511yescharacteristic, complemented
405D4—21no—
406D4—31no—
407D4—62no—
408D4—62no—
409D4—61no—
410D4—61no—
411D4—61no—
412D4—541no—
413D4—541no—
414D4—1084no—
415D4#619511yescharacteristic, complemented
416C2×C4—21no—
417C2×C4—62no—
418C2×C4—61no—
419C2×C4—181no—
420C2×C4C2×D3×D911yescharacteristic
421C8—181no—
422C8—541no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—81no—
430C2×C3—122no—
431C2×C3—121no—
432C2×C3#6351511yescharacteristic
433C2×C3#6397711yescharacteristic
434D3—42no—
435D3—41no—
436D3—722no—
437D3—721no—
438D3—721no—
439C22—21no—
440C22—61no—
441C22—62no—
442C22—122no—
443C22—242no—
444C22—1082no—
445C22—1081no—
446C22D3×#61111yescharacteristic
447C4—21no—
448C4—61no—
449C4—182no—
450C4C2×#619511yescharacteristic
451C4D3×#61111yescharacteristic
452C3—21no—
453C3#1058892211yescharacteristic, complemented
454C3#1059166611yescharacteristic
455C2—21no—
456C2—41no—
457C2—122no—
458C2—121no—
459C2—2161no—
460C2#1172128711yescharacteristic
subgroupG/H
centerC2#11721287
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×#41C22
socleC2×C32#953
hypercenter#41D3×D9
hypocenterC3×C9#453
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C9×#1009C2semidirect
2. #11721867C2semidirect
3. #11721040C2semidirect
4. #11720885C2semidirect
5. C3×#63396C2semidirect
6. #11720301C2semidirect
7. #11720296C2semidirect
11. C2×C4⋊(C3⋊D9)C22semidirect
22. #88532C22semidirect
23. C3×C4×Dic9C22semidirect
24. C3×#3667C22semidirect
40. #63396D3semidirect
45. C3×Q8×C9D4semidirect
47. C3×D4×C9D4semidirect
58. C9×D12D4semidirect
61. C9×Dic6D4semidirect
102. C4×Dic9C2×D3semidirect
105. #3667C2×D3semidirect
120. #1009D9semidirect
163. Q8×C9D12semidirect
173. D4×C9D12semidirect
275. C3×C9#453semidirect
285. C3×Q8#611semidirect
297. C3×D4#611semidirect
322. D12#611semidirect
325. Dic6#611semidirect
399. C9#4822semidirect
404. Q8#6195semidirect
415. D4#6195semidirect
453. C3#10588922semidirect

Maximal subgroups

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

subgroupindex
C9×#10092
#117218672
#117210402
#117208852
C3×#633962
#117203012
#117202962
#105916663
#105889223