Finite Group Databasebeta

← #13139520

Subgroups — #13139520

Order 1728.

524 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic
2C2×#88420C211yesmaximal, characteristic, complemented
3C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
4#11720571C211yesmaximal, characteristic, complemented
5C3×#63389C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
8#10591145—31nomaximal
9#10588912—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C9×#67C411yescharacteristic
12C2×C9×#67C411yescharacteristic
13C9×#235—22no—
14C9×#235—22no—
15C2×C9×D12C411yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C9×Dic6C411yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88420—22no—
24D3×C9⋊C8—22no—
25C2×C3×#235D311yescharacteristic
26C2×#3711—31no—
27C2×D3×C3⋊C8—31no—
28C2×C9×#41—31no—
29#63657—31no—
30C3×#830—31no—
31(C2×D3)⋊(C3⋊C8)—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3667—31no—
34C22×C9⋊C8—31no—
35#63389—31no—
36#63389—31no—
37#63389—31no—
38#63389—61no—
39#63389D311yescharacteristic, complemented
40C23×C3×C9C2×C411yescharacteristic
41C22×D3×C9C2×C411yescharacteristic
42C22×D3×C9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2×C411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C9×#67—22no—
48C9×#67—22no—
49C2×C9×Dic3C2×C411yescharacteristic
50C2×C9×Dic3C2×C411yescharacteristic
51C9×D12—22no—
52C4×D3×C9—21no—
53C4×D3×C9—21no—
54C4×D3×C9D412yes—
55C4×D3×C9D412yes—
56C9×Dic6—22no—
57C3⋊(C9⋊C8)—21no—
58C3⋊(C9⋊C8)—21no—
59C3×C9⋊C8—21no—
60C3×C9⋊C8—21no—
61#5234—91no—
62#4727—91no—
63C22×C32×C4C2×D311yescharacteristic
64C2×C3×#67Dic311yescharacteristic
65C2×C3×#67Dic311yescharacteristic
66C3×#235—22no—
67C3×#235—22no—
68C2×C3×D12Dic311yescharacteristic
69C2×C3×C4×D3C2×D311yescharacteristic
70C2×C3×C4×D3C2×D311yescharacteristic
71C2×C3×Dic6Dic311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76#3711—62no—
77D3×C3⋊C8—62no—
78C9×#41—62no—
79C9×#41—62no—
80C2×Q8×C9—31no—
81C2×D4×C9—31no—
82C2×D4×C9—31no—
83C2×D4×C9—31no—
84C22×C4×C9—21no—
85C22×C4×C9—31no—
86C22×C4×C9—31no—
87C22×C4×C9C2×D311yescharacteristic
88#3667—62no—
89C2×C9⋊C8—31no—
90C2×C9⋊C8—31no—
91C2×C9⋊C8—62no—
92C2×C9⋊C8—61no—
93C2×C9⋊C8—61no—
94C2×C9⋊C8C2×D311yescharacteristic
95C2×C9⋊C8C2×D311yescharacteristic
96C22×C3×C9—21no—
97C22×C3×C9—21no—
98C22×C3×C9—21no—
99C22×C3×C9C22×C411yescharacteristic
100C2×D3×C9—21no—
101C2×D3×C9—21no—
102C2×D3×C9—21no—
103C2×D3×C9—21no—
104C2×D3×C9—21no—
105C2×D3×C9C22⋊C412yes—
106C3×C4×C9—21no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C9×Dic3—21no—
110C9×Dic3C22⋊C412yes—
111C2×C3×#41—31no—
112C2×#235D911yescharacteristic
113C2×#208—91no—
114C22×C3⋊C8—91no—
115C2×ΓL(1,𝔽25)—91no—
116C2×D3×C8—91no—
117C3×C22⋊C8—91no—
118#830—31no—
119#830—91no—
120#830—91no—
121#830—91no—
122#830—91no—
123#830—181no—
124(C2×D3)⋊C8—91no—
125(C2×D3)⋊C8—91no—
126C23×C32C2×Dic311yescharacteristic
127C22×C3×D3C2×Dic311yescharacteristic
128C22×C3×D3C2×Dic311yescharacteristic
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4C2×Dic311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C3×#67—22no—
134C3×#67—22no—
135C2×C3×Dic3C2×Dic311yescharacteristic
136C2×C3×Dic3C2×Dic311yescharacteristic
137C3×D12—22no—
138C3×C4×D3—21no—
139C3×C4×D3—21no—
140C3×C4×D3#6712yes—
141C3×C4×D3#6712yes—
142C3×Dic6—22no—
143C23×C9—21no—
144C23×C9—31no—
145C23×C9—31no—
146C23×C9C4×D311yescharacteristic
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—61no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—61no—
151Q8×C9—62no—
152D4×C9—62no—
153D4×C9—62no—
154D4×C9—62no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9—21no—
158C2×C4×C9—21no—
159C2×C4×C9—31no—
160C2×C4×C9—32no—
161C2×C4×C9—32no—
162C2×C4×C9—31no—
163C2×C4×C9—41no—
164C2×C4×C9—41no—
165C2×C4×C9—61no—
166C2×C4×C9—61no—
167C2×C4×C9C4×D311yescharacteristic
168C2×C4×C9C22×D311yescharacteristic
169C9⋊C8—21no—
170C9⋊C8—21no—
171C9⋊C8—61no—
172C9⋊C8—61no—
173C9⋊C8—121no—
174C9⋊C8—121no—
175#408—271noSylow
176C2×C3×C9—41no—
177C2×C3×C9C2×C22⋊C411yescharacteristic
178C2×C3×C9#13111yescharacteristic
179C2×C3×C9#13111yescharacteristic
180D3×C9—22no—
181D3×C9—41no—
182C3×#41—62no—
183C3×#41—62no—
184C2×C3×Q8—31no—
185C2×C3×D4—31no—
186C2×C3×D4—31no—
187C2×C3×D4—31no—
188C22×C3×C4—21no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4C2×D911yescharacteristic
192C22×C3×C4D3211yescharacteristic
193C2×#67Dic911yescharacteristic
194C2×#67Dic911yescharacteristic
195#235—22no—
196#235—22no—
197C2×D12Dic911yescharacteristic
198C2×C4×D3C2×D911yescharacteristic
199C2×C4×D3C2×D911yescharacteristic
200C2×Dic6Dic911yescharacteristic
201C2×C3×C8—91no—
202C2×C3×C8—91no—
203#208—182no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—181no—
211C2×C3⋊C8—181no—
212C2×C3⋊C8—182no—
213ΓL(1,𝔽25)—182no—
214D3×C8—182no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32C22×Dic311yescharacteristic
219C2×C3×D3—21no—
220C2×C3×D3—21no—
221C2×C3×D3—21no—
222C2×C3×D3—21no—
223C2×C3×D3—21no—
224C2×C3×D3#21712yes—
225C32×C4—21no—
226C32×C4C2×#6711yescharacteristic
227C32×C4C2×#6711yescharacteristic
228C3×Dic3—21no—
229C3×Dic3#21712yes—
230C22×C9—22no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—21no—
234C22×C9—21no—
235C22×C9—32no—
236C22×C9—41no—
237C22×C9—41no—
238C22×C9—61no—
239C22×C9—61no—
240C22×C9—61no—
241C22×C9—61no—
242C22×C9—61no—
243C22×C9C2×C4×D311yescharacteristic
244C4×C9—21no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—32no—
248C4×C9—41no—
249C4×C9—61no—
250C4×C9D3×D411yescharacteristic
251C4×C9D3×D411yescharacteristic
252C2×#41—31no—
253C2×M16—271no—
254C22×C8—271no—
255C22⋊C8—91no—
256C22⋊C8—271no—
257C22⋊C8—271no—
258C22⋊C8—271no—
259C3×C9#40811yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—31no—
262C23×C3—31no—
263C23×C3C2×Dic911yescharacteristic
264C23×C3D3×Dic311yescharacteristic
265C22×D3C2×Dic911yescharacteristic
266C22×D3C2×Dic911yescharacteristic
267C3×Q8—62no—
268C3×D4—62no—
269C3×D4—62no—
270C3×D4—62no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—31no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—31no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4C2×Dic911yescharacteristic
286C2×C3×C4C22×D911yescharacteristic
287C2×C3×C4D3×Dic311yescharacteristic
288C2×C3×C4C2×D3211yescharacteristic
289#67—22no—
290#67—22no—
291C2×Dic3C2×Dic911yescharacteristic
292C2×Dic3C2×Dic911yescharacteristic
293D12—22no—
294C4×D3—21no—
295C4×D3—21no—
296C4×D3#61112yes—
297C4×D3#61112yes—
298Dic6—22no—
299C3×C8—181no—
300C3×C8—181no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—181no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C3⋊C8—361no—
308C3⋊C8—361no—
309C2×C32—41no—
310C2×C32#94811yescharacteristic
311C2×C32#94811yescharacteristic
312C2×C32C2×#21711yescharacteristic
313C3×D3—22no—
314C3×D3—41no—
315C2×C9—21no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—42no—
319C2×C9—41no—
320C2×C9—62no—
321C2×C9—121no—
322C2×C9D3×C22⋊C411yescharacteristic
323C2×C9#90111yescharacteristic
324C2×C9#90711yescharacteristic
325#41—62no—
326#41—62no—
327C2×Q8—31no—
328C2×D4—31no—
329C2×D4—31no—
330C2×D4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4D3×D911yescharacteristic
334M16—542no—
335C2×C8—91no—
336C2×C8—91no—
337C2×C8—271no—
338C2×C8—271no—
339C2×C8—542no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—32no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—61no—
356C22×C3C22×Dic911yescharacteristic
357C22×C3C2×D3×Dic311yescharacteristic
358C2×D3—21no—
359C2×D3—21no—
360C2×D3—21no—
361C2×D3—21no—
362C2×D3—21no—
363C2×D3#367612yes—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—32no—
369C3×C4—41no—
370C3×C4—61no—
371C3×C4C2×#61111yescharacteristic
372C3×C4C2×#61111yescharacteristic
373C3×C4D3×#6711yescharacteristic
374C3×C4D3×#6711yescharacteristic
375Dic3—21no—
376Dic3#367612yes—
377C32#523411yescharacteristic
378C9—21no—
379C9#472711yescharacteristic, complemented
380C23—31no—
381C23—31no—
382C23D3×Dic911yescharacteristic
383Q8—62no—
384D4—62no—
385D4—62no—
386D4—62no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—31no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—31no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4D3×Dic911yescharacteristic
396C2×C4C2×D3×D911yescharacteristic
397C8—181no—
398C8—181no—
399C8—541no—
400C8—541no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—42no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—62no—
408C2×C3—121no—
409C2×C3#6351011yescharacteristic
410C2×C3#6351011yescharacteristic
411C2×C3C2×#367611yescharacteristic
412C2×C3#6381411yescharacteristic
413C2×C3#6381511yescharacteristic
414C2×C3D3×#21711yescharacteristic
415D3—22no—
416D3—41no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—32no—
421C22—61no—
422C22—61no—
423C22—61no—
424C22—61no—
425C22—61no—
426C22C2×D3×Dic911yescharacteristic
427C4—21no—
428C4—32no—
429C4—61no—
430C4D3×#61111yescharacteristic
431C4D3×#61111yescharacteristic
432C3—21no—
433C3#1058891211yescharacteristic, complemented
434C3#1059114511yescharacteristic
435C2—41no—
436C2—62no—
437C2—121no—
438C2#1172087711yescharacteristic
439C2#1172087811yescharacteristic
440C2D3×#367611yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C22×Dic3
prime radicalC2×C3×#235D3
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#408
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88420C2semidirect
3. C2×D3×C9⋊C8C2semidirect
4. #11720571C2semidirect
5. C3×#63389C2semidirect
6. (C2×D3)⋊(C9⋊C8)C2semidirect
7. (C2×D3)⋊(C9⋊C8)C2semidirect
39. #63389D3semidirect
259. C3×C9#408semidirect
379. C9#4727semidirect
433. C3#10588912semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C2×#884202
C2×D3×C9⋊C82
#117205712
C3×#633892
(C2×D3)⋊(C9⋊C8)2
#105911453
#105889123