Finite Group Databasebeta

← #13137624

Subgroups — #13137624

Order 1728.

446 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#11721753C211yesmaximal, characteristic, complemented
2#11721007C211yesmaximal, characteristic
3C2×#88415C211yesmaximal, characteristic, complemented
4C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6C9×#808C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590507—31nomaximal
9#10588544—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×D36C2211yescharacteristic
13C2×#6192—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C4⋊(C3⋊Dic9)—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21#88415—22no—
22(C3⋊D9)⋊C8—22no—
23#64106—31no—
24#63916—31no—
25C2×#3716—31no—
26C2×#3710—31no—
27C3×#809—31no—
28C3×#808D311yescharacteristic
29#63567—31no—
30C2×#3664—31no—
31C2×C8×D9—31no—
32#63459—31no—
33#63459—61no—
34#63457—31no—
35C9×#106—31no—
36#63369D311yescharacteristic, complemented
37#63368—31no—
38C22×C3⋊D9D411yescharacteristic
39C22×C3×D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C4×C3⋊D9—21no—
45C4×C3⋊D9—21no—
46#6226—21no—
47#6226—21no—
48C2×C9×Dic3—21no—
49C3×D36—21no—
50C3×D36—21no—
51#6192—42no—
52C9×C3⋊C8D412yescomplemented
53C3×C9⋊C8—21no—
54#4864—91no—
55#4778—91no—
56C2×C4×C3⋊D3—31no—
57C2×#634—31no—
58C2×C3×D12—31no—
59C2×#626—61no—
60C32×C4⋊C4C2×D311yescharacteristic
61#3752—61no—
62C4⋊(C3⋊Dic3)—61no—
63C3×C4⋊Dic3C2×D311yescharacteristic
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66(C2×D3)⋊Dic3—61no—
67#3716—62no—
68#3710—62no—
69C2×D36—61no—
70C2×D36C2×D311yescharacteristic
71C2×C4×D9—31no—
72C2×C4×D9—61no—
73C2×Dic18—31no—
74C2×Dic18—61no—
75C2×C8×C9—31no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4—31no—
78C9×C4⋊C4C2×D311yescharacteristic
79(C2×D9)⋊C4—61no—
80(C2×D9)⋊C4—61no—
81(C2×D9)⋊C4—62no—
82C4⋊Dic9—61no—
83C4⋊Dic9—62no—
84C2×C9⋊C8C2×D311yescharacteristic
85#3664—62no—
86C8×D9—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—21no—
89C2×C3⋊D9—21no—
90C2×C3⋊D9SD1612yescomplemented
91C2×C3×D9—41no—
92C2×C3×D9—41no—
93C2×C3×D9—41no—
94C3×C4×C9—41no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9#4111yescharacteristic
97C3⋊Dic9—21no—
98C3⋊Dic9—41no—
99C9×Dic3—41no—
100#938—91no—
101C2×#214—91no—
102C2×#204—91no—
103C2×D3×C8—91no—
104C2×D3×C8—91no—
105#897—91no—
106#897—91no—
107#897—181no—
108#895—91no—
109C3×#106—31no—
110C3×#102—91no—
111#816—91no—
112#810—91no—
113#809—31no—
114#808D911yescharacteristic, complemented
115C22×C3⋊D3—31no—
116C22×C3×D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—61no—
121C4×C3⋊D3—61no—
122C4×C3⋊D3—61no—
123#634—61no—
124#634—61no—
125C2×C3×Dic3—21no—
126C3×D12—61no—
127C3×D12—61no—
128#626—122no—
129C22×D9—21no—
130C22×D9—31no—
131C22×D9—61no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8D1212yes—
134C2×C4×C9—22no—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9—31no—
140C2×Dic9—61no—
141C2×Dic9—62no—
142C2×Dic9—61no—
143D36—21no—
144D36—21no—
145D36—122no—
146C4×D9—61no—
147C4×D9—61no—
148C4×D9—121no—
149C4×D9—121no—
150Dic18—61no—
151Dic18—61no—
152Dic18—121no—
153Dic18—121no—
154C8×C9—32no—
155C9⋊C8—21no—
156#465—271noSylow
157C2×C3×C9#12111yescharacteristic
158C2×C3×C9C2×SD1611yescharacteristic
159C2×C3×C9#13511yescharacteristic
160C3⋊D9—22no—
161C3×D9—81no—
162C2×C3×D4—91no—
163C2×#67—181no—
164C2×D12—31no—
165C2×D12—181no—
166C2×C4×D3—91no—
167C2×C4×D3—91no—
168C2×C4×D3—181no—
169C2×Dic6—91no—
170C2×Dic6—91no—
171C2×Dic6—181no—
172C2×C3×C8—31no—
173C2×C3×C8—91no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4C2×D911yescharacteristic
177C3×C4⋊C4D3211yescharacteristic
178#217—181no—
179#214—182no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—182no—
184C4⋊Dic3—181no—
185C4⋊Dic3—181no—
186C4⋊Dic3—182no—
187C4⋊Dic3C2×D911yescharacteristic
188C2×C3⋊C8—31no—
189C2×C3⋊C8C2×D911yescharacteristic
190#204—182no—
191D3×C8—182no—
192D3×C8—182no—
193C22×C32D3×D411yescharacteristic
194C2×C3⋊D3—32no—
195C2×C3⋊D3—61no—
196C2×C3⋊D3—61no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—41no—
201C32×C4C2×D1211yescharacteristic
202C32×C4#23911yescharacteristic
203C3⋊Dic3—61no—
204C3⋊Dic3—121no—
205C3×Dic3—41no—
206C22×C9—21no—
207C22×C9D3×D411yescharacteristic
208C2×D9—32no—
209C2×D9—41no—
210C2×D9—41no—
211C2×D9—41no—
212C2×D9—62no—
213C2×D9—61no—
214C2×D9—61no—
215C2×D9—121no—
216C2×D9—121no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—42no—
220C4×C9—41no—
221C4×C9—121no—
222C4×C9#23511yescharacteristic
223C4×C9D3×D411yescharacteristic
224Dic9—61no—
225Dic9—121no—
226Dic9—122no—
227Dic9—121no—
228C2×SD16—271no—
229C22×C8—271no—
230C22⋊Q8—271no—
231#121—271no—
232#106—31no—
233#103—271no—
234#102—91no—
235C3×C9#46511yescharacteristic, complemented, Sylow
236C23×C3—181no—
237C22×D3—61no—
238C22×D3—91no—
239C22×D3—91no—
240C22×D3—181no—
241C3×D4—181no—
242C3×D4—181no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—61no—
248C2×C3×C4C22×D911yescharacteristic
249C2×C3×C4C2×D3211yescharacteristic
250#67—362no—
251C2×Dic3—21no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—181no—
255C2×Dic3—182no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258D12—61no—
259D12—61no—
260D12—362no—
261C4×D3—181no—
262C4×D3—181no—
263C4×D3—181no—
264C4×D3—181no—
265C4×D3—361no—
266C4×D3—361no—
267Dic6—181no—
268Dic6—181no—
269Dic6—181no—
270Dic6—181no—
271Dic6—361no—
272Dic6—361no—
273C3×C8—32no—
274C3×C8—181no—
275C3⋊C8—61no—
276C3⋊C8D3612yescomplemented
277C2×C32#89511yescharacteristic
278C2×C32D3×SD1611yescharacteristic
279C2×C32#91611yescharacteristic
280C3⋊D3—62no—
281C3×D3—241no—
282C2×C9—21no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9#88411yescharacteristic
286C2×C9#91211yescharacteristic
287C2×C9D3×SD1611yescharacteristic
288D9—62no—
289D9—81no—
290D9—122no—
291C2×Q8—271no—
292C2×D4—91no—
293C2×D4—541no—
294C22×C4—271no—
295SD16—542no—
296C2×C8—31no—
297C2×C8—91no—
298C2×C8—542no—
299C4⋊C4—31no—
300C4⋊C4—541no—
301C4⋊C4D3×D911yescharacteristic
302C22⋊C4—541no—
303C22⋊C4—541no—
304C22×C3—21no—
305C22×C3—361no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3D4×D911yescharacteristic
309C22×C3#381111yescharacteristic
310C2×D3—92no—
311C2×D3—92no—
312C2×D3—121no—
313C2×D3—121no—
314C2×D3—121no—
315C2×D3—182no—
316C2×D3—181no—
317C2×D3—181no—
318C2×D3—181no—
319C2×D3—181no—
320C2×D3—361no—
321C2×D3—361no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4—121no—
328C3×C4C2×D3611yescharacteristic
329C3×C4#370111yescharacteristic
330C3×C4#379911yescharacteristic
331C3×C4D3×D1211yescharacteristic
332Dic3—41no—
333Dic3—181no—
334Dic3—181no—
335Dic3—361no—
336Dic3—362no—
337Dic3—361no—
338Dic3—361no—
339C32#486411yescharacteristic
340C9—21no—
341C9#477811yescharacteristic, complemented
342C23—181no—
343C23—271no—
344Q8—541no—
345Q8—541no—
346D4—181no—
347D4—181no—
348D4—1082no—
349C2×C4—21no—
350C2×C4—61no—
351C2×C4—271no—
352C2×C4—541no—
353C2×C4—541no—
354C2×C4—541no—
355C2×C4C2×D3×D911yescharacteristic
356C8—32no—
357C8—181no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—721no—
362C2×C3#6345711yescharacteristic
363C2×C3SD16×D911yescharacteristic
364C2×C3#6347811yescharacteristic
365C2×C3#6391011yescharacteristic
366C2×C3#6393611yescharacteristic
367C2×C3#6394011yescharacteristic
368D3—182no—
369D3—182no—
370D3—241no—
371D3—362no—
372C22—272no—
373C22—361no—
374C22—361no—
375C22—361no—
376C22—541no—
377C22—541no—
378C22#8866811yescharacteristic
379C4—41no—
380C4—121no—
381C4—541no—
382C4—1081no—
383C4#8863511yescharacteristic
384C4D3×D3611yescharacteristic
385C3—21no—
386C3#1058854411yescharacteristic, complemented
387C3#1059050711yescharacteristic
388C2—542no—
389C2—721no—
390C2#1172100511yescharacteristic
391C2#1172118611yescharacteristic
392C2#1172121211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721007C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#465
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721753C2semidirect
3. C2×#88415C2semidirect
4. C2×(C3⋊D9)⋊C8C2semidirect
5. C3×#63369C2semidirect
6. C9×#808C2semidirect
7. #11720269C2semidirect
36. #63369D3semidirect
52. C9×C3⋊C8D4semidirect
90. C2×C3⋊D9SD16semidirect
114. #808D9semidirect
235. C3×C9#465semidirect
276. C3⋊C8D36semidirect
341. C9#4778semidirect
386. C3#10588544semidirect

Maximal subgroups

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

subgroupindex
#117217532
#117210072
C2×#884152
C2×(C3⋊D9)⋊C82
C3×#633692
C9×#8082
#117202692
#105905073
#105885443