Finite Group Databasebeta

← #13137656

Subgroups — #13137656

Order 1728.

496 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 97 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#11721749C211yesmaximal, characteristic, complemented
2#11720998C211yesmaximal, characteristic, complemented
3C2×#88417C211yesmaximal, characteristic
4C3×#63369C211yesmaximal, characteristic, complemented
5C9×#807C211yesmaximal, characteristic, complemented
6#11720271C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590459—31nomaximal
9#10588552—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×C3×D36C2211yescharacteristic
13C2×Dic3×D9—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21#88417C414yescomplemented
22#64102—31no—
23#63898—31no—
24C2×#3714—31no—
25C3×#809—31no—
26C3×#807D311yescharacteristic
27#63564—31no—
28#63553—31no—
29C2×D72—31no—
30#63455—31no—
31#63455—31no—
32#63455—61no—
33C9×#107—31no—
34C8×Dic9—31no—
35#63369—31no—
36#63369D311yescharacteristic, complemented
37C22×C3⋊D9—21no—
38C22×C3×D9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C4⋊(C3⋊D9)—21no—
44C4⋊(C3⋊D9)C2×C412yescomplemented
45C4×C3⋊D9—42no—
46C2×C9×Dic3—21no—
47C3×D36—21no—
48C3×D36C2×C412yescomplemented
49Dic3×D9—42no—
50C9×C3⋊C8C2×C412yescomplemented
51C3×C9⋊C8—21no—
52#4872—91no—
53#4756—91no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4×C3⋊D3—61no—
56C2×C3×D12—31no—
57C2×D3×Dic3—61no—
58C32×C4⋊C4C2×D311yescharacteristic
59#3752—61no—
60C4×C3⋊Dic3—31no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64(C2×D3)⋊Dic3—61no—
65#3714—34no—
66C2×D36—31no—
67C2×D36—61no—
68C2×D36C2×D311yescharacteristic
69C2×C4×D9—61no—
70C2×C4×D9—61no—
71C2×C4×D9—62no—
72C2×C8×C9—31no—
73C9×C4⋊C4—21no—
74C9×C4⋊C4—31no—
75C9×C4⋊C4C2×D311yescharacteristic
76(C2×D9)⋊C4—61no—
77(C2×D9)⋊C4—61no—
78(C2×D9)⋊C4—62no—
79C4×Dic9—31no—
80C4×Dic9—61no—
81C2×C9⋊C8C2×D311yescharacteristic
82D72—34no—
83C22×C3×C9C2×D411yescharacteristic
84C2×C3⋊D9—22no—
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C2×C3×D9—22no—
88C2×C3×D9—41no—
89C2×C3×D9—41no—
90C3×C4×C9—41no—
91C3×C4×C9C22×C411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—21no—
94C3⋊Dic9D812yescomplemented
95C9×Dic3—41no—
96D4×Dic3—91no—
97C2×#213—91no—
98C2×D24—91no—
99#893—91no—
100#893—91no—
101#893—91no—
102#893—181no—
103C3×#107—31no—
104C3×#102—91no—
105#821—91no—
106C8×Dic3—91no—
107C8×Dic3—91no—
108#809—31no—
109#809—91no—
110#807D911yescharacteristic, complemented
111C22×C3⋊D3—61no—
112C22×C3×D3—61no—
113C2×C32×C4—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C4⋊(C3⋊D3)—32no—
118C4⋊(C3⋊D3)—61no—
119C4×C3⋊D3—122no—
120C2×C3×Dic3—21no—
121C3×D12—32no—
122C3×D12—61no—
123D3×Dic3—122no—
124C22×D9—21no—
125C22×D9—62no—
126C22×D9—61no—
127C3×C3⋊C8—61no—
128C3×C3⋊C8C4×D312yes—
129C2×C4×C9—22no—
130C2×C4×C9—21no—
131C2×C4×C9—21no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—31no—
135C2×Dic9—31no—
136C2×Dic9—61no—
137C2×Dic9—61no—
138D36—21no—
139D36—32no—
140D36—62no—
141D36—62no—
142D36—61no—
143D36C4×D312yescomplemented
144C4×D9—122no—
145C4×D9—122no—
146C4×D9—124no—
147C8×C9—32no—
148C9⋊C8—21no—
149C4×D8—271noSylow
150C2×C3×C9C4×D411yescharacteristic
151C2×C3×C9C2×D811yescharacteristic
152C2×C3×C9#13511yescharacteristic
153C3⋊D9—42no—
154C3×D9—42no—
155C2×C3×D4—91no—
156C22×Dic3—181no—
157C2×D12—31no—
158C2×D12—91no—
159C2×D12—91no—
160C2×D12—181no—
161C2×C4×D3—181no—
162C2×C4×D3—181no—
163C2×C4×D3—181no—
164C2×C4×D3—182no—
165C2×C3×C8—31no—
166C2×C3×C8—91no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4C2×D911yescharacteristic
170C3×C4⋊C4D3211yescharacteristic
171#217—181no—
172#213—94no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—182no—
177C4⋊Dic3C2×D911yescharacteristic
178C4×Dic3—91no—
179C4×Dic3—91no—
180C4×Dic3—181no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8C2×D911yescharacteristic
183D24—94no—
184C22×C32D3×D411yescharacteristic
185C2×C3⋊D3—62no—
186C2×C3⋊D3—121no—
187C2×C3⋊D3—121no—
188C2×C3×D3—62no—
189C2×C3×D3—121no—
190C2×C3×D3—121no—
191C32×C4—41no—
192C32×C4C2×C4×D311yescharacteristic
193C32×C4#23911yescharacteristic
194C3⋊Dic3—32no—
195C3⋊Dic3—61no—
196C3×Dic3—41no—
197C22×C9—21no—
198C22×C9D3×D411yescharacteristic
199C2×D9—22no—
200C2×D9—41no—
201C2×D9—41no—
202C2×D9—64no—
203C2×D9—62no—
204C2×D9—122no—
205C2×D9—122no—
206C2×D9—121no—
207C2×D9—121no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—42no—
211C4×C9—41no—
212C4×C9—121no—
213C4×C9C2×C4×D311yescharacteristic
214C4×C9#23711yescharacteristic
215Dic9—32no—
216Dic9—62no—
217Dic9—61no—
218Dic9—121no—
219C2×D8—271no—
220C4×D4—271no—
221C4×D4—271no—
222#107—31no—
223#102—91no—
224#102—271no—
225C4×C8—271no—
226C3×C9C4×D811yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C22×D3—182no—
230C22×D3—181no—
231C22×D3—181no—
232C3×D4—92no—
233C3×D4—181no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—61no—
239C2×C3×C4C22×D911yescharacteristic
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—21no—
242C2×Dic3—91no—
243C2×Dic3—91no—
244C2×Dic3—91no—
245C2×Dic3—91no—
246C2×Dic3—181no—
247C2×Dic3—181no—
248C2×Dic3—362no—
249D12—32no—
250D12—61no—
251D12—92no—
252D12—92no—
253D12—182no—
254D12—182no—
255D12—181no—
256D12—181no—
257C4×D3—362no—
258C4×D3—362no—
259C4×D3—362no—
260C4×D3—364no—
261C3×C8—32no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8C4×D912yescomplemented
265C2×C32#89311yescharacteristic
266C2×C32D3×D811yescharacteristic
267C2×C32#91911yescharacteristic
268C3⋊D3—122no—
269C3×D3—122no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9#67⋊C411yescharacteristic
274C2×C9D3×D811yescharacteristic
275C2×C9#91611yescharacteristic
276D9—42no—
277D9—124no—
278D9—122no—
279C2×D4—91no—
280C2×D4—271no—
281C22×C4—541no—
282C22×C4—541no—
283D8—274no—
284C2×C8—31no—
285C2×C8—91no—
286C4⋊C4—31no—
287C4⋊C4D3×D911yescharacteristic
288C22⋊C4—541no—
289C22⋊C4—541no—
290C42—271no—
291C22×C3—21no—
292C22×C3—182no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3D4×D911yescharacteristic
296C22×C3#381111yescharacteristic
297C2×D3—62no—
298C2×D3—121no—
299C2×D3—121no—
300C2×D3—184no—
301C2×D3—182no—
302C2×D3—182no—
303C2×D3—362no—
304C2×D3—362no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—361no—
308C2×D3—361no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—121no—
315C3×C4C2×C4×D911yescharacteristic
316C3×C4#370111yescharacteristic
317C3×C4#379611yescharacteristic
318C3×C4C4×D3211yescharacteristic
319Dic3—41no—
320Dic3—92no—
321Dic3—92no—
322Dic3—182no—
323Dic3—181no—
324Dic3—181no—
325Dic3—361no—
326C32#487211yescharacteristic
327C9—21no—
328C9#475611yescharacteristic, complemented
329C23—181no—
330C23—541no—
331D4—92no—
332D4—181no—
333D4—272no—
334D4—541no—
335C2×C4—21no—
336C2×C4—61no—
337C2×C4—271no—
338C2×C4—271no—
339C2×C4—1082no—
340C2×C4—1082no—
341C2×C4C2×D3×D911yescharacteristic
342C8—32no—
343C8—181no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—362no—
348C2×C3#6345511yescharacteristic
349C2×C3D8×D911yescharacteristic
350C2×C3#6348111yescharacteristic
351C2×C3#6388111yescharacteristic
352C2×C3#6392611yescharacteristic
353C2×C3#6395011yescharacteristic
354D3—122no—
355D3—364no—
356D3—362no—
357D3—362no—
358C22—182no—
359C22—361no—
360C22—361no—
361C22—542no—
362C22—1081no—
363C22—1081no—
364C22#8866811yescharacteristic
365C4—41no—
366C4—121no—
367C4—272no—
368C4—541no—
369C4#8863411yescharacteristic
370C4C4×D3×D911yescharacteristic
371C3—21no—
372C3#1058855211yescharacteristic, complemented
373C3#1059045911yescharacteristic
374C2—362no—
375C2—1082no—
376C2#1172097811yescharacteristic
377C2#1172117411yescharacteristic
378C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88417C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×D8
LCSG ▷ 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. #11721749C2semidirect
2. #11720998C2semidirect
4. C3×#63369C2semidirect
5. C9×#807C2semidirect
6. #11720271C2semidirect
7. #11720267C2semidirect
21. #88417C4semidirect
36. #63369D3semidirect
44. C4⋊(C3⋊D9)C2×C4semidirect
48. C3×D36C2×C4semidirect
50. C9×C3⋊C8C2×C4semidirect
94. C3⋊Dic9D8semidirect
110. #807D9semidirect
143. D36C4×D3semidirect
226. C3×C9C4×D8semidirect
264. C3⋊C8C4×D9semidirect
328. C9#4756semidirect
372. C3#10588552semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117209982
C2×#884172
C3×#633692
C9×#8072
#117202712
#117202672
#105904593
#105885523