Finite Group Databasebeta

← #13138490

Subgroups — #13138490

Order 1728.

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

normal subgroups only

This group has 446 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
1D12×Dic9C211yesmaximal, characteristic, complemented
2#11720992C211yesmaximal, characteristic, complemented
3C2×#88422C211yesmaximal, characteristic
4#11720543C211yesmaximal, characteristic, complemented
5C9×#821C211yesmaximal, characteristic, complemented
6C3×#63375C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10590746—31nomaximal
9#10588465—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×D3×Dic9—21no—
13C2×(C3⋊D9)⋊C4—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic9—21no—
20#88445—21no—
21#88422C414yescomplemented
22#63894—31no—
23Dic3×D12—31no—
24C2×#3714—31no—
25#63648—31no—
26C3×#821D311yescharacteristic
27C3×#815—31no—
28#63573—31no—
29D4×Dic9—31no—
30C2×#3673—31no—
31#63455—31no—
32C9×#102—31no—
33#63380—31no—
34#63380—61no—
35#63375D311yescharacteristic, complemented
36#63367—31no—
37C22×C3⋊D9—21no—
38C22×D3×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)C2×C412yescomplemented
43C2×C9×Dic3—21no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C9×D12—21no—
47C9×D12C2×C412yescomplemented
48D3×Dic9—42no—
49(C3⋊D9)⋊C4—42no—
50C3×C8×C9—21no—
51C3×C9⋊C8C2×C412yescomplemented
52#4785—91no—
53#4711—91no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C3×D12C2×D311yescharacteristic
56C2×#624—61no—
57C2×D3×Dic3—61no—
58C2×C32×C8C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8—31no—
63#3722—61no—
64(C2×D3)⋊Dic3—61no—
65#3714—34no—
66C2×D4×C9—31no—
67C22×Dic9—61no—
68C2×D36—31no—
69C2×D36—61no—
70C2×C4×D9—61no—
71C2×C8×C9—21no—
72C2×C8×C9C2×D311yescharacteristic
73C9×C4⋊C4—31no—
74#3676—61no—
75#3673—34no—
76(C2×D9)⋊C4—61no—
77C4⋊Dic9—31no—
78C4⋊Dic9—61no—
79C4×Dic9C2×D311yescharacteristic
80C2×C9⋊C8C2×D311yescharacteristic
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C2×D3×C9—22no—
86C2×D3×C9—41no—
87C2×D3×C9—41no—
88C3×C4×C9C22×C411yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—41no—
91C9×Dic3—41no—
92C3×Dic9—21no—
93C3×Dic9—21no—
94D4×Dic3—91no—
95C2×#213—91no—
96C2×D24—91no—
97#893—91no—
98C4×D12—91no—
99C4×D12—91no—
100C3×#102—31no—
101C3×#97—91no—
102#821—91no—
103#821—91no—
104#821—181no—
105#821D911yescharacteristic, complemented
106#817—91no—
107#815—31no—
108#808—91no—
109C22×C3⋊D3—61no—
110C22×C3×D3—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—61no—
113C4⋊(C3⋊D3)—32no—
114C4⋊(C3⋊D3)—61no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C3×D12—21no—
119C3×D12C4×D312yes—
120#624—122no—
121D3×Dic3—122no—
122C23×C9—61no—
123C22×D9—61no—
124C22×D9—121no—
125C32×C8—21no—
126C3×C3⋊C8—32no—
127D4×C9—32no—
128D4×C9—61no—
129C2×C4×C9—21no—
130C2×C4×C9—61no—
131C2×C4×C9C22×D311yescharacteristic
132C2×Dic9—61no—
133C2×Dic9—121no—
134C2×Dic9—122no—
135C2×Dic9D1211yescharacteristic
136C2×Dic9D1211yescharacteristic
137D36—32no—
138D36—62no—
139D36—61no—
140D36—121no—
141C4×D9—122no—
142C8×C9—21no—
143C8×C9—41no—
144C9⋊C8C4×D312yescomplemented
145#442—271noSylow
146C2×C3×C9C4×D411yescharacteristic
147C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
148C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
149C3⋊D9—42no—
150D3×C9—42no—
151C2×C3×D4—31no—
152C22×Dic3—181no—
153C2×D12—91no—
154C2×D12—91no—
155C2×D12—181no—
156C2×D12C2×D911yescharacteristic
157C2×C4×D3—181no—
158C2×C4×D3—181no—
159C2×C4×D3—181no—
160C2×C3×C8—21no—
161C2×C3×C8—91no—
162C2×C3×C8C2×D911yescharacteristic
163C2×C3×C8D3211yescharacteristic
164C3×C4⋊C4—31no—
165C3×C42—91no—
166#217—181no—
167#213—94no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174C4⋊Dic3C2×D911yescharacteristic
175C4×Dic3—31no—
176C2×C3⋊C8—31no—
177D24—94no—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—62no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—121no—
182C2×C3×D3—22no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C32×C4C2×C4×D311yescharacteristic
186C32×C4#23711yescharacteristic
187C3⋊Dic3—121no—
188C3×Dic3—41no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C22×C9—21no—
192C22×C9—62no—
193C22×C9—121no—
194C22×C9—121no—
195C22×C9C2×D1211yescharacteristic
196C2×D9—62no—
197C2×D9—122no—
198C2×D9—121no—
199C2×D9—121no—
200C2×D9—241no—
201C2×D9—241no—
202C4×C9—21no—
203C4×C9—21no—
204C4×C9—121no—
205C4×C9C2×C4×D311yescharacteristic
206C4×C9#23511yescharacteristic
207Dic9—21no—
208Dic9—21no—
209Dic9—121no—
210Dic9—241no—
211C2×D8—271no—
212C4×D4—271no—
213C4×D4—271no—
214#106—271no—
215#102—31no—
216#102—271no—
217#97—91no—
218C3×C9#44211yescharacteristic, complemented, Sylow
219C23×C3—61no—
220C22×D3—21no—
221C22×D3—181no—
222C22×D3—181no—
223C22×D3—361no—
224C3×D4—32no—
225C3×D4—61no—
226C2×C3×C4—21no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—21no—
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C2×Dic3—361no—
238C2×Dic3—362no—
239D12—21no—
240D12—92no—
241D12—92no—
242D12—182no—
243D12—181no—
244D12—181no—
245D12—361no—
246D12C4×D912yescomplemented
247C4×D3—362no—
248C4×D3—362no—
249C4×D3—362no—
250C3×C8—21no—
251C3×C8—21no—
252C3×C8—41no—
253C3×C8—92no—
254C3⋊C8—32no—
255C2×C32#67⋊C411yescharacteristic
256C2×C32#91111yescharacteristic
257C2×C32#91411yescharacteristic
258C3⋊D3—122no—
259C3×D3—42no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—122no—
264C2×C9C4×D1211yescharacteristic
265C2×C9#90811yescharacteristic
266C2×C9#90811yescharacteristic
267D9—122no—
268D9—242no—
269C2×D4—31no—
270C2×D4—271no—
271C22×C4—541no—
272C22×C4—541no—
273D8—274no—
274C2×C8—91no—
275C2×C8D3×D911yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—271no—
278C22⋊C4—541no—
279C22⋊C4—541no—
280C42—91no—
281C22×C3—21no—
282C22×C3—62no—
283C22×C3—121no—
284C22×C3—121no—
285C22×C3D4×D911yescharacteristic
286C22×C3D3×D1211yescharacteristic
287C2×D3—22no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—182no—
291C2×D3—182no—
292C2×D3—362no—
293C2×D3—361no—
294C2×D3—361no—
295C2×D3—361no—
296C2×D3—361no—
297C2×D3—721no—
298C2×D3—721no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—121no—
302C3×C4—181no—
303C3×C4—181no—
304C3×C4C2×C4×D911yescharacteristic
305C3×C4#369911yescharacteristic
306C3×C4C4×D3211yescharacteristic
307C3×C4#380411yescharacteristic
308Dic3—41no—
309Dic3—61no—
310Dic3—61no—
311Dic3—361no—
312Dic3—361no—
313Dic3—721no—
314C32#478511yescharacteristic
315C9—21no—
316C9#471111yescharacteristic, complemented
317C23—61no—
318C23—541no—
319D4—32no—
320D4—61no—
321D4—272no—
322D4—541no—
323C2×C4—61no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—541no—
327C2×C4—1082no—
328C2×C4—1082no—
329C2×C4C2×D3×D911yescharacteristic
330C8—21no—
331C8—92no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—122no—
336C2×C3#611⋊C411yescharacteristic
337C2×C3#6347311yescharacteristic
338C2×C3#6347611yescharacteristic
339C2×C3#6379411yescharacteristic
340C2×C3#6379511yescharacteristic
341C2×C3#6388011yescharacteristic
342D3—42no—
343D3—362no—
344D3—362no—
345D3—722no—
346C22—62no—
347C22—121no—
348C22—121no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D9×D1211yescharacteristic
353C4—121no—
354C4—181no—
355C4—181no—
356C4—1081no—
357C4C4×D3×D911yescharacteristic
358C4#8866211yescharacteristic
359C3—21no—
360C3#1058846511yescharacteristic, complemented
361C3#1059074611yescharacteristic
362C2—122no—
363C2—1082no—
364C2#1172081511yescharacteristic
365C2#1172081611yescharacteristic
366C2#1172101311yescharacteristic
subgroupG/H
centerC22D9×D12
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88422C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#442
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. D12×Dic9C2semidirect
2. #11720992C2semidirect
4. #11720543C2semidirect
5. C9×#821C2semidirect
6. C3×#63375C2semidirect
7. #11720264C2semidirect
21. #88422C4semidirect
35. #63375D3semidirect
42. C4⋊(C3⋊D9)C2×C4semidirect
47. C9×D12C2×C4semidirect
51. C3×C9⋊C8C2×C4semidirect
105. #821D9semidirect
144. C9⋊C8C4×D3semidirect
218. C3×C9#442semidirect
246. D12C4×D9semidirect
316. C9#4711semidirect
360. C3#10588465semidirect

Maximal subgroups

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

subgroupindex
D12×Dic92
#117209922
C2×#884222
#117205432
C9×#8212
C3×#633752
#117202642
#105907463
#105884653