Finite Group Databasebeta

← #13137653

Subgroups — #13137653

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
1C3×#63455C211yesmaximal, characteristic, complemented
2#11720992C211yesmaximal, characteristic, complemented
3C2×#88417C211yesmaximal, characteristic
4#11720528C211yesmaximal, characteristic, complemented
5C9×#808C211yesmaximal, characteristic, complemented
6#11720274C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590589—31nomaximal
9#10588549—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9—21no—
13C2×(C3⋊D9)⋊C4—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20#88445—21no—
21#88417C414yescomplemented
22C3×#893—31no—
23#63894—31no—
24C2×#3714—31no—
25#63636—31no—
26C3×#808D311yescharacteristic
27#63562—31no—
28#63554—31no—
29C2×D72—31no—
30#63455—31no—
31#63455D311yescharacteristic, complemented
32C9×#106—31no—
33#63375—31no—
34#63369—31no—
35#63369—31no—
36#63369—61no—
37C22×C3⋊D9—21no—
38C22×C3×D9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)C2×C412yescomplemented
43C2×C9×Dic3—21no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C3×D36—21no—
47C3×D36C2×C412yescomplemented
48C3×C4×D9—42no—
49(C3⋊D9)⋊C4—42no—
50C3⋊(C9⋊C8)—21no—
51C9×C3⋊C8C2×C412yescomplemented
52#5014—91no—
53#4869—91no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C3×D12—31no—
56C2×C3×C4×D3—61no—
57C2×#624—61no—
58C32×C4⋊C4C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)—31no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C3×C4×Dic3—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64#3722—61no—
65#3714—34no—
66C2×D36—31no—
67C2×D36—61no—
68C2×D36C2×D311yescharacteristic
69C2×C4×D9—21no—
70C2×C4×D9—61no—
71C2×C8×C9—31no—
72C9×C4⋊C4—21no—
73C9×C4⋊C4—31no—
74C9×C4⋊C4C2×D311yescharacteristic
75(C2×D9)⋊C4—21no—
76(C2×D9)⋊C4—61no—
77C4×Dic9C2×D311yescharacteristic
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80D72—34no—
81C22×C3×C9C2×D411yescharacteristic
82C2×C3⋊D9—22no—
83C2×C3⋊D9—41no—
84C2×C3⋊D9—41no—
85C2×C3×D9—22no—
86C2×C3×D9—41no—
87C2×C3×D9—41no—
88C3×C4×C9—41no—
89C3×C4×C9C22×C411yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C9×Dic3—41no—
92C3×Dic9—21no—
93C3×Dic9—21no—
94C3×C4×D4—91no—
95C2×#213—91no—
96C2×D24—91no—
97#893—31no—
98#893—91no—
99C4×D12—91no—
100C3×#106—31no—
101#832—91no—
102#815—91no—
103#809—91no—
104#809—91no—
105#809—91no—
106#809—181no—
107#808D911yescharacteristic, complemented
108#803—91no—
109C22×C3⋊D3—61no—
110C22×C3×D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C4⋊(C3⋊D3)—32no—
114C4⋊(C3⋊D3)—61no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C3×D12—32no—
119C3×D12—61no—
120C3×C4×D3—122no—
121#624—122no—
122C22×D9—21no—
123C22×D9—61no—
124C22×D9—121no—
125C3⋊(C3⋊C8)—61no—
126C3×C3⋊C8C4×D312yes—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—41no—
130C2×C4×C9—61no—
131C2×C4×C9C22×D311yescharacteristic
132C2×Dic9#6711yescharacteristic
133C2×Dic9#6711yescharacteristic
134D36—21no—
135D36—32no—
136D36—62no—
137D36—61no—
138D36—121no—
139D36C4×D312yescomplemented
140C4×D9—42no—
141C4×D9—122no—
142C8×C9—32no—
143C9⋊C8—61no—
144C9⋊C8—121no—
145#442—271noSylow
146C2×C3×C9C4×D411yescharacteristic
147C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
148C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
149C3⋊D9—42no—
150C3×D9—42no—
151C2×C3×D4—91no—
152C22×C3×C4—181no—
153C2×D12—31no—
154C2×D12—91no—
155C2×D12—91no—
156C2×D12—181no—
157C2×C4×D3—61no—
158C2×C4×D3—181no—
159C2×C4×D3—181no—
160C2×C3×C8—31no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4C2×D911yescharacteristic
164C3×C4⋊C4D3211yescharacteristic
165C3×C22⋊C4—181no—
166C3×C42—91no—
167#213—94no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3C2×D911yescharacteristic
172C4×Dic3—31no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C2×C3⋊C8C2×D911yescharacteristic
177D24—94no—
178C22×C32D3×D411yescharacteristic
179C2×C3⋊D3—62no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—121no—
182C2×C3×D3—62no—
183C2×C3×D3—121no—
184C2×C3×D3—121no—
185C32×C4—41no—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4#23911yescharacteristic
188C3×Dic3—41no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C22×C9—21no—
192C22×C9C2×#6711yescharacteristic
193C2×D9—22no—
194C2×D9—41no—
195C2×D9—41no—
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—41no—
205C4×C9—81no—
206C4×C9—121no—
207C4×C9C2×C4×D311yescharacteristic
208C4×C9#23511yescharacteristic
209Dic9—21no—
210Dic9—21no—
211C2×D8—271no—
212C4×D4—91no—
213C4×D4—271no—
214#106—31no—
215#102—271no—
216#102—271no—
217#97—271no—
218C3×C9#44211yescharacteristic, complemented, Sylow
219C23×C3—181no—
220C22×D3—61no—
221C22×D3—181no—
222C22×D3—181no—
223C22×D3—361no—
224C3×D4—92no—
225C3×D4—181no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—61no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—362no—
234C2×C3×C4C22×D911yescharacteristic
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239D12—32no—
240D12—61no—
241D12—92no—
242D12—92no—
243D12—182no—
244D12—181no—
245D12—181no—
246D12—361no—
247C4×D3—122no—
248C4×D3—362no—
249C4×D3—362no—
250C3×C8—32no—
251C3⋊C8—181no—
252C3⋊C8—181no—
253C3⋊C8—361no—
254C3⋊C8C4×D912yescomplemented
255C2×C32#89311yescharacteristic
256C2×C32#91411yescharacteristic
257C2×C32#91411yescharacteristic
258C3⋊D3—122no—
259C3×D3—122no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9C4×#6711yescharacteristic
264C2×C9#93211yescharacteristic
265C2×C9#94911yescharacteristic
266D9—42no—
267D9—122no—
268D9—242no—
269C2×D4—91no—
270C2×D4—271no—
271C22×C4—181no—
272C22×C4—541no—
273D8—274no—
274C2×C8—31no—
275C2×C8—271no—
276C4⋊C4—31no—
277C4⋊C4D3×D911yescharacteristic
278C22⋊C4—181no—
279C22⋊C4—541no—
280C42—91no—
281C22×C3—21no—
282C22×C3—182no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3D4×D911yescharacteristic
286C22×C3D3×#6711yescharacteristic
287C2×D3—62no—
288C2×D3—121no—
289C2×D3—121no—
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—41no—
302C3×C4—41no—
303C3×C4—81no—
304C3×C4—121no—
305C3×C4—181no—
306C3×C4—181no—
307C3×C4C2×C4×D911yescharacteristic
308C3×C4#370111yescharacteristic
309C3×C4#379911yescharacteristic
310C3×C4C4×D3211yescharacteristic
311Dic3—41no—
312Dic3—61no—
313Dic3—61no—
314C32#486911yescharacteristic
315C9—21no—
316C9#501411yescharacteristic, complemented
317C23—181no—
318C23—541no—
319D4—92no—
320D4—181no—
321D4—272no—
322D4—541no—
323C2×C4—21no—
324C2×C4—61no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—362no—
328C2×C4—1082no—
329C2×C4C2×D3×D911yescharacteristic
330C8—32no—
331C8—541no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—362no—
336C2×C3#6345511yescharacteristic
337C2×C3#6347611yescharacteristic
338C2×C3#6347611yescharacteristic
339C2×C3#6390411yescharacteristic
340C2×C3#6393411yescharacteristic
341C2×C3#6393911yescharacteristic
342D3—122no—
343D3—362no—
344D3—362no—
345D3—722no—
346C22—182no—
347C22—361no—
348C22—361no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D9×#6711yescharacteristic
353C4—41no—
354C4—121no—
355C4—181no—
356C4—181no—
357C4#8863511yescharacteristic
358C4C4×D3×D911yescharacteristic
359C3—21no—
360C3#1058854911yescharacteristic, complemented
361C3#1059058911yescharacteristic
362C2—362no—
363C2—1082no—
364C2#1172100211yescharacteristic
365C2#1172118411yescharacteristic
366C2#1172121111yescharacteristic
subgroupG/H
centerC22D9×#67
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×C9#442
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. C3×#63455C2semidirect
2. #11720992C2semidirect
4. #11720528C2semidirect
5. C9×#808C2semidirect
6. #11720274C2semidirect
7. #11720266C2semidirect
21. #88417C4semidirect
31. #63455D3semidirect
42. C4⋊(C3⋊D9)C2×C4semidirect
47. C3×D36C2×C4semidirect
51. C9×C3⋊C8C2×C4semidirect
107. #808D9semidirect
139. D36C4×D3semidirect
218. C3×C9#442semidirect
254. C3⋊C8C4×D9semidirect
316. C9#5014semidirect
360. C3#10588549semidirect

Maximal subgroups

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

subgroupindex
C3×#634552
#117209922
C2×#884172
#117205282
C9×#8082
#117202742
#117202662
#105905893
#105885493