Finite Group Databasebeta

← #13137516

Subgroups — #13137516

Order 1728.

380 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C9×#890C211yesmaximal, characteristic, complemented
2#11721006C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic, complemented
5#11720267C211yesmaximal, characteristic, complemented
6#11720266C211yesmaximal, characteristic, complemented
7#11720258C211yesmaximal, characteristic, complemented
8#10590550—31nomaximal
9#10588748—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×#6192—21no—
13C2×#6192—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C4×C9×Dic3C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#890D311yescharacteristic
22#63911—31no—
23#63636—31no—
24C3×#809—31no—
25#63564—31no—
26#63562—31no—
27#63554—31no—
28C9×#125—31no—
29#63436—31no—
30#63369—31no—
31#63369—31no—
32#63369—31no—
33#63369—61no—
34#63369D311yescharacteristic, complemented
35#63362—31no—
36C22×C3⋊D9—21no—
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C4⋊(C3⋊D9)—21no—
41C4⋊(C3⋊D9)—21no—
42C2×C9×Dic3—21no—
43C2×C9×Dic3D411yescharacteristic
44C2×C9×Dic3D411yescharacteristic
45C3×D36—21no—
46C3×D36—21no—
47#6192—42no—
48#6192—42no—
49C3⋊(C9⋊C8)—21no—
50C3×C9⋊C8—21no—
51#5067—91no—
52#4764—91no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C3×D12—31no—
55C2×#626—61no—
56C2×#626—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C3×Dic3⋊C4—21no—
62C3×C4×Dic3C2×D311yescharacteristic
63C2×C3×C3⋊C8—31no—
64C2×D36—31no—
65C2×D36—61no—
66C2×D36—61no—
67C2×D36—61no—
68C2×D36C2×D311yescharacteristic
69C9×C4⋊C4—21no—
70C9×C4⋊C4—31no—
71C9×C4⋊C4—61no—
72C9×C4⋊C4—61no—
73C9×C4⋊C4C2×D311yescharacteristic
74C42×C9—31no—
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C2×C3×D9—41no—
83C2×C3×D9—41no—
84C2×C3×D9—41no—
85C3×C4×C9—41no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C9×Dic3—21no—
89C9×Dic3—21no—
90C9×Dic3—41no—
91C3×#125—31no—
92#940—91no—
93#890D911yescharacteristic, complemented
94#874—91no—
95C3×#102—91no—
96#832—91no—
97#821—91no—
98#815—91no—
99#809—31no—
100#809—91no—
101#809—91no—
102#809—91no—
103#809—91no—
104#809—181no—
105#803—91no—
106C22×C3⋊D3—61no—
107C22×C3×D3—61no—
108C2×C32×C4—21no—
109C2×C32×C4C22×D311yescharacteristic
110C4⋊(C3⋊D3)—61no—
111C4⋊(C3⋊D3)—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3#6711yescharacteristic
114C2×C3×Dic3#6711yescharacteristic
115C3×D12—61no—
116C3×D12—61no—
117#626—122no—
118#626—122no—
119C22×D9—21no—
120C22×D9—61no—
121C22×D9—121no—
122C3⋊(C3⋊C8)—61no—
123C3×C3⋊C8—61no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—31no—
127C2×C4×C9—31no—
128C2×C4×C9—41no—
129C2×C4×C9—61no—
130C2×C4×C9C22×D311yescharacteristic
131D36—21no—
132D36—21no—
133D36—62no—
134D36—61no—
135D36—61no—
136D36—121no—
137D36—122no—
138D36—122no—
139C9⋊C8—21no—
140C9⋊C8—61no—
141C9⋊C8—121no—
142#490—271noSylow
143C2×C3×C9#12411yescharacteristic
144C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C3⋊D9—81no—
147C3×D9—81no—
148C2×C3×D4—91no—
149C2×#67—181no—
150C2×#67—181no—
151C2×D12—31no—
152C2×D12—91no—
153C2×D12—91no—
154C2×D12—181no—
155C2×D12—181no—
156C2×D12—181no—
157C2×C3×C8—91no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4C2×D911yescharacteristic
163C3×C4⋊C4D3211yescharacteristic
164C3×C42—31no—
165C4⋊Dic3—21no—
166C4⋊Dic3C2×D911yescharacteristic
167Dic3⋊C4—21no—
168C4×Dic3C2×D911yescharacteristic
169C2×C3⋊C8—31no—
170C2×C3⋊C8—91no—
171C2×C3⋊C8—91no—
172C2×C3⋊C8—181no—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—121no—
175C2×C3⋊D3—121no—
176C2×C3⋊D3—121no—
177C2×C3×D3—121no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C32×C4—41no—
181C32×C4#23911yescharacteristic
182C32×C4#23911yescharacteristic
183C3×Dic3—21no—
184C3×Dic3—21no—
185C3×Dic3—41no—
186C22×C9—21no—
187C22×C9D3×D411yescharacteristic
188C2×D9—41no—
189C2×D9—41no—
190C2×D9—41no—
191C2×D9—122no—
192C2×D9—121no—
193C2×D9—121no—
194C2×D9—121no—
195C2×D9—241no—
196C2×D9—241no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—41no—
200C4×C9—61no—
201C4×C9—61no—
202C4×C9—81no—
203C4×C9—121no—
204C4×C9#23511yescharacteristic
205C4×C9#23711yescharacteristic
206#127—271no—
207#125—31no—
208#102—91no—
209#102—271no—
210#102—271no—
211#102—271no—
212#97—271no—
213C3×C9#49011yescharacteristic, complemented, Sylow
214C23×C3—181no—
215C22×D3—61no—
216C22×D3—181no—
217C22×D3—181no—
218C22×D3—361no—
219C3×D4—181no—
220C3×D4—181no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—41no—
227C2×C3×C4—61no—
228C2×C3×C4C22×D911yescharacteristic
229C2×C3×C4C2×D3211yescharacteristic
230#67—362no—
231#67—362no—
232C2×Dic3—21no—
233C2×Dic3#61111yescharacteristic
234C2×Dic3#61111yescharacteristic
235D12—61no—
236D12—61no—
237D12—182no—
238D12—181no—
239D12—181no—
240D12—181no—
241D12—181no—
242D12—361no—
243D12—362no—
244D12—362no—
245C3×C8—181no—
246C3⋊C8—61no—
247C3⋊C8—181no—
248C3⋊C8—181no—
249C3⋊C8—361no—
250C2×C32#94711yescharacteristic
251C2×C32#94911yescharacteristic
252C2×C32#94911yescharacteristic
253C3⋊D3—241no—
254C3×D3—241no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9#88511yescharacteristic
259C2×C9#91111yescharacteristic
260C2×C9#91411yescharacteristic
261D9—81no—
262D9—242no—
263D9—241no—
264C2×D4—91no—
265C2×D4—271no—
266C2×D4—541no—
267C2×D4—541no—
268C2×C8—91no—
269C2×C8—271no—
270C4⋊C4—31no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4D3×D911yescharacteristic
274C42—31no—
275C22×C3—21no—
276C22×C3—361no—
277C22×C3—361no—
278C22×C3—361no—
279C22×C3C2×#61111yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—362no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—361no—
289C2×D3—361no—
290C2×D3—361no—
291C2×D3—721no—
292C2×D3—721no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—81no—
300C3×C4—121no—
301C3×C4#370111yescharacteristic
302C3×C4#370111yescharacteristic
303C3×C4#379611yescharacteristic
304C3×C4#379911yescharacteristic
305Dic3—21no—
306Dic3—21no—
307Dic3—41no—
308C32#506711yescharacteristic
309C9—21no—
310C9#476411yescharacteristic, complemented
311C23—181no—
312C23—541no—
313D4—181no—
314D4—181no—
315D4—541no—
316D4—541no—
317D4—1082no—
318D4—1082no—
319C2×C4—21no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—61no—
323C2×C4C2×D3×D911yescharacteristic
324C8—181no—
325C8—541no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—721no—
330C2×C3#6350911yescharacteristic
331C2×C3#6351211yescharacteristic
332C2×C3#6351211yescharacteristic
333C2×C3#6390711yescharacteristic
334C2×C3#6392511yescharacteristic
335C2×C3#6393911yescharacteristic
336D3—241no—
337D3—722no—
338D3—721no—
339D3—721no—
340C22—361no—
341C22—361no—
342C22—361no—
343C22—1081no—
344C22—1081no—
345C22—1081no—
346C22D3×#61111yescharacteristic
347C4—41no—
348C4—61no—
349C4—61no—
350C4—121no—
351C4#8863411yescharacteristic
352C4#8863511yescharacteristic
353C3—21no—
354C3#1058874811yescharacteristic, complemented
355C3#1059055011yescharacteristic
356C2—721no—
357C2—2161no—
358C2#1172100311yescharacteristic
359C2#1172116711yescharacteristic
360C2#1172120511yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radical#11721006C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#490
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. C9×#890C2semidirect
3. #11720528C2semidirect
4. C3×#63369C2semidirect
5. #11720267C2semidirect
6. #11720266C2semidirect
7. #11720258C2semidirect
34. #63369D3semidirect
93. #890D9semidirect
213. C3×C9#490semidirect
310. C9#4764semidirect
354. C3#10588748semidirect

Maximal subgroups

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

subgroupindex
C9×#8902
#117210062
#117205282
C3×#633692
#117202672
#117202662
#117202582
#105905503
#105887483