Finite Group Databasebeta

← #13137514

Subgroups — #13137514

Order 1728.

404 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 95 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×#888C211yesmaximal, characteristic, complemented
2#11721006C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic, complemented
5#11720267C211yesmaximal, characteristic, complemented
6#11720266C211yesmaximal, characteristic, complemented
7Dic3×C9⋊C8C211yesmaximal, characteristic, complemented
8#10590548—31nomaximal
9#10588773—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6—21no—
13C2×#6192—21no—
14C2×#6192—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2211yescharacteristic
17C9×C4⋊Dic3C2211yescharacteristic
18C9×Dic3⋊C4—21no—
19C4×C9×Dic3C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21C3×#888D311yescharacteristic
22#63911—31no—
23#63636—31no—
24C3×#809—31no—
25#63564—31no—
26#63562—31no—
27Dic3×C3⋊C8—31no—
28C9×C4⋊Q8—31no—
29#63436—31no—
30#63369—31no—
31#63369—31no—
32#63369—31no—
33#63369—61no—
34#63369D311yescharacteristic, complemented
35C4×C9⋊C8—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—
47C9×Dic6—42no—
48#6192—42no—
49#6192—42no—
50C3⋊(C9⋊C8)—21no—
51C3×C9⋊C8—21no—
52#5088—91no—
53#4760—91no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C3×D12—31no—
56C2×C3×Dic6—21no—
57C2×#626—61no—
58C2×#626—61no—
59C32×C4⋊C4C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)—31no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3C2×D311yescharacteristic
64C2×C3×C3⋊C8—31no—
65C2×Q8×C9—61no—
66C2×D36—31no—
67C2×D36—61no—
68C2×D36—61no—
69C2×D36—61no—
70C2×D36C2×D311yescharacteristic
71C9×C4⋊C4—21no—
72C9×C4⋊C4—31no—
73C9×C4⋊C4—61no—
74C9×C4⋊C4C2×D311yescharacteristic
75C42×C9—31no—
76C2×C9⋊C8—31no—
77C2×C9⋊C8—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3⋊D9—41no—
81C2×C3⋊D9—41no—
82C2×C3⋊D9—41no—
83C2×C3×D9—41no—
84C2×C3×D9—41no—
85C2×C3×D9—41no—
86C3×C4×C9—41no—
87C3×C4×C9#4111yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C9×Dic3—41no—
90C9×Dic3D812yes—
91C9×Dic3SD1612yes—
92C3×C4⋊Q8—31no—
93#940—91no—
94#888D911yescharacteristic, complemented
95#874—91no—
96C3×#102—91no—
97#832—91no—
98#821—91no—
99C8×Dic3—91no—
100#809—31no—
101#809—91no—
102#809—91no—
103#809—91no—
104#809—91no—
105#809—181no—
106C4×C3⋊C8—91no—
107C22×C3⋊D3—61no—
108C22×C3×D3—61no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C4⋊(C3⋊D3)—61no—
112C4⋊(C3⋊D3)—61no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3#6711yescharacteristic
115C2×C3×Dic3#6711yescharacteristic
116C3×D12—61no—
117C3×D12—61no—
118C3×Dic6—42no—
119#626—122no—
120#626—122no—
121C22×D9—21no—
122C22×D9—61no—
123C22×D9—121no—
124C3⋊(C3⋊C8)—61no—
125C3×C3⋊C8—61no—
126Q8×C9—122no—
127C2×C4×C9—21no—
128C2×C4×C9—21no—
129C2×C4×C9—31no—
130C2×C4×C9—31no—
131C2×C4×C9—41no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134D36—21no—
135D36—21no—
136D36—62no—
137D36—61no—
138D36—61no—
139D36—121no—
140D36—122no—
141D36—122no—
142C9⋊C8—21no—
143C9⋊C8—61no—
144C9⋊C8—121no—
145#486—271noSylow
146C2×C3×C9#12411yescharacteristic
147C2×C3×C9C2×D811yescharacteristic
148C2×C3×C9C2×SD1611yescharacteristic
149C3⋊D9—81no—
150C3×D9—81no—
151C2×C3×Q8—61no—
152C2×C3×D4—91no—
153C2×#67—181no—
154C2×#67—181no—
155C2×D12—31no—
156C2×D12—91no—
157C2×D12—91no—
158C2×D12—181no—
159C2×D12—181no—
160C2×D12—181no—
161C2×Dic6—21no—
162C2×C3×C8—91no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4C2×D911yescharacteristic
167C3×C4⋊C4D3211yescharacteristic
168C3×C42—31no—
169C4⋊Dic3C2×D911yescharacteristic
170Dic3⋊C4—21no—
171C4×Dic3C2×D911yescharacteristic
172C2×C3⋊C8—31no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—181no—
176C22×C32C2×#6711yescharacteristic
177C2×C3⋊D3—121no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—41no—
184C32×C4#23911yescharacteristic
185C32×C4#23911yescharacteristic
186C3×Dic3—41no—
187C3×Dic3#21312yes—
188C3×Dic3#21512yes—
189C22×C9—21no—
190C22×C9D3×D411yescharacteristic
191C2×D9—41no—
192C2×D9—41no—
193C2×D9—41no—
194C2×D9—122no—
195C2×D9—121no—
196C2×D9—121no—
197C2×D9—121no—
198C2×D9—241no—
199C2×D9—241no—
200C4×C9—21no—
201C4×C9—21no—
202C4×C9—32no—
203C4×C9—32no—
204C4×C9—41no—
205C4×C9—81no—
206C4×C9—121no—
207C4×C9#23511yescharacteristic
208C4×C9#23711yescharacteristic
209C4⋊Q8—31no—
210#127—271no—
211#102—91no—
212#102—271no—
213#102—271no—
214#102—271no—
215C4×C8—271no—
216C3×C9#48611yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—181no—
220C22×D3—181no—
221C22×D3—361no—
222C3×Q8—122no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—41no—
231C2×C3×C4—61no—
232C2×C3×C4C22×D911yescharacteristic
233C2×C3×C4C2×D3211yescharacteristic
234#67—362no—
235#67—362no—
236C2×Dic3—21no—
237C2×Dic3#61111yescharacteristic
238C2×Dic3#61111yescharacteristic
239D12—61no—
240D12—61no—
241D12—182no—
242D12—181no—
243D12—181no—
244D12—181no—
245D12—181no—
246D12—361no—
247D12—362no—
248D12—362no—
249Dic6—42no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8—181no—
253C3⋊C8—181no—
254C3⋊C8—361no—
255C2×C32C2×#21311yescharacteristic
256C2×C32C2×#21511yescharacteristic
257C2×C32#94711yescharacteristic
258C3⋊D3—241no—
259C3×D3—241no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9#88511yescharacteristic
264C2×C9D3×D811yescharacteristic
265C2×C9D3×SD1611yescharacteristic
266D9—81no—
267D9—242no—
268D9—241no—
269C2×Q8—61no—
270C2×D4—91no—
271C2×D4—271no—
272C2×D4—541no—
273C2×D4—541no—
274C2×C8—91no—
275C2×C8—271no—
276C4⋊C4—31no—
277C4⋊C4—61no—
278C4⋊C4D3×D911yescharacteristic
279C42—31no—
280C22×C3—21no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3C2×#61111yescharacteristic
285C22×C3D3×#6711yescharacteristic
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—121no—
289C2×D3—362no—
290C2×D3—361no—
291C2×D3—361no—
292C2×D3—361no—
293C2×D3—361no—
294C2×D3—361no—
295C2×D3—361no—
296C2×D3—721no—
297C2×D3—721no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—32no—
301C3×C4—32no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—81no—
305C3×C4—121no—
306C3×C4#370111yescharacteristic
307C3×C4#370111yescharacteristic
308C3×C4#379611yescharacteristic
309C3×C4#379911yescharacteristic
310Dic3—41no—
311Dic3#367312yes—
312Dic3#367512yes—
313C32#508811yescharacteristic
314C9—21no—
315C9#476011yescharacteristic, complemented
316C23—181no—
317C23—541no—
318Q8—122no—
319D4—181no—
320D4—181no—
321D4—541no—
322D4—541no—
323D4—1082no—
324D4—1082no—
325C2×C4—21no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4C2×D3×D911yescharacteristic
330C8—181no—
331C8—541no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—721no—
336C2×C3C2×#367311yescharacteristic
337C2×C3C2×#367511yescharacteristic
338C2×C3#6350911yescharacteristic
339C2×C3#6390711yescharacteristic
340C2×C3D3×#21311yescharacteristic
341C2×C3D3×#21511yescharacteristic
342D3—241no—
343D3—722no—
344D3—721no—
345D3—721no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22—1081no—
350C22—1081no—
351C22—1081no—
352C22D3×#61111yescharacteristic
353C4—32no—
354C4—32no—
355C4—41no—
356C4—121no—
357C4#8863411yescharacteristic
358C4#8863511yescharacteristic
359C3—21no—
360C3#1058877311yescharacteristic, complemented
361C3#1059054811yescharacteristic
362C2—721no—
363C2—2161no—
364C2#1172100311yescharacteristic
365C2D3×#367311yescharacteristic
366C2D3×#367511yescharacteristic
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#486
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×#888C2semidirect
3. #11720528C2semidirect
4. C3×#63369C2semidirect
5. #11720267C2semidirect
6. #11720266C2semidirect
7. Dic3×C9⋊C8C2semidirect
34. #63369D3semidirect
94. #888D9semidirect
216. C3×C9#486semidirect
315. C9#4760semidirect
360. C3#10588773semidirect

Maximal subgroups

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

subgroupindex
C9×#8882
#117210062
#117205282
C3×#633692
#117202672
#117202662
Dic3×C9⋊C82
#105905483
#105887733