Finite Group Databasebeta

← #13137760

Subgroups — #13137760

Order 1728.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721746C211yesmaximal, characteristic, complemented
2#11721043C211yesmaximal, characteristic
3C9×#809C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720275C211yesmaximal, characteristic, complemented
8#10590554—31nomaximal
9#10588444—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×#6194—21no—
13C2×#6194—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17#88534—21no—
18C4×C3⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#64099—31no—
22#63914—31no—
23C3×#809—31no—
24C3×#809D311yescharacteristic
25#63563—31no—
26#63563—31no—
27#63555—31no—
28#63502—31no—
29#63452—31no—
30#63452—61no—
31C9×#102—31no—
32#63392—31no—
33#63380—31no—
34#63375—31no—
35#63369D311yescharacteristic, complemented
36C22×D3×C9—21no—
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C9×D12—21no—
44C9×D12—21no—
45C3×D36—21no—
46C3×D36—21no—
47#6194—42no—
48#6194—42no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8—21no—
51#4764—91no—
52#4764—91no—
53C2×C3×D12—31no—
54C2×C3×D12C2×D311yescharacteristic
55C2×#625—61no—
56C2×#625—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—61no—
60#3750—61no—
61C4×C3⋊Dic3—31no—
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64C2×D4×C9—31no—
65C2×#611—61no—
66C2×#611—61no—
67C2×D36C2×D311yescharacteristic
68C2×C8×C9—31no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4C2×D311yescharacteristic
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73C4⋊Dic9—61no—
74C4⋊Dic9—62no—
75Dic9⋊C4—61no—
76Dic9⋊C4—62no—
77C4×Dic9—31no—
78C4×Dic9—61no—
79C2×C9⋊C8C2×D311yescharacteristic
80C22×C3×C9C2×D411yescharacteristic
81C2×D3×C9—41no—
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C2×C3×D9—41no—
85C2×C3×D9—41no—
86C2×C3×D9—41no—
87C3×C4×C9—41no—
88C3×C4×C9#4111yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—21no—
91C3⋊Dic9—21no—
92C3⋊Dic9—41no—
93#940—91no—
94#940—91no—
95#890—91no—
96#890—91no—
97#890—181no—
98C3×#102—31no—
99C3×#102—91no—
100#832—91no—
101#832—91no—
102#821—91no—
103#821—91no—
104#815—91no—
105#815—91no—
106#809—31no—
107#809D911yescharacteristic, complemented
108C22×C3×D3—21no—
109C22×C3×D3—61no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—61no—
115C3×D12—21no—
116C3×D12—21no—
117C3×D12—61no—
118C3×D12—61no—
119#625—122no—
120#625—122no—
121C23×C9—61no—
122C22×D9—21no—
123C3×C3⋊C8—21no—
124C3×C3⋊C8—61no—
125D4×C9—61no—
126D4×C9—61no—
127C2×C4×C9—22no—
128C2×C4×C9—21no—
129C2×C4×C9—21no—
130C2×C4×C9C22×D311yescharacteristic
131#611—122no—
132#611—122no—
133C2×Dic9—31no—
134C2×Dic9—31no—
135C2×Dic9—61no—
136C2×Dic9—61no—
137C2×Dic9—62no—
138C2×Dic9—61no—
139D36—21no—
140D36—21no—
141C8×C9—61no—
142C9⋊C8—21no—
143#490—271noSylow
144C2×C3×C9#12411yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
147D3×C9—81no—
148C3×D9—81no—
149C2×C3×D4—31no—
150C2×C3×D4—91no—
151C2×#67—181no—
152C2×#67—181no—
153C2×#67—181no—
154C2×#67—181no—
155C2×D12—31no—
156C2×D12C2×D911yescharacteristic
157C2×C3×C8—31no—
158C2×C3×C8—91no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4C2×D911yescharacteristic
161C3×C4⋊C4D3211yescharacteristic
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—181no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167C4⋊Dic3—182no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—182no—
171C4×Dic3—91no—
172C4×Dic3—91no—
173C4×Dic3—181no—
174C2×C3⋊C8—31no—
175C2×C3⋊C8C2×D911yescharacteristic
176C22×C32D3×D411yescharacteristic
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C2×C3×D3—121no—
183C32×C4—41no—
184C32×C4#23511yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—61no—
187C3⋊Dic3—61no—
188C3⋊Dic3—121no—
189C22×C9—21no—
190C22×C9—121no—
191C22×C9—121no—
192C22×C9—121no—
193C22×C9D3×D411yescharacteristic
194C2×D9—41no—
195C2×D9—41no—
196C2×D9—41no—
197C4×C9—21no—
198C4×C9—21no—
199C4×C9—42no—
200C4×C9—41no—
201C4×C9#23511yescharacteristic
202C4×C9#23711yescharacteristic
203Dic9—61no—
204Dic9—61no—
205Dic9—121no—
206Dic9—121no—
207Dic9—122no—
208Dic9—121no—
209#127—271no—
210#125—271no—
211#102—31no—
212#102—91no—
213#102—271no—
214#102—271no—
215#97—271no—
216C3×C9#49011yescharacteristic, complemented, Sylow
217C23×C3—61no—
218C23×C3—181no—
219C22×D3—21no—
220C22×D3—61no—
221C3×D4—61no—
222C3×D4—61no—
223C3×D4—181no—
224C3×D4—181no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4C22×D911yescharacteristic
230C2×C3×C4C2×D3211yescharacteristic
231#67—362no—
232#67—362no—
233#67—362no—
234#67—362no—
235C2×Dic3—91no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—181no—
240C2×Dic3—181no—
241C2×Dic3—182no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244D12—21no—
245D12—21no—
246D12—61no—
247D12—61no—
248C3×C8—61no—
249C3×C8—181no—
250C3⋊C8—21no—
251C3⋊C8—61no—
252C2×C32#88511yescharacteristic
253C2×C32#91111yescharacteristic
254C2×C32#91411yescharacteristic
255C3×D3—81no—
256C3×D3—241no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—241no—
261C2×C9#88511yescharacteristic
262C2×C9#91111yescharacteristic
263C2×C9#91411yescharacteristic
264D9—81no—
265C2×D4—31no—
266C2×D4—91no—
267C2×D4—541no—
268C2×D4—541no—
269C2×C8—31no—
270C2×C8—91no—
271C4⋊C4—271no—
272C4⋊C4—541no—
273C4⋊C4—541no—
274C4⋊C4D3×D911yescharacteristic
275C42—271no—
276C22×C3—21no—
277C22×C3—121no—
278C22×C3—121no—
279C22×C3—121no—
280C22×C3—361no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3D4×D911yescharacteristic
284C22×C3#381111yescharacteristic
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—121no—
289C2×D3—121no—
290C2×D3—121no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—42no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4#369711yescharacteristic
297C3×C4#369911yescharacteristic
298C3×C4#379511yescharacteristic
299C3×C4#379511yescharacteristic
300Dic3—181no—
301Dic3—181no—
302Dic3—181no—
303Dic3—181no—
304Dic3—361no—
305Dic3—361no—
306Dic3—362no—
307Dic3—361no—
308Dic3—361no—
309C32#476411yescharacteristic
310C9—21no—
311C9#476411yescharacteristic, complemented
312C23—61no—
313C23—181no—
314D4—61no—
315D4—61no—
316D4—181no—
317D4—181no—
318D4—1082no—
319D4—1082no—
320C2×C4—21no—
321C2×C4—271no—
322C2×C4—271no—
323C2×C4—541no—
324C2×C4C2×D3×D911yescharacteristic
325C8—61no—
326C8—181no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—241no—
331C2×C3—721no—
332C2×C3#6344711yescharacteristic
333C2×C3#6347311yescharacteristic
334C2×C3#6347611yescharacteristic
335C2×C3#6391311yescharacteristic
336C2×C3#6392711yescharacteristic
337C2×C3#6394111yescharacteristic
338D3—81no—
339D3—241no—
340C22—121no—
341C22—121no—
342C22—121no—
343C22—361no—
344C22—361no—
345C22—361no—
346C22#8866811yescharacteristic
347C4—41no—
348C4—541no—
349C4—541no—
350C4—1081no—
351C4#8863811yescharacteristic
352C4#8864111yescharacteristic
353C3—21no—
354C3#1058844411yescharacteristic, complemented
355C3#1059055411yescharacteristic
356C2—241no—
357C2—721no—
358C2#1172104211yescharacteristic
359C2#1172117511yescharacteristic
360C2#1172121311yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721043C2
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. #11721746C2semidirect
3. C9×#809C2semidirect
4. C3×#63369C2semidirect
5. #11720295C2semidirect
6. #11720283C2semidirect
7. #11720275C2semidirect
35. #63369D3semidirect
107. #809D9semidirect
216. C3×C9#490semidirect
311. C9#4764semidirect
354. C3#10588444semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117210432
C9×#8092
C3×#633692
#117202952
#117202832
#117202752
#105905543
#105884443