Finite Group Databasebeta

← #13138549

Subgroups — #13138549

Order 1728.

392 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721045C211yesmaximal, characteristic
2#11721000C211yesmaximal, characteristic, complemented
3#11720542C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic, complemented
5C3×#63377C211yesmaximal, characteristic, complemented
6#11720294C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10590649—31nomaximal
9#10588457—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×#6195—21no—
13C2×(C3⋊D9)⋊C4—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C3×C4⋊Dic9C2211yescharacteristic
19(C2×D3)⋊Dic9—21no—
20#88442—21no—
21#63916—31no—
22#63900—31no—
23#63647—31no—
24C3×#821D311yescharacteristic
25C3×#817—31no—
26#63574—31no—
27#63562—31no—
28#63500—31no—
29C4⋊C4×D9—31no—
30C9×#102—31no—
31#63392—31no—
32(C2×D9)⋊C8—31no—
33(C2×D9)⋊C8—61no—
34#63377D311yescharacteristic, complemented
35#63367—31no—
36C22×C3⋊D9D411yescharacteristic
37C22×D3×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C4×C3⋊D9—21no—
41C4×C3⋊D9—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C9×D12—21no—
46C9×D12—21no—
47#6195—42no—
48(C3⋊D9)⋊C4—42no—
49C3×C8×C9—21no—
50C3⋊(C9⋊C8)—21no—
51#4863—91no—
52#4777—91no—
53C2×C4×C3⋊D3—31no—
54C2×C3×D12C2×D311yescharacteristic
55C2×#626—61no—
56C2×#624—61no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3C2×D311yescharacteristic
62#3724—61no—
63(C2×D3)⋊Dic3—61no—
64C2×D4×C9—31no—
65C2×#611—61no—
66C2×C4×D9—31no—
67C2×C4×D9—61no—
68C2×C4×D9—61no—
69C2×C8×C9—21no—
70C2×C8×C9C2×D311yescharacteristic
71C9×C4⋊C4—31no—
72#3676—61no—
73C4⋊Dic9C2×D311yescharacteristic
74C4⋊Dic9C2×D311yescharacteristic
75Dic9⋊C4—61no—
76C2×C9⋊C8—31no—
77C2×C9⋊C8—61no—
78C22×C3×C9C2×D411yescharacteristic
79C2×C3⋊D9—21no—
80C2×C3⋊D9—21no—
81C2×C3⋊D9SD1612yes—
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C2×D3×C9—41no—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—21no—
88C9×Dic3—41no—
89C3×Dic9—41no—
90C3×Dic9—41no—
91#938—91no—
92#895—91no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—91no—
95C3×#106—91no—
96C3×#102—31no—
97#832—91no—
98#821D911yescharacteristic, complemented
99(C2×D3)⋊C8—91no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—181no—
102#817—31no—
103#809—91no—
104#808—91no—
105#808—91no—
106C22×C3⋊D3—31no—
107C22×C3×D3—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—31no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C3×D12—21no—
116C3×D12—21no—
117#626—122no—
118#624—122no—
119C23×C9—61no—
120C22×D9—31no—
121C22×D9—61no—
122C32×C8—21no—
123C3⋊(C3⋊C8)—61no—
124D4×C9—61no—
125D4×C9—61no—
126C2×C4×C9—21no—
127C2×C4×C9—61no—
128C2×C4×C9C22×D311yescharacteristic
129#611—122no—
130C2×Dic9—21no—
131C2×Dic9—21no—
132C2×Dic9—31no—
133C2×Dic9—61no—
134C4×D9—61no—
135C4×D9—61no—
136C4×D9—121no—
137C4×D9—121no—
138C4×D9—122no—
139C8×C9—21no—
140C8×C9—41no—
141C9⋊C8—61no—
142C9⋊C8—121no—
143#481—271noSylow
144C2×C3×C9#12311yescharacteristic
145C2×C3×C9C2×SD1611yescharacteristic
146C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
147C3⋊D9—22no—
148D3×C9—81no—
149C2×C3×D4—31no—
150C2×#67—181no—
151C2×D12—181no—
152C2×D12C2×D911yescharacteristic
153C2×C4×D3—91no—
154C2×C4×D3—91no—
155C2×C4×D3—181no—
156C2×C4×D3—181no—
157C2×C4×D3—181no—
158C2×C3×C8—21no—
159C2×C3×C8C2×D911yescharacteristic
160C2×C3×C8D3211yescharacteristic
161C3×C4⋊C4—31no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164#217—181no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3C2×D911yescharacteristic
169Dic3⋊C4—181no—
170Dic3⋊C4—181no—
171C2×C3⋊C8—91no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—181no—
174C22×C32D3×D411yescharacteristic
175C2×C3⋊D3—32no—
176C2×C3⋊D3—61no—
177C2×C3⋊D3—61no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C32×C4#23511yescharacteristic
182C32×C4#23711yescharacteristic
183C3⋊Dic3—61no—
184C3×Dic3—41no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C22×C9—21no—
188C22×C9—121no—
189C22×C9—121no—
190C22×C9—121no—
191C22×C9D3×D411yescharacteristic
192C2×D9—32no—
193C2×D9—62no—
194C2×D9—61no—
195C2×D9—61no—
196C2×D9—121no—
197C2×D9—121no—
198C4×C9—21no—
199C4×C9—21no—
200C4×C9—121no—
201C4×C9#23511yescharacteristic
202C4×C9#23911yescharacteristic
203Dic9—41no—
204Dic9—41no—
205Dic9—61no—
206Dic9—121no—
207#121—271no—
208C2×C4⋊C4—271no—
209#106—91no—
210#106—271no—
211#102—31no—
212#102—271no—
213C22⋊C8—271no—
214C3×C9#48111yescharacteristic, complemented, Sylow
215C23×C3—61no—
216C22×D3—21no—
217C22×D3—91no—
218C22×D3—91no—
219C22×D3—181no—
220C3×D4—61no—
221C3×D4—61no—
222C2×C3×C4—21no—
223C2×C3×C4—61no—
224C2×C3×C4—181no—
225C2×C3×C4—181no—
226C2×C3×C4C22×D911yescharacteristic
227C2×C3×C4C2×D3211yescharacteristic
228#67—362no—
229C2×Dic3—21no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—181no—
235D12—21no—
236D12—21no—
237D12—362no—
238C4×D3—181no—
239C4×D3—181no—
240C4×D3—181no—
241C4×D3—181no—
242C4×D3—361no—
243C4×D3—361no—
244C4×D3—362no—
245C4×D3—362no—
246C3×C8—21no—
247C3×C8—21no—
248C3×C8—41no—
249C3⋊C8—181no—
250C3⋊C8—181no—
251C3⋊C8—361no—
252C2×C32#88311yescharacteristic
253C2×C32#91111yescharacteristic
254C2×C32D3×SD1611yescharacteristic
255C3⋊D3—62no—
256C3×D3—81no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9—241no—
261C2×C9#89411yescharacteristic
262C2×C9D3×SD1611yescharacteristic
263C2×C9#91411yescharacteristic
264D9—62no—
265D9—122no—
266C2×D4—31no—
267C2×D4—541no—
268C22×C4—271no—
269C22×C4—541no—
270C2×C8—271no—
271C2×C8D3×D911yescharacteristic
272C4⋊C4—31no—
273C4⋊C4—91no—
274C4⋊C4—91no—
275C4⋊C4—541no—
276C22⋊C4—541no—
277C22×C3—21no—
278C22×C3—121no—
279C22×C3—121no—
280C22×C3—121no—
281C22×C3D4×D911yescharacteristic
282C22×C3#380211yescharacteristic
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—92no—
287C2×D3—92no—
288C2×D3—182no—
289C2×D3—181no—
290C2×D3—181no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—361no—
294C2×D3—361no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—121no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4#369711yescharacteristic
301C3×C4#369911yescharacteristic
302C3×C4#379911yescharacteristic
303C3×C4#380411yescharacteristic
304Dic3—41no—
305Dic3—121no—
306Dic3—121no—
307Dic3—181no—
308Dic3—181no—
309Dic3—361no—
310C32#477711yescharacteristic
311C9—21no—
312C9#486311yescharacteristic, complemented
313C23—61no—
314C23—271no—
315D4—61no—
316D4—61no—
317D4—1082no—
318C2×C4—61no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—271no—
322C2×C4—541no—
323C2×C4—541no—
324C2×C4—1082no—
325C2×C4C2×D3×D911yescharacteristic
326C8—21no—
327C8—541no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—241no—
332C2×C3#6344511yescharacteristic
333C2×C3#6347311yescharacteristic
334C2×C3SD16×D911yescharacteristic
335C2×C3#6379611yescharacteristic
336C2×C3#6379811yescharacteristic
337C2×C3#6385411yescharacteristic
338D3—81no—
339D3—182no—
340D3—182no—
341D3—362no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—272no—
346C22—541no—
347C22—541no—
348C22#8864611yescharacteristic
349C4—121no—
350C4—361no—
351C4—361no—
352C4—541no—
353C4#8864211yescharacteristic
354C4#8866211yescharacteristic
355C3—21no—
356C3#1058845711yescharacteristic, complemented
357C3#1059064911yescharacteristic
358C2—241no—
359C2—542no—
360C2#1172081711yescharacteristic
361C2#1172081911yescharacteristic
362C2#1172095811yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radical#11721045C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#481
LCSG ▷ C2×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
2. #11721000C2semidirect
3. #11720542C2semidirect
4. C9×#821C2semidirect
5. C3×#63377C2semidirect
6. #11720294C2semidirect
7. #11720265C2semidirect
34. #63377D3semidirect
98. #821D9semidirect
214. C3×C9#481semidirect
312. C9#4863semidirect
356. C3#10588457semidirect

Maximal subgroups

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

subgroupindex
#117210452
#117210002
#117205422
C9×#8212
C3×#633772
#117202942
#117202652
#105906493
#105884573