Finite Group Databasebeta

← #13137571

Subgroups — #13137571

Order 1728.

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

normal subgroups only

This group has 386 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#11721753C211yesmaximal, characteristic
2#11721000C211yesmaximal, characteristic
3C9×#807C211yesmaximal, characteristic, complemented
4C3×#63366C211yesmaximal, characteristic, complemented
5#11720282C211yesmaximal, characteristic
6#11720277C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10590444—31nomaximal
9#10588564—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×(C3⋊D9)⋊C4—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14#88536—21no—
15C4⋊(C3⋊Dic9)—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64106—31no—
22#63900—31no—
23C3×#807—31no—
24C3×#807D311yescharacteristic
25#63567—31no—
26#63567—31no—
27#63557—31no—
28#63459—31no—
29#63459—61no—
30C4⋊C4×D9—31no—
31C9×#107—31no—
32(C2×D9)⋊C8—31no—
33#63376—31no—
34#63368—31no—
35#63366D311yescharacteristic, complemented
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C4×C3⋊D9—21no—
42C4×C3⋊D9—21no—
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47(C3⋊D9)⋊C4—42no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4884—91no—
51#4884—91no—
52C2×C4×C3⋊D3—31no—
53C2×#634—31no—
54C2×#624—61no—
55C32×C4⋊C4C2×D311yescharacteristic
56#3752—61no—
57C4⋊(C3⋊Dic3)—61no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62#3724—61no—
63C2×C4×D9—31no—
64C2×C4×D9—61no—
65C2×C4×D9—61no—
66C2×Dic18—31no—
67C2×Dic18—61no—
68C2×C8×C9—31no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4—31no—
71C9×C4⋊C4C2×D311yescharacteristic
72(C2×D9)⋊C4—61no—
73(C2×D9)⋊C4—62no—
74C4⋊Dic9—61no—
75C4⋊Dic9—62no—
76C4⋊Dic9C2×D311yescharacteristic
77Dic9⋊C4—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3⋊D9—21no—
81C2×C3⋊D9—21no—
82C2×C3⋊D9Q1612yes—
83C3×C4×C9—41no—
84C3×C4×C9#4111yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—21no—
87C3⋊Dic9—41no—
88C9×Dic3—41no—
89C3×Dic9—41no—
90#897—91no—
91#897—91no—
92#897—181no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—91no—
95C3×#107—31no—
96C3×#107—91no—
97(C2×D3)⋊C8—91no—
98(C2×D3)⋊C8—91no—
99#816—91no—
100#816—91no—
101#810—91no—
102#810—91no—
103#807—31no—
104#807D911yescharacteristic, complemented
105C22×C3⋊D3—31no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—61no—
112#634—61no—
113#634—61no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—61no—
116#624—122no—
117C22×D9—31no—
118C22×D9—61no—
119C3×C3⋊C8—21no—
120C3×C3⋊C8—61no—
121C2×C4×C9—22no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9—31no—
128C2×Dic9—61no—
129C2×Dic9—62no—
130C2×Dic9—61no—
131C4×D9—61no—
132C4×D9—61no—
133C4×D9—121no—
134C4×D9—121no—
135C4×D9—122no—
136Dic18—61no—
137Dic18—61no—
138Dic18—121no—
139Dic18—121no—
140C8×C9—61no—
141C9⋊C8—21no—
142#484—271noSylow
143C2×C3×C9#12311yescharacteristic
144C2×C3×C9C2×Q1611yescharacteristic
145C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
146C3⋊D9—22no—
147C2×C4×D3—91no—
148C2×C4×D3—91no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C2×C4×D3—181no—
152C2×Dic6—91no—
153C2×Dic6—91no—
154C2×Dic6—181no—
155C2×C3×C8—31no—
156C2×C3×C8—91no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4C2×D911yescharacteristic
161C3×C4⋊C4D3211yescharacteristic
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—182no—
165C4⋊Dic3—31no—
166C4⋊Dic3—181no—
167C4⋊Dic3—181no—
168C4⋊Dic3—182no—
169C4⋊Dic3C2×D911yescharacteristic
170Dic3⋊C4—181no—
171Dic3⋊C4—181no—
172C2×C3⋊C8—31no—
173C2×C3⋊C8C2×D911yescharacteristic
174C22×C32D3×D411yescharacteristic
175C2×C3⋊D3—32no—
176C2×C3⋊D3—61no—
177C2×C3⋊D3—61no—
178C32×C4—41no—
179C32×C4#23511yescharacteristic
180C32×C4#23911yescharacteristic
181C3⋊Dic3—61no—
182C3⋊Dic3—121no—
183C3×Dic3—41no—
184C3×Dic3—121no—
185C22×C9—21no—
186C22×C9D3×D411yescharacteristic
187C2×D9—32no—
188C2×D9—62no—
189C2×D9—61no—
190C2×D9—61no—
191C2×D9—121no—
192C2×D9—121no—
193C4×C9—21no—
194C4×C9—21no—
195C4×C9—42no—
196C4×C9—41no—
197C4×C9—121no—
198C4×C9#23511yescharacteristic
199C4×C9#23911yescharacteristic
200Dic9—41no—
201Dic9—61no—
202Dic9—121no—
203Dic9—122no—
204Dic9—121no—
205C22⋊Q8—271no—
206C2×C4⋊C4—271no—
207#107—31no—
208#107—91no—
209#103—271no—
210#103—271no—
211C22⋊C8—271no—
212C3×C9#48411yescharacteristic, complemented, Sylow
213C22×D3—91no—
214C22×D3—91no—
215C22×D3—181no—
216C2×C3×C4—22no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—61no—
221C2×C3×C4—181no—
222C2×C3×C4C22×D911yescharacteristic
223C2×C3×C4C2×D3211yescharacteristic
224C2×Dic3—21no—
225C2×Dic3—61no—
226C2×Dic3—91no—
227C2×Dic3—91no—
228C2×Dic3—181no—
229C2×Dic3—182no—
230C2×Dic3—181no—
231C2×Dic3—181no—
232C4×D3—181no—
233C4×D3—181no—
234C4×D3—181no—
235C4×D3—181no—
236C4×D3—361no—
237C4×D3—361no—
238C4×D3—362no—
239C4×D3—362no—
240Dic6—181no—
241Dic6—181no—
242Dic6—181no—
243Dic6—181no—
244Dic6—361no—
245Dic6—361no—
246C3×C8—61no—
247C3×C8—181no—
248C3⋊C8—21no—
249C3⋊C8—61no—
250C2×C32#89411yescharacteristic
251C2×C32#91111yescharacteristic
252C2×C32D3×Q1611yescharacteristic
253C3⋊D3—62no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9—21no—
257C2×C9#89411yescharacteristic
258C2×C9#91111yescharacteristic
259C2×C9D3×Q1611yescharacteristic
260D9—62no—
261D9—122no—
262C2×Q8—271no—
263C22×C4—271no—
264C22×C4—541no—
265C2×C8—31no—
266C2×C8—91no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C4⋊C4—541no—
270C4⋊C4—541no—
271C4⋊C4D3×D911yescharacteristic
272C22⋊C4—541no—
273C22×C3—21no—
274C22×C3D4×D911yescharacteristic
275C22×C3#381111yescharacteristic
276C2×D3—92no—
277C2×D3—92no—
278C2×D3—182no—
279C2×D3—181no—
280C2×D3—181no—
281C2×D3—181no—
282C2×D3—181no—
283C2×D3—361no—
284C2×D3—361no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—42no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—121no—
291C3×C4—361no—
292C3×C4#369711yescharacteristic
293C3×C4#370111yescharacteristic
294C3×C4#379911yescharacteristic
295C3×C4#379911yescharacteristic
296Dic3—41no—
297Dic3—121no—
298Dic3—181no—
299Dic3—181no—
300Dic3—361no—
301Dic3—362no—
302Dic3—361no—
303Dic3—361no—
304C32#488411yescharacteristic
305C9—21no—
306C9#488411yescharacteristic, complemented
307C23—271no—
308Q8—541no—
309Q8—541no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—181no—
313C2×C4—271no—
314C2×C4—541no—
315C2×C4—541no—
316C2×C4—541no—
317C2×C4—1082no—
318C2×C4C2×D3×D911yescharacteristic
319C8—61no—
320C8—181no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3#6345611yescharacteristic
325C2×C3#6347311yescharacteristic
326C2×C3Q16×D911yescharacteristic
327C2×C3#6389711yescharacteristic
328C2×C3#6392711yescharacteristic
329C2×C3#6394411yescharacteristic
330D3—182no—
331D3—182no—
332D3—362no—
333C22—272no—
334C22—541no—
335C22—541no—
336C22#8866811yescharacteristic
337C4—41no—
338C4—121no—
339C4—361no—
340C4—541no—
341C4—1081no—
342C4#8863511yescharacteristic
343C4#8864211yescharacteristic
344C3—21no—
345C3#1058856411yescharacteristic, complemented
346C3#1059044411yescharacteristic
347C2—542no—
348C2#1172099711yescharacteristic
349C2#1172117511yescharacteristic
350C2#1172122011yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#484
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
3. C9×#807C2semidirect
4. C3×#63366C2semidirect
6. #11720277C2semidirect
7. #11720269C2semidirect
35. #63366D3semidirect
104. #807D9semidirect
212. C3×C9#484semidirect
306. C9#4884semidirect
345. C3#10588564semidirect

Maximal subgroups

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

subgroupindex
#117217532
#117210002
C9×#8072
C3×#633662
#117202822
#117202772
#117202692
#105904443
#105885643