Finite Group Databasebeta

← #13137562

Subgroups — #13137562

Order 1728.

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

normal subgroups only

This group has 400 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#11721749C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic, complemented
3C9×#808C211yesmaximal, characteristic, complemented
4C3×#63366C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic
8#10590435—31nomaximal
9#10588551—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C3×C9×C4⋊C4C2211yescharacteristic
13#88536—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64102—31no—
22#63872—31no—
23C3×#808D311yescharacteristic
24C3×#807—31no—
25#63572—31no—
26#63564—31no—
27#63564—31no—
28#63455—31no—
29#63455—61no—
30#63452—31no—
31C9×#106—31no—
32#63380—31no—
33Dic9⋊C8—31no—
34#63369—31no—
35#63366D311yescharacteristic, complemented
36C22×C3⋊D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)Q812yes—
43C4×C3⋊D9—42no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C9×C3⋊C8—21no—
47C3×C9⋊C8—21no—
48#4891—91no—
49#4871—91no—
50C2×C4⋊(C3⋊D3)—31no—
51C2×C4×C3⋊D3—61no—
52C32×C4⋊C4C2×D311yescharacteristic
53#3752—61no—
54C4×C3⋊Dic3—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8C2×D311yescharacteristic
59Dic3⋊Dic3—61no—
60Dic3⋊Dic3—61no—
61C2×D36—31no—
62C2×D36—61no—
63C2×C4×D9—61no—
64C2×C4×D9—62no—
65C2×C8×C9—31no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4—31no—
68C9×C4⋊C4C2×D311yescharacteristic
69(C2×D9)⋊C4—61no—
70(C2×D9)⋊C4—62no—
71C4⋊Dic9—61no—
72C4⋊Dic9C2×D311yescharacteristic
73Dic9⋊C4—61no—
74C4×Dic9—31no—
75C4×Dic9—61no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C2×C3⋊D9—22no—
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C3×C4×C9—41no—
82C3×C4×C9C2×Q811yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—21no—
85C3⋊Dic9—21no—
86C9×Dic3—41no—
87C3×Dic9—41no—
88#893—91no—
89#893—91no—
90#893—181no—
91#890—91no—
92#890—91no—
93C3×#107—91no—
94C3×#106—31no—
95#821—91no—
96#821—91no—
97Dic3⋊C8—91no—
98Dic3⋊C8—91no—
99#809—91no—
100#809—91no—
101#808D911yescharacteristic, complemented
102#807—31no—
103C22×C3⋊D3—61no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—31no—
108C4⋊(C3⋊D3)—32no—
109C4⋊(C3⋊D3)—61no—
110C4×C3⋊D3—122no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—61no—
113C22×D9—62no—
114C22×D9—61no—
115C3×C3⋊C8—21no—
116C3×C3⋊C8—61no—
117C2×C4×C9—22no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—61no—
121C2×C4×C9C22×D311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—31no—
124C2×Dic9—31no—
125C2×Dic9—61no—
126C2×Dic9—61no—
127D36—32no—
128D36—62no—
129D36—62no—
130D36—61no—
131C4×D9—124no—
132C4×D9—122no—
133C8×C9—61no—
134C9⋊C8—21no—
135#478—271noSylow
136C2×C3×C9C22⋊Q811yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139C3⋊D9—42no—
140C2×D12—91no—
141C2×D12—91no—
142C2×D12—181no—
143C2×C4×D3—181no—
144C2×C4×D3—181no—
145C2×C4×D3—182no—
146C2×C3×C8—31no—
147C2×C3×C8—91no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4C2×D911yescharacteristic
152C3×C4⋊C4D3211yescharacteristic
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—182no—
156C4⋊Dic3—31no—
157C4⋊Dic3—181no—
158C4⋊Dic3—181no—
159C4⋊Dic3C2×D911yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162C4×Dic3—91no—
163C4×Dic3—91no—
164C4×Dic3—181no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8C2×D911yescharacteristic
167C22×C32D3×D411yescharacteristic
168C2×C3⋊D3—62no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C32×C4—41no—
172C32×C4#23511yescharacteristic
173C32×C4D3×Q811yescharacteristic
174C3⋊Dic3—61no—
175C3⋊Dic3—61no—
176C3×Dic3—41no—
177C3×Dic3—121no—
178C22×C9—21no—
179C22×C9D3×D411yescharacteristic
180C2×D9—64no—
181C2×D9—62no—
182C2×D9—122no—
183C2×D9—122no—
184C2×D9—121no—
185C2×D9—121no—
186C4×C9—21no—
187C4×C9—21no—
188C4×C9—42no—
189C4×C9—41no—
190C4×C9—121no—
191C4×C9#23511yescharacteristic
192C4×C9D3×Q811yescharacteristic
193Dic9—41no—
194Dic9—62no—
195Dic9—61no—
196Dic9—61no—
197Dic9—121no—
198#125—271no—
199C4×D4—271no—
200#107—91no—
201#106—31no—
202C4⋊C8—271no—
203#102—271no—
204#102—271no—
205C3×C9#47811yescharacteristic, complemented, Sylow
206C22×D3—182no—
207C22×D3—181no—
208C22×D3—181no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—61no—
214C2×C3×C4—181no—
215C2×C3×C4C22×D911yescharacteristic
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—61no—
219C2×Dic3—91no—
220C2×Dic3—91no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225D12—92no—
226D12—92no—
227D12—182no—
228D12—182no—
229D12—181no—
230D12—181no—
231C4×D3—362no—
232C4×D3—364no—
233C4×D3—362no—
234C3×C8—61no—
235C3×C8—181no—
236C3⋊C8—21no—
237C3⋊C8—61no—
238C2×C32#89611yescharacteristic
239C2×C32#91411yescharacteristic
240C2×C32#91611yescharacteristic
241C3⋊D3—122no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9#89611yescharacteristic
246C2×C9#91111yescharacteristic
247C2×C9#91911yescharacteristic
248D9—124no—
249D9—122no—
250C2×D4—271no—
251C22×C4—541no—
252C2×C8—31no—
253C2×C8—91no—
254C4⋊C4—31no—
255C4⋊C4—91no—
256C4⋊C4—541no—
257C4⋊C4—541no—
258C4⋊C4D3×D911yescharacteristic
259C22⋊C4—541no—
260C42—271no—
261C22×C3—21no—
262C22×C3D4×D911yescharacteristic
263C22×C3#381111yescharacteristic
264C2×D3—184no—
265C2×D3—182no—
266C2×D3—182no—
267C2×D3—362no—
268C2×D3—362no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—361no—
272C2×D3—361no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—42no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—121no—
279C3×C4—361no—
280C3×C4#369711yescharacteristic
281C3×C4Q8×D911yescharacteristic
282C3×C4#379711yescharacteristic
283C3×C4#379811yescharacteristic
284Dic3—41no—
285Dic3—121no—
286Dic3—182no—
287Dic3—181no—
288Dic3—181no—
289Dic3—181no—
290Dic3—181no—
291Dic3—361no—
292C32#487111yescharacteristic
293C9—21no—
294C9#489111yescharacteristic, complemented
295C23—541no—
296D4—272no—
297D4—541no—
298C2×C4—21no—
299C2×C4—61no—
300C2×C4—181no—
301C2×C4—271no—
302C2×C4—271no—
303C2×C4—1082no—
304C2×C4C2×D3×D911yescharacteristic
305C8—61no—
306C8—181no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3#6345811yescharacteristic
311C2×C3#6347611yescharacteristic
312C2×C3#6347811yescharacteristic
313C2×C3#6386511yescharacteristic
314C2×C3#6393711yescharacteristic
315C2×C3#6395011yescharacteristic
316D3—364no—
317D3—362no—
318D3—362no—
319C22—542no—
320C22—1081no—
321C22—1081no—
322C22#8866811yescharacteristic
323C4—41no—
324C4—121no—
325C4—361no—
326C4—541no—
327C4—541no—
328C4(C3⋊D9)⋊Q811yescharacteristic
329C4#8864011yescharacteristic
330C3—21no—
331C3#1058855111yescharacteristic, complemented
332C3#1059043511yescharacteristic
333C2—1082no—
334C2#1172092311yescharacteristic
335C2#1172118711yescharacteristic
336C2#1172121611yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#478
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
2. #11720937C2semidirect
3. C9×#808C2semidirect
4. C3×#63366C2semidirect
6. #11720273C2semidirect
35. #63366D3semidirect
101. #808D9semidirect
205. C3×C9#478semidirect
294. C9#4891semidirect
331. C3#10588551semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117209372
C9×#8082
C3×#633662
#117202842
#117202732
#117202672
#105904353
#105885513