Finite Group Databasebeta

← #13138293

Subgroups — #13138293

Order 1728.

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

normal subgroups only

This group has 362 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#11721000C211yesmaximal, characteristic
2#11720936C211yesmaximal, characteristic
3#11720542C211yesmaximal, characteristic
4C9×#817C211yesmaximal, characteristic, complemented
5C3×#63376C211yesmaximal, characteristic, complemented
6#11720268C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10590686—31nomaximal
9#10588542—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C2×(C3⋊D9)⋊C4—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18#88445—21no—
19Dic3⋊Dic9—21no—
20#88442—21no—
21#63900—31no—
22#63871—31no—
23#63647—31no—
24C3×#817D311yescharacteristic
25C3×#816—31no—
26#63574—31no—
27#63565—31no—
28#63459—31no—
29C4⋊C4×D9—31no—
30C9×#106—31no—
31(C2×D9)⋊C8—31no—
32(C2×D9)⋊C8—61no—
33#63376D311yescharacteristic, complemented
34#63368—31no—
35#63367—31no—
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9D411yescharacteristic
39C4×C3⋊D9—21no—
40C4×C3⋊D9—21no—
41C2×C9×Dic3—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C3×Dic18—21no—
46C3×Dic18—21no—
47(C3⋊D9)⋊C4—42no—
48C3×C8×C9—21no—
49C3⋊(C9⋊C8)—21no—
50#4862—91no—
51#4805—91no—
52C2×C4×C3⋊D3—31no—
53C2×C3×Dic6—31no—
54C2×#624—61no—
55C2×C32×C8C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C3×C4⋊Dic3C2×D311yescharacteristic
60#3724—61no—
61Dic3⋊Dic3—61no—
62#3722—61no—
63C2×C4×D9—31no—
64C2×C4×D9—61no—
65C2×C4×D9—61no—
66C2×Dic18C2×D311yescharacteristic
67C2×C8×C9—21no—
68C2×C8×C9C2×D311yescharacteristic
69C9×C4⋊C4—31no—
70C9×C4⋊C4—31no—
71(C2×D9)⋊C4—61no—
72C4⋊Dic9—61no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3⋊D9—21no—
79C2×C3⋊D9—21no—
80C2×C3⋊D9SD1612yes—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—21no—
84C9×Dic3—41no—
85C9×Dic3—41no—
86C3×Dic9—41no—
87C3×Dic9—41no—
88#946—91no—
89#897—91no—
90D3×C4⋊C4—91no—
91D3×C4⋊C4—91no—
92C3×#106—31no—
93C3×#103—91no—
94#835—91no—
95(C2×D3)⋊C8—91no—
96(C2×D3)⋊C8—91no—
97(C2×D3)⋊C8—181no—
98#817D911yescharacteristic, complemented
99#816—31no—
100#810—91no—
101#808—91no—
102#808—91no—
103C22×C3⋊D3—31no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—31no—
106C4×C3⋊D3—61no—
107C4×C3⋊D3—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C3×Dic6—61no—
113C3×Dic6—61no—
114#624—122no—
115C22×D9—31no—
116C22×D9—61no—
117C32×C8—21no—
118C3⋊(C3⋊C8)—61no—
119C2×C4×C9—21no—
120C2×C4×C9—61no—
121C2×C4×C9—61no—
122C2×C4×C9C22×D311yescharacteristic
123C2×Dic9—21no—
124C2×Dic9—21no—
125C2×Dic9—31no—
126C2×Dic9—61no—
127C4×D9—61no—
128C4×D9—61no—
129C4×D9—121no—
130C4×D9—121no—
131C4×D9—122no—
132Dic18—21no—
133Dic18—21no—
134C8×C9—21no—
135C8×C9—41no—
136C9⋊C8—61no—
137C9⋊C8—121no—
138#483—271noSylow
139C2×C3×C9#12311yescharacteristic
140C2×C3×C9C2×SD1611yescharacteristic
141C2×C3×C9#13711yescharacteristic
142C3⋊D9—22no—
143C2×C3×Q8—91no—
144C2×C4×D3—91no—
145C2×C4×D3—91no—
146C2×C4×D3—181no—
147C2×C4×D3—181no—
148C2×C4×D3—181no—
149C2×Dic6—31no—
150C2×C3×C8—21no—
151C2×C3×C8C2×D911yescharacteristic
152C2×C3×C8D3211yescharacteristic
153C3×C4⋊C4—31no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4—91no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158C4⋊Dic3—31no—
159C4⋊Dic3—181no—
160C4⋊Dic3C2×D911yescharacteristic
161C4⋊Dic3C2×D911yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—181no—
168C22×C32D3×D411yescharacteristic
169C2×C3⋊D3—32no—
170C2×C3⋊D3—61no—
171C2×C3⋊D3—61no—
172C32×C4#23511yescharacteristic
173C32×C4#23911yescharacteristic
174C3⋊Dic3—61no—
175C3×Dic3—41no—
176C3×Dic3—41no—
177C3×Dic3—121no—
178C3×Dic3—121no—
179C22×C9—21no—
180C22×C9D3×D411yescharacteristic
181C2×D9—32no—
182C2×D9—62no—
183C2×D9—61no—
184C2×D9—61no—
185C2×D9—121no—
186C2×D9—121no—
187C4×C9—21no—
188C4×C9—21no—
189C4×C9—121no—
190C4×C9—121no—
191C4×C9#23511yescharacteristic
192C4×C9#23711yescharacteristic
193Dic9—41no—
194Dic9—41no—
195Dic9—61no—
196Dic9—121no—
197C22⋊Q8—271no—
198C2×C4⋊C4—271no—
199#106—31no—
200#106—271no—
201#103—91no—
202#103—271no—
203C22⋊C8—271no—
204C3×C9#48311yescharacteristic, complemented, Sylow
205C22×D3—91no—
206C22×D3—91no—
207C22×D3—181no—
208C3×Q8—181no—
209C3×Q8—181no—
210C2×C3×C4—21no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4C22×D911yescharacteristic
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—181no—
224C4×D3—181no—
225C4×D3—181no—
226C4×D3—181no—
227C4×D3—181no—
228C4×D3—361no—
229C4×D3—361no—
230C4×D3—362no—
231C4×D3—362no—
232Dic6—61no—
233Dic6—61no—
234C3×C8—21no—
235C3×C8—21no—
236C3×C8—41no—
237C3⋊C8—181no—
238C3⋊C8—181no—
239C3⋊C8—361no—
240C2×C32#89411yescharacteristic
241C2×C32D3×SD1611yescharacteristic
242C2×C32#91511yescharacteristic
243C3⋊D3—62no—
244C2×C9—21no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9#88311yescharacteristic
248C2×C9D3×SD1611yescharacteristic
249C2×C9#91811yescharacteristic
250D9—62no—
251D9—122no—
252C2×Q8—91no—
253C22×C4—271no—
254C22×C4—541no—
255C2×C8—271no—
256C2×C8D3×D911yescharacteristic
257C4⋊C4—31no—
258C4⋊C4—31no—
259C4⋊C4—91no—
260C4⋊C4—541no—
261C4⋊C4—541no—
262C22⋊C4—541no—
263C22×C3—21no—
264C22×C3D4×D911yescharacteristic
265C22×C3#380211yescharacteristic
266C2×D3—92no—
267C2×D3—92no—
268C2×D3—182no—
269C2×D3—181no—
270C2×D3—181no—
271C2×D3—181no—
272C2×D3—181no—
273C2×D3—361no—
274C2×D3—361no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—121no—
278C3×C4—121no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4#369711yescharacteristic
282C3×C4#370111yescharacteristic
283C3×C4#379911yescharacteristic
284C3×C4#380411yescharacteristic
285Dic3—41no—
286Dic3—41no—
287Dic3—121no—
288Dic3—121no—
289Dic3—181no—
290Dic3—181no—
291Dic3—361no—
292C32#486211yescharacteristic
293C9—21no—
294C9#480511yescharacteristic, complemented
295C23—271no—
296Q8—181no—
297Q8—181no—
298C2×C4—61no—
299C2×C4—61no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—271no—
303C2×C4—541no—
304C2×C4—541no—
305C2×C4—1082no—
306C2×C4C2×D3×D911yescharacteristic
307C8—21no—
308C8—541no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#6345611yescharacteristic
313C2×C3SD16×D911yescharacteristic
314C2×C3#6347711yescharacteristic
315C2×C3#6379611yescharacteristic
316C2×C3#6381111yescharacteristic
317C2×C3#6385411yescharacteristic
318D3—182no—
319D3—182no—
320D3—362no—
321C22—272no—
322C22—541no—
323C22—541no—
324C22#8864611yescharacteristic
325C4—121no—
326C4—121no—
327C4—361no—
328C4—361no—
329C4—541no—
330C4#8863511yescharacteristic
331C4#8865811yescharacteristic
332C3—21no—
333C3#1058854211yescharacteristic, complemented
334C3#1059068611yescharacteristic
335C2—542no—
336C2#1172081711yescharacteristic
337C2#1172082511yescharacteristic
338C2#1172094211yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#483
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
4. C9×#817C2semidirect
5. C3×#63376C2semidirect
6. #11720268C2semidirect
7. #11720265C2semidirect
33. #63376D3semidirect
98. #817D9semidirect
204. C3×C9#483semidirect
294. C9#4805semidirect
333. C3#10588542semidirect

Maximal subgroups

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

subgroupindex
#117210002
#117209362
#117205422
C9×#8172
C3×#633762
#117202682
#117202652
#105906863
#105885423