Finite Group Databasebeta

← #13140323

Subgroups — #13140323

Order 1728.

418 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 93 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
1D4×C3⋊Dic9C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic
4C3×#63392C211yesmaximal, characteristic
5#11720279C211yesmaximal, characteristic, complemented
6#11720273C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10591494—31nomaximal
9#10588439—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×#6189—21no—
13#88551—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21D4×C3⋊Dic3—31no—
22#63919—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63575—31no—
26#63575—31no—
27#63572—31no—
28D4×Dic9—31no—
29D4×Dic9—61no—
30#63450—31no—
31C9×#102—31no—
32#63392D311yescharacteristic
33#63378—31no—
34Dic9⋊C8—31no—
35#63366—31no—
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9Q812yescomplemented
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9—42no—
42C2×C3⋊Dic9D411yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46#6189—42no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4759—91no—
50#4759—91no—
51C2×C32×D4C2×D311yescharacteristic
52C22×C3⋊Dic3—61no—
53C2×#627—61no—
54#3757—61no—
55C4⋊(C3⋊Dic3)—31no—
56C4×C3⋊Dic3—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3724—61no—
62C2×D4×C9—21no—
63C2×D4×C9C2×D311yescharacteristic
64C22×Dic9—61no—
65C22×Dic9—62no—
66C2×Dic18—61no—
67C2×C8×C9—31no—
68C9×C4⋊C4—31no—
69#3676—61no—
70#3676—62no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73C4⋊Dic9C2×D311yescharacteristic
74Dic9⋊C4—61no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9—22no—
79C22×C3×C9—41no—
80C22×C3×C9—41no—
81C22×C3×C9C2×D411yescharacteristic
82C3×C4×C9C2×Q811yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—21no—
85C3⋊Dic9—41no—
86C3⋊Dic9D812yes—
87C9×Dic3—41no—
88C3×Dic9—41no—
89D4×Dic3—91no—
90D4×Dic3—91no—
91D4×Dic3—181no—
92#888—91no—
93#888—91no—
94C3×#102—31no—
95C3×#102—91no—
96#832—31no—
97#832D911yescharacteristic
98#818—91no—
99#818—91no—
100Dic3⋊C8—91no—
101Dic3⋊C8—91no—
102#807—91no—
103#807—91no—
104C23×C32—21no—
105C32×D4—21no—
106C32×D4Dic612yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—122no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114#627—122no—
115C23×C9—22no—
116C23×C9—21no—
117C3×C3⋊C8—21no—
118C3×C3⋊C8—61no—
119D4×C9—22no—
120D4×C9—21no—
121D4×C9—22no—
122D4×C9Dic612yescomplemented
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9—31no—
128C2×Dic9—31no—
129C2×Dic9—61no—
130C2×Dic9—61no—
131C2×Dic9—62no—
132C2×Dic9—61no—
133C2×Dic9—122no—
134C2×Dic9—124no—
135Dic18—122no—
136C8×C9—61no—
137C9⋊C8—21no—
138#474—271noSylow
139C2×C3×C9—42no—
140C2×C3×C9C22⋊Q811yescharacteristic
141C2×C3×C9C2×D811yescharacteristic
142C2×C3×C9#13711yescharacteristic
143C2×C3×D4—21no—
144C2×C3×D4C2×D911yescharacteristic
145C2×C3×D4D3211yescharacteristic
146C22×Dic3—181no—
147C22×Dic3—181no—
148C22×Dic3—182no—
149C2×Dic6—181no—
150C2×Dic6—181no—
151C2×C3×C8—31no—
152C2×C3×C8—91no—
153C3×C4⋊C4—31no—
154C3×C4⋊C4—91no—
155#217—181no—
156#217—181no—
157#217—182no—
158C4⋊Dic3—31no—
159C4⋊Dic3—91no—
160C4⋊Dic3—91no—
161C4⋊Dic3—181no—
162C4⋊Dic3C2×D911yescharacteristic
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165C4×Dic3—91no—
166C4×Dic3—91no—
167C4×Dic3—181no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8C2×D911yescharacteristic
170C22×C32—22no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32D3×D411yescharacteristic
174C32×C4C2×Dic611yescharacteristic
175C32×C4#23711yescharacteristic
176C3⋊Dic3—32no—
177C3⋊Dic3—61no—
178C3⋊Dic3—121no—
179C3×Dic3—41no—
180C3×Dic3—121no—
181C22×C9—24no—
182C22×C9—22no—
183C22×C9—21no—
184C22×C9—42no—
185C22×C9—42no—
186C22×C9—41no—
187C22×C9—41no—
188C22×C9D3×D411yescharacteristic
189C4×C9—21no—
190C4×C9—21no—
191C4×C9—121no—
192C4×C9C2×Dic611yescharacteristic
193C4×C9#23711yescharacteristic
194Dic9—32no—
195Dic9—41no—
196Dic9—62no—
197Dic9—61no—
198Dic9—121no—
199Dic9—122no—
200Dic9—121no—
201C4⋊Q8—271no—
202C4×D4—271no—
203#107—271no—
204#107—271no—
205C4⋊C8—271no—
206#102—31no—
207#102—91no—
208C3×C9#47411yescharacteristic, complemented, Sylow
209C23×C3—22no—
210C23×C3—21no—
211C23×C3—21no—
212C3×D4—22no—
213C3×D4—21no—
214C3×D4—21no—
215C3×D4—22no—
216C3×D4Dic1812yescomplemented
217C3×D4#62712yes—
218C2×C3×C4—21no—
219C2×C3×C4—61no—
220C2×C3×C4—181no—
221C2×C3×C4C22×D911yescharacteristic
222C2×C3×C4C2×D3211yescharacteristic
223C2×Dic3—21no—
224C2×Dic3—61no—
225C2×Dic3—91no—
226C2×Dic3—91no—
227C2×Dic3—91no—
228C2×Dic3—91no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—182no—
232C2×Dic3—181no—
233C2×Dic3—181no—
234C2×Dic3—362no—
235C2×Dic3—362no—
236C2×Dic3—364no—
237Dic6—362no—
238Dic6—362no—
239C3×C8—61no—
240C3×C8—181no—
241C3⋊C8—21no—
242C3⋊C8—61no—
243C2×C32—42no—
244C2×C32#87811yescharacteristic
245C2×C32D3×D811yescharacteristic
246C2×C32#91511yescharacteristic
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—21no—
250C2×C9—44no—
251C2×C9—42no—
252C2×C9#87811yescharacteristic
253C2×C9D3×D811yescharacteristic
254C2×C9#91511yescharacteristic
255C2×Q8—541no—
256C2×D4D3×D911yescharacteristic
257C22×C4—541no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—31no—
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C22⋊C4—541no—
265C42—271no—
266C22×C3—24no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3D4×D911yescharacteristic
277C22×C3#381111yescharacteristic
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—121no—
281C3×C4—361no—
282C3×C4C2×Dic1811yescharacteristic
283C3×C4#369911yescharacteristic
284C3×C4#380511yescharacteristic
285C3×C4C2×#62711yescharacteristic
286Dic3—41no—
287Dic3—92no—
288Dic3—92no—
289Dic3—121no—
290Dic3—182no—
291Dic3—181no—
292Dic3—181no—
293Dic3—361no—
294Dic3—362no—
295Dic3—361no—
296Dic3—361no—
297C32#475911yescharacteristic
298C9—21no—
299C9#475911yescharacteristic, complemented
300C23—21no—
301Q8—1082no—
302D4—21no—
303D4#618912yescomplemented
304C2×C4—61no—
305C2×C4—181no—
306C2×C4—271no—
307C2×C4—271no—
308C2×C4—541no—
309C2×C4—1082no—
310C2×C4C2×D3×D911yescharacteristic
311C8—61no—
312C8—181no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—44no—
317C2×C3—42no—
318C2×C3—42no—
319C2×C3#6344011yescharacteristic
320C2×C3D8×D911yescharacteristic
321C2×C3#6347711yescharacteristic
322C2×C3#6392611yescharacteristic
323C2×C3#6393111yescharacteristic
324C2×C3#6398211yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22#8866811yescharacteristic
329C4—121no—
330C4—272no—
331C4—361no—
332C4—541no—
333C4—1081no—
334C4C2×#618911yescharacteristic
335C4#8866311yescharacteristic
336C3—21no—
337C3#1058843911yescharacteristic, complemented
338C3#1059149411yescharacteristic
339C2—42no—
340C2#1172117411yescharacteristic
341C2#1172118311yescharacteristic
342C2#1172126711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#474
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11720971C2semidirect
5. #11720279C2semidirect
6. #11720273C2semidirect
7. #11720262C2semidirect
38. C3×D4×C9Q8semidirect
122. D4×C9Dic6semidirect
208. C3×C9#474semidirect
216. C3×D4Dic18semidirect
299. C9#4759semidirect
303. D4#6189semidirect
337. C3#10588439semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
#117209712
C9×#8322
C3×#633922
#117202792
#117202732
#117202622
#105914943
#105884393