Finite Group Databasebeta

← #13161025

Subgroups — #13161025

Order 1728.

848 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C32×D4×Dic3C211yesmaximal, characteristic
2#11723534C211yesmaximal, characteristic, complemented
3#11722202C211yesmaximal, characteristic
4C3×#63659C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592707—31nomaximal
9#10591408—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C32×Dic3—21no—
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4×D4—31no—
22#64078—31no—
23C3×D4×Dic3D314yes—
24#63918—34no—
25#63659—31no—
26#63659—34no—
27#63659—64no—
28#63659D311yescharacteristic
29#63635—31no—
30#63635—31no—
31C4⋊(C3⋊(C3⋊C8))—31no—
32C3×#832—34no—
33#63574—34no—
34#63573—34no—
35Dic3⋊(C3⋊C8)—34no—
36C23×C33—21no—
37C33×D4—21no—
38C33×D4Q812yescomplemented
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—42no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46#6293—42no—
47C3⋊(C3⋊(C3⋊C8))—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5008—94no—
50#4761—91no—
51C2×C32×D4—24no—
52C2×C32×D4C2×D314yes—
53C2×C32×D4C2×D311yescharacteristic
54C22×C32×C4—61no—
55C2×#634—61no—
56C22×C3×Dic3—24no—
57C2×#627—64no—
58C32×C4⋊C4—31no—
59C32×C22⋊C4—61no—
60C32×C42—31no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—61no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—64no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×#217—24no—
71C3×C4⋊Dic3—34no—
72C3×C4⋊Dic3C2×D314yes—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75Dic3⋊Dic3—64no—
76C22×C33—22no—
77C22×C33—41no—
78C22×C33—41no—
79C22×C33C2×D411yescharacteristic
80C33×C4C2×Q811yescharacteristic
81C33×C4#4111yescharacteristic
82C3⋊(C3⋊Dic3)—41no—
83C3×C3⋊Dic3—41no—
84C32×Dic3—21no—
85C32×Dic3—41no—
86C32×Dic3SD1612yes—
87C3×C4×D4—34no—
88D4×Dic3C3⋊D311yescharacteristic
89#888—91no—
90#869—94no—
91C3×#102—91no—
92#832—34no—
93#832—94no—
94#832—91no—
95#832—184no—
96#817—91no—
97Dic3⋊C8—91no—
98#808—94no—
99#808—94no—
100#808—91no—
101C4⋊(C3⋊C8)—94no—
102C23×C32—21no—
103C23×C32—24no—
104C23×C32—44no—
105C32×D4—21no—
106C32×D4—28no—
107C32×D4—24no—
108C32×D4—44no—
109C32×D4Dic618yescomplemented
110C32×D4Dic612yescomplemented
111C2×C32×C4—24no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—61no—
115C2×C32×C4—122no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—124no—
122#634—122no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—48no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3#6714yes—
127C2×C3×Dic3#6714yes—
128#627—128no—
129C3⋊(C3⋊C8)—21no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3×C3⋊C8—64no—
134#476—271noSylow
135C2×C33—42no—
136C2×C33C22⋊Q811yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C2×C3×D4—24no—
140C2×C3×D4D3214yes—
141C2×C3×D4C2×C3⋊D311yescharacteristic
142C22×C3×C4—64no—
143C22×Dic3—21no—
144C2×Dic6—184no—
145C2×Dic6—181no—
146C2×C3×C8—91no—
147C3×C4⋊C4—34no—
148C3×C4⋊C4—91no—
149C3×C22⋊C4—64no—
150C3×C42—34no—
151#217—21no—
152C4⋊Dic3—34no—
153C4⋊Dic3—94no—
154C4⋊Dic3—91no—
155C4⋊Dic3—184no—
156C4⋊Dic3—184no—
157C4⋊Dic3C2×C3⋊D311yescharacteristic
158Dic3⋊C4—181no—
159C4×Dic3C2×C3⋊D311yescharacteristic
160C2×C3⋊C8—34no—
161C2×C3⋊C8—94no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C22×C32—22no—
165C22×C32—24no—
166C22×C32—28no—
167C22×C32—48no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—44no—
171C22×C32—44no—
172C22×C32—84no—
173C22×C32—84no—
174C22×C32D3×D411yescharacteristic
175C22×C32C2×#6714yes—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—32no—
179C32×C4—61no—
180C32×C4—121no—
181C32×C4C2×Dic614yes—
182C32×C4C2×Dic611yescharacteristic
183C32×C4#23514yes—
184C32×C4#23711yescharacteristic
185C3⋊Dic3—41no—
186C3⋊Dic3—121no—
187C3⋊Dic3—124no—
188C3⋊Dic3—244no—
189C3×Dic3—24no—
190C3×Dic3—44no—
191C3×Dic3—124no—
192C3×Dic3#21418yes—
193C4⋊Q8—271no—
194C4×D4—31no—
195#106—271no—
196#106—271no—
197C4⋊C8—271no—
198#102—91no—
199#102—271no—
200C33#47611yescharacteristic, complemented, Sylow
201C23×C3—24no—
202C23×C3—21no—
203C23×C3—44no—
204C3×D4—24no—
205C3×D4—21no—
206C3×D4—28no—
207C3×D4—44no—
208C3×D4#62718yescomplemented
209C3×D4#63412yescomplemented
210C2×C3×C4—24no—
211C2×C3×C4—34no—
212C2×C3×C4—34no—
213C2×C3×C4—64no—
214C2×C3×C4—128no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—42no—
220C2×Dic3—64no—
221C2×Dic3—184no—
222C2×Dic3—181no—
223C2×Dic3—364no—
224C2×Dic3#63811yescharacteristic
225C2×Dic3#63811yescharacteristic
226Dic6—362no—
227Dic6—368no—
228C3×C8—181no—
229C3⋊C8—64no—
230C3⋊C8—184no—
231C3⋊C8—181no—
232C3⋊C8—364no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—24no—
236C2×C32—42no—
237C2×C32—48no—
238C2×C32—88no—
239C2×C32#87811yescharacteristic
240C2×C32#91111yescharacteristic
241C2×C32D3×SD1611yescharacteristic
242C2×C32#92414yes—
243C2×C32C2×#21414yes—
244C2×C32#94914yes—
245C2×Q8—541no—
246C2×D4D3×C3⋊D311yescharacteristic
247C22×C4—61no—
248C2×C8—91no—
249C2×C8—271no—
250C4⋊C4—31no—
251C4⋊C4—91no—
252C4⋊C4—271no—
253C4⋊C4—541no—
254C22⋊C4—61no—
255C42—31no—
256C22×C3—22no—
257C22×C3—28no—
258C22×C3—24no—
259C22×C3—48no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—44no—
263C22×C3—44no—
264C22×C3—84no—
265C22×C3—84no—
266C22×C3D3×#6714yes—
267C22×C3C2×#63811yescharacteristic
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—38no—
271C3×C4—64no—
272C3×C4—124no—
273C3×C4—361no—
274C3×C4#380414yes—
275C3×C4C2×#62714yes—
276C3×C4C2×#63411yescharacteristic
277C3×C4#382811yescharacteristic
278Dic3—21no—
279Dic3—41no—
280Dic3—124no—
281Dic3—361no—
282Dic3—364no—
283Dic3—724no—
284Dic3#375412yes—
285C32—24no—
286C32#476111yescharacteristic, complemented
287C32#500814yescomplemented
288C23—21no—
289Q8—1082no—
290D4—21no—
291D4#629312yescomplemented
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—61no—
295C2×C4—122no—
296C2×C4—181no—
297C2×C4—541no—
298C2×C4C2×D3×C3⋊D311yescharacteristic
299C8—181no—
300C8—541no—
301C2×C3—24no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—48no—
305C2×C3—42no—
306C2×C3—88no—
307C2×C3#6392514yes—
308C2×C3D3×#21414yes—
309C2×C3#6396414yes—
310C2×C3#6413311yescharacteristic
311C2×C3C2×#375411yescharacteristic
312C2×C3#6415811yescharacteristic
313C22—22no—
314C22—41no—
315C22—41no—
316C22D3×#63811yescharacteristic
317C4—32no—
318C4—61no—
319C4—121no—
320C4—361no—
321C4—1081no—
322C4#8903111yescharacteristic
323C4C2×#629311yescharacteristic
324C3—24no—
325C3#1059140814yescomplemented
326C3#1059270711yescharacteristic, complemented
327C2—42no—
328C2#1172354411yescharacteristic
329C2D3×#375411yescharacteristic
330C2#1172364011yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723534C2semidirect
5. #11722054C2semidirect
6. #11722053C2semidirect
7. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
38. C33×D4Q8semidirect
109. C32×D4Dic6semidirect
110. C32×D4Dic6semidirect
200. C33#476semidirect
208. C3×D4#627semidirect
209. C3×D4#634semidirect
286. C32#4761semidirect
287. C32#5008semidirect
291. D4#6293semidirect
325. C3#10591408semidirect
326. C3#10592707semidirect

Maximal subgroups

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

subgroupindex
C32×D4×Dic32
#117235342
#117222022
C3×#636592
#117220542
#117220532
Dic3⋊(C3⋊(C3⋊C8))2
#105927073
#105914083