Finite Group Databasebeta

← #13161246

Subgroups — #13161246

Order 1728.

768 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C3×#64156C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic, complemented
3#11722205C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic
6#11722033C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592493—31nomaximal
9#10591624—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64156D311yescharacteristic
22#64097—31no—
23C3×#947—34no—
24#63918—34no—
25C32×#103—31no—
26#63662—31no—
27#63662—34no—
28#63662—64no—
29#63648—31no—
30#63642—31no—
31#63636—31no—
32C3×#835D314yescomplemented
33#63563—34no—
34#63562—34no—
35#63554—34no—
36C22×C3×C3⋊D3—21no—
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3—21no—
47#6293—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C32×C3⋊C8—21no—
50#5055—91no—
51#4794—94no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×#634—61no—
57C2×C3×D12—34no—
58C2×#627—64no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61#3752—21no—
62#3752—21no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66#3750—61no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×(C2×D3)⋊C4—64no—
72C3×(C2×D3)⋊C4—64no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—34no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C33×C4—41no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—41no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—21no—
87C32×Dic3—41no—
88C3×#124—91no—
89#947—34no—
90#888—94no—
91#869—91no—
92C3×#103—34no—
93#835—94no—
94#835—91no—
95#835—184no—
96#835C3⋊D311yescharacteristic, complemented
97#832—91no—
98#832—91no—
99#821—94no—
100#815—94no—
101#809—94no—
102#803—91no—
103C22×C3⋊D3—21no—
104C22×C3×D3—64no—
105C32×Q8—28no—
106C32×Q8—24no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—44no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—124no—
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3⋊Dic3#6711yescharacteristic
123C4⋊(C3⋊D3)—21no—
124C4⋊(C3⋊D3)—21no—
125#634—122no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—34no—
129C3×D12—64no—
130C3×D12—64no—
131#627—128no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—64no—
135C3⋊(C3⋊C8)—124no—
136C3×C3⋊C8—24no—
137#489—271noSylow
138C2×C33#12411yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C33#13711yescharacteristic
141C3×C3⋊D3—81no—
142C2×C3×Q8—24no—
143C2×C3×Q8D3214yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C2×C3×D4—91no—
146C2×D12—34no—
147C2×Dic6—181no—
148C2×Dic6—184no—
149C2×C3×C8—34no—
150C3×C4⋊C4—34no—
151C3×C22⋊C4—181no—
152C3×C22⋊C4—181no—
153C3×C42—91no—
154(C2×D3)⋊C4—64no—
155(C2×D3)⋊C4—64no—
156C4⋊Dic3—94no—
157C4⋊Dic3—91no—
158C4⋊Dic3—184no—
159C4⋊Dic3—181no—
160C4⋊Dic3C2×C3⋊D311yescharacteristic
161Dic3⋊C4—184no—
162C4×Dic3—34no—
163C2×C3⋊C8—94no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—184no—
166C2×C3⋊C8C2×C3⋊D311yescharacteristic
167C22×C32—24no—
168C22×C32D3×D414yes—
169C22×C32C2×#6711yescharacteristic
170C2×C3⋊D3—41no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3—41no—
173C2×C3×D3—124no—
174C2×C3×D3—124no—
175C2×C3×D3—124no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—48no—
179C32×C4—44no—
180C32×C4—41no—
181C32×C4—121no—
182C32×C4#23514yes—
183C32×C4#23714yes—
184C32×C4#23711yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—121no—
189C3⋊Dic3—124no—
190C3⋊Dic3—244no—
191C3×Dic3—44no—
192C3×Dic3—64no—
193C3×Dic3—64no—
194C4⋊Q8—271no—
195#124—91no—
196#103—31no—
197#103—271no—
198#102—271no—
199#102—271no—
200#97—271no—
201C33#48911yescharacteristic, complemented, Sylow
202C23×C3—181no—
203C22×D3—64no—
204C3×Q8—28no—
205C3×Q8—24no—
206C3×Q8—24no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8—44no—
210C3×D4—181no—
211C3×D4—181no—
212C2×C3×C4—24no—
213C2×C3×C4—21no—
214C2×C3×C4—24no—
215C2×C3×C4—44no—
216C2×C3×C4—64no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—34no—
223C2×Dic3—34no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226C2×Dic3—364no—
227D12—64no—
228D12—64no—
229Dic6—362no—
230Dic6—368no—
231C3×C8—64no—
232C3⋊C8—21no—
233C3⋊C8—184no—
234C3⋊C8—181no—
235C3⋊C8—364no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32#88514yes—
240C2×C32#91414yes—
241C2×C32#91814yes—
242C2×C32#93611yescharacteristic
243C2×C32#94911yescharacteristic
244C2×C32#95111yescharacteristic
245C3⋊D3—81no—
246C3×D3—244no—
247C2×Q8—541no—
248C2×Q8D3×C3⋊D311yescharacteristic
249C2×D4—91no—
250C2×C8—31no—
251C2×C8—271no—
252C4⋊C4—31no—
253C4⋊C4—271no—
254C4⋊C4—541no—
255C22⋊C4—181no—
256C22⋊C4—181no—
257C42—91no—
258C22×C3—24no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3D3×#6714yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—124no—
265C2×D3—124no—
266C2×D3—124no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—48no—
270C3×C4—41no—
271C3×C4—44no—
272C3×C4—124no—
273C3×C4—181no—
274C3×C4—181no—
275C3×C4#380414yes—
276C3×C4#380514yes—
277C3×C4#382811yescharacteristic
278C3×C4#383011yescharacteristic
279Dic3—41no—
280Dic3—64no—
281Dic3—64no—
282Dic3—361no—
283Dic3—364no—
284Dic3—724no—
285C32—24no—
286C32#479414yescomplemented
287C32#505511yescharacteristic, complemented
288C23—181no—
289Q8—21no—
290Q8—21no—
291Q8—1082no—
292D4—181no—
293D4—181no—
294C2×C4—21no—
295C2×C4—61no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—541no—
299C2×C4C2×D3×C3⋊D311yescharacteristic
300C8—61no—
301C8—541no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—721no—
306C2×C3#6393914yes—
307C2×C3#6394314yes—
308C2×C3#6395914yes—
309C2×C3#6409411yescharacteristic
310C2×C3#6412311yescharacteristic
311C2×C3#6412711yescharacteristic
312D3—244no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22C3⋊D3×#6711yescharacteristic
317C4—41no—
318C4—121no—
319C4—181no—
320C4—181no—
321C4—1081no—
322C4#8902911yescharacteristic
323C4#8903111yescharacteristic
324C3—24no—
325C3#1059162414yescomplemented
326C3#1059249311yescharacteristic, complemented
327C2—721no—
328C2#1172357011yescharacteristic
329C2#1172357611yescharacteristic
330C2#1172359111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#489
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11723535C2semidirect
3. #11722205C2semidirect
4. C32×#835C2semidirect
7. #11722020C2semidirect
32. C3×#835D3semidirect
96. #835C3⋊D3semidirect
201. C33#489semidirect
286. C32#4794semidirect
287. C32#5055semidirect
325. C3#10591624semidirect
326. C3#10592493semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
#117235352
#117222052
C32×#8352
#117220342
#117220332
#117220202
#105924933
#105916243