Finite Group Databasebeta

← #13161257

Subgroups — #13161257

Order 1728.

856 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 168 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
1Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722055C211yesmaximal, characteristic
7#11722050C211yesmaximal, characteristic
8#10592492—31nomaximal
9#10591631—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6293—21no—
12C4⋊(C3⋊(C3⋊Dic3))—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21Q8×C3⋊Dic3—31no—
22Q8×C3⋊Dic3—34no—
23Q8×C3⋊Dic3—64no—
24#64097—31no—
25#63919—34no—
26C32×#103—31no—
27#63662D311yescharacteristic
28#63645—31no—
29#63641—31no—
30#63634—31no—
31C3×#835—34no—
32C3×#835D314yes—
33#63575—34no—
34#63575—34no—
35#63572—34no—
36C33×Q8—21no—
37C33×Q8Q812yescomplemented
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3—21no—
45#6293—42no—
46C3×C3⋊(C3⋊C8)—21no—
47C32×C3⋊C8—21no—
48#4793—94no—
49#4793—91no—
50C2×C32×Q8—24no—
51C2×C32×Q8C2×D314yes—
52C2×C32×Q8C2×D311yescharacteristic
53C2×#634—61no—
54C2×#627—64no—
55C2×C32×C8—31no—
56C32×C4⋊C4—31no—
57C4⋊(C3⋊Dic3)—31no—
58C4⋊(C3⋊Dic3)—34no—
59C4⋊(C3⋊Dic3)—61no—
60C4⋊(C3⋊Dic3)—64no—
61C4⋊(C3⋊Dic3)—64no—
62C4⋊(C3⋊Dic3)—68no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64#3750—61no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3—64no—
69C4×C3⋊Dic3—64no—
70C4×C3⋊Dic3—68no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76#3724—64no—
77C22×C33C2×D411yescharacteristic
78C33×C4—22no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—21no—
82C3⋊(C3⋊Dic3)—41no—
83C3⋊(C3⋊Dic3)Q1612yes—
84C3×C3⋊Dic3—41no—
85C32×Dic3—41no—
86Q8×Dic3—94no—
87Q8×Dic3—91no—
88Q8×Dic3—184no—
89#888—94no—
90#888—91no—
91C3×#103—34no—
92C3×#103—91no—
93#835—34no—
94#835C3⋊D311yescharacteristic
95#818—94no—
96#818—91no—
97Dic3⋊C8—94no—
98Dic3⋊C8—91no—
99#807—94no—
100#807—91no—
101C32×Q8—28no—
102C32×Q8—24no—
103C32×Q8—21no—
104C32×Q8—28no—
105C32×Q8Dic618yescomplemented
106C32×Q8Dic612yescomplemented
107C2×C32×C4—28no—
108C2×C32×C4—24no—
109C2×C32×C4—21no—
110C2×C32×C4—24no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—68no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—61no—
124#634—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—64no—
127#627—128no—
128C32×C8—61no—
129C3⋊(C3⋊C8)—21no—
130C3×C3⋊C8—24no—
131C3×C3⋊C8—64no—
132#477—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×Q1611yescharacteristic
135C2×C33AGL(1,ℤ/8)11yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8D3214yes—
138C2×C3×Q8C2×C3⋊D311yescharacteristic
139C2×Dic6—181no—
140C2×Dic6—184no—
141C2×C3×C8—34no—
142C2×C3×C8—91no—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—91no—
145C4⋊Dic3—34no—
146C4⋊Dic3—94no—
147C4⋊Dic3—91no—
148C4⋊Dic3—184no—
149C4⋊Dic3—184no—
150C4⋊Dic3—181no—
151C4⋊Dic3—188no—
152C4⋊Dic3C2×C3⋊D311yescharacteristic
153Dic3⋊C4—181no—
154Dic3⋊C4—184no—
155C4×Dic3—94no—
156C4×Dic3—91no—
157C4×Dic3—184no—
158C4×Dic3—184no—
159C4×Dic3—181no—
160C4×Dic3—188no—
161C2×C3⋊C8—34no—
162C2×C3⋊C8C2×C3⋊D311yescharacteristic
163C22×C32—24no—
164C22×C32D3×D414yes—
165C22×C32D3×D411yescharacteristic
166C32×C4—216no—
167C32×C4—28no—
168C32×C4—22no—
169C32×C4—24no—
170C32×C4—24no—
171C32×C4—121no—
172C32×C4C2×Dic614yes—
173C32×C4C2×Dic611yescharacteristic
174C32×C4#23714yes—
175C32×C4#23711yescharacteristic
176C3⋊Dic3—32no—
177C3⋊Dic3—38no—
178C3⋊Dic3—41no—
179C3⋊Dic3—68no—
180C3⋊Dic3—61no—
181C3⋊Dic3—64no—
182C3⋊Dic3—124no—
183C3⋊Dic3—128no—
184C3⋊Dic3—124no—
185C3⋊Dic3—121no—
186C3×Dic3—44no—
187C3×Dic3—124no—
188C4⋊Q8—271no—
189C4×Q8—271no—
190#107—271no—
191#107—271no—
192C4⋊C8—271no—
193#103—31no—
194#103—91no—
195C33#47711yescharacteristic, complemented, Sylow
196C3×Q8—28no—
197C3×Q8—24no—
198C3×Q8—21no—
199C3×Q8—28no—
200C3×Q8#62718yescomplemented
201C3×Q8#63412yescomplemented
202C2×C3×C4—28no—
203C2×C3×C4—24no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—64no—
207C2×C3×C4—181no—
208C2×C3×C4C2×D3214yes—
209C2×C3×C4C22×C3⋊D311yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—64no—
212C2×Dic3—94no—
213C2×Dic3—91no—
214C2×Dic3—94no—
215C2×Dic3—91no—
216C2×Dic3—184no—
217C2×Dic3—184no—
218C2×Dic3—188no—
219C2×Dic3—184no—
220C2×Dic3—181no—
221Dic6—362no—
222Dic6—368no—
223C3×C8—64no—
224C3×C8—181no—
225C3⋊C8—21no—
226C3⋊C8—64no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#87814yes—
231C2×C32#87811yescharacteristic
232C2×C32#91414yes—
233C2×C32#91411yescharacteristic
234C2×C32D3×Q1614yes—
235C2×C32D3×Q1611yescharacteristic
236C2×Q8—541no—
237C2×Q8D3×C3⋊D311yescharacteristic
238C2×C8—31no—
239C2×C8—91no—
240C4⋊C4—31no—
241C4⋊C4—91no—
242C4⋊C4—271no—
243C4⋊C4—541no—
244C4⋊C4—541no—
245C42—271no—
246C42—541no—
247C22×C3—24no—
248C22×C3#381114yes—
249C22×C3D4×C3⋊D311yescharacteristic
250C3×C4—216no—
251C3×C4—22no—
252C3×C4—28no—
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—124no—
256C3×C4—361no—
257C3×C4#380514yes—
258C3×C4C2×#62714yes—
259C3×C4C2×#63411yescharacteristic
260C3×C4#383011yescharacteristic
261Dic3—41no—
262Dic3—92no—
263Dic3—98no—
264Dic3—124no—
265Dic3—188no—
266Dic3—181no—
267Dic3—184no—
268Dic3—364no—
269Dic3—368no—
270Dic3—361no—
271Dic3—364no—
272C32—24no—
273C32#479314yescomplemented
274C32#479311yescharacteristic, complemented
275Q8—21no—
276Q8—1082no—
277Q8#629312yescomplemented
278C2×C4—21no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—271no—
282C2×C4—271no—
283C2×C4—541no—
284C2×C4C2×D3×C3⋊D311yescharacteristic
285C8—61no—
286C8—181no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3#6394114yes—
291C2×C3#6394414yes—
292C2×C3#6398214yes—
293C2×C3#6408711yescharacteristic
294C2×C3#6412311yescharacteristic
295C2×C3Q16×C3⋊D311yescharacteristic
296C22#8903911yescharacteristic
297C4—22no—
298C4—121no—
299C4—272no—
300C4—361no—
301C4—541no—
302C4—1081no—
303C4#8902911yescharacteristic
304C4C2×#629311yescharacteristic
305C3—24no—
306C3#1059163114yescomplemented
307C3#1059249211yescharacteristic, complemented
308C2#1172357211yescharacteristic
309C2#1172357711yescharacteristic
310C2#1172364211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#477
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
37. C33×Q8Q8semidirect
105. C32×Q8Dic6semidirect
106. C32×Q8Dic6semidirect
195. C33#477semidirect
200. C3×Q8#627semidirect
201. C3×Q8#634semidirect
273. C32#4793semidirect
274. C32#4793semidirect
277. Q8#6293semidirect
306. C3#10591631semidirect
307. C3#10592492semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊Dic3)2
#117235362
C3×#636622
C32×#8352
#117220572
#117220552
#117220502
#105924923
#105916313