Finite Group Databasebeta

← #13147072

Subgroups — #13147072

Order 1728.

382 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×#88610C211yesmaximal, characteristic, complemented
3#11720738C211yesmaximal, characteristic, complemented
4C3×#5425—33nomaximal
5C3×#5239—31nomaximal
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8#88610—21no—
9#88540—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×C3×#41⋊C3D313yes—
12C2×C3×#226—33no—
13C2×C3×#216—31no—
14C2×C3×#214—31no—
15C3×#924—31no—
16C3×#859—33no—
17C3×#835—31no—
18C3×#832—31no—
19C3×#830—31no—
20C2×C4×H(3,𝔽3)—41no—
21C2×#1084—41no—
22C2×#1084—41no—
23Q8⋊H(3,𝔽3)D411yescharacteristic
24#5425—33no—
25C3×#452—91no—
26#5239—31no—
27C32×#41—21no—
28C32×#41—61no—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C2×C3×Dic6—31no—
33C3×#41⋊C3—23no—
34C2×C3×SL(2,𝔽3)C2×D313yes—
35C3×#226—63no—
36C3×#217—61no—
37C3×#216—61no—
38C3×#216—61no—
39C3×#214—62no—
40C3×C4⋊Dic3—31no—
41C3×Dic3⋊C4—61no—
42C3×Dic3⋊C4—123no—
43C2×C3×C3⋊C8—31no—
44C2×C3×C3⋊C8—31no—
45C22×H(3,𝔽3)—41no—
46C4×H(3,𝔽3)—81no—
47#1084—42no—
48#1084—81no—
49C2×C3×#41—31no—
50C2×C3×#41C3⋊D311yescharacteristic
51C2×#41⋊C3—33no—
52C2×#226—33no—
53C2×C3×Q16—91no—
54C2×C3×SD16—91no—
55C3×C22⋊Q8—91no—
56C2×#216—31no—
57C2×#214—31no—
58#924—31no—
59#859—33no—
60C3×#103—91no—
61C3×#102—91no—
62C3×C22⋊C8—91no—
63#835—31no—
64#832—31no—
65#830—31no—
66C23×C32—31no—
67C32×Q8—31no—
68C32×Q8#6711yescharacteristic
69C32×D4—32no—
70C32×D4—61no—
71C2×C32×C4—31no—
72C2×C32×C4—43no—
73C2×C32×C4—61no—
74C2×C32×C4—61no—
75C2×C32×C4S411yescharacteristic
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—123no—
79C2×C3×Dic3—123no—
80C3×Dic6—61no—
81C3×Dic6—61no—
82C3×SL(2,𝔽3)#6713yes—
83C3×C3⋊C8—61no—
84C3×C3⋊C8—61no—
85#452—91noSylow
86C2×H(3,𝔽3)—41no—
87C2×H(3,𝔽3)—41no—
88C2×H(3,𝔽3)—41no—
89C3×#41—21no—
90C3×#41—61no—
91C3×#41—62no—
92C3×#41—61no—
93C3×#41—61no—
94C2×C3×Q8—31no—
95C2×C3×Q8—91no—
96C2×C3×Q8C2×C3⋊D311yescharacteristic
97C2×C3×D4—31no—
98C2×C3×D4—31no—
99C2×C3×D4—61no—
100C22×C3×C4—31no—
101C22×C3×C4—31no—
102C22×C3×C4—61no—
103C2×Dic6—31no—
104#41⋊C3—63no—
105C2×SL(2,𝔽3)—33no—
106#226—63no—
107C3×Q16—181no—
108C3×Q16—181no—
109C3×SD16—182no—
110C2×C3×C8—91no—
111C2×C3×C8—91no—
112C3×C4⋊C4—91no—
113C3×C4⋊C4—181no—
114C3×C22⋊C4—181no—
115#217—61no—
116#216—61no—
117#216—61no—
118#214—62no—
119C4⋊Dic3—31no—
120Dic3⋊C4—61no—
121Dic3⋊C4—123no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C22×C32—43no—
125C22×C32—61no—
126C22×C32—61no—
127C22×C32—61no—
128C22×C32C2×S411yescharacteristic
129C32×C4—21no—
130C32×C4—31no—
131C32×C4—31no—
132C32×C4—83no—
133C3×Dic3—126no—
134C3×Dic3—121no—
135C3×Dic3—121no—
136C3×Dic3—243no—
137C2×#41#27211yescharacteristic
138C2×Q16—91no—
139C2×SD16—91no—
140C22⋊Q8—91no—
141#103—91no—
142#102—91no—
143C22⋊C8—91no—
144H(3,𝔽3)—41noSylow
145C23×C3—31no—
146C23×C3—31no—
147C23×C3—61no—
148C3×Q8—31no—
149C3×Q8—31no—
150C3×Q8—31no—
151C3×Q8—61no—
152C3×Q8—181no—
153C3×Q8—181no—
154C3×Q8#63811yescharacteristic
155C3×D4—32no—
156C3×D4—32no—
157C3×D4—62no—
158C3×D4—61no—
159C3×D4—61no—
160C3×D4—62no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—61no—
168C2×C3×C4—61no—
169C2×C3×C4—62no—
170C2×C3×C4—62no—
171C2×C3×C4—123no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×C3×C4C3⋊S411yescharacteristic
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—123no—
178C2×Dic3—123no—
179Dic6—61no—
180Dic6—61no—
181SL(2,𝔽3)—33no—
182C3×C8—181no—
183C3×C8—181no—
184C3⋊C8—61no—
185C3⋊C8—61no—
186C2×C32—43no—
187C2×C32—43no—
188C2×C32—43no—
189C2×C32—121no—
190C2×C32#98411yescharacteristic
191C2×C32#98511yescharacteristic
192C2×C32GL(2,ℤ/4)11yescharacteristic
193#41—21no—
194#41—61no—
195C2×Q8—91no—
196C2×Q8C2×#27211yescharacteristic
197C2×D4—31no—
198C22×C4—31no—
199Q16—181no—
200Q16—181no—
201SD16—182no—
202C2×C8—91no—
203C2×C8—91no—
204C4⋊C4—91no—
205C4⋊C4—181no—
206C22⋊C4—181no—
207C22×C3—31no—
208C22×C3—62no—
209C22×C3—61no—
210C22×C3—61no—
211C22×C3—61no—
212C22×C3—61no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—121no—
216C22×C3—121no—
217C22×C3—123no—
218C22×C3C2×C3⋊S411yescharacteristic
219C3×C4—21no—
220C3×C4—31no—
221C3×C4—31no—
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—61no—
225C3×C4—61no—
226C3×C4—61no—
227C3×C4—243no—
228C3×C4—361no—
229C3×C4—361no—
230Dic3—121no—
231Dic3—121no—
232Dic3—126no—
233Dic3—243no—
234C32—43no—
235C32#542511yescharacteristic, complemented
236C23—31no—
237Q8—31no—
238Q8—181no—
239Q8—181no—
240Q8#623511yescharacteristic
241D4—32no—
242D4—61no—
243C2×C4—31no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4#625811yescharacteristic
249C8—181no—
250C8—181no—
251C2×C3—31no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—121no—
255C2×C3—122no—
256C2×C3—121no—
257C2×C3—123no—
258C2×C3—123no—
259C2×C3—123no—
260C2×C3#6426511yescharacteristic
261C2×C3#6426611yescharacteristic
262C2×C3#6426911yescharacteristic
263C22—61no—
264C22—61no—
265C22—61no—
266C22C2×#625811yescharacteristic
267C4—21no—
268C4—31no—
269C4—31no—
270C4—361no—
271C4—361no—
272C3—31no—
273C3—123no—
274C3#1059365911yescharacteristic
275C2—121no—
276C2#1172239211yescharacteristic
277C2#1172239311yescharacteristic
278C2#1172239811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C2×C3⋊S4
commutator, cosocleC2×Q8⋊H(3,𝔽3)C22
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
hypercenterC2×C3×C4C3⋊S4
hypocenterQ8⋊H(3,𝔽3)D4
LCSG ▷ C2×Q8⋊H(3,𝔽3) ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88610C2semidirect
3. #11720738C2semidirect
235. C32#5425semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
C2×#886102
#117207382
C3×#54253
C3×#52393
#885404