Finite Group Databasebeta

← #13144506

Subgroups — #13144506

Order 1728.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2C2×#88549C211yesmaximal, characteristic, complemented
3C2×#88545C211yesmaximal, characteristic
4C4×C4⋊#272C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720578C211yesmaximal, characteristic, complemented
7C4⋊#6180C211yesmaximal, characteristic, complemented
8C3×#5057—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272—21no—
13C22⋊C4×H(3,𝔽3)—21no—
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#88549—22no—
17#88545—22no—
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×#124D314yescomplemented
23C2×C3×#215—34no—
24C2×C3×#213—34no—
25C3×C4×D12—34no—
26C3×#835—34no—
27C3×#832—34no—
28C3×C4⋊(C3⋊C8)—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C4⋊#272—21no—
41C4⋊#272D412yes—
42C4×#272—42no—
43#6180—21no—
44#6180—21no—
45C3×#463—91no—
46#5057—34no—
47C2×C32×Q8C2×D314yes—
48C2×C32×D4C2×D314yes—
49C2×C3×D12—34no—
50C2×C3×C4×D3—64no—
51C32×C22⋊C4—24no—
52C32×C22⋊C4—24no—
53C32×C42C2×D314yes—
54C3×#215—68no—
55C3×#213—68no—
56C3×(C2×D3)⋊C4—64no—
57C3×C4⋊Dic3—34no—
58C2×C3×C3⋊C8—34no—
59C2×C3×C3⋊C8—34no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×#272—22no—
65C2×#272—41no—
66C2×#272—41no—
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)—41no—
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)#4111yescharacteristic
72#1084—41no—
73C2×C3×SD16—91no—
74C2×C3×D8—91no—
75C3×#124—34no—
76C3×#124C3⋊D311yescharacteristic
77C3×C4×D4—91no—
78C2×#215—34no—
79C2×#213—34no—
80C4×D12—34no—
81C3×C4⋊C8—91no—
82C3×#103—91no—
83C3×#102—91no—
84#835—34no—
85#832—34no—
86C4⋊(C3⋊C8)—34no—
87C23×C32—24no—
88C22×C3×D3—64no—
89C32×Q8—24no—
90C32×Q8—24no—
91C32×D4—24no—
92C32×D4—24no—
93C2×C32×C4—24no—
94C2×C32×C4#6714yes—
95C2×C32×C4#6714yes—
96C2×C32×C4C22×D314yes—
97C2×C3×Dic3—64no—
98C3×D12—38no—
99C3×D12—64no—
100C3×C4×D3—128no—
101C3×C3⋊C8—64no—
102C3×C3⋊C8—64no—
103#463—91noSylow
104C2×H(3,𝔽3)—81no—
105C2×H(3,𝔽3)#12111yescharacteristic
106C2×H(3,𝔽3)#13511yescharacteristic
107C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
108#272—42no—
109C2×C3×Q8—34no—
110C2×C3×Q8C2×C3⋊D311yescharacteristic
111C2×C3×D4—34no—
112C2×C3×D4—91no—
113C2×C3×D4C2×C3⋊D311yescharacteristic
114C22×C3×C4—181no—
115C2×D12—34no—
116C2×C4×D3—64no—
117C3×SD16—182no—
118C3×D8—182no—
119C2×C3×C8—91no—
120C2×C3×C8—91no—
121C3×C4⋊C4—91no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—64no—
125C3×C22⋊C4—64no—
126C3×C22⋊C4—181no—
127C3×C42—34no—
128C3×C42C2×C3⋊D311yescharacteristic
129#215—68no—
130#213—68no—
131(C2×D3)⋊C4—64no—
132C4⋊Dic3—34no—
133C2×C3⋊C8—34no—
134C2×C3⋊C8—34no—
135C22×C32—44no—
136C22×C32—44no—
137C22×C32—44no—
138C22×C32C2×#6714yes—
139C2×C3×D3—68no—
140C2×C3×D3—124no—
141C2×C3×D3—124no—
142C32×C4—24no—
143C32×C4—24no—
144C32×C4—44no—
145C32×C4D3×D414yes—
146C32×C4#23714yes—
147C3×Dic3—124no—
148C2×SD16—91no—
149C2×D8—91no—
150#124#27211yescharacteristic, complemented
151C4×D4—91no—
152C4⋊C8—91no—
153#103—91no—
154#102—91no—
155H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
156C23×C3—21no—
157C23×C3—64no—
158C23×C3—181no—
159C22×D3—64no—
160C3×Q8—21no—
161C3×Q8—21no—
162C3×Q8—64no—
163C3×Q8—64no—
164C3×D4—21no—
165C3×D4—21no—
166C3×D4—64no—
167C3×D4—64no—
168C3×D4—92no—
169C3×D4—181no—
170C2×C3×C4—21no—
171C2×C3×C4—34no—
172C2×C3×C4—34no—
173C2×C3×C4—34no—
174C2×C3×C4—64no—
175C2×C3×C4—181no—
176C2×C3×C4—362no—
177C2×C3×C4#63811yescharacteristic
178C2×C3×C4#63811yescharacteristic
179C2×C3×C4C22×C3⋊D311yescharacteristic
180C2×Dic3—64no—
181D12—38no—
182D12—64no—
183C4×D3—128no—
184C3×C8—181no—
185C3×C8—181no—
186C3⋊C8—64no—
187C3⋊C8—64no—
188C2×C32—84no—
189C2×C32#93814yes—
190C2×C32#94914yes—
191C2×C32#95014yes—
192C3×D3—128no—
193C2×Q8C2×#27211yescharacteristic
194C2×D4—91no—
195C2×D4C2×#27211yescharacteristic
196C22×C4—181no—
197SD16—182no—
198D8—182no—
199C2×C8—91no—
200C2×C8—91no—
201C4⋊C4—91no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—181no—
205C42C2×#27211yescharacteristic
206C22×C3—34no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—124no—
211C22×C3—124no—
212C22×C3—124no—
213C22×C3—182no—
214C22×C3—361no—
215C22×C3—361no—
216C22×C3C2×#63811yescharacteristic
217C2×D3—68no—
218C2×D3—124no—
219C2×D3—124no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—34no—
223C3×C4—34no—
224C3×C4—41no—
225C3×C4—64no—
226C3×C4—64no—
227C3×C4—124no—
228C3×C4—361no—
229C3×C4D4×C3⋊D311yescharacteristic
230C3×C4#383011yescharacteristic
231Dic3—124no—
232C32#505714yescomplemented
233C23—21no—
234C23—181no—
235Q8—21no—
236Q8—21no—
237D4—21no—
238D4—21no—
239D4—92no—
240D4—181no—
241C2×C4—21no—
242C2×C4—181no—
243C2×C4—362no—
244C2×C4#623511yescharacteristic
245C2×C4#623511yescharacteristic
246C2×C4C22×#27211yescharacteristic
247C8—181no—
248C8—181no—
249C2×C3—34no—
250C2×C3—34no—
251C2×C3—34no—
252C2×C3—81no—
253C2×C3—244no—
254C2×C3—362no—
255C2×C3#6414711yescharacteristic
256C2×C3#6415811yescharacteristic
257C2×C3#6415911yescharacteristic
258D3—128no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—182no—
263C22—361no—
264C22—361no—
265C22C2×#623511yescharacteristic
266C4—21no—
267C4—21no—
268C4—41no—
269C4—361no—
270C4D4×#27211yescharacteristic
271C4#8874411yescharacteristic
272C3—34no—
273C3#1059275611yescharacteristic
274C2—81no—
275C2—362no—
276C2#1172184811yescharacteristic
277C2#1172187011yescharacteristic
278C2#1172187111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×#88545C2
hypercenterC3×#124C3⋊D3
hypocenterH(3,𝔽3)#463
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. C2×#88549C2semidirect
4. C4×C4⋊#272C2semidirect
5. #11720583C2semidirect
6. #11720578C2semidirect
7. C4⋊#6180C2semidirect
22. C32×#124D3semidirect
150. #124#272semidirect
155. H(3,𝔽3)#463semidirect
232. C32#5057semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#885492
C2×#885452
C4×C4⋊#2722
#117205832
#117205782
C4⋊#61802
C3×#50573