Finite Group Databasebeta

← #13140590

Subgroups — #13140590

Order 1728.

675 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#11721850C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C212yesmaximal, complemented
3#11721144C211yesmaximal, characteristic, complemented
4C2×#88436C212yesmaximal, complemented
5#11720302C211yesmaximal, characteristic, complemented
6#10591482—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#6235—22no—
9C2×C4⋊#272C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×#6198—22no—
12C4×#1084C2211yescharacteristic
13#3753⋊C3—24no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88436—24no—
16C2×#3753D312yescomplemented
17C3×#940—32no—
18C3×#940—61no—
19C2×C3×#213—32no—
20#63911—32no—
21C2×#3714—32no—
22C2×#3713—32no—
23Dic3×C3⋊C8—32no—
24C23×H(3,𝔽3)—21no—
25C22×#272—21no—
26C22×(C3⋊D3)⋊C3—22no—
27D4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29#6235—44no—
30C2×#1084D412yes—
31C4⋊#272—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#6198—44no—
35(C3⋊(C3⋊C8))⋊C3D414yescomplemented
36#5151—92no—
37#5073—91no—
38C2×C32×D4—21no—
39C2×C32×D4C2×D312yes—
40C2×C4⋊(C3⋊D3)C2×D312yes—
41C2×C3×#67—64no—
42C2×C3×#67—64no—
43C2×C3×D12—32no—
44C2×C3×D12—32no—
45C2×C3×D12—61no—
46C2×#626—64no—
47#3753—24no—
48C2×C3⋊(C3⋊C8)C2×D312yes—
49C3×#213—64no—
50C3×C4×Dic3—32no—
51C3×C4×Dic3—61no—
52C2×C3×C3⋊C8—32no—
53#3714—64no—
54#3713—64no—
55C22×H(3,𝔽3)—41no—
56C22×H(3,𝔽3)—42no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×#272—41no—
59C2×#272—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—44no—
62C4×H(3,𝔽3)C2×D412yes—
63#1084D814yescomplemented
64C2×C3×D8—92no—
65C3×#127—91no—
66#940—32no—
67#940—61no—
68#940—92no—
69C2×#213—32no—
70C2×#213—92no—
71C2×#213—92no—
72C2×#213—92no—
73C2×D24—92no—
74#874—91no—
75C8×Dic3—92no—
76C4×C3⋊C8—91no—
77C23×C32—22no—
78C23×C32—22no—
79C22×C3⋊D3—22no—
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C22×C3×D3—62no—
83C32×D4—22no—
84C32×D4—22no—
85C32×D4—42no—
86C2×C32×C4—21no—
87C2×C32×C4C22×D312yes—
88C4⋊(C3⋊D3)—22no—
89C4⋊(C3⋊D3)—22no—
90C3×#67—128no—
91C3×#67—128no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—62no—
94C3×D12—62no—
95C3×D12—62no—
96C3×D12—62no—
97C3×D12—62no—
98C3×D12—122no—
99#626—128no—
100C3⋊(C3⋊C8)#6714yescomplemented
101C3×C3⋊C8—34no—
102#493—271noSylow
103C2×H(3,𝔽3)—81no—
104C2×H(3,𝔽3)#12711yescharacteristic
105C2×H(3,𝔽3)C2×D812yes—
106#272—81no—
107(C3⋊D3)⋊C3—82no—
108C2×C3×D4—32no—
109C2×C3×D4—61no—
110C2×C3×D4—92no—
111C2×C3×D4—91no—
112C2×C3×D4—182no—
113C2×C3×D4D3211yescharacteristic
114C2×#67—64no—
115C2×#67—64no—
116C2×#67—184no—
117C2×D12—32no—
118C2×D12—32no—
119C2×D12—61no—
120C2×D12—92no—
121C2×D12—182no—
122C3×D8—184no—
123C2×C3×C8—92no—
124C3×C42—91no—
125#213—64no—
126#213—184no—
127#213—184no—
128#213—184no—
129C4×Dic3—32no—
130C4×Dic3—61no—
131C2×C3⋊C8—32no—
132C2×C3⋊C8—92no—
133D24—184no—
134C22×C32—21no—
135C22×C32—44no—
136C22×C32—42no—
137C22×C32—44no—
138C22×C32—42no—
139C22×C32D3×D412yes—
140C2×C3⋊D3—44no—
141C2×C3⋊D3—42no—
142C2×C3×D3—122no—
143C2×C3×D3—124no—
144C2×C3×D3—124no—
145C2×C3×D3—124no—
146C2×C3×D3—122no—
147C2×C3×D3—122no—
148C32×C4—22no—
149C32×C4D3×D412yes—
150C32×C4C2×#6712yes—
151C3×Dic3—38no—
152C3×Dic3—64no—
153C2×D8—92no—
154C2×D8—272no—
155#127—91no—
156#127—271no—
157C4×C8—271no—
158H(3,𝔽3)#49311yescharacteristic, complemented, Sylow
159C23×C3—21no—
160C23×C3—62no—
161C23×C3—62no—
162C23×C3—182no—
163C23×C3—181no—
164C22×D3—62no—
165C22×D3—62no—
166C22×D3—62no—
167C22×D3—182no—
168C3×D4—22no—
169C3×D4—62no—
170C3×D4—62no—
171C3×D4—122no—
172C3×D4—182no—
173C3×D4—182no—
174C3×D4—182no—
175C3×D4—364no—
176C2×C3×C4—32no—
177C2×C3×C4—61no—
178C2×C3×C4—92no—
179C2×C3×C4C2×D3211yescharacteristic
180#67—128no—
181#67—128no—
182#67—368no—
183C2×Dic3—34no—
184C2×Dic3—62no—
185D12—62no—
186D12—62no—
187D12—62no—
188D12—62no—
189D12—122no—
190D12—182no—
191D12—182no—
192D12—364no—
193C3×C8—94no—
194C3⋊C8—34no—
195C3⋊C8—94no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32—82no—
199C2×C32—82no—
200C2×C32D3×D814yes—
201C2×C32#94012yes—
202C3⋊D3—82no—
203C3×D3—242no—
204C3×D3—242no—
205C3×D3—242no—
206C2×D4—91no—
207C2×D4—92no—
208C2×D4—182no—
209C2×D4—542no—
210C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
211D8—184no—
212D8—544no—
213C2×C8—92no—
214C42—91no—
215C22×C3—32no—
216C22×C3—42no—
217C22×C3—41no—
218C22×C3—61no—
219C22×C3—124no—
220C22×C3—122no—
221C22×C3—124no—
222C22×C3—122no—
223C22×C3—364no—
224C22×C3—362no—
225C22×C3—362no—
226C22×C3—361no—
227C22×C3#381111yescharacteristic
228C2×D3—122no—
229C2×D3—124no—
230C2×D3—122no—
231C2×D3—124no—
232C2×D3—124no—
233C2×D3—122no—
234C2×D3—362no—
235C2×D3—364no—
236C3×C4—32no—
237C3×C4—32no—
238C3×C4—62no—
239C3×C4—94no—
240C3×C4D3×#6712yes—
241Dic3—38no—
242Dic3—64no—
243C32—21no—
244C32#515112yescomplemented
245C23—21no—
246C23—181no—
247C23—182no—
248D4—22no—
249D4—182no—
250D4—182no—
251D4—182no—
252D4—364no—
253D4—1084no—
254C2×C4—92no—
255C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
256C8—94no—
257C2×C3—32no—
258C2×C3—34no—
259C2×C3—61no—
260C2×C3—62no—
261C2×C3—81no—
262C2×C3—242no—
263C2×C3—242no—
264C2×C3—722no—
265C2×C3—721no—
266C2×C3#6392612yes—
267C2×C3#6397911yescharacteristic
268D3—242no—
269D3—242no—
270D3—242no—
271D3—722no—
272C22—41no—
273C22—42no—
274C22—361no—
275C22—362no—
276C22—362no—
277C22—364no—
278C22#8867711yescharacteristic
279C4—94no—
280C4#8867612yes—
281C3—32no—
282C3—61no—
283C3#1059148411yescharacteristic
284C2—81no—
285C2—721no—
286C2—722no—
287C2#1172119012yes—
288C2#1172131711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#493
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721850C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721144C2semidirect
4. C2×#88436C2semidirect
5. #11720302C2semidirect
16. C2×#3753D3semidirect
35. (C3⋊(C3⋊C8))⋊C3D4semidirect
63. #1084D8semidirect
100. C3⋊(C3⋊C8)#67semidirect
158. H(3,𝔽3)#493semidirect
244. C32#5151semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#3753⋊C32
#117211442
C2×#884362
#117203022
#105914823