Finite Group Databasebeta

← #13137885

Subgroups — #13137885

Order 1728.

584 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721761C211yesmaximal, characteristic, complemented
2#11721144C211yesmaximal, characteristic, complemented
3C2×#88436C212yesmaximal, complemented
4#63636⋊C3C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10590547—32nomaximal
7C2×C4⋊#272C2211yescharacteristic
8C2×C4×#272—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6198—21no—
11C2×#6198—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13#88542—21no—
14C4×#1084C2211yescharacteristic
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16#88436—24no—
17C3×#893—32no—
18C3×#893—61no—
19#63911—32no—
20C2×#3714—32no—
21C2×#3713—32no—
22#63636D312yescomplemented
23C3×#809—32no—
24Dic3⋊(C3⋊C8)—32no—
25C22×#272—21no—
26C22×(C3⋊D3)⋊C3—22no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084D411yescharacteristic
30C2×#1084D411yescharacteristic
31C4⋊#272—21no—
32C4⋊#272D412yescomplemented
33C4×#272—42no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36#6198—42no—
37#6198—42no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5015—91no—
40#4786—92no—
41C2×C4⋊(C3⋊D3)C2×D312yes—
42C2×C3×D12—32no—
43C2×C3×D12—32no—
44C2×C3×D12—61no—
45C2×C3×C4×D3—62no—
46C2×C3×C4×D3—62no—
47C2×#626—62no—
48C2×#626—62no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4C2×D312yes—
51C2×C3⋊(C3⋊C8)C2×D312yes—
52C3×(C2×D3)⋊C4—62no—
53C3×(C2×D3)⋊C4—62no—
54C3×C4×Dic3—32no—
55C3×C4×Dic3—61no—
56C2×C3×C3⋊C8—32no—
57#3714—64no—
58#3713—64no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C2×#272—22no—
61C2×#272—41no—
62C2×#272—41no—
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3—42no—
65C2×(C3⋊D3)⋊C3—42no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)#4111yescharacteristic
69#1084—21no—
70#1084D812yescomplemented
71C3×C4×D4—91no—
72#940—92no—
73C2×#213—92no—
74C2×#213—92no—
75C2×D24—92no—
76#893—32no—
77#893—61no—
78#874—91no—
79C3×#102—92no—
80Dic3⋊C8—92no—
81#809—32no—
82#809—92no—
83C4⋊(C3⋊C8)—91no—
84C22×C3⋊D3—22no—
85C22×C3×D3—62no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4C22×D312yes—
92C4⋊(C3⋊D3)—22no—
93C4⋊(C3⋊D3)—22no—
94C2×C3×Dic3—32no—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C3×D12—34no—
99C3×D12—62no—
100C3×D12—62no—
101C3×D12—62no—
102C3×D12—62no—
103C3×D12—62no—
104C3×C4×D3—124no—
105C3×C4×D3—124no—
106#626—124no—
107#626—124no—
108C3⋊(C3⋊C8)—22no—
109C3×C3⋊C8—62no—
110#459—271noSylow
111C2×H(3,𝔽3)#12111yescharacteristic
112C2×H(3,𝔽3)C2×D811yescharacteristic
113C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
114#272—42no—
115(C3⋊D3)⋊C3—82no—
116C2×C3×D4—92no—
117C2×C3×D4—91no—
118C22×C3×C4—181no—
119C2×#67—182no—
120C2×#67—182no—
121C2×D12—32no—
122C2×D12—32no—
123C2×D12—61no—
124C2×D12—92no—
125C2×D12—181no—
126C2×D12—181no—
127C2×C4×D3—62no—
128C2×C4×D3—62no—
129C2×C3×C8—92no—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—61no—
132C3×C4⋊C4D3211yescharacteristic
133C3×C22⋊C4—181no—
134C3×C42—91no—
135#213—184no—
136#213—184no—
137(C2×D3)⋊C4—62no—
138(C2×D3)⋊C4—62no—
139C4×Dic3—32no—
140C4×Dic3—61no—
141C2×C3⋊C8—32no—
142C2×C3⋊C8—92no—
143D24—184no—
144C22×C32—21no—
145C22×C32D3×D412yes—
146C2×C3⋊D3—42no—
147C2×C3⋊D3—42no—
148C2×C3⋊D3—42no—
149C2×C3×D3—64no—
150C2×C3×D3—64no—
151C2×C3×D3—122no—
152C2×C3×D3—122no—
153C2×C3×D3—122no—
154C2×C3×D3—122no—
155C2×C3×D3—122no—
156C2×C3×D3—122no—
157C2×C3×D3—122no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—42no—
161C32×C4—42no—
162C32×C4#23512yes—
163C32×C4D3×D412yes—
164C3×Dic3—34no—
165C3×Dic3—62no—
166C3×Dic3—62no—
167C3×Dic3—121no—
168C2×D8—272no—
169#127—271no—
170C4×D4—91no—
171C4⋊C8—271no—
172#102—92no—
173H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
174C23×C3—182no—
175C23×C3—181no—
176C22×D3—62no—
177C22×D3—62no—
178C22×D3—62no—
179C22×D3—182no—
180C3×D4—92no—
181C3×D4—182no—
182C3×D4—182no—
183C3×D4—181no—
184C2×C3×C4—21no—
185C2×C3×C4—32no—
186C2×C3×C4—62no—
187C2×C3×C4—62no—
188C2×C3×C4—61no—
189C2×C3×C4—91no—
190C2×C3×C4—91no—
191C2×C3×C4—362no—
192C2×C3×C4C2×D3211yescharacteristic
193#67—364no—
194#67—364no—
195C2×Dic3—32no—
196C2×Dic3—32no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199D12—34no—
200D12—62no—
201D12—62no—
202D12—62no—
203D12—62no—
204D12—62no—
205D12—182no—
206D12—182no—
207D12—362no—
208D12—362no—
209C4×D3—124no—
210C4×D3—124no—
211C3×C8—182no—
212C3⋊C8—62no—
213C3⋊C8—182no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32#88412yes—
218C2×C32D3×D812yes—
219C2×C32#91412yes—
220C3⋊D3—82no—
221C3×D3—124no—
222C3×D3—124no—
223C3×D3—242no—
224C2×D4—91no—
225C2×D4—92no—
226C2×D4—541no—
227C2×D4—541no—
228C22×C4—181no—
229D8—544no—
230C2×C8—92no—
231C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
232C22⋊C4—181no—
233C42—91no—
234C22×C3—32no—
235C22×C3—61no—
236C22×C3—182no—
237C22×C3—362no—
238C22×C3—362no—
239C22×C3—362no—
240C22×C3—361no—
241C22×C3—361no—
242C22×C3#381111yescharacteristic
243C2×D3—64no—
244C2×D3—64no—
245C2×D3—122no—
246C2×D3—122no—
247C2×D3—122no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—122no—
251C2×D3—122no—
252C2×D3—362no—
253C2×D3—362no—
254C2×D3—362no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—41no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—92no—
261C3×C4—122no—
262C3×C4—122no—
263C3×C4—181no—
264C3×C4#379811yescharacteristic
265C3×C4#380211yescharacteristic
266Dic3—34no—
267Dic3—62no—
268Dic3—62no—
269Dic3—121no—
270C32—21no—
271C32#478612yescomplemented
272C23—181no—
273C23—182no—
274D4—92no—
275D4—181no—
276D4—182no—
277D4—182no—
278D4—1082no—
279D4—1082no—
280C2×C4—21no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—362no—
284C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C8—182no—
286C2×C3—32no—
287C2×C3—32no—
288C2×C3—32no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—362no—
293C2×C3—722no—
294C2×C3#6391211yescharacteristic
295C2×C3#6392611yescharacteristic
296C2×C3#6394111yescharacteristic
297D3—124no—
298D3—124no—
299D3—242no—
300D3—722no—
301C22—182no—
302C22—361no—
303C22—361no—
304C22—362no—
305C22—362no—
306C22—362no—
307C22#8867711yescharacteristic
308C4—41no—
309C4—92no—
310C4—181no—
311C4#8865011yescharacteristic
312C4#8865511yescharacteristic
313C3—32no—
314C3—61no—
315C3#1059055111yescharacteristic
316C2—362no—
317C2—722no—
318C2#1172114211yescharacteristic
319C2#1172119011yescharacteristic
320C2#1172122911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#459
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721761C2semidirect
2. #11721144C2semidirect
3. C2×#88436C2semidirect
4. #63636⋊C3C2semidirect
5. #11720304C2semidirect
22. #63636D3semidirect
32. C4⋊#272D4semidirect
70. #1084D8semidirect
173. H(3,𝔽3)#459semidirect
271. C32#4786semidirect

Maximal subgroups

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

subgroupindex
#117217612
#117211442
C2×#884362
#63636⋊C32
#117203042
#105905473