Finite Group Databasebeta

← #13140527

Subgroups — #13140527

Order 1728.

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

normal subgroups only

This group has 584 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
1D4×#1084C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C212yesmaximal, complemented
3#11721144C211yesmaximal, characteristic, complemented
4#11720308C212yesmaximal, complemented
5#11720304C211yesmaximal, characteristic, complemented
6#10591481—32nomaximal
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084—21no—
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6198—21no—
11C2×#6198—21no—
12#88552—21no—
13C4⋊#1084C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15#3753⋊C3—24no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×#3753D312yescomplemented
18C3×D4×Dic3—32no—
19C3×D4×Dic3—61no—
20C2×C3×#213—32no—
21#63911—32no—
22Dic3⋊(C3⋊C8)—32no—
23#63564—32no—
24#63562—32no—
25C23×H(3,𝔽3)—21no—
26C22×(C3⋊D3)⋊C3—22no—
27D4×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)D412yescomplemented
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—21no—
31C2×#1084—42no—
32C2×#1084D411yescharacteristic
33C2×#1084D411yescharacteristic
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—22no—
36#6198—42no—
37#6198—42no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5150—92no—
40#5015—91no—
41C2×C32×D4—21no—
42C2×C32×D4C2×D312yes—
43C2×C4⋊(C3⋊D3)C2×D312yes—
44C22×C3×Dic3—62no—
45C22×C3×Dic3—62no—
46C2×C3×D12—32no—
47C2×#626—62no—
48C2×#626—62no—
49#3753—24no—
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×#217—62no—
52C3×#217—62no—
53C3×#213—64no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—61no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C2×C3×C3⋊C8—32no—
59C22×H(3,𝔽3)—22no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—42no—
64C2×(C3⋊D3)⋊C3—42no—
65C2×(C3⋊D3)⋊C3—42no—
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—21no—
69#1084—41no—
70#1084D812yescomplemented
71C2×C3×D8—92no—
72C3×C4×D4—91no—
73#940—92no—
74D4×Dic3—32no—
75D4×Dic3—61no—
76C2×#213—32no—
77C2×#213—92no—
78#874—91no—
79#832—92no—
80#821—92no—
81Dic3⋊C8—92no—
82#809—92no—
83C4⋊(C3⋊C8)—91no—
84C23×C32—22no—
85C23×C32—22no—
86C22×C3⋊D3—22no—
87C22×C3×D3—62no—
88C32×D4—22no—
89C32×D4—22no—
90C32×D4—22no—
91C32×D4#6714yescomplemented
92C2×C32×C4—21no—
93C2×C32×C4C22×D312yes—
94C4⋊(C3⋊D3)—22no—
95C4⋊(C3⋊D3)—22no—
96C2×C3×Dic3—32no—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—124no—
103C2×C3×Dic3—124no—
104C3×D12—62no—
105C3×D12—62no—
106#626—124no—
107#626—124no—
108C3⋊(C3⋊C8)—22no—
109C3×C3⋊C8—62no—
110#459—271noSylow
111C2×H(3,𝔽3)—42no—
112C2×H(3,𝔽3)#12111yescharacteristic
113C2×H(3,𝔽3)C2×D811yescharacteristic
114C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
115(C3⋊D3)⋊C3—82no—
116C2×C3×D4—32no—
117C2×C3×D4—61no—
118C2×C3×D4—92no—
119C2×C3×D4D3211yescharacteristic
120C22×C3×C4—181no—
121C2×#67—182no—
122C2×#67—182no—
123C22×Dic3—62no—
124C22×Dic3—62no—
125C2×D12—32no—
126C2×D12—92no—
127C2×D12—181no—
128C2×D12—181no—
129C3×D8—184no—
130C2×C3×C8—92no—
131C3×C4⋊C4—91no—
132C3×C22⋊C4—181no—
133C3×C42—91no—
134#217—62no—
135#217—62no—
136#213—64no—
137#213—184no—
138C4⋊Dic3—32no—
139C4⋊Dic3—61no—
140C4×Dic3—32no—
141C4×Dic3—61no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—92no—
144C22×C32—24no—
145C22×C32—24no—
146C22×C32—21no—
147C22×C32—42no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32—42no—
151C22×C32D3×D412yes—
152C2×C3⋊D3—42no—
153C2×C3⋊D3—42no—
154C2×C3⋊D3—42no—
155C2×C3×D3—122no—
156C2×C3×D3—122no—
157C2×C3×D3—122no—
158C32×C4—21no—
159C32×C4—21no—
160C32×C4#23712yes—
161C32×C4C2×#6712yes—
162C3×Dic3—34no—
163C3×Dic3—62no—
164C3×Dic3—62no—
165C3×Dic3—121no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C2×D8—92no—
169#127—271no—
170C4×D4—91no—
171C4⋊C8—271no—
172#102—272no—
173H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
174C23×C3—21no—
175C23×C3—62no—
176C23×C3—62no—
177C23×C3—182no—
178C22×D3—62no—
179C22×D3—182no—
180C3×D4—21no—
181C3×D4—34no—
182C3×D4—62no—
183C3×D4—62no—
184C3×D4—62no—
185C3×D4—182no—
186C3×D4—182no—
187C3×D4#62512yes—
188C2×C3×C4—32no—
189C2×C3×C4—61no—
190C2×C3×C4—91no—
191C2×C3×C4—91no—
192C2×C3×C4—181no—
193C2×C3×C4—362no—
194C2×C3×C4C2×D3211yescharacteristic
195#67—364no—
196#67—364no—
197C2×Dic3—32no—
198C2×Dic3—32no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3—124no—
204C2×Dic3—124no—
205D12—62no—
206D12—62no—
207D12—182no—
208D12—182no—
209D12—362no—
210D12—362no—
211C3×C8—182no—
212C3⋊C8—62no—
213C3⋊C8—182no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—44no—
218C2×C32—44no—
219C2×C32D3×D812yes—
220C2×C32#91112yes—
221C2×C32#93912yes—
222C3⋊D3—82no—
223C3×D3—242no—
224C2×D4—92no—
225C2×D4—541no—
226C2×D4—541no—
227C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C22×C4—181no—
229D8—184no—
230C2×C8—92no—
231C4⋊C4—91no—
232C22⋊C4—181no—
233C42—91no—
234C22×C3—22no—
235C22×C3—32no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—64no—
239C22×C3—64no—
240C22×C3—61no—
241C22×C3—122no—
242C22×C3—122no—
243C22×C3—122no—
244C22×C3—122no—
245C22×C3—362no—
246C22×C3—362no—
247C22×C3—362no—
248C22×C3#381111yescharacteristic
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—61no—
258C3×C4—61no—
259C3×C4—92no—
260C3×C4—181no—
261C3×C4—361no—
262C3×C4#380511yescharacteristic
263C3×C4C2×#62511yescharacteristic
264Dic3—34no—
265Dic3—62no—
266Dic3—62no—
267Dic3—121no—
268Dic3—122no—
269Dic3—122no—
270C32—21no—
271C32#515012yescomplemented
272C23—21no—
273C23—182no—
274D4—21no—
275D4—182no—
276D4—182no—
277D4—1082no—
278D4—1082no—
279D4#619812yescomplemented
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—181no—
283C2×C4—362no—
284C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C8—182no—
286C2×C3—32no—
287C2×C3—32no—
288C2×C3—32no—
289C2×C3—42no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—61no—
293C2×C3—124no—
294C2×C3—124no—
295C2×C3—722no—
296C2×C3#6392611yescharacteristic
297C2×C3#6392711yescharacteristic
298C2×C3#6398011yescharacteristic
299D3—242no—
300D3—722no—
301C22—22no—
302C22—41no—
303C22—41no—
304C22—362no—
305C22—362no—
306C22—362no—
307C22#8867711yescharacteristic
308C4—92no—
309C4—181no—
310C4—361no—
311C4#8867211yescharacteristic
312C4C2×#619811yescharacteristic
313C3—32no—
314C3—61no—
315C3#1059148611yescharacteristic
316C2—42no—
317C2—722no—
318C2#1172119011yescharacteristic
319C2#1172119111yescharacteristic
320C2#1172131811yescharacteristic
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)#459
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. D4×#1084C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721144C2semidirect
4. #11720308C2semidirect
5. #11720304C2semidirect
17. C2×#3753D3semidirect
28. D4×H(3,𝔽3)D4semidirect
70. #1084D8semidirect
91. C32×D4#67semidirect
173. H(3,𝔽3)#459semidirect
271. C32#5150semidirect
279. D4#6198semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
C2×#3753⋊C32
#117211442
#117203082
#117203042
#105914813