Finite Group Databasebeta

← C2×#11720699 #13146555

Subgroups — C2×#11720699 #13146555

Order 1728.

583 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88655C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic, complemented
3C2×#88435C211yesmaximal, characteristic, complemented
4#11720699C214yesmaximal, direct product factor
5C22×((C3⋊D3)⋊C3)⋊C2C412yes—
6C2×C4×#272C2211yescharacteristic
7C2×C4⋊((C3⋊D3)⋊C3)—21no—
8C2×#6198—21no—
9#88655—21no—
10#88655—21no—
11#88655C412yes—
12#88655C2212yes—
13C2×#6188—21no—
14#88627—21no—
15#88627C2212yescomplemented
16C2×(C3⋊(C3⋊C8))⋊C3—21no—
17#88435—21no—
18#88435C2212yescomplemented
19C2×D3×D12—61no—
20C2×#3711—61no—
21C22×#272C2×C411yescharacteristic
22C22×(C3⋊D3)⋊C3—22no—
23C2×((C3⋊D3)⋊C3)⋊C2—22no—
24C2×((C3⋊D3)⋊C3)⋊C2—44no—
25C2×((C3⋊D3)⋊C3)⋊C2C2×C414yes—
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×#1084D411yescharacteristic
28C4×#272C2×C411yescharacteristic
29C4×#272D411yescharacteristic
30C4×#272D411yescharacteristic
31C4×#272C2311yescharacteristic
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#6198—22no—
35#6198—22no—
36#6188—22no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38C2×#823—91nomaximal
39C22×D32—62no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C3×D12—61no—
42C2×C3×C4×D3—61no—
43C2×C3×C4×D3—61no—
44C2×#626—62no—
45D3×D12—62no—
46D3×D12—62no—
47D3×D12—62no—
48D3×D12—62no—
49C2×C3⋊(C3⋊C8)—21no—
50C2×C3×C3⋊C8—61no—
51#3711—62no—
52#3711—62no—
53C22×H(3,𝔽3)C22⋊C411yescharacteristic
54C2×#272—22no—
55C2×#272C22⋊C411yescharacteristic
56C2×#272C22×C411yescharacteristic
57C2×(C3⋊D3)⋊C3—24no—
58C2×(C3⋊D3)⋊C3—44no—
59((C3⋊D3)⋊C3)⋊C2—44no—
60((C3⋊D3)⋊C3)⋊C2—44no—
61C4×H(3,𝔽3)C22⋊C411yescharacteristic
62C4×H(3,𝔽3)C2×D411yescharacteristic
63#1084C22⋊C411yescharacteristic
64#1084C2×D411yescharacteristic
65C2×D3×D4—181no—
66C22×D12—91no—
67C2×C3×M16—91no—
68C2×#208—91no—
69C2×ΓL(1,𝔽25)—181no—
70#823—94no—
71C22×C3⋊D3—22no—
72C22×C3×D3—61no—
73C22×C3×D3—61no—
74C22×C3×D3—62no—
75C2×D32—64no—
76C2×D32—64no—
77C2×D32—128no—
78C2×C32×C4—21no—
79C2×C32×C4—21no—
80C4⋊(C3⋊D3)—22no—
81C4⋊(C3⋊D3)—22no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C3×D12—62no—
85C3×D12—62no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94#626—64no—
95#626—64no—
96C3⋊(C3⋊C8)—22no—
97C3×C3⋊C8—62no—
98C2×#100—271noSylow
99C2×H(3,𝔽3)#10012yescomplemented
100C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
101#272—21no—
102#272—21no—
103(C3⋊D3)⋊C3—44no—
104C23×D3—92no—
105C23×D3—182no—
106C2×C3×D4—181no—
107C22×C3×C4—91no—
108C2×#67—182no—
109D3×D4—182no—
110D3×D4—182no—
111D3×D4—182no—
112D3×D4—182no—
113C2×D12—61no—
114C2×D12—92no—
115C2×D12—92no—
116C2×D12—181no—
117C2×D12—181no—
118C2×D12—181no—
119C2×D12—181no—
120C2×C4×D3—61no—
121C2×C4×D3—61no—
122C3×M16—92no—
123C3×M16—181no—
124C2×C3×C8—181no—
125C2×C3×C8—181no—
126#208—92no—
127#208—181no—
128C2×C3⋊C8—61no—
129C2×C3⋊C8—181no—
130ΓL(1,𝔽25)—182no—
131ΓL(1,𝔽25)—182no—
132C22×C32—21no—
133C22×C32—21no—
134C2×C3⋊D3—24no—
135C2×C3⋊D3—44no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—64no—
141C2×C3×D3—64no—
142C2×C3×D3—122no—
143C2×C3×D3—124no—
144D32—128no—
145D32—128no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C32×C4—21no—
150C3×Dic3—61no—
151C3×Dic3—61no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C22×D4—271no—
155C2×M16—91no—
156C2×M16—271no—
157#100—274no—
158H(3,𝔽3)C2×#10011yescharacteristic, complemented, Sylow
159C23×C3—91no—
160C23×C3—182no—
161C22×D3—61no—
162C22×D3—61no—
163C22×D3—62no—
164C22×D3—94no—
165C22×D3—182no—
166C22×D3—184no—
167C22×D3—182no—
168C22×D3—184no—
169C22×D3—368no—
170C22×D3—364no—
171C3×D4—182no—
172C3×D4—182no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4GO+(4,𝔽2)11yescharacteristic
181#67—184no—
182#67—184no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185D12—62no—
186D12—62no—
187D12—182no—
188D12—182no—
189D12—182no—
190D12—182no—
191C4×D3—61no—
192C4×D3—61no—
193C4×D3—61no—
194C4×D3—61no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—61no—
198C4×D3—61no—
199C3×C8—182no—
200C3×C8—182no—
201C3⋊C8—62no—
202C3⋊C8—182no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—22no—
206C2×C32—22no—
207C3⋊D3—44no—
208C3×D3—62no—
209C3×D3—62no—
210C3×D3—121no—
211C3×D3—121no—
212C3×D3—124no—
213C24—272no—
214C2×D4—181no—
215C2×D4—272no—
216C2×D4—272no—
217C2×D4—541no—
218C2×D4—541no—
219C2×D4—541no—
220C22×C4—91no—
221M16—92no—
222M16—181no—
223M16—272no—
224M16—541no—
225C2×C8—181no—
226C2×C8—181no—
227C22×C3—61no—
228C22×C3—61no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—184no—
232C22×C3—182no—
233C22×C3—364no—
234C22×C3D32⋊C411yescharacteristic
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—61no—
239C2×D3—64no—
240C2×D3—64no—
241C2×D3—122no—
242C2×D3—124no—
243C2×D3—184no—
244C2×D3—364no—
245C2×D3—368no—
246C2×D3—368no—
247C2×D3—364no—
248C2×D3—364no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—91no—
254C3×C4—91no—
255C3×C4D32⋊C411yescharacteristic
256C3×C4C2×GO+(4,𝔽2)11yescharacteristic
257Dic3—61no—
258Dic3—61no—
259Dic3—61no—
260Dic3—61no—
261C32—21no—
262C32—21no—
263C23—91no—
264C23—182no—
265C23—274no—
266C23—542no—
267C23—1084no—
268D4—182no—
269D4—182no—
270D4—542no—
271D4—542no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4#1088⋊C211yescharacteristic
278C8—182no—
279C8—182no—
280C2×C3—61no—
281C2×C3—61no—
282C2×C3—62no—
283C2×C3—62no—
284C2×C3—181no—
285C2×C3—181no—
286C2×C3—364no—
287C2×C3#6372712yes—
288C2×C3C2×D32⋊C411yescharacteristic
289D3—62no—
290D3—62no—
291D3—121no—
292D3—121no—
293D3—124no—
294D3—364no—
295C22—91no—
296C22—91no—
297C22—182no—
298C22—184no—
299C22—364no—
300C22—1084no—
301C22—1084no—
302C22(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
303C4—91no—
304C4—91no—
305C4(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
306C4C2×#1088⋊C211yescharacteristic
307C3—61no—
308C3—61no—
309C3C2×#6372711yescharacteristic
310C2—181no—
311C2—181no—
312C2—364no—
313C2#1172069912yesdirect product factor
314C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
centerC22(((C3⋊D3)⋊C3)⋊C2)⋊C4
commutatorC2×#272C22×C4
FrattiniC3×C4C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
socleC22×C3D32⋊C4
prime radicalC2×#88655C2
cosocleC4×#272C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)C2×#100
LCSG ▷ C2×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88627C2semidirect
3. C2×#88435C2semidirect
4. #11720699C2direct (N × H)
15. #88627C22semidirect
18. #88435C22semidirect
99. C2×H(3,𝔽3)#100semidirect
158. H(3,𝔽3)C2×#100semidirect
313. C2#11720699direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886272
C2×#884352
#117206992
C2×#8239