Finite Group Databasebeta

← C2×#11720842 #13149819

Subgroups — C2×#11720842 #13149819

Order 1728.

1007 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 203 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×#88650C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3C2×#3743⋊C3C212yesmaximal, complemented
4C2×#88435C211yesmaximal, characteristic, complemented
5C2×((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
6#11720842C218yesmaximal, direct product factor
7C2×#63804—32nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C2212yes—
10C2×#6200C412yes—
11C2×#6198C411yescharacteristic
12C2×#6196C411yescharacteristic
13#88650C414yes—
14#88650C2214yes—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17C8×#272C2214yescomplemented
18C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
19#3743⋊C3C2218yescomplemented
20#88435C2214yescomplemented
21((C3⋊D3)⋊C3)⋊C8C2218yescomplemented
22C2×#3798—32no—
23C2×#3743D312yescomplemented
24C2×C3×ΓL(1,𝔽25)—32no—
25C2×C3×D3×C8—32no—
26C2×C3×D3×C8—61no—
27C2×#3711—32no—
28C2×#3710—32no—
29C2×D3×C3⋊C8—32no—
30#63804—316no—
31C22×#272C2×C411yescharacteristic
32C22×(C3⋊D3)⋊C3C2×C412yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084C2×C411yescharacteristic
35C4×#272C2×C412yes—
36C4×#272C2312yes—
37C2×(C3⋊Dic3)⋊C3C2×C412yes—
38C4×(C3⋊D3)⋊C3C2×C414yes—
39C4×(C3⋊D3)⋊C3C2314yes—
40#6200C2×C418yes—
41#6198C2×C414yes—
42#6196C2×C414yes—
43C8×H(3,𝔽3)C2312yes—
44#6180C2312yes—
45(C3⋊(C3⋊C8))⋊C3C2314yes—
46C2×#907—92no—
47C2×#901—91no—
48C2×C4×C3⋊D3C2×D312yes—
49C2×C3×C4×D3—32no—
50C2×C3×C4×D3—32no—
51C2×C3×C4×D3—61no—
52C2×#627—32no—
53C2×#626—32no—
54C2×#626—32no—
55C2×#625—32no—
56#3798—38no—
57#3798—38no—
58C2×C32×C8—21no—
59C2×C32×C8C2×D312yes—
60C2×C3⋊(C3⋊C8)C2×D312yes—
61#3743C2×D318yescomplemented
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—61no—
65C3×ΓL(1,𝔽25)—38no—
66C3×D3×C8—38no—
67C3×D3×C8—64no—
68#3711—38no—
69#3710—38no—
70D3×C3⋊C8—38no—
71C22×H(3,𝔽3)C22×C411yescharacteristic
72C2×#272—22no—
73C2×#272C22×C412yes—
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3C22×C414yes—
76C4×H(3,𝔽3)C22×C411yescharacteristic
77C4×H(3,𝔽3)C2411yescharacteristic
78#1084C22×C412yes—
79(C3⋊Dic3)⋊C3C22×C414yes—
80C2×#235—91no—
81C2×#235—92no—
82C2×C3×M16—92no—
83C22×C3×C8—91no—
84C2×#208—91no—
85C22×C3⋊C8—92no—
86#907—916no—
87#901—98no—
88C2×ΓL(1,𝔽25)—32no—
89C2×ΓL(1,𝔽25)—92no—
90C2×ΓL(1,𝔽25)—92no—
91C2×D3×C8—32no—
92C2×D3×C8—61no—
93C2×D3×C8—92no—
94C2×D3×C8—92no—
95C22×C3⋊D3C4×D312yes—
96C22×C3×D3—32no—
97C22×C3×D3—32no—
98C22×C3×D3—61no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D312yes—
101C2×C3⋊Dic3C4×D312yes—
102C4×C3⋊D3C4×D314yes—
103C4×C3⋊D3C22×D314yes—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—61no—
107C3×C4×D3—34no—
108C3×C4×D3—34no—
109C3×C4×D3—34no—
110C3×C4×D3—34no—
111C3×C4×D3—62no—
112C3×C4×D3—62no—
113#627—38no—
114#626—38no—
115#626—38no—
116#625—38no—
117C32×C8—22no—
118C32×C8C22×D314yes—
119C3⋊(C3⋊C8)C22×D314yes—
120C3×C3⋊C8—34no—
121C3×C3⋊C8—34no—
122C3×C3⋊C8—62no—
123C2×#131—271noSylow
124C2×H(3,𝔽3)#13112yescomplemented
125C2×H(3,𝔽3)C23×C411yescharacteristic
126#272—22no—
127(C3⋊D3)⋊C3—24no—
128C22×C3×C4—92no—
129C22×C3×C4—91no—
130C2×#67—92no—
131C2×#67—92no—
132C2×#67—92no—
133#235—94no—
134#235—94no—
135#235—98no—
136#235—98no—
137C2×D12—91no—
138C2×D12—92no—
139C2×C4×D3—32no—
140C2×C4×D3—32no—
141C2×C4×D3—61no—
142C2×C4×D3—92no—
143C2×Dic6—91no—
144C2×Dic6—92no—
145C3×M16—98no—
146C2×C3×C8—32no—
147C2×C3×C8—61no—
148C2×C3×C8—92no—
149C2×C3×C8—91no—
150C2×C3×C8—94no—
151C2×C3×C8D3211yescharacteristic
152#208—94no—
153C2×C3⋊C8—32no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—92no—
157C2×C3⋊C8—98no—
158ΓL(1,𝔽25)—38no—
159ΓL(1,𝔽25)—98no—
160ΓL(1,𝔽25)—98no—
161D3×C8—38no—
162D3×C8—64no—
163D3×C8—98no—
164D3×C8—98no—
165C22×C32—21no—
166C22×C32C2×C4×D312yes—
167C2×C3⋊D3—24no—
168C2×C3⋊D3C2×C4×D314yes—
169C2×C3×D3—34no—
170C2×C3×D3—34no—
171C2×C3×D3—62no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C2×C3×D3—64no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4C2×C4×D312yes—
178C32×C4C23×D312yes—
179C3⋊Dic3C2×C4×D314yes—
180C3×Dic3—34no—
181C3×Dic3—34no—
182C3×Dic3—62no—
183C2×#41—271no—
184#131—278no—
185C2×M16—92no—
186C2×M16—271no—
187C22×C8—91no—
188C22×C8—272no—
189H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
190C23×C3—92no—
191C23×C3—91no—
192C22×D3—32no—
193C22×D3—32no—
194C22×D3—61no—
195C22×D3—92no—
196C2×C3×C4—32no—
197C2×C3×C4—61no—
198C2×C3×C4—94no—
199C2×C3×C4—94no—
200C2×C3×C4—92no—
201C2×C3×C4—91no—
202C2×C3×C4—92no—
203C2×C3×C4—92no—
204C2×C3×C4C2×D3211yescharacteristic
205#67—98no—
206#67—98no—
207#67—98no—
208C2×Dic3—32no—
209C2×Dic3—32no—
210C2×Dic3—61no—
211C2×Dic3—92no—
212D12—94no—
213D12—98no—
214C4×D3—34no—
215C4×D3—34no—
216C4×D3—34no—
217C4×D3—34no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—94no—
221C4×D3—94no—
222Dic6—94no—
223Dic6—98no—
224C3×C8—34no—
225C3×C8—62no—
226C3×C8—94no—
227C3×C8—92no—
228C3×C8C2×D3212yes—
229C3⋊C8—34no—
230C3⋊C8—34no—
231C3⋊C8—62no—
232C3⋊C8—94no—
233C2×C32—21no—
234C2×C32—22no—
235C2×C32#90714yescomplemented
236C2×C32C22×C4×D312yes—
237C3⋊D3—24no—
238C3×D3—64no—
239C3×D3—64no—
240C3×D3—64no—
241#41—274no—
242#41—274no—
243C2×Q8—271no—
244C2×D4—271no—
245C2×D4—272no—
246C22×C4—91no—
247C22×C4—92no—
248M16—98no—
249M16—274no—
250C2×C8—91no—
251C2×C8—92no—
252C2×C8—94no—
253C2×C8—278no—
254C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C22×C3—32no—
256C22×C3—61no—
257C22×C3—94no—
258C22×C3—92no—
259C22×C3—184no—
260C22×C3—182no—
261C22×C3C4×D3211yescharacteristic
262C2×D3—34no—
263C2×D3—34no—
264C2×D3—62no—
265C2×D3—64no—
266C2×D3—64no—
267C2×D3—64no—
268C2×D3—94no—
269C2×D3—184no—
270C3×C4—32no—
271C3×C4—32no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—94no—
275C3×C4—92no—
276C3×C4C4×D3211yescharacteristic
277C3×C4C22×D3211yescharacteristic
278Dic3—34no—
279Dic3—34no—
280Dic3—62no—
281Dic3—94no—
282C32—21no—
283C32C2×#90712yescomplemented
284C23—91no—
285C23—92no—
286Q8—274no—
287D4—274no—
288D4—278no—
289C2×C4—91no—
290C2×C4—92no—
291C2×C4—92no—
292C2×C4—94no—
293C2×C4—92no—
294C2×C4—94no—
295C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C8—92no—
297C8—94no—
298C8C2×((C3⋊D3)⋊C3)⋊C212yes—
299C2×C3—34no—
300C2×C3—32no—
301C2×C3—62no—
302C2×C3—61no—
303C2×C3—184no—
304C2×C3—182no—
305C2×C3#6380512yes—
306C2×C3C2×C4×D3211yescharacteristic
307D3—64no—
308D3—64no—
309D3—64no—
310D3—184no—
311C22—92no—
312C22—94no—
313C22—182no—
314C22—184no—
315C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
316C4—92no—
317C4—94no—
318C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C3—32no—
321C3—61no—
322C3C2×#6380511yescharacteristic
323C2—182no—
324C2—184no—
325C2#1172084212yesdirect product factor
326C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC2×#88650C2
cosocleC4×H(3,𝔽3)C24
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. C2×#3743⋊C3C2semidirect
4. C2×#88435C2semidirect
5. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
6. #11720842C2direct (N × H)
17. C8×#272C22semidirect
19. #3743⋊C3C22semidirect
20. #88435C22semidirect
21. ((C3⋊D3)⋊C3)⋊C8C22semidirect
23. C2×#3743D3semidirect
61. #3743C2×D3semidirect
124. C2×H(3,𝔽3)#131semidirect
189. H(3,𝔽3)C2×#131semidirect
235. C2×C32#907semidirect
283. C32C2×#907semidirect
325. C2#11720842direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886502
C2×C8×#2722
C2×#3743⋊C32
C2×#884352
C2×((C3⋊D3)⋊C3)⋊C82
#117208422
C2×#638043