Finite Group Databasebeta

← C2×#11720843 #13149820

Subgroups — C2×#11720843 #13149820

Order 1728.

1007 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88649C212yesmaximal, complemented
2C2×C8×#272C211yesmaximal, characteristic, complemented
3C2×#3744⋊C3C212yesmaximal, complemented
4C2×#88433C211yesmaximal, characteristic, complemented
5C2×#88432C211yesmaximal, characteristic, complemented
6#11720843C218yesmaximal, direct product factor
7C2×#63806—32nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#634⋊C3C2212yes—
11C2×#6200—22no—
12C2×((C3⋊D3)⋊C3)⋊C4—22no—
13#88649—24no—
14#88649C2218yescomplemented
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17C8×#272C2214yescomplemented
18#3744⋊C3C2218yescomplemented
19#88433C2214yescomplemented
20#88432C2214yescomplemented
21C2×#3799—32no—
22C2×#3795—32no—
23C2×#3744D312yescomplemented
24C2×C3×#204—32no—
25C2×C3×D3×C8—32no—
26C2×C3×D3×C8—61no—
27C2×#3719—32no—
28C2×#3714—32no—
29#63806—316no—
30C22×#272D411yescharacteristic
31C22×(C3⋊D3)⋊C3—22no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084D411yescharacteristic
34C4×#272D412yes—
35C4×#272C2312yes—
36C2×(C3⋊Dic3)⋊C3—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)C2314yes—
39#634⋊C3—22no—
40#634⋊C3C2314yes—
41#6200—28no—
42((C3⋊D3)⋊C3)⋊C4—28no—
43C8×H(3,𝔽3)C2312yes—
44#6180C2312yes—
45C2×#916—92no—
46C2×#904—91no—
47C2×C4⋊(C3⋊D3)C2×D312yes—
48C2×#634C2×D312yes—
49C2×C3×D12—32no—
50C2×C3×C4×D3—32no—
51C2×C3×C4×D3—61no—
52C2×C3×Dic6—32no—
53C2×#626—62no—
54C2×#625—62no—
55C2×#624—62no—
56C2×D3×Dic3—62no—
57#3799—38no—
58#3799—64no—
59#3795—38no—
60#3795—64no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D312yes—
63#3744C2×D318yescomplemented
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—61no—
66C3×#204—38no—
67C3×D3×C8—38no—
68C3×D3×C8—64no—
69#3719—38no—
70#3714—38no—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×#272—22no—
73C2×#272C2×D412yes—
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3—44no—
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2411yescharacteristic
78#1084C2×D412yes—
79(C3⋊Dic3)⋊C3—24no—
80C2×#239—92no—
81C2×#237—92no—
82C2×#235—92no—
83C2×C3×SD16—92no—
84C22×C3×C8—91no—
85C2×#216—92no—
86C2×#213—92no—
87#916—916no—
88C2×Dic12—91no—
89#904—98no—
90C2×D24—91no—
91C2×#204—32no—
92C2×#204—92no—
93C2×D3×C8—32no—
94C2×D3×C8—61no—
95C22×C3⋊D3—22no—
96C22×C3×D3—32no—
97C22×C3×D3—61no—
98C22×C3×D3—62no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D312yes—
101C2×C3⋊Dic3—22no—
102C4⋊(C3⋊D3)—22no—
103C4⋊(C3⋊D3)C22×D314yes—
104#634—22no—
105#634C22×D314yes—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—62no—
109C3×D12—34no—
110C3×D12—62no—
111C3×C4×D3—34no—
112C3×C4×D3—34no—
113C3×C4×D3—62no—
114C3×C4×D3—62no—
115C3×Dic6—34no—
116C3×Dic6—62no—
117#626—68no—
118#625—68no—
119#624—68no—
120D3×Dic3—68no—
121C32×C8—22no—
122C32×C8C22×D314yes—
123C3×C3⋊C8—34no—
124C3×C3⋊C8—62no—
125C2×#135—271noSylow
126C2×H(3,𝔽3)#13512yescomplemented
127C2×H(3,𝔽3)C22×D411yescharacteristic
128#272—22no—
129(C3⋊D3)⋊C3—44no—
130C2×C3×Q8—92no—
131C2×C3×D4—92no—
132C22×C3×C4—91no—
133C2×#67—182no—
134C2×#67—182no—
135C22×Dic3—182no—
136#239—98no—
137#239—184no—
138#237—98no—
139#237—184no—
140#235—98no—
141#235—184no—
142C2×D12—32no—
143C2×D12—92no—
144C2×D12—182no—
145C2×C4×D3—32no—
146C2×C4×D3—61no—
147C2×C4×D3—182no—
148C2×C4×D3—182no—
149C2×Dic6—32no—
150C2×Dic6—92no—
151C3×SD16—98no—
152C2×C3×C8—32no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C2×C3×C8—94no—
156C2×C3×C8D3211yescharacteristic
157#216—98no—
158#213—98no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—61no—
161Dic12—94no—
162D24—94no—
163#204—38no—
164#204—98no—
165D3×C8—38no—
166D3×C8—64no—
167C22×C32—21no—
168C22×C32D3×D412yes—
169C2×C3⋊D3—24no—
170C2×C3⋊D3—44no—
171C2×C3×D3—34no—
172C2×C3×D3—62no—
173C2×C3×D3—64no—
174C2×C3×D3—64no—
175C2×C3×D3—64no—
176C2×C3×D3—124no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4D3×D412yes—
180C32×C4C23×D312yes—
181C3⋊Dic3—24no—
182C3×Dic3—34no—
183C3×Dic3—62no—
184C3×Dic3—64no—
185C2×#41—272no—
186#135—278no—
187C2×Q16—271no—
188C2×SD16—92no—
189C2×D8—271no—
190C22×C8—91no—
191H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
192C23×C3—91no—
193C23×C3—182no—
194C22×D3—32no—
195C22×D3—61no—
196C22×D3—62no—
197C22×D3—182no—
198C3×Q8—94no—
199C3×Q8—182no—
200C3×D4—94no—
201C3×D4—182no—
202C2×C3×C4—32no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—92no—
206C2×C3×C4—92no—
207C2×C3×C4—182no—
208C2×C3×C4C2×D3211yescharacteristic
209#67—188no—
210#67—188no—
211C2×Dic3—32no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C2×Dic3—182no—
215C2×Dic3—188no—
216D12—34no—
217D12—62no—
218D12—94no—
219D12—182no—
220D12—188no—
221C4×D3—34no—
222C4×D3—34no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—188no—
226C4×D3—188no—
227Dic6—34no—
228Dic6—62no—
229Dic6—94no—
230Dic6—182no—
231C3×C8—34no—
232C3×C8—62no—
233C3×C8—92no—
234C3×C8C2×D3212yes—
235C3⋊C8—34no—
236C3⋊C8—62no—
237C2×C32—21no—
238C2×C32—22no—
239C2×C32#91614yescomplemented
240C2×C32C2×D3×D412yes—
241C3⋊D3—44no—
242C3×D3—64no—
243C3×D3—64no—
244C3×D3—124no—
245#41—278no—
246#41—544no—
247C2×Q8—92no—
248C2×D4—92no—
249C2×D4—542no—
250C22×C4—91no—
251C22×C4—542no—
252Q16—274no—
253SD16—98no—
254D8—274no—
255C2×C8—91no—
256C2×C8—94no—
257C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C22×C3—32no—
259C22×C3—61no—
260C22×C3—92no—
261C22×C3—184no—
262C22×C3—182no—
263C22×C3—364no—
264C22×C3#380211yescharacteristic
265C2×D3—34no—
266C2×D3—62no—
267C2×D3—64no—
268C2×D3—64no—
269C2×D3—64no—
270C2×D3—124no—
271C2×D3—184no—
272C2×D3—364no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—92no—
278C3×C4—184no—
279C3×C4#380211yescharacteristic
280C3×C4C22×D3211yescharacteristic
281Dic3—34no—
282Dic3—62no—
283Dic3—64no—
284Dic3—184no—
285C32—21no—
286C32C2×#91612yescomplemented
287C23—91no—
288C23—182no—
289Q8—94no—
290Q8—182no—
291D4—94no—
292D4—182no—
293D4—548no—
294C2×C4—91no—
295C2×C4—92no—
296C2×C4—92no—
297C2×C4—182no—
298C2×C4—548no—
299C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
300C8—92no—
301C8C2×((C3⋊D3)⋊C3)⋊C212yes—
302C2×C3—34no—
303C2×C3—32no—
304C2×C3—62no—
305C2×C3—61no—
306C2×C3—182no—
307C2×C3—364no—
308C2×C3#6380912yes—
309C2×C3C2×#380211yescharacteristic
310D3—64no—
311D3—64no—
312D3—124no—
313D3—364no—
314C22—92no—
315C22—182no—
316C22—184no—
317C22—364no—
318C22#8865511yescharacteristic
319C4—92no—
320C4—184no—
321C4#8865511yescharacteristic
322C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C3—32no—
324C3—61no—
325C3C2×#6380911yescharacteristic
326C2—182no—
327C2—364no—
328C2#1172084312yesdirect product factor
329C2C2×#8865511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88649C2semidirect
2. C2×C8×#272C2semidirect
3. C2×#3744⋊C3C2semidirect
4. C2×#88433C2semidirect
5. C2×#88432C2semidirect
6. #11720843C2direct (N × H)
14. #88649C22semidirect
17. C8×#272C22semidirect
18. #3744⋊C3C22semidirect
19. #88433C22semidirect
20. #88432C22semidirect
23. C2×#3744D3semidirect
63. #3744C2×D3semidirect
126. C2×H(3,𝔽3)#135semidirect
191. H(3,𝔽3)C2×#135semidirect
239. C2×C32#916semidirect
286. C32C2×#916semidirect
328. C2#11720843direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886492
C2×C8×#2722
C2×#3744⋊C32
C2×#884332
C2×#884322
#117208432
C2×#638063