Finite Group Databasebeta

← C2×#11721060 #13150594

Subgroups — C2×#11721060 #13150594

Order 1728.

1007 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 179 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×#3828⋊C3C212yesmaximal, complemented
2C22×#6180C211yesmaximal, characteristic, complemented
3C2×#88433C211yesmaximal, characteristic, complemented
4#11721060C218yesmaximal, direct product factor
5C2×#88432C211yesmaximal, characteristic, complemented
6C2×#88431C212yesmaximal, complemented
7C2×#63829—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—22no—
10#3828⋊C3—24no—
11#3828⋊C3C2218yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4×(C3⋊D3)⋊C3—22no—
14C2×#634⋊C3C2212yes—
15C2×#6180C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17C2×#6180C2214yescomplemented
18#88433C2214yescomplemented
19#88432C2214yescomplemented
20#88431C2218yescomplemented
21C2×#3828D312yescomplemented
22C2×C3×#235—32no—
23C22×C3×C3⋊C8—32no—
24C22×C3×C3⋊C8—61no—
25C2×#3719—32no—
26C2×#3717—32no—
27#63829—316no—
28C2×#3716—32no—
29C2×#3714—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—22no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×C4×H(3,𝔽3)C2312yes—
36#638⋊C3—28no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)C2314yes—
40C4×(C3⋊D3)⋊C3—28no—
41#634⋊C3—22no—
42#634⋊C3C2314yes—
43#6180C2312yes—
44#6180C2312yes—
45C2×#950—92no—
46C2×#904—91no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D312yes—
49C2×#638—22no—
50#3828—24no—
51#3828C2×D318yescomplemented
52C2×C4⋊(C3⋊D3)C2×D312yes—
53C2×C4×C3⋊D3—22no—
54C2×#634C2×D312yes—
55C2×C3×#67—62no—
56C3×#235—38no—
57C3×#235—64no—
58C2×C3×D12—32no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—38no—
64C2×C3×C3⋊C8—61no—
65C2×C3×C3⋊C8—61no—
66C2×C3×C3⋊C8—64no—
67#3719—38no—
68#3717—38no—
69#3716—38no—
70#3714—38no—
71C22×H(3,𝔽3)—22no—
72C22×H(3,𝔽3)C2×D412yes—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3—44no—
76C4×H(3,𝔽3)C2×D411yescharacteristic
77C4×H(3,𝔽3)C2×D412yes—
78C4×H(3,𝔽3)C2411yescharacteristic
79(C3⋊Dic3)⋊C3—24no—
80C2×C3×#41—92no—
81C2×#235—32no—
82C2×#235—92no—
83C22×C3×C8—91no—
84#950—916no—
85C2×#216—92no—
86C2×#215—92no—
87C2×#214—92no—
88C2×#213—92no—
89C22×C3⋊C8—32no—
90C22×C3⋊C8—61no—
91C2×Dic12—91no—
92#904—98no—
93C2×D24—91no—
94C2×#204—92no—
95C23×C32—21no—
96C23×C32#6712yes—
97C22×C3⋊D3—22no—
98C22×C3×D3—62no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4#6712yes—
104C2×C32×C4#6714yes—
105C2×C32×C4C22×D312yes—
106C2×C32×C4C22×D314yes—
107#638—28no—
108C2×C3⋊Dic3—22no—
109C4⋊(C3⋊D3)—22no—
110C4⋊(C3⋊D3)C22×D314yes—
111C4×C3⋊D3—28no—
112#634—22no—
113#634C22×D314yes—
114C3×#67—68no—
115C2×C3×Dic3—62no—
116C3×D12—34no—
117C3×D12—62no—
118C3×C4×D3—68no—
119C3×Dic6—34no—
120C3×Dic6—62no—
121C3×C3⋊C8—34no—
122C3×C3⋊C8—34no—
123C3×C3⋊C8—62no—
124C3×C3⋊C8—62no—
125C2×#135—271noSylow
126C2×H(3,𝔽3)—22no—
127C2×H(3,𝔽3)#13512yescomplemented
128C2×H(3,𝔽3)C22×D411yescharacteristic
129(C3⋊D3)⋊C3—44no—
130C3×#41—98no—
131C3×#41—184no—
132C2×C3×Q8—92no—
133C2×C3×D4—92no—
134C2×C3×D4—182no—
135C22×C3×C4—32no—
136C22×C3×C4—61no—
137C22×C3×C4—182no—
138C22×C3×C4D3211yescharacteristic
139C2×#67—62no—
140C2×#67—182no—
141#235—38no—
142#235—64no—
143#235—98no—
144#235—184no—
145C2×D12—32no—
146C2×D12—92no—
147C2×C4×D3—62no—
148C2×C4×D3—182no—
149C2×Dic6—32no—
150C2×Dic6—92no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C2×C3×C8—94no—
154#216—98no—
155#215—98no—
156#214—98no—
157#213—98no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—38no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—61no—
163C2×C3⋊C8—64no—
164Dic12—94no—
165D24—94no—
166#204—98no—
167C22×C32—24no—
168C22×C32—24no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32C2×#6714yes—
172C22×C32C2×#6712yes—
173C2×C3⋊D3—24no—
174C2×C3⋊D3—44no—
175C2×C3×D3—64no—
176C2×C3×D3—124no—
177C32×C4—21no—
178C32×C4—21no—
179C32×C4—22no—
180C32×C4C2×#6712yes—
181C32×C4C2×#6714yes—
182C32×C4C23×D312yes—
183C3⋊Dic3—24no—
184C3×Dic3—64no—
185C2×#41—92no—
186#135—278no—
187C2×Q16—271no—
188C2×SD16—272no—
189C2×D8—271no—
190C22×C8—91no—
191H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
192C23×C3—32no—
193C23×C3—61no—
194C23×C3—182no—
195C23×C3#62511yescharacteristic
196C22×D3—62no—
197C22×D3—182no—
198C3×Q8—94no—
199C3×Q8—182no—
200C3×D4—94no—
201C3×D4—182no—
202C3×D4—188no—
203C2×C3×C4—34no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—34no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—188no—
212C2×C3×C4—182no—
213C2×C3×C4#62512yes—
214C2×C3×C4#62511yescharacteristic
215C2×C3×C4C2×D3211yescharacteristic
216C2×C3×C4C2×D3212yes—
217#67—68no—
218#67—188no—
219C2×Dic3—62no—
220C2×Dic3—182no—
221D12—34no—
222D12—62no—
223D12—94no—
224D12—182no—
225C4×D3—68no—
226C4×D3—188no—
227Dic6—34no—
228Dic6—62no—
229Dic6—94no—
230Dic6—182no—
231C3×C8—92no—
232C3×C8—92no—
233C3⋊C8—34no—
234C3⋊C8—34no—
235C3⋊C8—62no—
236C3⋊C8—62no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—21no—
240C2×C32—22no—
241C2×C32#95014yescomplemented
242C2×C32C22×#6712yes—
243C3⋊D3—44no—
244C3×D3—124no—
245#41—98no—
246#41—184no—
247C2×Q8—92no—
248C2×D4—92no—
249C2×D4—182no—
250C22×C4—182no—
251C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
252Q16—274no—
253SD16—278no—
254D8—274no—
255C2×C8—91no—
256C2×C8—91no—
257C2×C8—94no—
258C22×C3—22no—
259C22×C3—32no—
260C22×C3—34no—
261C22×C3—64no—
262C22×C3—64no—
263C22×C3—62no—
264C22×C3—61no—
265C22×C3—184no—
266C22×C3—364no—
267C22×C3C2×#62512yes—
268C22×C3C2×#62511yescharacteristic
269C2×D3—64no—
270C2×D3—124no—
271C2×D3—184no—
272C2×D3—364no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—34no—
276C3×C4—61no—
277C3×C4—62no—
278C3×C4—61no—
279C3×C4—184no—
280C3×C4C2×#62511yescharacteristic
281C3×C4C2×#62512yes—
282C3×C4C22×D3211yescharacteristic
283Dic3—64no—
284Dic3—184no—
285C32—21no—
286C32C2×#95012yescomplemented
287C23—182no—
288C23#619811yescharacteristic
289Q8—94no—
290Q8—182no—
291D4—94no—
292D4—188no—
293D4—182no—
294C2×C4—182no—
295C2×C4—188no—
296C2×C4#619812yes—
297C2×C4#619811yescharacteristic
298C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
300C8—92no—
301C8—92no—
302C2×C3—22no—
303C2×C3—32no—
304C2×C3—34no—
305C2×C3—64no—
306C2×C3—64no—
307C2×C3—61no—
308C2×C3—62no—
309C2×C3—364no—
310C2×C3#6382212yes—
311C2×C3C22×#62511yescharacteristic
312D3—124no—
313D3—364no—
314C22—22no—
315C22—184no—
316C22—364no—
317C22C2×#619812yes—
318C22C2×#619811yescharacteristic
319C4—184no—
320C4C2×#619812yes—
321C4C2×#619811yescharacteristic
322C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C3—32no—
324C3—61no—
325C3C2×#6382211yescharacteristic
326C2—22no—
327C2—364no—
328C2#1172106012yesdirect product factor
329C2C22×#619811yescharacteristic
subgroupG/H
centerC2×C4#6198
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. C22×#6180C2semidirect
3. C2×#88433C2semidirect
4. #11721060C2direct (N × H)
5. C2×#88432C2semidirect
6. C2×#88431C2semidirect
11. #3828⋊C3C22semidirect
17. C2×#6180C22semidirect
18. #88433C22semidirect
19. #88432C22semidirect
20. #88431C22semidirect
21. C2×#3828D3semidirect
51. #3828C2×D3semidirect
127. C2×H(3,𝔽3)#135semidirect
191. H(3,𝔽3)C2×#135semidirect
241. C2×C32#950semidirect
286. C32C2×#950semidirect
328. C2#11721060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×#61802
C2×#884332
#117210602
C2×#884322
C2×#884312
C2×#638293