Finite Group Databasebeta

← C2×#10593200 #11922097

Subgroups — C2×#10593200 #11922097

Order 1152.

729 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×C3⋊#859C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4#10593200C214yesmaximal, direct product factor
5C2×#5426—31nomaximal
6C2×#4803—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊(#41⋊C3)C414yescomplemented
9C2×C3⋊(#41⋊C3)C2212yes—
10C2×Dic3⋊Dic3—41nomaximal
11C3⋊#859C2212yescomplemented
12C3×#859C2212yescomplemented
13C22×#239D311yescharacteristic
14C22×#41⋊C3—31no—
15C22×ΓL(1,𝔽25)—31no—
16C2×D3×C4⋊C4—31no—
17#5426—34no—
18C2×#859—31no—
19C2×#859—61no—
20C2×#859D311yescharacteristic, complemented
21C2×C3×#103—31no—
22C2×#835—31no—
23C2×#821—31no—
24C2×#809—31no—
25#4803—34no—
26#4803—34no—
27C22×C3⋊Dic3—41no—
28C22×C3×Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊(#41⋊C3)C2×C414yescomplemented
33Dic3⋊Dic3—44no—
34C2×#417—91noSylow
35C22×C3×Q8C2×D311yescharacteristic
36C2×#239—34no—
37C2×#239—32no—
38C2×#239Dic314yescomplemented
39C2×#239C2×D312yes—
40C22×D12—31no—
41C22×C4×D3—31no—
42C22×C4×D3—61no—
43C2×#41⋊C3—34no—
44C2×#41⋊C3—32no—
45C22×SL(2,𝔽3)—21no—
46C22×SL(2,𝔽3)C2×D311yescharacteristic
47C22×C3×C8—31no—
48C2×C3×C4⋊C4—31no—
49C2×C4⋊Dic3—31no—
50C2×C4⋊Dic3—121no—
51C2×Dic3⋊C4—61no—
52C22×C3⋊C8—31no—
53C2×ΓL(1,𝔽25)—34no—
54C2×ΓL(1,𝔽25)—64no—
55D3×C4⋊C4—34no—
56D3×C4⋊C4—62no—
57#859—32no—
58#859—62no—
59#859C2×D312yescomplemented
60C3×#103—32no—
61C3×#103—32no—
62#835—32no—
63#835—32no—
64#821—34no—
65#809—34no—
66C23×C32—41no—
67C2×C3⋊Dic3—42no—
68C2×C3⋊Dic3—82no—
69C2×C3×Dic3—42no—
70C2×C3×Dic3—44no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3—82no—
73C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
74C22×#41—31no—
75C22×M16—91no—
76C22×C4⋊C4—91no—
77#417—94no—
78#417—94no—
79C2×#103—31no—
80C2×#103—91no—
81C2×#102—91no—
82C2×#102—91no—
83C23×D3—31no—
84C2×C3×Q8—31no—
85C2×C3×Q8—32no—
86C2×C3×Q8C2×Dic312yes—
87C2×C3×Q8C22×D311yescharacteristic
88C22×C3×C4—31no—
89C22×C3×C4—61no—
90C22×C3×C4—121no—
91C22×Dic3—41no—
92C22×Dic3—61no—
93C22×Dic3—121no—
94C22×Dic3—241no—
95C22×Dic3S411yescharacteristic
96#239—38no—
97#239—34no—
98#239—64no—
99#239C2×Dic314yescomplemented
100C2×D12—32no—
101C2×D12—34no—
102C2×D12—62no—
103C2×D12—61no—
104C2×C4×D3—34no—
105C2×C4×D3—34no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—64no—
109C2×C4×D3—124no—
110#41⋊C3—34no—
111C2×SL(2,𝔽3)—22no—
112C2×SL(2,𝔽3)—21no—
113C2×SL(2,𝔽3)C4×D312yescomplemented
114C2×SL(2,𝔽3)C22×D311yescharacteristic
115C2×C3×C8—32no—
116C2×C3×C8—62no—
117C3×C4⋊C4—32no—
118C3×C4⋊C4—61no—
119C4⋊Dic3—32no—
120C4⋊Dic3—61no—
121C4⋊Dic3—124no—
122Dic3⋊C4—64no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—62no—
125ΓL(1,𝔽25)—68no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—41no—
130C3⋊Dic3—82no—
131C3×Dic3—44no—
132C3×Dic3—82no—
133C2×#41—34no—
134C2×#41—32no—
135C2×#41—92no—
136C2×#41—94no—
137C22×Q8D3211yescharacteristic
138C22×D4—91no—
139C23×C4—91no—
140C23×C4—181no—
141C2×M16—94no—
142C2×M16—184no—
143C22×C8—31no—
144C22×C8—91no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—94no—
147C2×C4⋊C4—91no—
148C2×C4⋊C4—182no—
149C2×C4⋊C4—181no—
150#103—32no—
151#103—32no—
152#103—92no—
153#103—92no—
154#102—94no—
155#102—94no—
156C23×C3—41no—
157C23×C3—81no—
158C23×C3C2×S411yescharacteristic
159C22×D3—34no—
160C22×D3—32no—
161C22×D3—62no—
162C22×D3—62no—
163C3×Q8—32no—
164C3×Q8—31no—
165C3×Q8—61no—
166C3×Q8C22×Dic311yescharacteristic
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—62no—
172C2×C3×C4—122no—
173C2×C3×C4—124no—
174C2×C3×C4—122no—
175C2×Dic3—42no—
176C2×Dic3—62no—
177C2×Dic3—82no—
178C2×Dic3—122no—
179C2×Dic3—122no—
180C2×Dic3—242no—
181C2×Dic3—242no—
182C2×Dic3—244no—
183C2×Dic3SL(2,ℤ/4)14yes—
184C2×Dic3C2×S412yes—
185D12—34no—
186D12—62no—
187D12—64no—
188C4×D3—34no—
189C4×D3—34no—
190C4×D3—38no—
191C4×D3—128no—
192SL(2,𝔽3)—21no—
193SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
194C3×C8—62no—
195C3⋊C8—62no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C2×C32—44no—
200C24—91no—
201#41—34no—
202#41—94no—
203#41—98no—
204#41—184no—
205C2×Q8—31no—
206C2×Q8—32no—
207C2×Q8D3×Dic312yes—
208C2×Q8C2×D3211yescharacteristic
209C2×D4—92no—
210C2×D4—94no—
211C2×D4—181no—
212C2×D4—182no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—61no—
216C22×C4—92no—
217C22×C4—92no—
218C22×C4—94no—
219C22×C4—94no—
220C22×C4—184no—
221C22×C4—181no—
222C22×C4—364no—
223M16—188no—
224C2×C8—32no—
225C2×C8—62no—
226C2×C8—92no—
227C2×C8—182no—
228C4⋊C4—32no—
229C4⋊C4—61no—
230C4⋊C4—92no—
231C4⋊C4—184no—
232C4⋊C4—181no—
233C22×C3—41no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—42no—
237C22×C3—81no—
238C22×C3—82no—
239C22×C3—82no—
240C22×C3—82no—
241C22×C3#98412yes—
242C22×C3#98612yes—
243C22×C3C2×SL(2,ℤ/4)12yes—
244C22×C3C22×S411yescharacteristic
245C2×D3—34no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—68no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—32no—
252C3×C4—124no—
253C3×C4—122no—
254Dic3—82no—
255Dic3—122no—
256Dic3—242no—
257Dic3—244no—
258Dic3C2×SL(2,ℤ/4)14yes—
259C32—41noSylow
260C23—92no—
261C23—94no—
262C23—182no—
263C23—182no—
264C23D3×S411yescharacteristic
265Q8—32no—
266Q8—31no—
267Q8—61no—
268Q8C2×D3×Dic311yescharacteristic
269D4—94no—
270D4—182no—
271D4—184no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—34no—
278C2×C4—62no—
279C2×C4—94no—
280C2×C4—94no—
281C2×C4—98no—
282C2×C4—122no—
283C2×C4—182no—
284C2×C4—368no—
285C2×C4—362no—
286C8—62no—
287C8—182no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—44no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—84no—
296C2×C3#542614yescomplemented
297C2×C3C2×#98411yescharacteristic
298C2×C3C2×#98611yescharacteristic
299C2×C3C22×SL(2,ℤ/4)11yescharacteristic
300D3—64no—
301C22—94no—
302C22—182no—
303C22—182no—
304C22—188no—
305C22#6419612yes—
306C22#6419912yes—
307C22D3×SL(2,ℤ/4)12yes—
308C22C2×D3×S411yescharacteristic
309C4—31no—
310C4—31no—
311C4—32no—
312C4—34no—
313C4—122no—
314C4—362no—
315C3—41no—
316C3—81no—
317C3C2×#542611yescharacteristic, complemented
318C2—184no—
319C2#1059320014yesdirect product factor
320C2C2×#6419611yescharacteristic
321C2C2×#6419911yescharacteristic
322C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radicalC22×C3⋊(#41⋊C3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#859C2semidirect
3. C2×C3×#859C2semidirect
4. #10593200C2direct (N × H)
8. C2×C3⋊(#41⋊C3)C4semidirect
11. C3⋊#859C22semidirect
12. C3×#859C22semidirect
20. C2×#859D3semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C3⋊(#41⋊C3)C2×C4semidirect
38. C2×#239Dic3semidirect
59. #859C2×D3semidirect
73. C3×SL(2,𝔽3)C22×C4semidirect
99. #239C2×Dic3semidirect
113. C2×SL(2,𝔽3)C4×D3semidirect
193. SL(2,𝔽3)C2×C4×D3semidirect
296. C2×C3#5426semidirect
317. C3C2×#5426semidirect
319. C2#10593200direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊(#41⋊C3)2
C2×C3⋊#8592
C2×C3×#8592
#105932002
C2×#54263
C2×#48033
C2×Dic3⋊Dic34