Finite Group Databasebeta

← C2×#5574⋊C3 #11923174

Subgroups — C2×#5574⋊C3 #11923174

Order 1152.

825 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 825 conjugacy classes of subgroups — too many to draw. Only its 127 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5574⋊C3C214yesmaximal, direct product factor
5C2×#5574C311yesmaximal, characteristic, complemented
6C2×#520⋊C3—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3⋊(#41⋊C3)C414yescomplemented
9C2×C3⋊(#41⋊C3)C2212yes—
10C2×C3×Dic3⋊C4—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yescomplemented
12Dic3×SL(2,𝔽3)C2212yescomplemented
13C22×#239C2×C311yescharacteristic
14C22×#41⋊C3—31no—
15C2×C3×C4×Q8C2×C311yescharacteristic
16C2×Q8×Dic3C2×C311yescharacteristic
17#5574—34no—
18#5574C2×C314yescomplemented
19C2×#893—31no—
20C2×D3×C4⋊C4—31no—
21C2×C4×D12—31no—
22C2×#872—31no—
23#520⋊C3—34no—
24C2×C4×SL(2,𝔽3)—21no—
25C2×C4×SL(2,𝔽3)—31no—
26C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
27C22×C32×C4—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
33C3×Dic3⋊C4—44no—
34C2×#520—31noSylow
35C22×C3×Q8C22×C311yescharacteristic
36C2×#239—34no—
37C2×#239—32no—
38C2×#239C3×C414yescomplemented
39C2×#239C22×C312yes—
40C22×D12—31no—
41C22×C4×D3—31no—
42C22×C4×D3—32no—
43C2×#41⋊C3—34no—
44C2×#41⋊C3—32no—
45C22×SL(2,𝔽3)—21no—
46C22×SL(2,𝔽3)C2×D311yescharacteristic
47C3×C4×Q8—32no—
48C3×C4×Q8C22×C312yes—
49C2×C3×C4⋊C4—31no—
50C2×C3×C4⋊C4—121no—
51C2×C3×C42—31no—
52Q8×Dic3—32no—
53Q8×Dic3C22×C312yes—
54C2×(C2×D3)⋊C4—31no—
55C2×(C2×D3)⋊C4—31no—
56C2×C4⋊Dic3—31no—
57C2×Dic3⋊C4—31no—
58C2×Dic3⋊C4A411yescharacteristic
59C2×C4×Dic3—31no—
60#893—34no—
61#893—34no—
62D3×C4⋊C4—34no—
63D3×C4⋊C4—34no—
64C4×D12—34no—
65C4×D12—34no—
66#872—34no—
67#872—34no—
68C4×SL(2,𝔽3)—22no—
69C4×SL(2,𝔽3)—32no—
70C4×SL(2,𝔽3)C2×D312yescomplemented
71C23×C32—41no—
72C2×C32×C4—42no—
73C2×C32×C4—82no—
74C2×C3×Dic3—44no—
75C2×C3×Dic3—42no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—82no—
78C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
79C22×#41—31no—
80#520—34no—
81#520—94no—
82C2×C4×Q8—31no—
83C2×C4×Q8C3×D311yescharacteristic
84C2×C4×D4—91no—
85C2×C4×D4—91no—
86C2×#117—91no—
87C22×C4⋊C4—91no—
88C23×D3—31no—
89C2×C3×Q8—31no—
90C2×C3×Q8—32no—
91C2×C3×Q8C2×C3×C412yes—
92C2×C3×Q8C23×C311yescharacteristic
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—41no—
96C22×C3×C4—81no—
97C22×C3×C4—121no—
98C22×C3×C4—121no—
99C22×C3×C4C2×A411yescharacteristic
100C22×Dic3—31no—
101C22×Dic3C2×A411yescharacteristic
102C22×Dic3C2×A411yescharacteristic
103#239—38no—
104#239—38no—
105#239—34no—
106#239C2×C3×C414yescomplemented
107C2×D12—32no—
108C2×D12—34no—
109C2×D12—32no—
110C2×D12—34no—
111C2×C4×D3—34no—
112C2×C4×D3—34no—
113C2×C4×D3—32no—
114C2×C4×D3—32no—
115C2×C4×D3—38no—
116C2×C4×D3—124no—
117#41⋊C3—34no—
118C2×SL(2,𝔽3)—22no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)C4×D312yescomplemented
121C2×SL(2,𝔽3)C22×D311yescharacteristic
122C3×C4⋊C4—32no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—124no—
125C3×C42—32no—
126C3×C42—32no—
127(C2×D3)⋊C4—34no—
128(C2×D3)⋊C4—34no—
129C4⋊Dic3—32no—
130C4⋊Dic3—32no—
131Dic3⋊C4—34no—
132Dic3⋊C4C2×A414yes—
133C4×Dic3—32no—
134C4×Dic3—32no—
135C22×C32—42no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—41no—
139C32×C4—82no—
140C3×Dic3—44no—
141C3×Dic3—82no—
142C2×#41—34no—
143C2×#41—32no—
144C2×#41—92no—
145C2×#41—94no—
146C22×Q8C2×C3×D311yescharacteristic
147C22×D4—91no—
148C23×C4—91no—
149C23×C4—92no—
150C4×Q8—32no—
151C4×Q8—32no—
152C4×Q8—92no—
153C4×Q8C2×C3×D312yes—
154C4×D4—94no—
155C4×D4—94no—
156C4×D4—94no—
157C4×D4—94no—
158#117—94no—
159#117—94no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—94no—
163C2×C4⋊C4—94no—
164C2×C4⋊C4—91no—
165C2×C4⋊C4—91no—
166C2×C22⋊C4—91no—
167C2×C22⋊C4—91no—
168C2×C42—31no—
169C2×C42—91no—
170C23×C3—41no—
171C23×C3—81no—
172C23×C3C22×A411yescharacteristic
173C22×D3—34no—
174C22×D3—32no—
175C22×D3—62no—
176C22×D3—62no—
177C3×Q8—32no—
178C3×Q8—31no—
179C3×Q8—32no—
180C3×Q8C22×C3×C411yescharacteristic
181C2×C3×C4—22no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—42no—
188C2×C3×C4—62no—
189C2×C3×C4—82no—
190C2×C3×C4—84no—
191C2×C3×C4—82no—
192C2×C3×C4—122no—
193C2×C3×C4—124no—
194C2×C3×C4—122no—
195C2×C3×C4—242no—
196C2×C3×C4C22×A412yes—
197C2×Dic3—22no—
198C2×Dic3—32no—
199C2×Dic3—62no—
200C2×Dic3C4×A414yes—
201C2×Dic3C22×A412yes—
202C2×Dic3C22×A412yes—
203D12—34no—
204D12—34no—
205D12—38no—
206C4×D3—34no—
207C4×D3—38no—
208C4×D3—34no—
209C4×D3—128no—
210SL(2,𝔽3)—21no—
211SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—44no—
216C24—91no—
217#41—34no—
218#41—98no—
219#41—98no—
220#41—94no—
221C2×Q8—31no—
222C2×Q8—32no—
223C2×Q8C3×C4×D312yes—
224C2×Q8C22×C3×D311yescharacteristic
225C2×D4—92no—
226C2×D4—94no—
227C2×D4—92no—
228C2×D4—94no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—92no—
234C22×C4—92no—
235C22×C4—94no—
236C22×C4—94no—
237C22×C4—98no—
238C22×C4—91no—
239C22×C4—364no—
240C22×C4D3×A411yescharacteristic
241C4⋊C4—32no—
242C4⋊C4—32no—
243C4⋊C4—34no—
244C4⋊C4—94no—
245C4⋊C4—92no—
246C4⋊C4—92no—
247C22⋊C4—94no—
248C22⋊C4—94no—
249C42—32no—
250C42—32no—
251C42—92no—
252C42—92no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—41no—
257C22×C3—82no—
258C22×C3—82no—
259C22×C3—82no—
260C22×C3—81no—
261C22×C3C2×C4×A412yes—
262C22×C3#99412yes—
263C22×C32−1+4⋊C312yes—
264C22×C3C23×A411yescharacteristic
265C2×D3—34no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—68no—
269C3×C4—22no—
270C3×C4—31no—
271C3×C4—32no—
272C3×C4—31no—
273C3×C4—62no—
274C3×C4—82no—
275C3×C4—84no—
276C3×C4—124no—
277C3×C4—242no—
278Dic3—22no—
279Dic3—62no—
280Dic3C2×C4×A414yes—
281C32—41noSylow
282C23—94no—
283C23—92no—
284C23—182no—
285C23—182no—
286C23C2×D3×A411yescharacteristic
287Q8—32no—
288Q8—31no—
289Q8—32no—
290Q8C2×C3×C4×D311yescharacteristic
291D4—94no—
292D4—94no—
293D4—98no—
294C2×C4—22no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—34no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—32no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—94no—
306C2×C4—94no—
307C2×C4—98no—
308C2×C4—92no—
309C2×C4—182no—
310C2×C4—368no—
311C2×C4C2×D3×A412yes—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—44no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3—81no—
319C2×C3—84no—
320C2×C3#520⋊C314yescomplemented
321C2×C3C22×C4×A411yescharacteristic
322C2×C3C2×#99411yescharacteristic
323C2×C3C2×2−1+4⋊C311yescharacteristic
324D3—64no—
325C22—94no—
326C22—182no—
327C22—182no—
328C22—188no—
329C22C4×D3×A412yes—
330C22#1007⋊C312yes—
331C22C3⋊#99412yes—
332C22C22×D3×A411yescharacteristic
333C4—22no—
334C4—31no—
335C4—32no—
336C4—31no—
337C4—34no—
338C4—62no—
339C4—62no—
340C4—182no—
341C3—41no—
342C3—81no—
343C3C2×#520⋊C311yescharacteristic, complemented
344C2—184no—
345C2#5574⋊C314yesdirect product factor
346C2C2×C4×D3×A411yescharacteristic
347C2C2×#1007⋊C311yescharacteristic
348C2C2×C3⋊#99411yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3⋊(#41⋊C3)C2
cosocleC2×C3×Q8C23×C3
hypercenterC22×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×SL(2,𝔽3)C2semidirect
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. #5574⋊C3C2direct (N × H)
5. C2×#5574C3semidirect
8. C2×C3⋊(#41⋊C3)C4semidirect
11. C3×C4×SL(2,𝔽3)C22semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
18. #5574C2×C3semidirect
26. C2×C4×SL(2,𝔽3)D3semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C3⋊(#41⋊C3)C2×C4semidirect
38. C2×#239C3×C4semidirect
70. C4×SL(2,𝔽3)C2×D3semidirect
78. C3×SL(2,𝔽3)C22×C4semidirect
106. #239C2×C3×C4semidirect
120. C2×SL(2,𝔽3)C4×D3semidirect
211. SL(2,𝔽3)C2×C4×D3semidirect
320. C2×C3#520⋊C3semidirect
343. C3C2×#520⋊C3semidirect
345. C2#5574⋊C3direct (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×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
#5574⋊C32
C2×#55743
C2×#520⋊C33
C2×C3×Dic3⋊C44