Finite Group Databasebeta

← C2×#5572⋊C3 #11923184

Subgroups — C2×#5572⋊C3 #11923184

Order 1152.

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

normal subgroups only

This group has 841 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5572⋊C3C214yesmaximal, direct product factor
5C2×#5572C311yesmaximal, characteristic, complemented
6C2×#519⋊C3—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×D3×SL(2,𝔽3)C414yescomplemented
9C2×D3×SL(2,𝔽3)C2212yes—
10C2×C3×(C2×D3)⋊C4—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yescomplemented
12Dic3×SL(2,𝔽3)C2212yescomplemented
13C22×D3×Q8C2×C311yescharacteristic
14C23×SL(2,𝔽3)—31no—
15C2×C3×C4×Q8C2×C311yescharacteristic, complemented
16C2×Q8×Dic3C2×C311yescharacteristic, complemented
17#5572—34no—
18#5572C2×C314yescomplemented
19C2×#892—31no—
20C2×#887—31no—
21C2×#872—31no—
22C2×C4×Dic6—31no—
23#519⋊C3—34no—
24C2×C4×SL(2,𝔽3)—21no—
25C2×C4×SL(2,𝔽3)—31no—
26C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
27C23×C3×D3—41no—
28C22×C32×C4—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yes—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32D3×SL(2,𝔽3)C2×C414yescomplemented
33C3×(C2×D3)⋊C4—44no—
34C2×#519—31noSylow
35C22×C3×Q8C22×C311yescharacteristic
36C2×D3×Q8—34no—
37C2×D3×Q8—32no—
38C2×D3×Q8C3×C414yescomplemented
39C2×D3×Q8C22×C312yes—
40C22×C4×D3—31no—
41C22×Dic6—31no—
42C22×SL(2,𝔽3)—21no—
43C22×SL(2,𝔽3)—34no—
44C22×SL(2,𝔽3)—32no—
45C22×SL(2,𝔽3)C2×D311yescharacteristic
46C3×C4×Q8—32no—
47C3×C4×Q8C22×C312yescomplemented
48C2×C3×C4⋊C4—31no—
49C2×C3×C22⋊C4—121no—
50C2×C3×C42—31no—
51Q8×Dic3—32no—
52Q8×Dic3C22×C312yescomplemented
53C2×(C2×D3)⋊C4—31no—
54C2×(C2×D3)⋊C4A411yescharacteristic
55C2×C4⋊Dic3—31no—
56C2×Dic3⋊C4—31no—
57C2×Dic3⋊C4—31no—
58C2×C4×Dic3—32no—
59C2×C4×Dic3—31no—
60#892—34no—
61#892—34no—
62#887—34no—
63#887—34no—
64#872—34no—
65#872—34no—
66C4×Dic6—34no—
67C4×Dic6—34no—
68C4×SL(2,𝔽3)—22no—
69C4×SL(2,𝔽3)—32no—
70C4×SL(2,𝔽3)C2×D312yescomplemented
71C23×C32—41no—
72C22×C3×D3—44no—
73C22×C3×D3—42no—
74C22×C3×D3—82no—
75C22×C3×D3—82no—
76C2×C32×C4—42no—
77C2×C32×C4—82no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—82no—
80C3×SL(2,𝔽3)C22×C411yescharacteristic
81C23×Q8—31no—
82#519—34no—
83#519—94no—
84C2×C4×Q8—31no—
85C2×C4×Q8—91no—
86C2×C4×Q8—91no—
87C2×C4×Q8C3×D311yescharacteristic, complemented
88C2×#117—91no—
89C2×#117—91no—
90C24×C3—121no—
91C23×D3C2×A411yescharacteristic
92C2×C3×Q8—31no—
93C2×C3×Q8—32no—
94C2×C3×Q8C2×C3×C412yes—
95C2×C3×Q8C23×C311yescharacteristic
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—41no—
99C22×C3×C4—81no—
100C22×C3×C4—121no—
101C22×C3×C4C2×A411yescharacteristic
102C22×Dic3—31no—
103C22×Dic3—31no—
104C22×Dic3C2×A411yescharacteristic
105D3×Q8—38no—
106D3×Q8—38no—
107D3×Q8—34no—
108D3×Q8C2×C3×C414yescomplemented
109C2×C4×D3—34no—
110C2×C4×D3—34no—
111C2×C4×D3—32no—
112C2×C4×D3—32no—
113C2×Dic6—32no—
114C2×Dic6—34no—
115C2×Dic6—32no—
116C2×Dic6—34no—
117C2×SL(2,𝔽3)—22no—
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)—34no—
120C2×SL(2,𝔽3)C4×D312yes—
121C2×SL(2,𝔽3)C22×D311yescharacteristic
122C3×C4⋊C4—32no—
123C3×C4⋊C4—32no—
124C3×C22⋊C4—124no—
125C3×C42—32no—
126C3×C42—32no—
127(C2×D3)⋊C4—34no—
128(C2×D3)⋊C4C2×A414yes—
129C4⋊Dic3—32no—
130C4⋊Dic3—32no—
131Dic3⋊C4—34no—
132Dic3⋊C4—34no—
133C4×Dic3—38no—
134C4×Dic3—32no—
135C4×Dic3—32no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32—41no—
140C2×C3×D3—44no—
141C2×C3×D3—82no—
142C2×C3×D3—82no—
143C2×C3×D3—88no—
144C32×C4—82no—
145C3×Dic3—82no—
146C22×Q8—34no—
147C22×Q8—32no—
148C22×Q8—91no—
149C22×Q8—92no—
150C22×Q8—94no—
151C22×Q8C2×C3×D311yescharacteristic
152C23×C4—91no—
153C4×Q8—32no—
154C4×Q8—32no—
155C4×Q8—94no—
156C4×Q8—94no—
157C4×Q8—94no—
158C4×Q8—94no—
159C4×Q8—92no—
160C4×Q8C2×C3×D312yescomplemented
161#117—94no—
162#117—94no—
163#117—94no—
164#117—94no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—91no—
167C2×C4⋊C4—91no—
168C2×C4⋊C4—91no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—91no—
171C2×C42—31no—
172C2×C42—91no—
173C2×C42—92no—
174C23×C3—41no—
175C23×C3—81no—
176C23×C3—122no—
177C23×C3—124no—
178C23×C3—242no—
179C23×C3—242no—
180C23×C3C22×A411yescharacteristic
181C22×D3—22no—
182C22×D3—22no—
183C22×D3C4×A414yes—
184C22×D3C22×A412yes—
185C3×Q8—32no—
186C3×Q8—31no—
187C3×Q8—32no—
188C3×Q8C22×C3×C411yescharacteristic
189C2×C3×C4—22no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—42no—
196C2×C3×C4—62no—
197C2×C3×C4—82no—
198C2×C3×C4—84no—
199C2×C3×C4—82no—
200C2×C3×C4—122no—
201C2×C3×C4—242no—
202C2×C3×C4C22×A412yes—
203C2×Dic3—22no—
204C2×Dic3—34no—
205C2×Dic3—32no—
206C2×Dic3—32no—
207C2×Dic3—62no—
208C2×Dic3C22×A412yes—
209C4×D3—34no—
210C4×D3—38no—
211C4×D3—34no—
212Dic6—38no—
213Dic6—34no—
214Dic6—34no—
215SL(2,𝔽3)—21no—
216SL(2,𝔽3)C2×C4×D311yescharacteristic
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—44no—
221C3×D3—84no—
222C24—31no—
223C2×Q8—31no—
224C2×Q8—32no—
225C2×Q8—34no—
226C2×Q8—92no—
227C2×Q8—94no—
228C2×Q8—92no—
229C2×Q8—94no—
230C2×Q8—98no—
231C2×Q8—98no—
232C2×Q8—94no—
233C2×Q8C3×C4×D312yes—
234C2×Q8C22×C3×D311yescharacteristic
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—92no—
239C22×C4—92no—
240C22×C4—94no—
241C22×C4—94no—
242C22×C4—91no—
243C22×C4—91no—
244C22×C4D3×A411yescharacteristic
245C4⋊C4—32no—
246C4⋊C4—32no—
247C4⋊C4—94no—
248C4⋊C4—92no—
249C4⋊C4—92no—
250C4⋊C4—94no—
251C22⋊C4—34no—
252C22⋊C4—94no—
253C42—32no—
254C42—32no—
255C42—92no—
256C42—92no—
257C42—98no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—82no—
263C22×C3—82no—
264C22×C3—82no—
265C22×C3—81no—
266C22×C3—124no—
267C22×C3—242no—
268C22×C3—242no—
269C22×C3—248no—
270C22×C3C2×C4×A412yes—
271C22×C32−1+4⋊C312yes—
272C22×C32−1+4⋊C312yes—
273C22×C3C23×A411yescharacteristic
274C2×D3—22no—
275C2×D3—22no—
276C2×D3—28no—
277C2×D3C2×C4×A414yes—
278C3×C4—22no—
279C3×C4—31no—
280C3×C4—32no—
281C3×C4—31no—
282C3×C4—62no—
283C3×C4—82no—
284C3×C4—84no—
285C3×C4—242no—
286Dic3—22no—
287Dic3—34no—
288Dic3—62no—
289C32—41noSylow
290C23—34no—
291C23—32no—
292C23—62no—
293C23—62no—
294C23C2×D3×A411yescharacteristic
295Q8—32no—
296Q8—31no—
297Q8—32no—
298Q8—94no—
299Q8—98no—
300Q8—94no—
301Q8C2×C3×C4×D311yescharacteristic
302C2×C4—22no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—94no—
312C2×C4—94no—
313C2×C4—98no—
314C2×C4—94no—
315C2×C4—92no—
316C2×C4—92no—
317C2×C4—182no—
318C2×C4C2×D3×A412yes—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—44no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3—81no—
326C2×C3—84no—
327C2×C3—244no—
328C2×C3#519⋊C314yescomplemented
329C2×C3C22×C4×A411yescharacteristic
330C2×C3C2×2−1+4⋊C311yescharacteristic
331C2×C3C2×2−1+4⋊C311yescharacteristic
332D3—24no—
333C22—34no—
334C22—62no—
335C22—62no—
336C22—68no—
337C22C4×D3×A412yes—
338C22#1007⋊C312yes—
339C22#1010⋊C312yes—
340C22C22×D3×A411yescharacteristic
341C4—22no—
342C4—31no—
343C4—32no—
344C4—31no—
345C4—62no—
346C4—62no—
347C4—94no—
348C4—182no—
349C3—41no—
350C3—81no—
351C3C2×#519⋊C311yescharacteristic, complemented
352C2—64no—
353C2#5572⋊C314yesdirect product factor
354C2C2×C4×D3×A411yescharacteristic
355C2C2×#1007⋊C311yescharacteristic
356C2C2×#1010⋊C311yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×D3×SL(2,𝔽3)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. #5572⋊C3C2direct (N × H)
5. C2×#5572C3semidirect
8. C2×D3×SL(2,𝔽3)C4semidirect
11. C3×C4×SL(2,𝔽3)C22semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
15. C2×C3×C4×Q8C2×C3semidirect
16. C2×Q8×Dic3C2×C3semidirect
18. #5572C2×C3semidirect
26. C2×C4×SL(2,𝔽3)D3semidirect
32. D3×SL(2,𝔽3)C2×C4semidirect
38. C2×D3×Q8C3×C4semidirect
47. C3×C4×Q8C22×C3semidirect
52. Q8×Dic3C22×C3semidirect
70. C4×SL(2,𝔽3)C2×D3semidirect
87. C2×C4×Q8C3×D3semidirect
108. D3×Q8C2×C3×C4semidirect
160. C4×Q8C2×C3×D3semidirect
328. C2×C3#519⋊C3semidirect
351. C3C2×#519⋊C3semidirect
353. C2#5572⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×SL(2,𝔽3)2
C2×C3×C4×SL(2,𝔽3)2
C2×Dic3×SL(2,𝔽3)2
#5572⋊C32
C2×#55723
C2×#519⋊C33
C2×C3×(C2×D3)⋊C44