Finite Group Databasebeta

← C2×#5576⋊C3 #11923186

Subgroups — C2×#5576⋊C3 #11923186

Order 1152.

885 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 111 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, complemented
2C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5576⋊C3C214yesmaximal, direct product factor
5C2×#5576C311yesmaximal, characteristic, complemented
6C2×#548⋊C3—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×D3×SL(2,𝔽3)—22no—
9C2×D3×SL(2,𝔽3)C2212yescomplemented
10C2×C3⋊(#41⋊C3)—22no—
11C2×C3⋊(#41⋊C3)C2212yescomplemented
12C2×C3×(C2×D3)⋊C4—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14C22×#239C2×C311yescharacteristic, complemented
15C22×D3×Q8C2×C311yescharacteristic
16C22×#41⋊C3—31no—
17C23×SL(2,𝔽3)—31no—
18C2×C3×C4×Q8C2×C311yescharacteristic, complemented
19#5576—34no—
20#5576C2×C314yescomplemented
21C2×#897—31no—
22C2×#895—31no—
23C2×#875—31no—
24C2×C4×D12—31no—
25#548⋊C3—34no—
26C2×C4×SL(2,𝔽3)—21no—
27C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
28C23×C3×D3—41no—
29C22×C32×C4—41no—
30C22×C3×Dic3—41no—
31C2×C3×SL(2,𝔽3)D412yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33D3×SL(2,𝔽3)—22no—
34C3⋊(#41⋊C3)—22no—
35C3×(C2×D3)⋊C4—44no—
36C2×#548—31noSylow
37C22×C3×Q8C22×C311yescharacteristic
38C2×#239—22no—
39C2×#239—32no—
40C2×#239—62no—
41C2×#239C22×C312yescomplemented
42C2×D3×Q8—22no—
43C2×D3×Q8—32no—
44C2×D3×Q8—62no—
45C2×D3×Q8C22×C312yes—
46C22×D12—31no—
47C22×D12—31no—
48C22×C4×D3—31no—
49C22×C4×D3—31no—
50C22×Dic6—31no—
51C2×#41⋊C3—32no—
52C2×#41⋊C3—62no—
53C22×SL(2,𝔽3)—21no—
54C22×SL(2,𝔽3)—32no—
55C22×SL(2,𝔽3)—62no—
56C22×SL(2,𝔽3)C2×D311yescharacteristic
57C3×C4×Q8—32no—
58C3×C4×Q8C22×C312yescomplemented
59C2×C3×C4⋊C4—31no—
60C2×C3×C22⋊C4—121no—
61C2×C3×C42—31no—
62C2×(C2×D3)⋊C4—32no—
63C2×(C2×D3)⋊C4—31no—
64C2×(C2×D3)⋊C4A411yescharacteristic
65C2×C4⋊Dic3—31no—
66#897—34no—
67#897—34no—
68#895—34no—
69#895—34no—
70#875—34no—
71#875—34no—
72C4×D12—34no—
73C4×D12—34no—
74C4×SL(2,𝔽3)—22no—
75C4×SL(2,𝔽3)C2×D312yescomplemented
76C23×C32—41no—
77C22×C3×D3—44no—
78C22×C3×D3—42no—
79C22×C3×D3—82no—
80C22×C3×D3—82no—
81C2×C32×C4—42no—
82C2×C32×C4—82no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—82no—
85C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
86C22×#41—31no—
87C23×Q8—31no—
88#548—34no—
89#548—94no—
90C2×#124—91no—
91C2×C22⋊Q8—91no—
92C2×#121—91no—
93C2×C4×Q8C3×D311yescharacteristic, complemented
94C2×C4×D4—91no—
95C24×C3—121no—
96C23×D3—31no—
97C23×D3C2×A411yescharacteristic
98C2×C3×Q8—31no—
99C2×C3×Q8—32no—
100C2×C3×Q8C3×D412yes—
101C2×C3×Q8C23×C311yescharacteristic
102C22×C3×C4—31no—
103C22×C3×C4—31no—
104C22×C3×C4—41no—
105C22×C3×C4—81no—
106C22×C3×C4—121no—
107C22×C3×C4C2×A411yescharacteristic
108C22×Dic3—31no—
109C22×Dic3C2×A411yescharacteristic
110#239—22no—
111#239—64no—
112#239—64no—
113#239—62no—
114D3×Q8—22no—
115D3×Q8—64no—
116D3×Q8—64no—
117D3×Q8—62no—
118C2×D12—32no—
119C2×D12—32no—
120C2×D12—34no—
121C2×D12—62no—
122C2×D12—62no—
123C2×D12—64no—
124C2×C4×D3—32no—
125C2×C4×D3—32no—
126C2×C4×D3—32no—
127C2×C4×D3—32no—
128C2×C4×D3—62no—
129C2×C4×D3—62no—
130C2×C4×D3—62no—
131C2×C4×D3—62no—
132C2×Dic6—32no—
133C2×Dic6—32no—
134C2×Dic6—62no—
135C2×Dic6—62no—
136#41⋊C3—62no—
137C2×SL(2,𝔽3)—22no—
138C2×SL(2,𝔽3)—21no—
139C2×SL(2,𝔽3)—62no—
140C2×SL(2,𝔽3)D1212yescomplemented
141C2×SL(2,𝔽3)C22×D311yescharacteristic
142C3×C4⋊C4—32no—
143C3×C4⋊C4—32no—
144C3×C22⋊C4—124no—
145C3×C42—32no—
146C3×C42—32no—
147(C2×D3)⋊C4—38no—
148(C2×D3)⋊C4—34no—
149(C2×D3)⋊C4C2×A414yes—
150C4⋊Dic3—34no—
151C22×C32—42no—
152C22×C32—42no—
153C22×C32—42no—
154C22×C32—41no—
155C2×C3×D3—44no—
156C2×C3×D3—82no—
157C2×C3×D3—88no—
158C2×C3×D3—82no—
159C32×C4—82no—
160C3×Dic3—82no—
161C2×#41—32no—
162C2×#41—62no—
163C2×#41—92no—
164C2×#41—182no—
165C22×Q8—32no—
166C22×Q8—62no—
167C22×Q8—91no—
168C22×Q8—92no—
169C22×Q8—182no—
170C22×Q8C2×C3×D311yescharacteristic
171C22×D4—91no—
172C22×D4—91no—
173C23×C4—91no—
174C23×C4—91no—
175#124—94no—
176#124—94no—
177C22⋊Q8—94no—
178C22⋊Q8—94no—
179#121—94no—
180#121—94no—
181C4×Q8—32no—
182C4×Q8C2×C3×D312yescomplemented
183C4×D4—94no—
184C4×D4—94no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—91no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—92no—
189C2×C22⋊C4—91no—
190C2×C42—31no—
191C23×C3—41no—
192C23×C3—81no—
193C23×C3—122no—
194C23×C3—124no—
195C23×C3—242no—
196C23×C3—242no—
197C23×C3C22×A411yescharacteristic
198C22×D3—22no—
199C22×D3—22no—
200C22×D3—32no—
201C22×D3—62no—
202C22×D3—62no—
203C22×D3—62no—
204C22×D3#41⋊C314yescomplemented
205C22×D3C22×A412yes—
206C3×Q8—32no—
207C3×Q8—31no—
208C3×Q8—32no—
209C3×Q8C2×C3×D411yescharacteristic
210C2×C3×C4—22no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—32no—
216C2×C3×C4—42no—
217C2×C3×C4—62no—
218C2×C3×C4—82no—
219C2×C3×C4—84no—
220C2×C3×C4—82no—
221C2×C3×C4—122no—
222C2×C3×C4—242no—
223C2×C3×C4C22×A412yes—
224C2×Dic3—22no—
225C2×Dic3—32no—
226C2×Dic3—62no—
227C2×Dic3C22×A412yes—
228D12—62no—
229D12—62no—
230D12—64no—
231D12—68no—
232C4×D3—62no—
233C4×D3—64no—
234C4×D3—62no—
235C4×D3—62no—
236C4×D3—64no—
237C4×D3—62no—
238Dic6—64no—
239Dic6—62no—
240Dic6—62no—
241SL(2,𝔽3)—21no—
242SL(2,𝔽3)C2×D1211yescharacteristic, complemented
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—44no—
247C3×D3—84no—
248C24—31no—
249C24—91no—
250#41—62no—
251#41—184no—
252#41—184no—
253#41—182no—
254C2×Q8—31no—
255C2×Q8—32no—
256C2×Q8—62no—
257C2×Q8—92no—
258C2×Q8—92no—
259C2×Q8—182no—
260C2×Q8—182no—
261C2×Q8—184no—
262C2×Q8—182no—
263C2×Q8—184no—
264C2×Q8C3×D1212yes—
265C2×Q8C22×C3×D311yescharacteristic
266C2×D4—94no—
267C2×D4—92no—
268C2×D4—92no—
269C2×D4—182no—
270C2×D4—182no—
271C2×D4—184no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—92no—
276C22×C4—92no—
277C22×C4—92no—
278C22×C4—92no—
279C22×C4—91no—
280C22×C4—182no—
281C22×C4—182no—
282C22×C4—182no—
283C22×C4—182no—
284C22×C4D3×A411yescharacteristic
285C4⋊C4—32no—
286C4⋊C4—32no—
287C4⋊C4—94no—
288C22⋊C4—34no—
289C22⋊C4—98no—
290C22⋊C4—94no—
291C42—32no—
292C42—32no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—41no—
297C22×C3—82no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—81no—
301C22×C3—124no—
302C22×C3—242no—
303C22×C3—248no—
304C22×C3—242no—
305C22×C3D4×A412yes—
306C22×C3C2×#41⋊C312yes—
307C22×C32−1+4⋊C312yes—
308C22×C3C23×A411yescharacteristic
309C2×D3—22no—
310C2×D3—22no—
311C2×D3—28no—
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—124no—
316C2×D3C2×#41⋊C314yescomplemented
317C3×C4—22no—
318C3×C4—32no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—62no—
322C3×C4—82no—
323C3×C4—84no—
324C3×C4—242no—
325Dic3—22no—
326Dic3—62no—
327C32—41noSylow
328C23—34no—
329C23—32no—
330C23—62no—
331C23—62no—
332C23—92no—
333C23—182no—
334C23—182no—
335C23—182no—
336C23C2×D3×A411yescharacteristic
337Q8—32no—
338Q8—31no—
339Q8—32no—
340Q8—182no—
341Q8—182no—
342Q8—184no—
343Q8C2×C3×D1211yescharacteristic
344D4—182no—
345D4—182no—
346D4—184no—
347D4—188no—
348C2×C4—22no—
349C2×C4—32no—
350C2×C4—32no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—92no—
358C2×C4—182no—
359C2×C4—182no—
360C2×C4—184no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—184no—
364C2×C4—182no—
365C2×C4C2×D3×A412yes—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—44no—
370C2×C3—81no—
371C2×C3—81no—
372C2×C3—81no—
373C2×C3—84no—
374C2×C3—244no—
375C2×C3#548⋊C314yescomplemented
376C2×C3C2×D4×A411yescharacteristic
377C2×C3C22×#41⋊C311yescharacteristic
378C2×C3C2×2−1+4⋊C311yescharacteristic
379D3—24no—
380D3—122no—
381C22—34no—
382C22—62no—
383C22—62no—
384C22—68no—
385C22—182no—
386C22—182no—
387C22—182no—
388C22—364no—
389C22A4×D1212yes—
390C22#1007⋊C312yes—
391C22D3×#41⋊C312yes—
392C22C22×D3×A411yescharacteristic
393C4—22no—
394C4—31no—
395C4—32no—
396C4—31no—
397C4—62no—
398C4—62no—
399C4—182no—
400C3—41no—
401C3—81no—
402C3C2×#548⋊C311yescharacteristic, complemented
403C2—64no—
404C2—362no—
405C2#5576⋊C314yesdirect product factor
406C2C2×A4×D1211yescharacteristic
407C2C2×#1007⋊C311yescharacteristic
408C2C2×D3×#41⋊C311yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×D3×SL(2,𝔽3)C2semidirect
2. C22×C3⋊(#41⋊C3)C2semidirect
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. #5576⋊C3C2direct (N × H)
5. C2×#5576C3semidirect
9. C2×D3×SL(2,𝔽3)C22semidirect
11. C2×C3⋊(#41⋊C3)C22semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. C22×#239C2×C3semidirect
18. C2×C3×C4×Q8C2×C3semidirect
20. #5576C2×C3semidirect
27. C2×C4×SL(2,𝔽3)D3semidirect
31. C2×C3×SL(2,𝔽3)D4semidirect
41. C2×#239C22×C3semidirect
58. C3×C4×Q8C22×C3semidirect
75. C4×SL(2,𝔽3)C2×D3semidirect
85. C3×SL(2,𝔽3)C2×D4semidirect
93. C2×C4×Q8C3×D3semidirect
140. C2×SL(2,𝔽3)D12semidirect
182. C4×Q8C2×C3×D3semidirect
204. C22×D3#41⋊C3semidirect
242. SL(2,𝔽3)C2×D12semidirect
316. C2×D3C2×#41⋊C3semidirect
375. C2×C3#548⋊C3semidirect
402. C3C2×#548⋊C3semidirect
405. C2#5576⋊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
C22×C3⋊(#41⋊C3)2
C2×C3×C4×SL(2,𝔽3)2
#5576⋊C32
C2×#55763
C2×#548⋊C33
C2×C3×(C2×D3)⋊C44