Finite Group Databasebeta

← C2×#5817⋊C3 #11923362

Subgroups — C2×#5817⋊C3 #11923362

Order 1152.

933 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 933 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
3#5817⋊C3C214yesmaximal, direct product factor
4C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#5817C311yesmaximal, characteristic, complemented
6C2×#548⋊C3—31nomaximal
7C22×C3×SL(2,𝔽3)—22no—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C22×C3×SL(2,𝔽3)C2212yescomplemented
10C2×C3⋊(#41⋊C3)—22no—
11C2×C3⋊(#41⋊C3)C2212yescomplemented
12C2×C3×#217—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C23×C3×Q8C2×C311yescharacteristic
15C22×#239C2×C311yescharacteristic, complemented
16C22×#41⋊C3—31no—
17C23×SL(2,𝔽3)—21no—
18C23×SL(2,𝔽3)D311yescharacteristic, complemented
19#5817—34no—
20#5817C2×C314yescomplemented
21C2×#947—31no—
22C2×#946—31no—
23C2×Q8×Dic3C2×C311yescharacteristic, complemented
24C2×#930—31no—
25C2×C4×#67—31no—
26#548⋊C3—34no—
27C2×C4×SL(2,𝔽3)—31no—
28C24×C32—41no—
29C22×C3×Dic3—41no—
30C22×C3×Dic3—41no—
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)D412yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊(#41⋊C3)—22no—
35C3×#217—44no—
36C2×#548—31noSylow
37C22×C3×Q8—22no—
38C22×C3×Q8—31no—
39C22×C3×Q8—32no—
40C22×C3×Q8—62no—
41C22×C3×Q8C22×C312yes—
42C22×C3×Q8C22×C311yescharacteristic
43C23×C3×C4—31no—
44C22×#67—31no—
45C2×#239—22no—
46C2×#239—32no—
47C2×#239—62no—
48C2×#239C22×C312yescomplemented
49C22×D12—31no—
50C22×C4×D3—31no—
51C2×#41⋊C3—32no—
52C2×#41⋊C3—62no—
53C22×SL(2,𝔽3)—24no—
54C22×SL(2,𝔽3)—22no—
55C22×SL(2,𝔽3)—21no—
56C22×SL(2,𝔽3)—22no—
57C22×SL(2,𝔽3)C2×D311yescharacteristic
58C22×SL(2,𝔽3)C2×D312yescomplemented
59C2×C3×C22⋊C4—121no—
60C2×#217A411yescharacteristic
61#947—34no—
62#947—34no—
63#946—34no—
64#946—34no—
65Q8×Dic3—32no—
66Q8×Dic3C22×C312yescomplemented
67#930—34no—
68#930—34no—
69C4×#67—34no—
70C4×#67—34no—
71C2×(C2×D3)⋊C4—32no—
72C2×(C2×D3)⋊C4—31no—
73C2×C4⋊Dic3—31no—
74C2×Dic3⋊C4—31no—
75C2×C4×Dic3—31no—
76C4×SL(2,𝔽3)—32no—
77C23×C32—44no—
78C23×C32—41no—
79C23×C32—42no—
80C23×C32—82no—
81C23×C32—82no—
82C2×C3×Dic3—42no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—82no—
85C2×C3×Dic3—82no—
86C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
87C22×#41—31no—
88C23×Q8C3×D311yescharacteristic
89#548—34no—
90#548—94no—
91C2×#124—91no—
92C2×C22⋊Q8—91no—
93C2×#121—91no—
94C2×C4×Q8—31no—
95C2×C4×D4—91no—
96C24×C3—41no—
97C24×C3—81no—
98C24×C3C2×A411yescharacteristic
99C23×D3—31no—
100C2×C3×Q8—22no—
101C2×C3×Q8—32no—
102C2×C3×Q8—31no—
103C2×C3×Q8—32no—
104C2×C3×Q8—32no—
105C2×C3×Q8—62no—
106C2×C3×Q8—62no—
107C2×C3×Q8—64no—
108C2×C3×Q8—62no—
109C2×C3×Q8—64no—
110C2×C3×Q8C3×D412yes—
111C2×C3×Q8C23×C311yescharacteristic
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—62no—
117C22×C3×C4—62no—
118C22×C3×C4—121no—
119C22×C3×C4—121no—
120C2×#67—34no—
121C2×#67—64no—
122C22×Dic3—31no—
123C22×Dic3C2×A411yescharacteristic
124C22×Dic3C2×A411yescharacteristic
125#239—22no—
126#239—64no—
127#239—64no—
128#239—62no—
129C2×D12—32no—
130C2×D12—32no—
131C2×D12—62no—
132C2×D12—62no—
133C2×C4×D3—32no—
134C2×C4×D3—32no—
135C2×C4×D3—62no—
136C2×C4×D3—62no—
137#41⋊C3—62no—
138C2×SL(2,𝔽3)—24no—
139C2×SL(2,𝔽3)—22no—
140C2×SL(2,𝔽3)—22no—
141C2×SL(2,𝔽3)—21no—
142C2×SL(2,𝔽3)#6712yescomplemented
143C2×SL(2,𝔽3)C22×D311yescharacteristic
144C3×C22⋊C4—124no—
145#217C2×A414yes—
146(C2×D3)⋊C4—38no—
147(C2×D3)⋊C4—34no—
148C4⋊Dic3—32no—
149C4⋊Dic3—32no—
150Dic3⋊C4—34no—
151C4×Dic3—32no—
152C4×Dic3—32no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32—44no—
156C22×C32—42no—
157C22×C32—41no—
158C22×C32—82no—
159C22×C32—88no—
160C22×C32—82no—
161C3×Dic3—82no—
162C3×Dic3—82no—
163C2×#41—32no—
164C2×#41—62no—
165C2×#41—92no—
166C2×#41—182no—
167C22×Q8—22no—
168C22×Q8—31no—
169C22×Q8—32no—
170C22×Q8—62no—
171C22×Q8C2×C3×D312yes—
172C22×Q8C2×C3×D311yescharacteristic
173C22×D4—91no—
174C22×D4—91no—
175C23×C4—31no—
176C23×C4—91no—
177#124—94no—
178#124—94no—
179C22⋊Q8—94no—
180C22⋊Q8—94no—
181#121—94no—
182#121—94no—
183C4×Q8—32no—
184C4×Q8—92no—
185C4×D4—94no—
186C4×D4—94no—
187C2×C4⋊C4—91no—
188C2×C4⋊C4—91no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—91no—
191C2×C22⋊C4—92no—
192C2×C42—91no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—42no—
196C23×C3—41no—
197C23×C3—44no—
198C23×C3—82no—
199C23×C3—82no—
200C23×C3—84no—
201C23×C3—84no—
202C23×C3—82no—
203C23×C3—81no—
204C23×C3—84no—
205C23×C3#41⋊C314yescomplemented
206C23×C3C22×A412yes—
207C23×C3C22×A411yescharacteristic
208C22×D3—32no—
209C22×D3—62no—
210C22×D3—62no—
211C22×D3—62no—
212C3×Q8—32no—
213C3×Q8—31no—
214C3×Q8—32no—
215C3×Q8—62no—
216C3×Q8—62no—
217C3×Q8—64no—
218C3×Q8C2×C3×D411yescharacteristic
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—32no—
224C2×C3×C4—62no—
225C2×C3×C4—64no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—122no—
229C2×C3×C4—122no—
230C2×C3×C4—242no—
231C2×C3×C4—242no—
232#67—68no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—32no—
236C2×Dic3—62no—
237C2×Dic3C22×A412yes—
238C2×Dic3C22×A412yes—
239D12—62no—
240D12—62no—
241D12—64no—
242C4×D3—62no—
243C4×D3—64no—
244C4×D3—62no—
245SL(2,𝔽3)—21no—
246SL(2,𝔽3)C2×#6711yescharacteristic, complemented
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—44no—
251C2×C32—84no—
252C24—91no—
253C24D3×A411yescharacteristic
254#41—62no—
255#41—184no—
256#41—184no—
257#41—182no—
258C2×Q8—22no—
259C2×Q8—32no—
260C2×Q8—31no—
261C2×Q8—32no—
262C2×Q8—32no—
263C2×Q8—62no—
264C2×Q8—62no—
265C2×Q8—64no—
266C2×Q8—62no—
267C2×Q8—64no—
268C2×Q8C3×#6712yes—
269C2×Q8C22×C3×D311yescharacteristic
270C2×D4—92no—
271C2×D4—92no—
272C2×D4—94no—
273C2×D4—182no—
274C2×D4—182no—
275C2×D4—184no—
276C22×C4—31no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—92no—
285C22×C4—92no—
286C22×C4—91no—
287C22×C4—182no—
288C22×C4—182no—
289C4⋊C4—94no—
290C4⋊C4—92no—
291C4⋊C4—92no—
292C22⋊C4—34no—
293C22⋊C4—94no—
294C22⋊C4—98no—
295C42—92no—
296C42—92no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—28no—
300C22×C3—42no—
301C22×C3—44no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—41no—
305C22×C3—82no—
306C22×C3—88no—
307C22×C3—82no—
308C22×C3—84no—
309C22×C3—816no—
310C22×C3—84no—
311C22×C3—82no—
312C22×C3—84no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3D4×A412yes—
317C22×C3C2×#41⋊C312yes—
318C22×C3C2×#41⋊C314yescomplemented
319C22×C32−1+4⋊C312yes—
320C22×C3C23×A411yescharacteristic
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—62no—
324C2×D3—124no—
325C3×C4—31no—
326C3×C4—32no—
327C3×C4—31no—
328C3×C4—62no—
329C3×C4—242no—
330C3×C4—242no—
331Dic3—22no—
332Dic3—22no—
333Dic3—62no—
334C32—41noSylow
335C23—22no—
336C23—22no—
337C23—92no—
338C23—182no—
339C23—182no—
340C23—182no—
341C23C3⋊(#41⋊C3)14yescomplemented
342C23C2×D3×A412yes—
343C23C2×D3×A411yescharacteristic
344Q8—32no—
345Q8—31no—
346Q8—32no—
347Q8—62no—
348Q8—62no—
349Q8—64no—
350Q8C2×C3×#6711yescharacteristic
351D4—182no—
352D4—182no—
353D4—184no—
354D4—188no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—32no—
360C2×C4—32no—
361C2×C4—32no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—64no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—92no—
369C2×C4—182no—
370C2×C4—182no—
371C2×C4—184no—
372C2×C4—182no—
373C2×C3—24no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—44no—
378C2×C3—84no—
379C2×C3—88no—
380C2×C3—81no—
381C2×C3—81no—
382C2×C3—81no—
383C2×C3—84no—
384C2×C3#548⋊C314yescomplemented
385C2×C3C2×D4×A411yescharacteristic
386C2×C3C22×#41⋊C311yescharacteristic
387C2×C3C2×2−1+4⋊C311yescharacteristic
388D3—122no—
389C22—22no—
390C22—22no—
391C22—28no—
392C22—182no—
393C22—182no—
394C22—182no—
395C22—364no—
396C22C2×C3⋊(#41⋊C3)12yes—
397C22C2×C3⋊(#41⋊C3)14yescomplemented
398C22#1007⋊C312yes—
399C22A4×#6712yes—
400C22C22×D3×A411yescharacteristic
401C4—31no—
402C4—32no—
403C4—31no—
404C4—62no—
405C4—62no—
406C4—62no—
407C4—182no—
408C3—41no—
409C3—81no—
410C3C2×#548⋊C311yescharacteristic, complemented
411C2—24no—
412C2—362no—
413C2#5817⋊C314yesdirect product factor
414C2C22×C3⋊(#41⋊C3)11yescharacteristic
415C2C2×#1007⋊C311yescharacteristic
416C2C2×A4×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC23×C3×Q8C2×C3
socleC23×C3C22×A4
hypercenterC24D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C22×C3⋊(#41⋊C3)C2semidirect
3. #5817⋊C3C2direct (N × H)
4. C2×Dic3×SL(2,𝔽3)C2semidirect
5. C2×#5817C3semidirect
9. C22×C3×SL(2,𝔽3)C22semidirect
11. C2×C3⋊(#41⋊C3)C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
15. C22×#239C2×C3semidirect
18. C23×SL(2,𝔽3)D3semidirect
20. #5817C2×C3semidirect
23. C2×Q8×Dic3C2×C3semidirect
32. C2×C3×SL(2,𝔽3)D4semidirect
48. C2×#239C22×C3semidirect
58. C22×SL(2,𝔽3)C2×D3semidirect
66. Q8×Dic3C22×C3semidirect
86. C3×SL(2,𝔽3)C2×D4semidirect
142. C2×SL(2,𝔽3)#67semidirect
205. C23×C3#41⋊C3semidirect
246. SL(2,𝔽3)C2×#67semidirect
318. C22×C3C2×#41⋊C3semidirect
341. C23C3⋊(#41⋊C3)semidirect
384. C2×C3#548⋊C3semidirect
397. C22C2×C3⋊(#41⋊C3)semidirect
410. C3C2×#548⋊C3semidirect
413. C2#5817⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C22×C3⋊(#41⋊C3)2
#5817⋊C32
C2×Dic3×SL(2,𝔽3)2
C2×#58173
C2×#548⋊C33
C2×C3×#2174