Finite Group Databasebeta

← C2×SL(2,𝔽3)×#67 #11923369

Subgroups — C2×SL(2,𝔽3)×#67 #11923369

Order 1152.

1089 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 159 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#67C214yesmaximal, direct product factor
4C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×Q8×#67C311yesmaximal, characteristic, complemented
6C2×D4×SL(2,𝔽3)—31nomaximal
7C22×C3×#67—41nomaximal
8C22×C3×SL(2,𝔽3)—22no—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yescomplemented
11C2×D3×SL(2,𝔽3)—22no—
12C2×D3×SL(2,𝔽3)C2212yescomplemented
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C23×C3×Q8C2×C311yescharacteristic, complemented
15C22×D3×Q8C2×C311yescharacteristic, complemented
16C23×SL(2,𝔽3)—21no—
17C23×SL(2,𝔽3)—31no—
18C23×SL(2,𝔽3)D311yescharacteristic, complemented
19Q8×#67—34no—
20Q8×#67C2×C314yescomplemented
21C2×#946—31no—
22C2×Q8×Dic3C2×C311yescharacteristic, complemented
23C2×#944—31no—
24C2×C4×#67—31no—
25C2×#924—31no—
26D4×SL(2,𝔽3)—34no—
27C2×C4×SL(2,𝔽3)—31no—
28C24×C32—41no—
29C23×C3×D3—41no—
30C2×C3×#67—48no—
31C2×C3×#67—44no—
32C22×C3×Dic3—41no—
33C2×C3×SL(2,𝔽3)—22no—
34C2×C3×SL(2,𝔽3)D412yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36D3×SL(2,𝔽3)—22no—
37C2×D4×Q8—31noSylow
38C22×C3×Q8—22no—
39C22×C3×Q8—31no—
40C22×C3×Q8—32no—
41C22×C3×Q8—62no—
42C22×C3×Q8C22×C311yescharacteristic
43C22×C3×Q8C22×C312yescomplemented
44C22×C3×D4—121no—
45C23×C3×C4—31no—
46C22×#67A411yescharacteristic
47C2×D3×Q8—22no—
48C2×D3×Q8—32no—
49C2×D3×Q8—62no—
50C2×D3×Q8C22×C312yescomplemented
51C22×C4×D3—31no—
52C22×Dic6—31no—
53C22×SL(2,𝔽3)—24no—
54C22×SL(2,𝔽3)—22no—
55C22×SL(2,𝔽3)—21no—
56C22×SL(2,𝔽3)—22no—
57C22×SL(2,𝔽3)—32no—
58C22×SL(2,𝔽3)—62no—
59C22×SL(2,𝔽3)C2×D311yescharacteristic
60C22×SL(2,𝔽3)C2×D312yescomplemented
61C2×#217—31no—
62#946—34no—
63#946—34no—
64Q8×Dic3—32no—
65Q8×Dic3C22×C312yescomplemented
66#944—34no—
67#944—34no—
68C4×#67—34no—
69C4×#67—34no—
70C2×(C2×D3)⋊C4—31no—
71C2×C4⋊Dic3—31no—
72#924—34no—
73#924—34no—
74C2×Dic3⋊C4—32no—
75C2×Dic3⋊C4—31no—
76C2×C4×Dic3—31no—
77C4×SL(2,𝔽3)—32no—
78C23×C32—44no—
79C23×C32—41no—
80C23×C32—42no—
81C23×C32—82no—
82C23×C32—82no—
83C22×C3×D3—44no—
84C22×C3×D3—42no—
85C22×C3×D3—82no—
86C22×C3×D3—82no—
87C3×#67—416no—
88C2×C3×Dic3—44no—
89C2×C3×Dic3—42no—
90C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
91C23×Q8—31no—
92C23×Q8C3×D311yescharacteristic, complemented
93D4×Q8—34no—
94D4×Q8—94no—
95C2×C4⋊Q8—91no—
96C2×C22⋊Q8—91no—
97C2×C22⋊Q8—91no—
98C2×C4×Q8—31no—
99C2×C4×D4—91no—
100C24×C3—41no—
101C24×C3—81no—
102C24×C3—121no—
103C24×C3C2×A411yescharacteristic
104C23×D3C2×A411yescharacteristic
105C2×C3×Q8—22no—
106C2×C3×Q8—32no—
107C2×C3×Q8—31no—
108C2×C3×Q8—32no—
109C2×C3×Q8—32no—
110C2×C3×Q8—62no—
111C2×C3×Q8—62no—
112C2×C3×Q8—64no—
113C2×C3×Q8—62no—
114C2×C3×Q8—64no—
115C2×C3×Q8C3×D412yescomplemented
116C2×C3×Q8C23×C311yescharacteristic
117C2×C3×D4—124no—
118C2×C3×D4—128no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C22×C3×C4—32no—
122C22×C3×C4—32no—
123C22×C3×C4—62no—
124C22×C3×C4—62no—
125C22×C3×C4—121no—
126C2×#67SL(2,𝔽3)18yesdirect product factor
127C2×#67C2×A414yes—
128C22×Dic3—31no—
129C22×Dic3—31no—
130C22×Dic3C2×A411yescharacteristic
131D3×Q8—22no—
132D3×Q8—64no—
133D3×Q8—64no—
134D3×Q8—62no—
135C2×C4×D3—32no—
136C2×C4×D3—32no—
137C2×C4×D3—62no—
138C2×C4×D3—62no—
139C2×Dic6—32no—
140C2×Dic6—32no—
141C2×Dic6—62no—
142C2×Dic6—62no—
143C2×SL(2,𝔽3)—24no—
144C2×SL(2,𝔽3)—22no—
145C2×SL(2,𝔽3)—22no—
146C2×SL(2,𝔽3)—21no—
147C2×SL(2,𝔽3)—62no—
148C2×SL(2,𝔽3)#6712yesdirect product factor
149C2×SL(2,𝔽3)C22×D311yescharacteristic
150#217—34no—
151(C2×D3)⋊C4—34no—
152C4⋊Dic3—32no—
153C4⋊Dic3—32no—
154Dic3⋊C4—38no—
155Dic3⋊C4—34no—
156C4×Dic3—32no—
157C4×Dic3—32no—
158C22×C32—42no—
159C22×C32—42no—
160C22×C32—44no—
161C22×C32—42no—
162C22×C32—41no—
163C22×C32—82no—
164C22×C32—88no—
165C22×C32—82no—
166C2×C3×D3—44no—
167C2×C3×D3—82no—
168C2×C3×D3—88no—
169C2×C3×D3—82no—
170C3×Dic3—44no—
171C22×Q8—22no—
172C22×Q8—31no—
173C22×Q8—32no—
174C22×Q8—32no—
175C22×Q8—62no—
176C22×Q8—62no—
177C22×Q8—91no—
178C22×Q8—92no—
179C22×Q8—182no—
180C22×Q8C2×C3×D311yescharacteristic
181C22×Q8C2×C3×D312yescomplemented
182C22×D4—31no—
183C23×C4—31no—
184C23×C4—91no—
185C4⋊Q8—94no—
186C4⋊Q8—94no—
187C22⋊Q8—94no—
188C22⋊Q8—94no—
189C22⋊Q8—94no—
190C22⋊Q8—94no—
191C4×Q8—32no—
192C4×Q8—92no—
193C4×D4—94no—
194C4×D4—94no—
195C2×C4⋊C4—91no—
196C2×C4⋊C4—91no—
197C2×C4⋊C4—92no—
198C2×C22⋊C4—91no—
199C2×C22⋊C4—91no—
200C2×C42—91no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—42no—
204C23×C3—41no—
205C23×C3—44no—
206C23×C3—82no—
207C23×C3—82no—
208C23×C3—84no—
209C23×C3—84no—
210C23×C3—82no—
211C23×C3—81no—
212C23×C3—84no—
213C23×C3—122no—
214C23×C3—124no—
215C23×C3—242no—
216C23×C3—242no—
217C23×C3C2×SL(2,𝔽3)14yescomplemented
218C23×C3C22×A412yes—
219C23×C3C22×A411yescharacteristic
220C22×D3—22no—
221C22×D3—22no—
222C22×D3C2×SL(2,𝔽3)14yescomplemented
223C22×D3C22×A412yes—
224C3×Q8—32no—
225C3×Q8—31no—
226C3×Q8—32no—
227C3×Q8—62no—
228C3×Q8—62no—
229C3×Q8—64no—
230C3×Q8C2×C3×D411yescharacteristic, complemented
231C3×D4—1216no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—32no—
237C2×C3×C4—62no—
238C2×C3×C4—64no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—122no—
242C2×C3×C4—124no—
243#67C2×SL(2,𝔽3)116yesdirect product factor
244C2×Dic3—32no—
245C2×Dic3—32no—
246C2×Dic3—62no—
247C2×Dic3—62no—
248C2×Dic3C2×SL(2,𝔽3)14yescomplemented
249C2×Dic3C22×A412yes—
250C4×D3—62no—
251C4×D3—64no—
252C4×D3—62no—
253Dic6—64no—
254Dic6—62no—
255Dic6—62no—
256SL(2,𝔽3)—21no—
257SL(2,𝔽3)C2×#6711yescharacteristic, direct product factor
258C2×C32—41no—
259C2×C32—41no—
260C2×C32—41no—
261C2×C32—44no—
262C2×C32—84no—
263C3×D3—84no—
264C24—31no—
265C24D3×A411yescharacteristic
266C2×Q8—22no—
267C2×Q8—32no—
268C2×Q8—31no—
269C2×Q8—32no—
270C2×Q8—32no—
271C2×Q8—62no—
272C2×Q8—62no—
273C2×Q8—64no—
274C2×Q8—62no—
275C2×Q8—64no—
276C2×Q8—62no—
277C2×Q8—92no—
278C2×Q8—92no—
279C2×Q8—182no—
280C2×Q8—182no—
281C2×Q8—184no—
282C2×Q8—182no—
283C2×Q8—184no—
284C2×Q8C3×#6712yescomplemented
285C2×Q8C22×C3×D311yescharacteristic
286C2×D4—34no—
287C2×D4—38no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—32no—
291C22×C4—32no—
292C22×C4—31no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—92no—
296C22×C4—92no—
297C22×C4—91no—
298C22×C4—91no—
299C22×C4—182no—
300C22×C4—182no—
301C4⋊C4—94no—
302C4⋊C4—92no—
303C4⋊C4—92no—
304C4⋊C4—98no—
305C22⋊C4—94no—
306C22⋊C4—94no—
307C42—92no—
308C42—92no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—28no—
312C22×C3—42no—
313C22×C3—44no—
314C22×C3—42no—
315C22×C3—42no—
316C22×C3—41no—
317C22×C3—82no—
318C22×C3—88no—
319C22×C3—82no—
320C22×C3—84no—
321C22×C3—816no—
322C22×C3—84no—
323C22×C3—82no—
324C22×C3—84no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—81no—
328C22×C3—124no—
329C22×C3—242no—
330C22×C3—248no—
331C22×C3—242no—
332C22×C3D4×A412yes—
333C22×C3C22×SL(2,𝔽3)12yes—
334C22×C3C22×SL(2,𝔽3)14yescomplemented
335C22×C32−1+4⋊C312yes—
336C22×C3C23×A411yescharacteristic
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—28no—
340C2×D3C22×SL(2,𝔽3)14yescomplemented
341C3×C4—32no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—62no—
345C3×C4—124no—
346Dic3—62no—
347Dic3—62no—
348Dic3C22×SL(2,𝔽3)14yescomplemented
349C32—41noSylow
350C23—22no—
351C23—22no—
352C23—34no—
353C23—32no—
354C23—62no—
355C23—62no—
356C23D3×SL(2,𝔽3)14yescomplemented
357C23C2×D3×A412yes—
358C23C2×D3×A411yescharacteristic
359Q8—32no—
360Q8—31no—
361Q8—32no—
362Q8—62no—
363Q8—62no—
364Q8—64no—
365Q8—182no—
366Q8—182no—
367Q8—184no—
368Q8C2×C3×#6711yescharacteristic, complemented
369D4—316no—
370C2×C4—34no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—64no—
380C2×C4—62no—
381C2×C4—92no—
382C2×C4—92no—
383C2×C4—182no—
384C2×C4—182no—
385C2×C4—184no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C3—24no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—44no—
393C2×C3—84no—
394C2×C3—88no—
395C2×C3—81no—
396C2×C3—81no—
397C2×C3—81no—
398C2×C3—84no—
399C2×C3—244no—
400C2×C3D4×SL(2,𝔽3)14yescomplemented
401C2×C3C2×D4×A411yescharacteristic
402C2×C3C23×SL(2,𝔽3)11yescharacteristic
403C2×C3C2×2−1+4⋊C311yescharacteristic
404D3—24no—
405C22—22no—
406C22—22no—
407C22—28no—
408C22—34no—
409C22—62no—
410C22—62no—
411C22—68no—
412C22C2×D3×SL(2,𝔽3)12yes—
413C22C2×D3×SL(2,𝔽3)14yescomplemented
414C22#1007⋊C312yes—
415C22A4×#6712yes—
416C22C22×D3×A411yescharacteristic
417C4—34no—
418C4—31no—
419C4—32no—
420C4—31no—
421C4—62no—
422C4—182no—
423C4—182no—
424C3—41no—
425C3—81no—
426C3C2×D4×SL(2,𝔽3)11yescharacteristic, complemented
427C2—24no—
428C2—64no—
429C2SL(2,𝔽3)×#6714yesdirect product factor
430C2C22×D3×SL(2,𝔽3)11yescharacteristic
431C2C2×#1007⋊C311yescharacteristic
432C2C2×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×D3×SL(2,𝔽3)C2semidirect
3. SL(2,𝔽3)×#67C2direct (N × H)
4. C2×Dic3×SL(2,𝔽3)C2semidirect
5. C2×Q8×#67C3semidirect
10. C22×C3×SL(2,𝔽3)C22semidirect
12. C2×D3×SL(2,𝔽3)C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
14. C23×C3×Q8C2×C3semidirect
15. C22×D3×Q8C2×C3semidirect
18. C23×SL(2,𝔽3)D3semidirect
20. Q8×#67C2×C3semidirect
22. C2×Q8×Dic3C2×C3semidirect
34. C2×C3×SL(2,𝔽3)D4semidirect
43. C22×C3×Q8C22×C3semidirect
50. C2×D3×Q8C22×C3semidirect
60. C22×SL(2,𝔽3)C2×D3semidirect
65. Q8×Dic3C22×C3semidirect
90. C3×SL(2,𝔽3)C2×D4semidirect
92. C23×Q8C3×D3semidirect
115. C2×C3×Q8C3×D4semidirect
126. C2×#67SL(2,𝔽3)direct (N × H)
148. C2×SL(2,𝔽3)#67direct (N × H)
181. C22×Q8C2×C3×D3semidirect
217. C23×C3C2×SL(2,𝔽3)semidirect
222. C22×D3C2×SL(2,𝔽3)semidirect
230. C3×Q8C2×C3×D4semidirect
243. #67C2×SL(2,𝔽3)direct (N × H)
248. C2×Dic3C2×SL(2,𝔽3)semidirect
257. SL(2,𝔽3)C2×#67direct (N × H)
284. C2×Q8C3×#67semidirect
334. C22×C3C22×SL(2,𝔽3)semidirect
340. C2×D3C22×SL(2,𝔽3)semidirect
348. Dic3C22×SL(2,𝔽3)semidirect
356. C23D3×SL(2,𝔽3)semidirect
368. Q8C2×C3×#67semidirect
400. C2×C3D4×SL(2,𝔽3)semidirect
413. C22C2×D3×SL(2,𝔽3)semidirect
426. C3C2×D4×SL(2,𝔽3)semidirect
429. C2SL(2,𝔽3)×#67direct (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×D3×SL(2,𝔽3)2
SL(2,𝔽3)×#672
C2×Dic3×SL(2,𝔽3)2
C2×Q8×#673
C2×D4×SL(2,𝔽3)3
C22×C3×#674