Finite Group Databasebeta

← SL(2,𝔽3)×(C2×D3)⋊C4 #11921269

Subgroups — SL(2,𝔽3)×(C2×D3)⋊C4 #11921269

Order 1152.

585 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 91 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
4Q8×(C2×D3)⋊C4C311yesmaximal, characteristic, complemented
5C22⋊C4×SL(2,𝔽3)—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×D3×SL(2,𝔽3)—21no—
8C2×D3×SL(2,𝔽3)—21no—
9C2×D3×SL(2,𝔽3)C412yescomplemented
10C2×C3×(C2×D3)⋊C4—41nomaximal
11C3×C4×SL(2,𝔽3)—21no—
12Dic3×SL(2,𝔽3)—21no—
13C22×D3×Q8C2×C311yescharacteristic
14C23×SL(2,𝔽3)—31no—
15C2×C3×C4×Q8C2×C311yescharacteristic, complemented
16C2×Q8×Dic3C2×C311yescharacteristic, complemented
17C2×C4×SL(2,𝔽3)—21no—
18C2×C4×SL(2,𝔽3)—31no—
19C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
20#4987—31no—
21#4973—31no—
22C4×(C2×D3)⋊C4—31no—
23#4929—31no—
24C23×C3×D3—41no—
25C22×C32×C4—41no—
26C22×C3×Dic3—41no—
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28C2×C3×SL(2,𝔽3)D411yescharacteristic
29C2×C3×SL(2,𝔽3)D411yescharacteristic
30D3×SL(2,𝔽3)—22no—
31C3×(C2×D3)⋊C4—44no—
32Q8×C22⋊C4—31noSylow
33C22×C3×Q8C22×C311yescharacteristic
34C2×D3×Q8—21no—
35C2×D3×Q8—21no—
36C2×D3×Q8—32no—
37C2×D3×Q8—61no—
38C2×D3×Q8—61no—
39C2×D3×Q8C3×C412yescomplemented
40C22×C4×D3—31no—
41C22×Dic6—31no—
42C22×SL(2,𝔽3)—21no—
43C22×SL(2,𝔽3)—32no—
44C22×SL(2,𝔽3)—61no—
45C22×SL(2,𝔽3)—61no—
46C22×SL(2,𝔽3)C2×D311yescharacteristic
47C3×C4×Q8—21no—
48C3×C4×Q8—61no—
49C2×C3×C4⋊C4—31no—
50C2×C3×C22⋊C4—121no—
51C2×C3×C42—31no—
52Q8×Dic3—21no—
53Q8×Dic3—61no—
54C2×(C2×D3)⋊C4—31no—
55C2×(C2×D3)⋊C4A411yescharacteristic
56C2×C4⋊Dic3—31no—
57C2×C4×Dic3—31no—
58C4×SL(2,𝔽3)—22no—
59C4×SL(2,𝔽3)—21no—
60C4×SL(2,𝔽3)—61no—
61C3⋊#95—32no—
62C3⋊#95—31no—
63C3⋊#95—31no—
64C23×C32—41no—
65C22×C3×D3—42no—
66C22×C3×D3—42no—
67C22×C3×D3—42no—
68C22×C3×D3—81no—
69C22×C3×D3—81no—
70C22×C3×D3—81no—
71C22×C3×D3—81no—
72C2×C32×C4—42no—
73C2×C32×C4—81no—
74C2×C32×C4—81no—
75C2×C3×Dic3—42no—
76C2×C3×Dic3—81no—
77C2×C3×Dic3—81no—
78C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
79C23×Q8—31no—
80C2×C4×Q8—31no—
81C2×C4×Q8C3×D311yescharacteristic, complemented
82#391—91no—
83#391—91no—
84#380—91no—
85C4×C22⋊C4—91no—
86C24×C3—121no—
87C23×D3C2×A411yescharacteristic
88C2×C3×Q8—31no—
89C2×C3×Q8—31no—
90C2×C3×Q8—31no—
91C2×C3×Q8C2×C3×C411yescharacteristic
92C2×C3×Q8C3×D411yescharacteristic
93C2×C3×Q8C3×D411yescharacteristic
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—41no—
97C22×C3×C4—81no—
98C22×C3×C4—121no—
99C22×C3×C4C2×A411yescharacteristic
100C22×Dic3—31no—
101C22×Dic3—31no—
102C22×Dic3C2×A411yescharacteristic
103D3×Q8—22no—
104D3×Q8—64no—
105D3×Q8—62no—
106D3×Q8—62no—
107D3×Q8—62no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—61no—
111C2×C4×D3—61no—
112C2×C4×D3—61no—
113C2×C4×D3—61no—
114C2×Dic6—32no—
115C2×Dic6—32no—
116C2×Dic6—61no—
117C2×Dic6—61no—
118C2×Dic6—61no—
119C2×Dic6—61no—
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)—21no—
123C2×SL(2,𝔽3)—62no—
124C2×SL(2,𝔽3)C4×D311yescharacteristic
125C2×SL(2,𝔽3)D1211yescharacteristic
126C2×SL(2,𝔽3)#6711yescharacteristic
127C3×C4⋊C4—61no—
128C3×C4⋊C4—61no—
129C3×C22⋊C4—124no—
130C3×C42—61no—
131C3×C42—61no—
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4SL(2,𝔽3)14yesdirect product factor
134C4⋊Dic3—61no—
135C4⋊Dic3—61no—
136C4×Dic3—61no—
137C4×Dic3—61no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—41no—
141C22×C32—41no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32—41no—
145C2×C3×D3—44no—
146C2×C3×D3—82no—
147C2×C3×D3—82no—
148C2×C3×D3—82no—
149C2×C3×D3—82no—
150C2×C3×D3—82no—
151C2×C3×D3—82no—
152C32×C4—82no—
153C3×Dic3—82no—
154C22×Q8—32no—
155C22×Q8—61no—
156C22×Q8—61no—
157C22×Q8—91no—
158C22×Q8—92no—
159C22×Q8—181no—
160C22×Q8—181no—
161C22×Q8C2×C3×D311yescharacteristic
162C23×C4—91no—
163C4×Q8—21no—
164C4×Q8—61no—
165C4×Q8—61no—
166C4×Q8—181no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—91no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—91no—
171C2×C42—31no—
172C2×C42—91no—
173#95—91no—
174#95—92no—
175#95—91no—
176C23×C3—41no—
177C23×C3—81no—
178C23×C3—122no—
179C23×C3—122no—
180C23×C3—122no—
181C23×C3—241no—
182C23×C3—241no—
183C23×C3—241no—
184C23×C3—241no—
185C23×C3C22×A411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3C4×A412yes—
191C22×D3C2×SL(2,𝔽3)12yes—
192C22×D3#41⋊C312yes—
193C3×Q8—31no—
194C3×Q8—31no—
195C3×Q8—31no—
196C3×Q8—32no—
197C3×Q8C3×C22⋊C411yescharacteristic, complemented
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—42no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4—82no—
215C2×C3×C4—122no—
216C2×C3×C4—241no—
217C2×C3×C4—241no—
218C2×C3×C4C2×SL(2,𝔽3)12yescomplemented
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—32no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3C2×SL(2,𝔽3)12yescomplemented
228C4×D3—62no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—62no—
232Dic6—62no—
233Dic6—62no—
234Dic6—62no—
235Dic6—62no—
236SL(2,𝔽3)—21no—
237SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic, direct product factor
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C3×D3—84no—
246C24—31no—
247C2×Q8—31no—
248C2×Q8—31no—
249C2×Q8—31no—
250C2×Q8—62no—
251C2×Q8—92no—
252C2×Q8—92no—
253C2×Q8—181no—
254C2×Q8—181no—
255C2×Q8—181no—
256C2×Q8—181no—
257C2×Q8—184no—
258C2×Q8—182no—
259C2×Q8—182no—
260C2×Q8—182no—
261C2×Q8C3×C4×D311yescharacteristic
262C2×Q8C3×D1211yescharacteristic
263C2×Q8C3×#6711yescharacteristic
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—92no—
268C22×C4—92no—
269C22×C4—91no—
270C22×C4—91no—
271C22×C4—181no—
272C22×C4—181no—
273C22×C4—181no—
274C22×C4—181no—
275C22×C4D3×A411yescharacteristic
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—181no—
279C4⋊C4—181no—
280C22⋊C4—34no—
281C22⋊C4—182no—
282C42—61no—
283C42—61no—
284C42—181no—
285C42—181no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—124no—
301C22×C3—242no—
302C22×C3—242no—
303C22×C3—242no—
304C22×C3—242no—
305C22×C3—242no—
306C22×C3—242no—
307C22×C3C2×C4×A411yescharacteristic
308C22×C3D4×A411yescharacteristic
309C22×C3D4×A411yescharacteristic
310C22×C3C22×SL(2,𝔽3)11yescharacteristic
311C22×C3C2×#41⋊C311yescharacteristic
312C22×C32−1+4⋊C311yescharacteristic
313C22×C32−1+4⋊C311yescharacteristic
314C2×D3—22no—
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3C4×SL(2,𝔽3)14yescomplemented
321C3×C4—22no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—82no—
327C3×C4—84no—
328C3×C4—121no—
329C3×C4—242no—
330Dic3—22no—
331Dic3—62no—
332Dic3—121no—
333C32—41noSylow
334C23—32no—
335C23—32no—
336C23—32no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23C2×D3×A411yescharacteristic
342Q8—31no—
343Q8—31no—
344Q8—31no—
345Q8—32no—
346Q8—182no—
347Q8—182no—
348Q8—182no—
349Q8—182no—
350Q8C3×(C2×D3)⋊C411yescharacteristic, complemented
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—32no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—92no—
366C2×C4—182no—
367C2×C4—182no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4D3×SL(2,𝔽3)12yescomplemented
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—81no—
384C2×C3—81no—
385C2×C3—81no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—81no—
389C2×C3—81no—
390C2×C3—244no—
391C2×C3A4×C22⋊C411yescharacteristic
392C2×C3C2×C4×SL(2,𝔽3)11yescharacteristic
393C2×C3#519⋊C311yescharacteristic
394C2×C3#548⋊C311yescharacteristic
395C2×C3#548⋊C311yescharacteristic
396C2×C3D4×SL(2,𝔽3)11yescharacteristic
397C2×C3D4×SL(2,𝔽3)11yescharacteristic
398D3—24no—
399C22—34no—
400C22—62no—
401C22—62no—
402C22—62no—
403C22—62no—
404C22—62no—
405C22—62no—
406C22C4×D3×A411yescharacteristic
407C22A4×D1211yescharacteristic
408C22C2×D3×SL(2,𝔽3)11yescharacteristic
409C22#1007⋊C311yescharacteristic
410C22D3×#41⋊C311yescharacteristic
411C22#1010⋊C311yescharacteristic
412C22A4×#6711yescharacteristic
413C4—22no—
414C4—31no—
415C4—31no—
416C4—31no—
417C4—31no—
418C4—62no—
419C4—121no—
420C4—182no—
421C4—361no—
422C3—41no—
423C3—81no—
424C3C22⋊C4×SL(2,𝔽3)11yescharacteristic, complemented
425C2—64no—
426C2C4×D3×SL(2,𝔽3)11yescharacteristic
427C2SL(2,𝔽3)×D1211yescharacteristic
428C2#5572⋊C311yescharacteristic
429C2#5576⋊C311yescharacteristic
430C2A4×(C2×D3)⋊C411yescharacteristic
431C2SL(2,𝔽3)×#6711yescharacteristic
432C2#5831⋊C311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×D3×SL(2,𝔽3)C2
cosocleC22×C3×Q8C22×C3
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ 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. Q8×(C2×D3)⋊C4C3semidirect
9. C2×D3×SL(2,𝔽3)C4semidirect
15. C2×C3×C4×Q8C2×C3semidirect
16. C2×Q8×Dic3C2×C3semidirect
19. C2×C4×SL(2,𝔽3)D3semidirect
39. C2×D3×Q8C3×C4semidirect
78. C3×SL(2,𝔽3)C22⋊C4semidirect
81. C2×C4×Q8C3×D3semidirect
133. (C2×D3)⋊C4SL(2,𝔽3)direct (N × H)
197. C3×Q8C3×C22⋊C4semidirect
218. C2×C3×C4C2×SL(2,𝔽3)semidirect
227. C2×Dic3C2×SL(2,𝔽3)semidirect
237. SL(2,𝔽3)(C2×D3)⋊C4direct (N × H)
320. C2×D3C4×SL(2,𝔽3)semidirect
350. Q8C3×(C2×D3)⋊C4semidirect
375. C2×C4D3×SL(2,𝔽3)semidirect
424. C3C22⋊C4×SL(2,𝔽3)semidirect

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
Q8×(C2×D3)⋊C43
C22⋊C4×SL(2,𝔽3)3
C2×C3×(C2×D3)⋊C44