Finite Group Databasebeta

← D3×#519⋊C3 #11923212

Subgroups — D3×#519⋊C3 #11923212

Order 1152.

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

normal subgroups only

This group has 831 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
1C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#519⋊C3C211yesmaximal, characteristic, complemented
3#5812⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C212yesmaximal, complemented
5C4×D3×SL(2,𝔽3)C212yesmaximal, complemented
6D3×#519C311yesmaximal, characteristic, complemented
7C2×#519⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×D3×SL(2,𝔽3)C414yescomplemented
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C3×D3×C22⋊C4—41nomaximal
13C3×C4×SL(2,𝔽3)C2212yescomplemented
14Dic3×SL(2,𝔽3)C2212yescomplemented
15C22×D3×Q8C2×C311yescharacteristic
16C23×SL(2,𝔽3)—31no—
17C3×#519C2×C311yescharacteristic, complemented
18#5812C2×C311yescharacteristic, complemented
19#5572—32no—
20#5572C2×C312yescomplemented
21C4×D3×Q8—32no—
22C4×D3×Q8C2×C312yescomplemented
23D3×#117—31no—
24D3×#117—31no—
25#5516—31no—
26#5516—31no—
27#519⋊C3—21no—
28#519⋊C3—31no—
29#519⋊C3—32no—
30#519⋊C3D311yescharacteristic, direct product factor
31C2×C4×SL(2,𝔽3)—32no—
32C23×C3×D3—41no—
33C2×C3×C4×D3—42no—
34C2×C3×SL(2,𝔽3)C2×C412yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36D3×SL(2,𝔽3)C2×C412yes—
37D3×SL(2,𝔽3)C2×C412yescomplemented
38C32×C22⋊C4—41no—
39C3×#217—41no—
40C3×(C2×D3)⋊C4—42no—
41C2×#519—31noSylow
42C22×C3×Q8C22×C311yescharacteristic
43C2×D3×Q8—34no—
44C2×D3×Q8—31no—
45C2×D3×Q8—31no—
46C2×D3×Q8C3×C414yescomplemented
47C2×D3×Q8C22×C311yescharacteristic
48C2×D3×Q8C22×C311yescharacteristic
49C22×C4×D3—31no—
50C22×Dic6—31no—
51C22×SL(2,𝔽3)—21no—
52C22×SL(2,𝔽3)—34no—
53C22×SL(2,𝔽3)—31no—
54C22×SL(2,𝔽3)—31no—
55C22×SL(2,𝔽3)C2×D311yescharacteristic
56C3×C4×Q8—32no—
57C3×C4×Q8C22×C312yescomplemented
58C3×#117—31no—
59C3×#117—31no—
60C2×C3×C22⋊C4—121no—
61Q8×Dic3—32no—
62Q8×Dic3C22×C312yescomplemented
63#926—31no—
64#926—31no—
65#892—32no—
66#892—32no—
67D3×C4⋊C4—32no—
68D3×C4⋊C4—32no—
69#887—34no—
70#887—32no—
71#887—32no—
72D3×C22⋊C4—31no—
73D3×C22⋊C4A411yescharacteristic
74#877—32no—
75#877—31no—
76#877—31no—
77#872—32no—
78#872—32no—
79C42×D3—32no—
80C42×D3—32no—
81C4×Dic6—34no—
82C4×Dic6—32no—
83C4×Dic6—32no—
84C4×SL(2,𝔽3)—22no—
85C4×SL(2,𝔽3)—32no—
86C4×SL(2,𝔽3)C2×D312yescomplemented
87C23×C32—41no—
88C22×C3×D3—44no—
89C22×C3×D3—41no—
90C22×C3×D3—41no—
91C22×C3×D3—84no—
92C2×C32×C4—42no—
93C2×C3×Dic3—42no—
94C3×C4×D3—84no—
95C3×SL(2,𝔽3)C22×C411yescharacteristic
96C23×Q8—31no—
97#519—31no—
98#519—32no—
99#519—91no—
100#519—91no—
101#519—92no—
102#519C3×D311yescharacteristic, complemented
103C2×C4×Q8—32no—
104C2×C4×Q8—92no—
105C2×#117—91no—
106C2×#117—91no—
107C24×C3—121no—
108C23×D3C2×A411yescharacteristic
109C2×C3×Q8—31no—
110C2×C3×Q8—32no—
111C2×C3×Q8C2×C3×C412yescomplemented
112C2×C3×Q8C23×C311yescharacteristic
113C22×C3×C4—31no—
114C22×C3×C4—122no—
115C22×Dic3—31no—
116D3×Q8—34no—
117D3×Q8—34no—
118D3×Q8—34no—
119D3×Q8—34no—
120D3×Q8—32no—
121D3×Q8—32no—
122D3×Q8C2×C3×C412yes—
123D3×Q8C2×C3×C412yescomplemented
124C2×C4×D3—34no—
125C2×C4×D3—34no—
126C2×C4×D3—31no—
127C2×C4×D3—31no—
128C2×C4×D3—31no—
129C2×C4×D3—31no—
130C2×C4×D3—32no—
131C2×C4×D3C2×A412yes—
132C2×Dic6—32no—
133C2×Dic6—34no—
134C2×Dic6—31no—
135C2×Dic6—31no—
136C2×Dic6—34no—
137C2×SL(2,𝔽3)—22no—
138C2×SL(2,𝔽3)—21no—
139C2×SL(2,𝔽3)—32no—
140C2×SL(2,𝔽3)—32no—
141C2×SL(2,𝔽3)C4×D312yescomplemented
142C2×SL(2,𝔽3)C22×D311yescharacteristic
143C3×C4⋊C4—32no—
144C3×C4⋊C4—32no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—81no—
148C3×C22⋊C4—122no—
149C3×C22⋊C4—121no—
150C3×C22⋊C4C2×A411yescharacteristic
151C3×C42—32no—
152C3×C42—32no—
153#217—31no—
154#217C2×A411yescharacteristic
155(C2×D3)⋊C4—32no—
156(C2×D3)⋊C4C2×A412yes—
157C4⋊Dic3—32no—
158C4⋊Dic3—32no—
159Dic3⋊C4—34no—
160Dic3⋊C4—32no—
161Dic3⋊C4—32no—
162C4×Dic3—34no—
163C4×Dic3—32no—
164C4×Dic3—32no—
165C4×Dic3—32no—
166C4×Dic3—32no—
167C22×C32—42no—
168C22×C32—41no—
169C22×C32—82no—
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C2×C3×D3—44no—
173C2×C3×D3—88no—
174C2×C3×D3—82no—
175C32×C4—82no—
176C3×Dic3—82no—
177C22×Q8—34no—
178C22×Q8—31no—
179C22×Q8—31no—
180C22×Q8—91no—
181C22×Q8—91no—
182C22×Q8—91no—
183C22×Q8—94no—
184C22×Q8C2×C3×D311yescharacteristic
185C23×C4—91no—
186C4×Q8—32no—
187C4×Q8—32no—
188C4×Q8—94no—
189C4×Q8—92no—
190C4×Q8—92no—
191C4×Q8—94no—
192C4×Q8—92no—
193C4×Q8—92no—
194C4×Q8—92no—
195C4×Q8C2×C3×D312yescomplemented
196#117—31no—
197#117—31no—
198#117—92no—
199#117—92no—
200#117—91no—
201#117—92no—
202#117—91no—
203#117—92no—
204#117—91no—
205#117—92no—
206#117—91no—
207C2×C4⋊C4—92no—
208C2×C4⋊C4—92no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—91no—
211C2×C42—92no—
212C2×C42—92no—
213C23×C3—41no—
214C23×C3—81no—
215C23×C3—121no—
216C23×C3—121no—
217C23×C3—124no—
218C23×C3—244no—
219C23×C3C22×A411yescharacteristic
220C22×D3—24no—
221C22×D3C4×A414yes—
222C22×D3C22×A411yescharacteristic
223C22×D3C22×A411yescharacteristic
224C3×Q8—32no—
225C3×Q8—31no—
226C3×Q8—32no—
227C3×Q8C22×C3×C411yescharacteristic
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—32no—
233C2×C3×C4—42no—
234C2×C3×C4—82no—
235C2×C3×C4—122no—
236C2×C3×C4—244no—
237C2×C3×C4C22×A412yes—
238C2×Dic3—34no—
239C2×Dic3—31no—
240C2×Dic3—31no—
241C2×Dic3—32no—
242C2×Dic3C22×A412yes—
243C4×D3—24no—
244C4×D3—32no—
245C4×D3—34no—
246C4×D3—32no—
247C4×D3—34no—
248C4×D3—32no—
249C4×D3—32no—
250C4×D3—64no—
251Dic6—38no—
252Dic6—32no—
253Dic6—32no—
254Dic6—32no—
255Dic6—32no—
256SL(2,𝔽3)—21no—
257SL(2,𝔽3)C2×C4×D311yescharacteristic
258C2×C32—41no—
259C2×C32—42no—
260C2×C32—82no—
261C3×D3—44no—
262C3×D3—82no—
263C24—31no—
264C2×Q8—31no—
265C2×Q8—32no—
266C2×Q8—32no—
267C2×Q8—32no—
268C2×Q8—92no—
269C2×Q8—94no—
270C2×Q8—91no—
271C2×Q8—91no—
272C2×Q8—94no—
273C2×Q8—94no—
274C2×Q8—94no—
275C2×Q8—94no—
276C2×Q8—94no—
277C2×Q8—92no—
278C2×Q8—92no—
279C2×Q8C3×C4×D312yescomplemented
280C2×Q8C22×C3×D311yescharacteristic
281C22×C4—31no—
282C22×C4—32no—
283C22×C4—91no—
284C22×C4—91no—
285C22×C4—91no—
286C22×C4—91no—
287C22×C4—94no—
288C22×C4—94no—
289C22×C4—92no—
290C22×C4—91no—
291C4⋊C4—32no—
292C4⋊C4—32no—
293C4⋊C4—92no—
294C4⋊C4—92no—
295C4⋊C4—92no—
296C4⋊C4—94no—
297C4⋊C4—92no—
298C22⋊C4—31no—
299C22⋊C4—31no—
300C22⋊C4—32no—
301C22⋊C4—92no—
302C22⋊C4—91no—
303C22⋊C4D3×A411yescharacteristic
304C42—32no—
305C42—32no—
306C42—92no—
307C42—92no—
308C42—92no—
309C42—94no—
310C42—92no—
311C22×C3—22no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—81no—
317C22×C3—124no—
318C22×C3—122no—
319C22×C3—122no—
320C22×C3—162no—
321C22×C3—242no—
322C22×C3—248no—
323C22×C3C2×C4×A412yes—
324C22×C3C23×A411yescharacteristic
325C2×D3—22no—
326C2×D3—28no—
327C2×D3C2×C4×A412yes—
328C2×D3C2×C4×A412yes—
329C2×D32−1+4⋊C314yes—
330C3×C4—22no—
331C3×C4—31no—
332C3×C4—32no—
333C3×C4—31no—
334C3×C4—62no—
335C3×C4—82no—
336C3×C4—162no—
337C3×C4—242no—
338Dic3—22no—
339Dic3—32no—
340Dic3—32no—
341Dic3—62no—
342C32—41noSylow
343C23—31no—
344C23—34no—
345C23—31no—
346C23—64no—
347C23C2×D3×A411yescharacteristic
348Q8—32no—
349Q8—31no—
350Q8—32no—
351Q8—92no—
352Q8—92no—
353Q8—98no—
354Q8—92no—
355Q8—92no—
356Q8C2×C3×C4×D311yescharacteristic
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—32no—
362C2×C4—32no—
363C2×C4—64no—
364C2×C4—92no—
365C2×C4—92no—
366C2×C4—92no—
367C2×C4—94no—
368C2×C4—92no—
369C2×C4—94no—
370C2×C4—94no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—92no—
374C2×C4—184no—
375C2×C4C2×D3×A412yes—
376C2×C3—22no—
377C2×C3—41no—
378C2×C3—42no—
379C2×C3—82no—
380C2×C3—81no—
381C2×C3—82no—
382C2×C3—124no—
383C2×C3—162no—
384C2×C3—242no—
385C2×C3C22×C4×A411yescharacteristic
386C2×C3C2×2−1+4⋊C312yes—
387D3—22no—
388D3#519⋊C314yesdirect product factor
389C22—22no—
390C22—32no—
391C22—34no—
392C22—32no—
393C22—68no—
394C22—62no—
395C22C4×D3×A412yes—
396C22C22×D3×A411yescharacteristic
397C4—22no—
398C4—31no—
399C4—32no—
400C4—31no—
401C4—62no—
402C4—62no—
403C4—92no—
404C4—92no—
405C4—182no—
406C3—41no—
407C3—81no—
408C3C2×#519⋊C311yescharacteristic, complemented
409C2—22no—
410C2—34no—
411C2—62no—
412C2C2×C4×D3×A411yescharacteristic
413C2D3×2−1+4⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FittingC3×#519C2×C3
socleC22×C3C23×A4
prime radicalC22×D3×SL(2,𝔽3)C2
cosocleC2×C3×Q8C23×C3
hypercenterC22⋊C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#519⋊C3C2semidirect
3. #5812⋊C3C2semidirect
4. #5572⋊C3C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. D3×#519C3semidirect
9. C2×D3×SL(2,𝔽3)C4semidirect
13. C3×C4×SL(2,𝔽3)C22semidirect
14. Dic3×SL(2,𝔽3)C22semidirect
17. C3×#519C2×C3semidirect
18. #5812C2×C3semidirect
20. #5572C2×C3semidirect
22. C4×D3×Q8C2×C3semidirect
30. #519⋊C3D3direct (N × H)
34. C2×C3×SL(2,𝔽3)C2×C4semidirect
37. D3×SL(2,𝔽3)C2×C4semidirect
46. C2×D3×Q8C3×C4semidirect
57. C3×C4×Q8C22×C3semidirect
62. Q8×Dic3C22×C3semidirect
86. C4×SL(2,𝔽3)C2×D3semidirect
102. #519C3×D3semidirect
111. C2×C3×Q8C2×C3×C4semidirect
123. D3×Q8C2×C3×C4semidirect
141. C2×SL(2,𝔽3)C4×D3semidirect
195. C4×Q8C2×C3×D3semidirect
279. C2×Q8C3×C4×D3semidirect
388. D3#519⋊C3direct (N × H)
408. C3C2×#519⋊C3semidirect

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
C3×#519⋊C32
#5812⋊C32
#5572⋊C32
C4×D3×SL(2,𝔽3)2
D3×#5193
C2×#519⋊C33
C3×D3×C22⋊C44