Finite Group Databasebeta

← D3×#557⋊C3 #11923281

Subgroups — D3×#557⋊C3 #11923281

Order 1152.

585 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#557⋊C3C211yesmaximal, characteristic, complemented
2C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic
3C4×D3×SL(2,𝔽3)C212yesmaximal
4#5570⋊C3C211yesmaximal, characteristic, complemented
5#5569⋊C3C212yesmaximal, complemented
6D3×#557C311yesmaximal, characteristic, complemented
7C2×#557⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C3×D3×C4⋊C4—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×#557C2×C311yescharacteristic, complemented
15D3×C4⋊Q8—31no—
16#5721—31no—
17D3×#125—32no—
18#5685—32no—
19C4×D3×Q8—31no—
20C4×D3×Q8C2×C312yes—
21C4×D3×Q8C2×C311yescharacteristic
22#5570C2×C311yescharacteristic, complemented
23#5569C2×C312yescomplemented
24#5567—31no—
25#557⋊C3—21no—
26#557⋊C3—32no—
27#557⋊C3—31no—
28#557⋊C3D311yescharacteristic, direct product factor
29C2×C4×SL(2,𝔽3)—31no—
30C2×C4×SL(2,𝔽3)—32no—
31C2×C3×C4×D3—41no—
32C2×C3×C4×D3—42no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)Q812yescomplemented
35C32×C4⋊C4—41no—
36C3×C4⋊Dic3—41no—
37C3×Dic3⋊C4—42no—
38C2×#557—31noSylow
39C2×D3×Q8—31no—
40C2×D3×Q8C22×C311yescharacteristic
41C22×SL(2,𝔽3)—31no—
42C3×C4⋊Q8—31no—
43C3×#125—32no—
44C3×C4×Q8—31no—
45C3×C4×Q8C22×C312yes—
46C3×C4×Q8C22×C311yescharacteristic
47C2×C3×C4⋊C4—121no—
48Q8×Dic3—31no—
49Q8×Dic3C22×C311yescharacteristic
50Q8×Dic3C22×C312yes—
51#944—32no—
52D3×C4⋊C4—31no—
53D3×C4⋊C4—32no—
54D3×C4⋊C4—32no—
55D3×C4⋊C4—32no—
56D3×C4⋊C4A411yescharacteristic
57#890—32no—
58#890—32no—
59#889—34no—
60#889—34no—
61#888—32no—
62#888—32no—
63#887—31no—
64#887—32no—
65#887—31no—
66#887—32no—
67C42×D3—32no—
68C42×D3—31no—
69#870—32no—
70#869—31no—
71C4×Dic6—31no—
72C4×Dic6—32no—
73C4×Dic6—32no—
74C4×Dic6—31no—
75C4×SL(2,𝔽3)—21no—
76C4×SL(2,𝔽3)—22no—
77C4×SL(2,𝔽3)—31no—
78C4×SL(2,𝔽3)—32no—
79C4×SL(2,𝔽3)C2×D311yescharacteristic
80C4×SL(2,𝔽3)C2×D312yes—
81C22×C3×D3—41no—
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C3×C4×D3—44no—
87C3×C4×D3—84no—
88C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
89#557—32no—
90#557—31no—
91#557—91no—
92#557—92no—
93#557—91no—
94#557C3×D311yescharacteristic, complemented
95C2×C4⋊Q8—91no—
96C2×#125—92no—
97C2×C4×Q8—32no—
98C2×C4×Q8—31no—
99C2×C4×Q8—91no—
100C2×C3×Q8—31no—
101C2×C3×Q8C23×C311yescharacteristic
102C22×C3×C4—121no—
103C22×C3×C4—122no—
104D3×Q8—32no—
105D3×Q8—64no—
106D3×Q8C3×Q812yes—
107C2×C4×D3—31no—
108C2×C4×D3—31no—
109C2×C4×D3—32no—
110C2×C4×D3C2×A412yes—
111C2×C4×D3C2×A411yescharacteristic
112C2×Dic6—31no—
113C2×Dic6—31no—
114C2×Dic6—32no—
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)—32no—
117C2×SL(2,𝔽3)C22×D311yescharacteristic
118C3×C4⋊C4—32no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—81no—
124C3×C4⋊C4—122no—
125C3×C4⋊C4—121no—
126C3×C4⋊C4C2×A411yescharacteristic
127C3×C42—31no—
128C3×C42—32no—
129C4⋊Dic3—32no—
130C4⋊Dic3—32no—
131C4⋊Dic3—32no—
132C4⋊Dic3—31no—
133C4⋊Dic3C2×A411yescharacteristic
134Dic3⋊C4—34no—
135Dic3⋊C4—32no—
136Dic3⋊C4—34no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139Dic3⋊C4C2×A412yes—
140C4×Dic3—32no—
141C4×Dic3—31no—
142C4×Dic3—31no—
143C4×Dic3—32no—
144C4×Dic3—31no—
145C4×Dic3—32no—
146C22×C32—41no—
147C2×C3×D3—42no—
148C2×C3×D3—42no—
149C2×C3×D3—42no—
150C32×C4—42no—
151C32×C4—82no—
152C3×Dic3—42no—
153C3×Dic3—82no—
154C22×Q8—31no—
155C22×Q8—91no—
156C4⋊Q8—31no—
157C4⋊Q8—91no—
158C4⋊Q8—92no—
159C4⋊Q8—92no—
160C4⋊Q8—92no—
161#125—32no—
162#125—92no—
163#125—92no—
164#125—92no—
165#125—94no—
166#125—94no—
167C4×Q8—31no—
168C4×Q8—32no—
169C4×Q8—31no—
170C4×Q8—92no—
171C4×Q8—91no—
172C4×Q8—92no—
173C4×Q8—91no—
174C4×Q8—92no—
175C4×Q8—91no—
176C4×Q8—91no—
177C4×Q8—92no—
178C4×Q8—91no—
179C4×Q8C2×C3×D312yes—
180C4×Q8C2×C3×D311yescharacteristic
181C2×C4⋊C4—31no—
182C2×C4⋊C4—92no—
183C2×C4⋊C4—91no—
184C2×C4⋊C4—92no—
185C2×C4⋊C4—92no—
186C2×C42—92no—
187C2×C42—91no—
188C23×C3—121no—
189C22×D3C22×A411yescharacteristic
190C3×Q8—31no—
191C3×Q8—62no—
192C3×Q8C2×C3×Q811yescharacteristic
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—82no—
200C2×C3×C4—124no—
201C2×C3×C4—121no—
202C2×C3×C4—122no—
203C2×C3×C4—244no—
204C2×C3×C4C22×A412yes—
205C2×C3×C4C22×A411yescharacteristic
206C2×Dic3—31no—
207C2×Dic3—31no—
208C2×Dic3—32no—
209C2×Dic3C22×A412yes—
210C2×Dic3C22×A411yescharacteristic
211C4×D3—24no—
212C4×D3—32no—
213C4×D3—32no—
214C4×D3—64no—
215C4×D3—62no—
216C4×D3#41⋊C314yes—
217Dic6—32no—
218Dic6—32no—
219Dic6—62no—
220Dic6—64no—
221SL(2,𝔽3)—21no—
222SL(2,𝔽3)D3×Q811yescharacteristic, complemented
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C3×D3—44no—
227C2×Q8—31no—
228C2×Q8—32no—
229C2×Q8—92no—
230C2×Q8—91no—
231C2×Q8—91no—
232C2×Q8—92no—
233C2×Q8—184no—
234C2×Q8C22×C3×D311yescharacteristic
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—91no—
238C22×C4—91no—
239C22×C4—92no—
240C4⋊C4—31no—
241C4⋊C4—32no—
242C4⋊C4—32no—
243C4⋊C4—32no—
244C4⋊C4—31no—
245C4⋊C4—32no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—91no—
249C4⋊C4—92no—
250C4⋊C4—92no—
251C4⋊C4—94no—
252C4⋊C4—94no—
253C4⋊C4—92no—
254C4⋊C4—92no—
255C4⋊C4D3×A411yescharacteristic
256C42—32no—
257C42—31no—
258C42—92no—
259C42—91no—
260C42—92no—
261C42—91no—
262C42—92no—
263C42—91no—
264C22×C3—41no—
265C22×C3—81no—
266C22×C3—122no—
267C22×C3—122no—
268C22×C3—122no—
269C22×C3C23×A411yescharacteristic
270C2×D3Q8×A412yes—
271C2×D3C2×#41⋊C312yes—
272C2×D32−1+4⋊C312yes—
273C3×C4—22no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—42no—
277C3×C4—62no—
278C3×C4—61no—
279C3×C4—82no—
280C3×C4—82no—
281C3×C4—122no—
282C3×C4—162no—
283C3×C4—242no—
284C3×C4C2×#41⋊C312yes—
285Dic3—22no—
286Dic3—32no—
287Dic3—62no—
288Dic3—61no—
289Dic3C2×#41⋊C312yes—
290C32—41noSylow
291C23—31no—
292Q8—31no—
293Q8—62no—
294Q8—92no—
295Q8—92no—
296Q8—184no—
297Q8—182no—
298Q8C3×D3×Q811yescharacteristic
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—34no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—64no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—91no—
309C2×C4—91no—
310C2×C4—92no—
311C2×C4—184no—
312C2×C4—182no—
313C2×C4C2×D3×A412yes—
314C2×C4C2×D3×A411yescharacteristic
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3—81no—
321C2×C3—124no—
322C2×C3C2×Q8×A411yescharacteristic
323C2×C3C22×#41⋊C311yescharacteristic
324C2×C3C2×2−1+4⋊C311yescharacteristic
325D3#557⋊C314yesdirect product factor
326C22—32no—
327C22—32no—
328C22—32no—
329C22C22×D3×A411yescharacteristic
330C4—22no—
331C4—31no—
332C4—31no—
333C4—32no—
334C4—62no—
335C4—62no—
336C4—61no—
337C4—92no—
338C4—182no—
339C4—181no—
340C4D3×#41⋊C312yes—
341C3—41no—
342C3—81no—
343C3C2×#557⋊C311yescharacteristic, complemented
344C2—34no—
345C2D3×Q8×A411yescharacteristic
346C2C2×D3×#41⋊C311yescharacteristic
347C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×C3
socleC22×C3C23×A4
prime radicalC2×D3×SL(2,𝔽3)C22
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
4. #5570⋊C3C2semidirect
5. #5569⋊C3C2semidirect
6. D3×#557C3semidirect
14. C3×#557C2×C3semidirect
22. #5570C2×C3semidirect
23. #5569C2×C3semidirect
28. #557⋊C3D3direct (N × H)
34. D3×SL(2,𝔽3)Q8semidirect
88. C3×SL(2,𝔽3)C2×Q8semidirect
94. #557C3×D3semidirect
222. SL(2,𝔽3)D3×Q8semidirect
325. D3#557⋊C3direct (N × H)
343. C3C2×#557⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C4×D3×SL(2,𝔽3)2
#5570⋊C32
#5569⋊C32
D3×#5573
C2×#557⋊C33
C3×D3×C4⋊C44