Finite Group Databasebeta

← Dic3×#986 #11922371

Subgroups — Dic3×#986 #11922371

Order 1152.

629 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C2×C3×#986C211yesmaximal, characteristic
2Dic3×#41⋊C3C211yesmaximal, characteristic, complemented
3#10593660C211yesmaximal, characteristic, complemented
4Dic3×GL(2,𝔽3)C212yesmaximal, complemented
5#10593204C212yesmaximal, complemented
6C4×#986—31nomaximal
7Dic3×AGL(1,ℤ/8)—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#986C414yescomplemented
10C2×C3×GL(2,𝔽3)C2212yes—
11Dic3×D12—41nomaximal
12Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
13Dic3×SL(2,𝔽3)C2211yescharacteristic, complemented
14C3⋊#859C2212yescomplemented
15C2×#986D311yescharacteristic
16C2×C3×AGL(1,ℤ/8)—31no—
17Dic3×#41D311yescharacteristic, complemented
18C2×D4×Dic3—31no—
19C4×#41⋊C3—31no—
20#5426—31no—
21#5426—61no—
22#5394—32no—
23C4×GL(2,𝔽3)—32no—
24#5232—31no—
25#5215—31no—
26#5164—32no—
27Dic3×SD16—32no—
28#5152—32no—
29Dic3×D8—32no—
30#5121—31no—
31Dic3×M16—31no—
32C2×C3×D12—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×D3×Dic3—42no—
36C3×GL(2,𝔽3)C2×C414yescomplemented
37C4⋊(C3⋊Dic3)—41no—
38C3×C4×Dic3—41no—
39(C2×D3)⋊Dic3—42no—
40C4×AGL(1,ℤ/8)—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42C22×C3×D4—31no—
43C23×Dic3—61no—
44C2×#41⋊C3—21no—
45C2×#41⋊C3C2×D311yescharacteristic
46#986Dic314yesdirect product factor
47C2×GL(2,𝔽3)C2×D312yes—
48C3×AGL(1,ℤ/8)—34no—
49C3×AGL(1,ℤ/8)—34no—
50C2×C3×SD16—32no—
51C2×C3×D8—32no—
52C2×C3×M16—31no—
53C2×#217—61no—
54Q8×Dic3C2×D311yescharacteristic
55Q8×Dic3C2×D311yescharacteristic
56D4×Dic3—31no—
57D4×Dic3—31no—
58D4×Dic3—32no—
59D4×Dic3—32no—
60D4×Dic3—61no—
61D4×Dic3—61no—
62#926—31no—
63C2×C4⋊Dic3—31no—
64C2×C4×Dic3—31no—
65C2×#208—31no—
66C4×D12—121no—
67C4×SL(2,𝔽3)—31no—
68C4×SL(2,𝔽3)—31no—
69#859—32no—
70#859—62no—
71#835—32no—
72#832—32no—
73#832—32no—
74#832—32no—
75#818—32no—
76#817—32no—
77#815—32no—
78C8×Dic3—32no—
79C22×C3×D3—42no—
80C2×C32×C4—41no—
81C2×C3⋊Dic3—42no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—41no—
84C3×D12—44no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
86D3×Dic3—84no—
87C2×AGL(1,ℤ/8)—31no—
88C4×#41—31no—
89C2×C4×D4—91no—
90#442—92no—
91#440—92no—
92C4×SD16—92no—
93C4×D8—92no—
94#428—91no—
95#418—91no—
96#417—91no—
97C4×M16—91no—
98C24×C3—61no—
99C3×#41—32no—
100C3×#41C4×D312yescomplemented
101C2×C3×Q8C22×D311yescharacteristic
102C2×C3×D4—31no—
103C2×C3×D4—32no—
104C2×C3×D4—34no—
105C2×C3×D4—62no—
106C2×C3×D4—61no—
107C22×C3×C4—31no—
108C22×Dic3—31no—
109C22×Dic3—31no—
110C22×Dic3—62no—
111C22×Dic3—61no—
112C22×Dic3—62no—
113C22×Dic3—124no—
114C2×D12—41no—
115C2×C4×D3—122no—
116#41⋊C3—22no—
117#41⋊C3C2×Dic312yescomplemented
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)C22×D311yescharacteristic
120GL(2,𝔽3)C2×Dic314yescomplemented
121C3×SD16—34no—
122C3×SD16—34no—
123C3×D8—38no—
124C3×M16—34no—
125C2×C3×C8—32no—
126C3×C42—121no—
127#217—61no—
128#217—62no—
129#217—62no—
130(C2×D3)⋊C4—122no—
131C4⋊Dic3—31no—
132C4⋊Dic3—31no—
133C4⋊Dic3—32no—
134C4⋊Dic3—61no—
135C4⋊Dic3—121no—
136C4⋊Dic3—241no—
137C4×Dic3—31no—
138C4×Dic3—31no—
139C4×Dic3—31no—
140C4×Dic3S411yescharacteristic
141#208—62no—
142C2×C3⋊C8—32no—
143C22×C32—41no—
144C2×C3×D3—44no—
145C2×C3×D3—82no—
146C2×C3×D3—82no—
147C32×C4—42no—
148C3⋊Dic3—82no—
149C3×Dic3—42no—
150C3×Dic3—81no—
151C2×#41D3211yescharacteristic
152C22×D4—31no—
153C23×C4—181no—
154AGL(1,ℤ/8)—34no—
155AGL(1,ℤ/8)—34no—
156C2×SD16—32no—
157C2×D8—32no—
158C2×M16—31no—
159C2×M16—91no—
160C4×Q8—31no—
161C4×Q8—31no—
162C4×D4—91no—
163C4×D4—91no—
164C4×D4—92no—
165C4×D4—92no—
166C4×D4—181no—
167C4×D4—181no—
168#117—91no—
169C2×C4⋊C4—91no—
170C2×C22⋊C4—181no—
171C2×C42—91no—
172#107—92no—
173#106—92no—
174#103—92no—
175#102—92no—
176#102—92no—
177#102—92no—
178#97—92no—
179C4×C8—92no—
180C23×C3—31no—
181C23×C3—64no—
182C23×C3—62no—
183C23×C3—122no—
184C23×C3—122no—
185C22×D3—42no—
186C3×Q8—31no—
187C3×Q8C2×C4×D311yescharacteristic
188C3×D4—32no—
189C3×D4—34no—
190C3×D4—61no—
191C3×D4—62no—
192C3×D4—64no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—81no—
198C2×C3×C4—121no—
199C2×C3×C4—121no—
200C2×C3×C4C2×S411yescharacteristic
201C2×Dic3—31no—
202C2×Dic3—31no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—122no—
206C2×Dic3—121no—
207C2×Dic3—124no—
208C2×Dic3—122no—
209C2×Dic3—124no—
210C2×Dic3—242no—
211C2×Dic3C2×S411yescharacteristic
212C2×Dic3C2×S411yescharacteristic
213D12—44no—
214C4×D3—244no—
215SL(2,𝔽3)—21no—
216SL(2,𝔽3)C22×Dic311yescharacteristic, complemented
217C3×C8—34no—
218C3⋊C8—62no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C3×D3—84no—
223C24—61no—
224#41—32no—
225#41D3×Dic312yescomplemented
226C2×Q8C2×D3211yescharacteristic
227C2×D4—31no—
228C2×D4—32no—
229C2×D4—34no—
230C2×D4—62no—
231C2×D4—61no—
232C22×C4—31no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4—182no—
236C22×C4—182no—
237C22×C4—181no—
238C22×C4—364no—
239SD16—34no—
240SD16—34no—
241D8—38no—
242M16—34no—
243M16—182no—
244C2×C8—32no—
245C2×C8—92no—
246C4⋊C4—91no—
247C4⋊C4—91no—
248C4⋊C4—92no—
249C4⋊C4—181no—
250C22⋊C4—182no—
251C22⋊C4—181no—
252C22⋊C4—182no—
253C42—31no—
254C42—91no—
255C42—91no—
256C42—91no—
257C22×C3—32no—
258C22×C3—41no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—64no—
262C22×C3—81no—
263C22×C3—128no—
264C22×C3—122no—
265C22×C3—122no—
266C22×C3C22×S411yescharacteristic
267C2×D3—44no—
268C2×D3—82no—
269C2×D3—82no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—42no—
273C3×C4—82no—
274C3×C4—122no—
275C3×C4—241no—
276C3×C4C4×S412yes—
277Dic3—21no—
278Dic3—61no—
279Dic3—61no—
280Dic3—122no—
281Dic3—242no—
282Dic3—482no—
283Dic3#98612yesdirect product factor
284C32—41noSylow
285C23—31no—
286C23—64no—
287C23—62no—
288C23—122no—
289C23—122no—
290Q8—31no—
291Q8C2×D3×Dic311yescharacteristic
292D4—32no—
293D4—34no—
294D4—61no—
295D4—62no—
296D4—64no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—31no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—364no—
307C2×C4—361no—
308C2×C4—364no—
309C2×C4—362no—
310C2×C4D3×S411yescharacteristic
311C8—34no—
312C8—182no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—62no—
317C2×C3—81no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3—124no—
321C2×C3C2×C4×S411yescharacteristic
322C2×C3C2×#98611yescharacteristic
323C2×C3#592311yescharacteristic
324D3—84no—
325C22—32no—
326C22—61no—
327C22—61no—
328C22—64no—
329C22—128no—
330C22—122no—
331C22—122no—
332C22C2×D3×S411yescharacteristic
333C4—32no—
334C4—31no—
335C4—31no—
336C4—61no—
337C4—181no—
338C4—181no—
339C4—362no—
340C4Dic3×S412yes—
341C3—41no—
342C3—81no—
343C3C4×#98611yescharacteristic, complemented
344C2—62no—
345C2—124no—
346C2#1059645511yescharacteristic
347C2D3×#98611yescharacteristic
348C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3×#986C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. Dic3×#41⋊C3C2semidirect
3. #10593660C2semidirect
4. Dic3×GL(2,𝔽3)C2semidirect
5. #10593204C2semidirect
9. C3×#986C4semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
14. C3⋊#859C22semidirect
17. Dic3×#41D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. C3×GL(2,𝔽3)C2×C4semidirect
46. #986Dic3direct (N × H)
85. C3×SL(2,𝔽3)C22×C4semidirect
100. C3×#41C4×D3semidirect
117. #41⋊C3C2×Dic3semidirect
120. GL(2,𝔽3)C2×Dic3semidirect
216. SL(2,𝔽3)C22×Dic3semidirect
225. #41D3×Dic3semidirect
283. Dic3#986direct (N × H)
343. C3C4×#986semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9862
Dic3×#41⋊C32
#105936602
Dic3×GL(2,𝔽3)2
#105932042
C4×#9863
Dic3×AGL(1,ℤ/8)3
Dic3×D124