Finite Group Databasebeta

← D12×#226 #11921692

Subgroups — D12×#226 #11921692

Order 1152.

571 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 63 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×D3×#226C212yesmaximal, complemented
2SL(2,𝔽3)×D12C211yesmaximal, characteristic
3C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
4C3×C4×#226C211yesmaximal, characteristic, complemented
5#10593211C212yesmaximal, complemented
6D4×#226—31nomaximal
7Q16×D12—31nomaximal
8C2×D3×SL(2,𝔽3)C2212yes—
9C2×C3⋊#226C2212yes—
10C2×C3×#226C2211yescharacteristic
11D3×#226—24no—
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×D12—41nomaximal
14C3×#859C2211yescharacteristic
15C22×#226—32no—
16C2×D3×Q16—32no—
17Q8×D12—31no—
18Q8×D12D311yescharacteristic
19D4×SL(2,𝔽3)—31no—
20#5419—32no—
21C4⋊#226—31no—
22C4⋊#226—61no—
23C4×#226D311yescharacteristic, complemented
24C3×C4×Q16—31no—
25#5031—31no—
26#5031—31no—
27#4887—32no—
28#4809—32no—
29#4809—32no—
30#4699—31no—
31C8×D12—31no—
32C2×C3×D12—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C2×D3×Dic3—42no—
35D3×SL(2,𝔽3)—22no—
36C3⋊#226—22no—
37C3×#226D412yescomplemented
38C4⋊(C3⋊Dic3)—41no—
39C3×C4×Dic3—41no—
40(C2×D3)⋊Dic3—42no—
41D4×Q16—91noSylow
42C2×D3×Q8—32no—
43C2×D3×Q8C2×D312yes—
44C22×SL(2,𝔽3)—32no—
45C2×#226—32no—
46C2×#226—62no—
47C2×#226—64no—
48C2×#226C2×D311yescharacteristic
49C2×C3×Q16—31no—
50C3×C4×Q8—31no—
51C3×C4×Q8C2×D311yescharacteristic
52C2×#216—32no—
53C2×#216—32no—
54D4×Dic3—121no—
55D3×Q16—64no—
56D3×Q16—64no—
57C2×Dic12—32no—
58C2×Dic12—32no—
59C2×D3×C8—32no—
60#897—32no—
61#897—32no—
62#897—62no—
63C4×D12—31no—
64C4×D12—61no—
65#869—31no—
66#869—31no—
67#869—61no—
68C4×SL(2,𝔽3)—21no—
69C4×SL(2,𝔽3)C2×D311yescharacteristic
70#859C2×D311yescharacteristic
71C3×#107—31no—
72C3×#103—31no—
73C3×#103—31no—
74C3×C4×C8—31no—
75(C2×D3)⋊C8—32no—
76#810—32no—
77#810—32no—
78C4⋊(C3⋊C8)—31no—
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)C2×D411yescharacteristic
86D3×Dic3—84no—
87C22×Q16—92no—
88D4×Q8—31no—
89D4×Q8—91no—
90#494—91no—
91#467—92no—
92#462—91no—
93#462—91no—
94#451—92no—
95#451—92no—
96C4×Q16—31no—
97C8×D4—91no—
98C2×C3×Q8—31no—
99C2×C3×Q8C22×D311yescharacteristic
100C2×C3×D4—121no—
101C22×Dic3—122no—
102D3×Q8—22no—
103D3×Q8—62no—
104D3×Q8—64no—
105D3×Q8—122no—
106C2×D12S411yescharacteristic
107C2×C4×D3—32no—
108C2×C4×D3—62no—
109C2×Dic6—32no—
110C2×Dic6—32no—
111C2×Dic6—62no—
112C2×SL(2,𝔽3)—21no—
113C2×SL(2,𝔽3)—62no—
114C2×SL(2,𝔽3)C22×D311yescharacteristic
115#226—62no—
116#226—122no—
117#226D1212yesdirect product factor
118C3×Q16—32no—
119C3×Q16—32no—
120C2×C3×C8—31no—
121C2×C3×C8—31no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—61no—
125C3×C42—31no—
126C3×C42—61no—
127#217—122no—
128#216—64no—
129#216—62no—
130#216—62no—
131(C2×D3)⋊C4—32no—
132(C2×D3)⋊C4—62no—
133C4⋊Dic3—31no—
134C4⋊Dic3—61no—
135C4⋊Dic3—62no—
136C4⋊Dic3—61no—
137C4⋊Dic3—61no—
138C4⋊Dic3—121no—
139C4⋊Dic3—241no—
140C4×Dic3—41no—
141C2×C3⋊C8—32no—
142Dic12—64no—
143Dic12—122no—
144D3×C8—64no—
145C22×C32—41no—
146C2×C3×D3—44no—
147C2×C3×D3—82no—
148C2×C3×D3—82no—
149C32×C4—42no—
150C3⋊Dic3—82no—
151C3×Dic3—42no—
152C3×Dic3—81no—
153C22×Q8—32no—
154C22×Q8—92no—
155C2×Q16—31no—
156C2×Q16—92no—
157C2×Q16—92no—
158C2×Q16—92no—
159C2×Q16—92no—
160C2×Q16—184no—
161C2×Q16—184no—
162C22×C8—92no—
163C4⋊Q8—91no—
164C4⋊Q8—91no—
165C4⋊Q8—181no—
166C22⋊Q8—92no—
167C22⋊Q8—92no—
168C22⋊Q8—182no—
169C4×Q8—31no—
170C4×Q8D3211yescharacteristic
171C4×D4—91no—
172C4×D4—181no—
173#107—31no—
174C4⋊C8—91no—
175#103—31no—
176#103—31no—
177#103—92no—
178#103—92no—
179C22⋊C8—92no—
180C4×C8—31no—
181C23×C3—122no—
182C22×D3C2×S412yes—
183C3×Q8—31no—
184C3×Q8—32no—
185C3×Q8—61no—
186C3×Q8D3×D411yescharacteristic
187C3×D4—124no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—41no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—81no—
194C2×C3×C4C2×S411yescharacteristic
195C2×Dic3—32no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—62no—
199C2×Dic3—122no—
200C2×Dic3—242no—
201C2×Dic3—244no—
202D12#22614yesdirect product factor
203C4×D3—62no—
204C4×D3—62no—
205C4×D3—124no—
206Dic6—62no—
207Dic6—62no—
208Dic6—62no—
209Dic6—62no—
210Dic6—124no—
211SL(2,𝔽3)—21no—
212SL(2,𝔽3)C2×D1211yescharacteristic
213C3×C8—32no—
214C3×C8—61no—
215C3⋊C8—62no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C3×D3—84no—
220C2×Q8—31no—
221C2×Q8—62no—
222C2×Q8—92no—
223C2×Q8—92no—
224C2×Q8—182no—
225C2×Q8—182no—
226C2×Q8—184no—
227C2×Q8—362no—
228C2×Q8C2×D3211yescharacteristic
229C2×D4—31no—
230C22×C4—92no—
231C22×C4—182no—
232Q16—32no—
233Q16—32no—
234Q16—184no—
235Q16—182no—
236Q16—184no—
237Q16—182no—
238Q16—362no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—92no—
242C2×C8—184no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—61no—
246C4⋊C4—91no—
247C4⋊C4—181no—
248C4⋊C4—181no—
249C4⋊C4—182no—
250C4⋊C4—181no—
251C22⋊C4—92no—
252C22⋊C4—182no—
253C42—31no—
254C42—61no—
255C22×C3—41no—
256C22×C3—81no—
257C22×C3—124no—
258C22×C3—242no—
259C22×C3—242no—
260C22×C3C22×S411yescharacteristic
261C2×D3—22no—
262C2×D3—22no—
263C2×D3C2×#22614yescomplemented
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—42no—
267C3×C4—62no—
268C3×C4—61no—
269C3×C4—82no—
270C3×C4—121no—
271C3×C4C2×#22612yescomplemented
272Dic3—42no—
273Dic3—62no—
274Dic3—81no—
275Dic3—122no—
276Dic3—242no—
277Dic3—482no—
278C32—41noSylow
279C23—32no—
280Q8—31no—
281Q8—32no—
282Q8—61no—
283Q8—182no—
284Q8—182no—
285Q8—182no—
286Q8—182no—
287Q8—364no—
288Q8D3×D1211yescharacteristic
289D4—34no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—92no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4—182no—
298C2×C4—364no—
299C2×C4D3×S411yescharacteristic
300C8—32no—
301C8—61no—
302C8—182no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—81no—
307C2×C3—81no—
308C2×C3—81no—
309C2×C3—244no—
310C2×C3D4×S411yescharacteristic
311C2×C3C22×#22611yescharacteristic
312C2×C3#592711yescharacteristic
313D3—24no—
314C22—34no—
315C22—62no—
316C22—62no—
317C22C2×D3×S411yescharacteristic
318C4—31no—
319C4—31no—
320C4—62no—
321C4—61no—
322C4—121no—
323C4—182no—
324C4—362no—
325C4D3×#22612yescomplemented
326C3—41no—
327C3—81no—
328C3D4×#22611yescharacteristic, complemented
329C2—64no—
330C2D12×S411yescharacteristic
331C2C2×D3×#22611yescharacteristic
332C2#1059645211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
prime radicalSL(2,𝔽3)×D12C2
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×#226C2semidirect
3. C4⋊(C3⋊#226)C2semidirect
4. C3×C4×#226C2semidirect
5. #10593211C2semidirect
23. C4×#226D3semidirect
37. C3×#226D4semidirect
117. #226D12direct (N × H)
202. D12#226direct (N × H)
263. C2×D3C2×#226semidirect
271. C3×C4C2×#226semidirect
325. C4D3×#226semidirect
328. C3D4×#226semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#2262
SL(2,𝔽3)×D122
C4⋊(C3⋊#226)2
C3×C4×#2262
#105932112
D4×#2263
Q16×D123
Dic3×D124