Finite Group Databasebeta

← C2×D3×#859 #11922213

Subgroups — C2×D3×#859 #11922213

Order 1152.

913 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 913 conjugacy classes of subgroups — too many to draw. Only its 139 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⋊#859C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4D3×#859C214yesmaximal, direct product factor
5C22×#859—31nomaximal
6C2×D3×#103—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C22×D3×Dic3—41nomaximal
9C2×D3×SL(2,𝔽3)C414yescomplemented
10C2×D3×SL(2,𝔽3)C2212yes—
11C3⋊#859C2212yescomplemented
12C3×#859C2212yescomplemented
13C22×D3×Q8D311yescharacteristic
14C23×SL(2,𝔽3)—31no—
15C22×D3×C8—31no—
16C2×D3×C4⋊C4—31no—
17C2×#859—31no—
18C2×#859—34no—
19C2×#859—61no—
20C2×#859D311yescharacteristic, direct product factor
21C2×C3×#103—31no—
22C2×#835—31no—
23C2×#816—31no—
24C2×#810—31no—
25D3×#103—34no—
26D3×#103—34no—
27C23×C3×D3—41no—
28C22×C3⋊Dic3—41no—
29C22×C3×Dic3—41no—
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C2×D3×Dic3—44no—
33C2×D3×Dic3—48no—
34D3×SL(2,𝔽3)C2×C414yescomplemented
35C22×#103—91noSylow
36C22×C3×Q8C2×D311yescharacteristic
37C23×Dic3—121no—
38C2×D3×Q8—34no—
39C2×D3×Q8—32no—
40C2×D3×Q8Dic314yescomplemented
41C2×D3×Q8C2×D312yes—
42C22×C4×D3—31no—
43C22×C4×D3—61no—
44C22×Dic6—31no—
45C22×SL(2,𝔽3)—21no—
46C22×SL(2,𝔽3)—34no—
47C22×SL(2,𝔽3)—32no—
48C22×SL(2,𝔽3)C2×D311yescharacteristic
49C22×C3×C8—31no—
50C2×C3×C4⋊C4—31no—
51C2×C4⋊Dic3—31no—
52C2×Dic3⋊C4—61no—
53C22×C3⋊C8—31no—
54C2×D3×C8—34no—
55C2×D3×C8—64no—
56D3×C4⋊C4—34no—
57D3×C4⋊C4—62no—
58#859—32no—
59#859—62no—
60#859C2×D312yesdirect product factor
61C3×#103—32no—
62C3×#103—32no—
63#835—32no—
64#835—32no—
65#816—34no—
66#810—34no—
67C23×C32—41no—
68C22×C3×D3—44no—
69C22×C3×D3—44no—
70C22×C3×D3—44no—
71C22×C3×D3—42no—
72C2×C3⋊Dic3—42no—
73C2×C3⋊Dic3—44no—
74C2×C3×Dic3—42no—
75C2×C3×Dic3—44no—
76C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
77D3×Dic3—416no—
78C23×Q8—31no—
79C23×C8—91no—
80C22×C4⋊C4—91no—
81C2×#103—31no—
82C2×#103—94no—
83C2×#103—94no—
84C2×#103—91no—
85C2×#103—91no—
86C2×#103—91no—
87C24×C3—121no—
88C23×D3S411yescharacteristic
89C2×C3×Q8—31no—
90C2×C3×Q8—32no—
91C2×C3×Q8C2×Dic312yescomplemented
92C2×C3×Q8C22×D311yescharacteristic
93C22×C3×C4—31no—
94C22×C3×C4—61no—
95C22×Dic3—31no—
96C22×Dic3—41no—
97C22×Dic3—61no—
98C22×Dic3—124no—
99C22×Dic3—121no—
100C22×Dic3—128no—
101C22×Dic3—241no—
102D3×Q8—38no—
103D3×Q8—34no—
104D3×Q8—64no—
105D3×Q8C2×Dic314yescomplemented
106C2×C4×D3—34no—
107C2×C4×D3—34no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—64no—
111C2×C4×D3—124no—
112C2×Dic6—32no—
113C2×Dic6—34no—
114C2×Dic6—62no—
115C2×Dic6—61no—
116C2×SL(2,𝔽3)—22no—
117C2×SL(2,𝔽3)—21no—
118C2×SL(2,𝔽3)—34no—
119C2×SL(2,𝔽3)C4×D312yescomplemented
120C2×SL(2,𝔽3)C22×D311yescharacteristic
121C2×C3×C8—32no—
122C2×C3×C8—62no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—61no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127Dic3⋊C4—64no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—62no—
130D3×C8—68no—
131C22×C32—42no—
132C22×C32—42no—
133C22×C32—42no—
134C22×C32—41no—
135C2×C3×D3—416no—
136C2×C3×D3—44no—
137C2×C3×D3—44no—
138C2×C3×D3—44no—
139C3⋊Dic3—44no—
140C3×Dic3—44no—
141C22×Q8—34no—
142C22×Q8—32no—
143C22×Q8—91no—
144C22×Q8—92no—
145C22×Q8—94no—
146C22×Q8D3211yescharacteristic
147C23×C4—91no—
148C23×C4—181no—
149C22×C8—31no—
150C22×C8—94no—
151C22×C8—91no—
152C22×C8—184no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—94no—
155C2×C4⋊C4—91no—
156C2×C4⋊C4—182no—
157C2×C4⋊C4—181no—
158#103—32no—
159#103—32no—
160#103—94no—
161#103—94no—
162#103—92no—
163#103—92no—
164C23×C3—41no—
165C23×C3—81no—
166C23×C3—122no—
167C23×C3—124no—
168C23×C3—124no—
169C23×C3—124no—
170C23×C3C2×S411yescharacteristic
171C22×D3#22614yes—
172C22×D3GL(2,𝔽3)14yes—
173C22×D3SL(2,ℤ/4)14yes—
174C22×D3C2×S412yes—
175C3×Q8—32no—
176C3×Q8—31no—
177C3×Q8—61no—
178C3×Q8C22×Dic311yescharacteristic, complemented
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—62no—
184C2×C3×C4—122no—
185C2×Dic3—34no—
186C2×Dic3—32no—
187C2×Dic3—42no—
188C2×Dic3—44no—
189C2×Dic3—62no—
190C2×Dic3—122no—
191C2×Dic3—1216no—
192C2×Dic3—124no—
193C2×Dic3—122no—
194C2×Dic3—242no—
195C2×Dic3—244no—
196C4×D3—34no—
197C4×D3—34no—
198C4×D3—38no—
199C4×D3—128no—
200Dic6—34no—
201Dic6—62no—
202Dic6—64no—
203SL(2,𝔽3)—21no—
204SL(2,𝔽3)C2×C4×D311yescharacteristic, complemented
205C3×C8—62no—
206C3⋊C8—62no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—44no—
211C3×D3—48no—
212C24—31no—
213C2×Q8—31no—
214C2×Q8—32no—
215C2×Q8—34no—
216C2×Q8—92no—
217C2×Q8—94no—
218C2×Q8—94no—
219C2×Q8—98no—
220C2×Q8—181no—
221C2×Q8—182no—
222C2×Q8—184no—
223C2×Q8D3×Dic312yescomplemented
224C2×Q8C2×D3211yescharacteristic
225C22×C4—31no—
226C22×C4—61no—
227C22×C4—92no—
228C22×C4—92no—
229C22×C4—94no—
230C22×C4—94no—
231C22×C4—91no—
232C22×C4—184no—
233C22×C4—181no—
234C22×C4—364no—
235C2×C8—32no—
236C2×C8—62no—
237C2×C8—92no—
238C2×C8—188no—
239C2×C8—182no—
240C4⋊C4—32no—
241C4⋊C4—61no—
242C4⋊C4—92no—
243C4⋊C4—181no—
244C4⋊C4—184no—
245C22×C3—41no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—81no—
250C22×C3—82no—
251C22×C3—82no—
252C22×C3—82no—
253C22×C3—124no—
254C22×C3—124no—
255C22×C3—1216no—
256C22×C3—124no—
257C22×C3C2×#22612yes—
258C22×C3C2×GL(2,𝔽3)12yes—
259C22×C3C2×SL(2,ℤ/4)12yes—
260C22×C3C22×S411yescharacteristic
261C2×D3#859116yesdirect product factor
262C2×D3C2×#22614yes—
263C2×D3C2×GL(2,𝔽3)14yes—
264C2×D3C2×SL(2,ℤ/4)14yes—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—32no—
268C3×C4—122no—
269Dic3—34no—
270Dic3—44no—
271Dic3—124no—
272Dic3—122no—
273Dic3—244no—
274C32—41noSylow
275C23—32no—
276C23—34no—
277C23—34no—
278C23—34no—
279C23D3×S411yescharacteristic
280Q8—32no—
281Q8—31no—
282Q8—61no—
283Q8—94no—
284Q8—182no—
285Q8—184no—
286Q8C2×D3×Dic311yescharacteristic, complemented
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—32no—
290C2×C4—32no—
291C2×C4—62no—
292C2×C4—94no—
293C2×C4—94no—
294C2×C4—98no—
295C2×C4—92no—
296C2×C4—94no—
297C2×C4—122no—
298C2×C4—182no—
299C2×C4—368no—
300C2×C4—362no—
301C8—62no—
302C8—182no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—44no—
307C2×C3—81no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3—84no—
311C2×C3—128no—
312C2×C3C2×#85914yescomplemented
313C2×C3C22×#22611yescharacteristic
314C2×C3C22×GL(2,𝔽3)11yescharacteristic
315C2×C3C22×SL(2,ℤ/4)11yescharacteristic
316D3C2×#85918yesdirect product factor
317C22—34no—
318C22—34no—
319C22—34no—
320C22—316no—
321C22D3×#22612yes—
322C22D3×GL(2,𝔽3)12yes—
323C22D3×SL(2,ℤ/4)12yes—
324C22C2×D3×S411yescharacteristic
325C4—31no—
326C4—31no—
327C4—32no—
328C4—94no—
329C4—122no—
330C4—362no—
331C3—41no—
332C3—81no—
333C3C22×#85911yescharacteristic, complemented
334C2—38no—
335C2D3×#85914yesdirect product factor
336C2C2×D3×#22611yescharacteristic
337C2C2×D3×GL(2,𝔽3)11yescharacteristic
338C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FrattiniC22C2×D3×S4
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radicalC22×D3×SL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#859C2semidirect
3. C2×C3×#859C2semidirect
4. D3×#859C2direct (N × H)
9. C2×D3×SL(2,𝔽3)C4semidirect
11. C3⋊#859C22semidirect
12. C3×#859C22semidirect
20. C2×#859D3direct (N × H)
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
34. D3×SL(2,𝔽3)C2×C4semidirect
40. C2×D3×Q8Dic3semidirect
60. #859C2×D3direct (N × H)
76. C3×SL(2,𝔽3)C22×C4semidirect
91. C2×C3×Q8C2×Dic3semidirect
105. D3×Q8C2×Dic3semidirect
119. C2×SL(2,𝔽3)C4×D3semidirect
178. C3×Q8C22×Dic3semidirect
204. SL(2,𝔽3)C2×C4×D3semidirect
223. C2×Q8D3×Dic3semidirect
261. C2×D3#859direct (N × H)
286. Q8C2×D3×Dic3semidirect
312. C2×C3C2×#859semidirect
316. D3C2×#859direct (N × H)
333. C3C22×#859semidirect
335. C2D3×#859direct (N × H)

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⋊#8592
C2×C3×#8592
D3×#8592
C22×#8593
C2×D3×#1033
C22×D3×Dic34