Finite Group Databasebeta

← D3×#5386 #11921672

Subgroups — D3×#5386 #11921672

Order 1152.

559 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2#10593620C211yesmaximal, characteristic, complemented
3C3×#5386C211yesmaximal, characteristic, complemented
4D3×#859C212yesmaximal
5#10593212C212yesmaximal, complemented
6C2×#5386—31nomaximal
7D3×#475—31nomaximal
8C2×D3×Dic6—41nomaximal
9C2×D3×SL(2,𝔽3)C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11Dic3×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2212yes—
13C3×#859C2212yes—
14D3×C4⋊Q8—31no—
15C4×D3×Q8D311yescharacteristic
16C2×C4×SL(2,𝔽3)—31no—
17C2×#859—32no—
18#5386—31no—
19#5386—32no—
20#5386—61no—
21#5386D311yescharacteristic, direct product factor
22C3×#475—31no—
23#5094—31no—
24#5020—31no—
25D3×#106—32no—
26#4854—32no—
27#4851—31no—
28D3×C4⋊C8—31no—
29D3×#103—32no—
30#4791—32no—
31C2×#634—41no—
32C2×C3×C4×D3—41no—
33C2×C3×Dic6—41no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×#627—42no—
36C2×D3×Dic3—42no—
37D3×Dic6—48no—
38D3×SL(2,𝔽3)Q812yescomplemented
39C2×#475—91noSylow
40C2×D3×Q8—61no—
41C2×D3×Q8C2×D311yescharacteristic
42C22×Dic6—121no—
43C22×SL(2,𝔽3)—31no—
44C3×C4⋊Q8—31no—
45C3×C4×Q8C2×D311yescharacteristic
46Q8×Dic3C2×D311yescharacteristic
47#944—61no—
48C2×D3×C8—32no—
49D3×C4⋊C4—31no—
50D3×C4⋊C4—32no—
51D3×C4⋊C4—61no—
52#888—32no—
53#888—61no—
54#887—31no—
55C42×D3—31no—
56#869—31no—
57C4×Dic6—31no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)—31no—
60C4×SL(2,𝔽3)C2×D311yescharacteristic
61#859—32no—
62#859—62no—
63#859C2×D312yes—
64C3×#106—32no—
65C3×C4⋊C8—31no—
66C3×#103—32no—
67#835—32no—
68#817—32no—
69#816—32no—
70Dic3⋊C8—32no—
71#810—32no—
72#808—32no—
73#808—32no—
74C4⋊(C3⋊C8)—31no—
75C22×C3×D3—41no—
76C2×C32×C4—41no—
77C2×C3⋊Dic3—42no—
78#634—44no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—41no—
81C3×C4×D3—44no—
82C3×Dic6—44no—
83C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
84#627—48no—
85D3×Dic3—48no—
86C2×C4⋊Q8—91no—
87C2×C4×Q8—31no—
88#475—31no—
89#475—92no—
90#475—91no—
91#475—92no—
92#475—91no—
93#475—91no—
94C2×#106—92no—
95C2×C4⋊C8—91no—
96C2×#103—92no—
97C2×C3×Q8—61no—
98C2×C3×Q8C22×D311yescharacteristic
99C22×C3×C4—121no—
100C22×Dic3—122no—
101D3×Q8—32no—
102D3×Q8—124no—
103D3×Q8Dic612yescomplemented
104C2×C4×D3—31no—
105C2×C4×D3—31no—
106C2×C4×D3—62no—
107C2×C4×D3S411yescharacteristic
108C2×Dic6—31no—
109C2×Dic6—41no—
110C2×Dic6—61no—
111C2×Dic6—62no—
112C2×Dic6—122no—
113C2×Dic6—121no—
114C2×Dic6—128no—
115C2×Dic6—241no—
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)—32no—
118C2×SL(2,𝔽3)C22×D311yescharacteristic
119C2×C3×C8—32no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—61no—
123C3×C42—31no—
124C4⋊Dic3—31no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127Dic3⋊C4—61no—
128Dic3⋊C4—62no—
129Dic3⋊C4—62no—
130C4×Dic3—31no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C2×C3⋊C8—32no—
134D3×C8—64no—
135C22×C32—41no—
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C32×C4—42no—
140C3⋊Dic3—44no—
141C3×Dic3—44no—
142C3×Dic3—42no—
143C22×Q8—31no—
144C22×Q8—181no—
145C22×C8—92no—
146C4⋊Q8—31no—
147C4⋊Q8—91no—
148C4⋊Q8—92no—
149C4⋊Q8—181no—
150C4⋊Q8—181no—
151C4×Q8—31no—
152C4×Q8—91no—
153C4×Q8—91no—
154C4×Q8D3211yescharacteristic
155C2×C4⋊C4—91no—
156C2×C4⋊C4—92no—
157C2×C4⋊C4—181no—
158C2×C42—91no—
159#106—32no—
160#106—92no—
161#106—92no—
162#106—92no—
163C4⋊C8—31no—
164C4⋊C8—91no—
165C4⋊C8—92no—
166#103—32no—
167#103—92no—
168#103—92no—
169#103—92no—
170C23×C3—121no—
171C22×D3C2×S411yescharacteristic
172C3×Q8—31no—
173C3×Q8—122no—
174C3×Q8C2×Dic611yescharacteristic, complemented
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—41no—
178C2×C3×C4—62no—
179C2×C3×C4—81no—
180C2×C3×C4—124no—
181C2×C3×C4—121no—
182C2×C3×C4C2×S411yescharacteristic
183C2×Dic3—31no—
184C2×Dic3—31no—
185C2×Dic3—42no—
186C2×Dic3—62no—
187C2×Dic3—122no—
188C2×Dic3—128no—
189C2×Dic3—242no—
190C2×Dic3C2×S411yescharacteristic
191C4×D3—32no—
192C4×D3—32no—
193C4×D3—62no—
194C4×D3—124no—
195C4×D3GL(2,𝔽3)14yes—
196Dic6—32no—
197Dic6—44no—
198Dic6—61no—
199Dic6—128no—
200Dic6—124no—
201Dic6—122no—
202Dic6—124no—
203Dic6—244no—
204SL(2,𝔽3)—21no—
205SL(2,𝔽3)D3×Q811yescharacteristic, complemented
206C3×C8—62no—
207C3⋊C8—62no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C3×D3—44no—
212C2×Q8—32no—
213C2×Q8—61no—
214C2×Q8—91no—
215C2×Q8—92no—
216C2×Q8—182no—
217C2×Q8—181no—
218C2×Q8—364no—
219C2×Q8C2×D3211yescharacteristic
220C22×C4—31no—
221C22×C4—91no—
222C22×C4—91no—
223C22×C4—182no—
224C2×C8—32no—
225C2×C8—92no—
226C2×C8—184no—
227C4⋊C4—31no—
228C4⋊C4—32no—
229C4⋊C4—61no—
230C4⋊C4—91no—
231C4⋊C4—92no—
232C4⋊C4—181no—
233C4⋊C4—181no—
234C4⋊C4—182no—
235C4⋊C4—182no—
236C42—31no—
237C42—91no—
238C42—91no—
239C42—91no—
240C22×C3—41no—
241C22×C3—81no—
242C22×C3—122no—
243C22×C3—122no—
244C22×C3—122no—
245C22×C3C22×S411yescharacteristic
246C2×D3A4⋊Q812yes—
247C2×D3C2×GL(2,𝔽3)12yes—
248C2×D3#98412yes—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—42no—
252C3×C4—61no—
253C3×C4—82no—
254C3×C4—122no—
255C3×C4—122no—
256C3×C4C2×GL(2,𝔽3)12yes—
257Dic3—32no—
258Dic3—44no—
259Dic3—61no—
260Dic3—124no—
261Dic3—122no—
262Dic3—244no—
263Dic3C2×GL(2,𝔽3)12yes—
264C32—41noSylow
265C23—31no—
266Q8—31no—
267Q8—92no—
268Q8—122no—
269Q8—181no—
270Q8—364no—
271Q8—362no—
272Q8D3×Dic611yescharacteristic, complemented
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—34no—
276C2×C4—31no—
277C2×C4—62no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—364no—
285C2×C4D3×S411yescharacteristic
286C8—62no—
287C8—182no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—124no—
295C2×C3C2×A4⋊Q811yescharacteristic
296C2×C3C22×GL(2,𝔽3)11yescharacteristic
297C2×C3C2×#98411yescharacteristic
298D3#538614yesdirect product factor
299C22—32no—
300C22—32no—
301C22—32no—
302C22C2×D3×S411yescharacteristic
303C4—31no—
304C4—31no—
305C4—32no—
306C4—61no—
307C4—92no—
308C4—122no—
309C4—181no—
310C4—362no—
311C4D3×GL(2,𝔽3)12yes—
312C3—41no—
313C3—81no—
314C3C2×#538611yescharacteristic, complemented
315C2—34no—
316C2D3×A4⋊Q811yescharacteristic
317C2C2×D3×GL(2,𝔽3)11yescharacteristic
318C2D3×#98411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
prime radicalC2×D3×SL(2,𝔽3)C22
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×Q8
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
2. #10593620C2semidirect
3. C3×#5386C2semidirect
5. #10593212C2semidirect
21. #5386D3direct (N × H)
38. D3×SL(2,𝔽3)Q8semidirect
83. C3×SL(2,𝔽3)C2×Q8semidirect
103. D3×Q8Dic6semidirect
174. C3×Q8C2×Dic6semidirect
205. SL(2,𝔽3)D3×Q8semidirect
272. Q8D3×Dic6semidirect
298. D3#5386direct (N × H)
314. C3C2×#5386semidirect

Maximal subgroups

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

subgroupindex
C4×D3×SL(2,𝔽3)2
#105936202
C3×#53862
D3×#8592
#105932122
C2×#53863
D3×#4753
C2×D3×Dic64