Finite Group Databasebeta

← #11921701

Subgroups — #11921701

Order 1152.

475 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 59 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
1SL(2,𝔽3)×D12C211yesmaximal, characteristic
2C4×C3⋊#226C211yesmaximal, characteristic, complemented
3C3×#5391C211yesmaximal, characteristic, complemented
4D3×#859C212yesmaximal, complemented
5#10593211C212yesmaximal, complemented
6#85393—31nomaximal
7#81119—31nomaximal
8C2×D3×SL(2,𝔽3)C2212yes—
9C2×C3⋊#226C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63898—41nomaximal
12C3⋊#859C2211yescharacteristic
13C3×#859C2212yes—
14#5730—31no—
15Q8×D12D311yescharacteristic
16D4×SL(2,𝔽3)—31no—
17#5419—32no—
18C2×#859—32no—
19#5391D311yescharacteristic, complemented
20C4×#226—31no—
21C4×#226—61no—
22C3×#477—31no—
23#5093—31no—
24C4×#216—31no—
25#4887—32no—
26#4884—32no—
27#4849—31no—
28#4834—31no—
29#4809—32no—
30D3×#103—32no—
31C2×C3×D12—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C2×D3×Dic3—42no—
34D3×SL(2,𝔽3)—22no—
35C3⋊#226—21no—
36C4×C3⋊Dic3—41no—
37C3×C4⋊Dic3—41no—
38(C2×D3)⋊Dic3—42no—
39#3299—91noSylow
40C2×D3×Q8C2×D312yes—
41C22×SL(2,𝔽3)—32no—
42C2×#226—31no—
43C2×#226—61no—
44C3×C4⋊Q8—31no—
45C3×C4×Q8C2×D311yescharacteristic
46#946—62no—
47C2×#216—31no—
48D4×Dic3—121no—
49C2×Dic12—32no—
50C2×D3×C8—32no—
51#897—32no—
52#897—32no—
53#893—61no—
54D3×C4⋊C4—32no—
55#890—61no—
56C4×D12—31no—
57#869—31no—
58C4×Dic6—31no—
59C4×SL(2,𝔽3)—21no—
60C4×SL(2,𝔽3)C2×D311yescharacteristic
61#859—31no—
62#859—61no—
63#859C2×D312yes—
64C3×#107—32no—
65C3×C4⋊C8—31no—
66C3×#103—32no—
67#835—31no—
68(C2×D3)⋊C8—32no—
69#816—32no—
70#810—32no—
71#810—31no—
72#810—32no—
73#807—31no—
74C4×C3⋊C8—31no—
75C22×C3×D3—42no—
76C2×C32×C4—41no—
77C2×C3⋊Dic3—41no—
78C2×C3⋊Dic3—41no—
79C2×C3×Dic3—42no—
80C3×D12—44no—
81C3×SL(2,𝔽3)#4111yescharacteristic
82D3×Dic3—84no—
83#554—91no—
84D4×Q8—31no—
85#487—91no—
86#484—92no—
87#477—31no—
88#467—92no—
89#462—91no—
90#451—92no—
91C4×Q16—91no—
92C8×D4—91no—
93C2×#103—92no—
94C2×C3×Q8—61no—
95C2×C3×Q8C22×D311yescharacteristic
96C2×C3×D4—121no—
97C22×Dic3—122no—
98D3×Q8—22no—
99D3×Q8—62no—
100C2×D12S411yescharacteristic
101C2×C4×D3—32no—
102C2×C4×D3—62no—
103C2×Dic6—32no—
104C2×Dic6—31no—
105C2×SL(2,𝔽3)—21no—
106C2×SL(2,𝔽3)—62no—
107C2×SL(2,𝔽3)C22×D311yescharacteristic
108#226—62no—
109#226—61no—
110C2×C3×C8—32no—
111C3×C4⋊C4—31no—
112C3×C4⋊C4—32no—
113C3×C4⋊C4—61no—
114C3×C42—31no—
115#217—122no—
116#216—61no—
117#216—61no—
118(C2×D3)⋊C4—32no—
119(C2×D3)⋊C4—62no—
120C4⋊Dic3—31no—
121C4⋊Dic3—31no—
122C4⋊Dic3—41no—
123C4⋊Dic3—61no—
124C4⋊Dic3—62no—
125Dic3⋊C4—62no—
126Dic3⋊C4—61no—
127C4×Dic3—61no—
128C4×Dic3—121no—
129C4×Dic3—241no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—31no—
132Dic12—122no—
133D3×C8—122no—
134C22×C32—41no—
135C2×C3×D3—44no—
136C2×C3×D3—82no—
137C2×C3×D3—82no—
138C32×C4—42no—
139C3⋊Dic3—42no—
140C3⋊Dic3—81no—
141C3×Dic3—82no—
142C22×Q8—32no—
143C2×Q16—92no—
144C2×Q16—91no—
145C22×C8—92no—
146C4⋊Q8—31no—
147C4⋊Q8—91no—
148#125—181no—
149C22⋊Q8—92no—
150C22⋊Q8—92no—
151C22⋊Q8—182no—
152C4×Q8—91no—
153C4×Q8D3211yescharacteristic
154C4×D4—91no—
155C4×D4—181no—
156C2×C4⋊C4—92no—
157#107—32no—
158#107—91no—
159C4⋊C8—31no—
160#103—32no—
161#103—92no—
162#103—91no—
163#103—92no—
164#103—92no—
165#103—91no—
166C22⋊C8—92no—
167C4×C8—91no—
168C23×C3—122no—
169C22×D3C2×S412yes—
170C3×Q8—31no—
171C3×Q8—241no—
172C3×Q8#23711yescharacteristic
173C3×D4—124no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—41no—
177C2×C3×C4—62no—
178C2×C3×C4—81no—
179C2×C3×C4C2×S411yescharacteristic
180C2×Dic3—32no—
181C2×Dic3—42no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—121no—
185C2×Dic3—121no—
186C2×Dic3—241no—
187C2×Dic3—241no—
188C2×Dic3—244no—
189D12#22614yescomplemented
190C4×D3—62no—
191C4×D3—62no—
192C4×D3—242no—
193Dic6—62no—
194Dic6—62no—
195Dic6—61no—
196Dic6—61no—
197SL(2,𝔽3)—21no—
198SL(2,𝔽3)#23911yescharacteristic
199C3×C8—62no—
200C3⋊C8—61no—
201C3⋊C8—61no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C3×D3—84no—
206C2×Q8—61no—
207C2×Q8—62no—
208C2×Q8—92no—
209C2×Q8—91no—
210C2×Q8—182no—
211C2×Q8C2×D3211yescharacteristic
212C2×D4—31no—
213C22×C4—92no—
214C22×C4—182no—
215Q16—181no—
216Q16—181no—
217Q16—362no—
218C2×C8—32no—
219C2×C8—91no—
220C2×C8—91no—
221C2×C8—362no—
222C4⋊C4—31no—
223C4⋊C4—32no—
224C4⋊C4—61no—
225C4⋊C4—91no—
226C4⋊C4—91no—
227C4⋊C4—181no—
228C4⋊C4—181no—
229C4⋊C4—182no—
230C4⋊C4—182no—
231C22⋊C4—92no—
232C22⋊C4—182no—
233C42—31no—
234C42—181no—
235C22×C3—41no—
236C22×C3—81no—
237C22×C3—124no—
238C22×C3—242no—
239C22×C3—242no—
240C22×C3C22×S411yescharacteristic
241C2×D3—22no—
242C2×D3—22no—
243C2×D3C2×#22614yes—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—42no—
247C3×C4—61no—
248C3×C4—82no—
249C3×C4—122no—
250C3×C4C2×#22612yes—
251Dic3—62no—
252Dic3—82no—
253Dic3—122no—
254Dic3—121no—
255Dic3—121no—
256Dic3—241no—
257Dic3—242no—
258Dic3—242no—
259C32—41noSylow
260C23—32no—
261Q8—31no—
262Q8—182no—
263Q8—182no—
264Q8—181no—
265Q8—181no—
266Q8—241no—
267Q8#379611yescharacteristic
268D4—34no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—62no—
272C2×C4—92no—
273C2×C4—182no—
274C2×C4—182no—
275C2×C4—181no—
276C2×C4—181no—
277C2×C4—722no—
278C2×C4D3×S411yescharacteristic
279C8—62no—
280C8—181no—
281C8—181no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—81no—
286C2×C3—81no—
287C2×C3—81no—
288C2×C3—244no—
289C2×C3#591411yescharacteristic
290C2×C3C22×#22611yescharacteristic
291C2×C3#592611yescharacteristic
292D3—24no—
293C22—34no—
294C22—62no—
295C22—62no—
296C22C2×D3×S411yescharacteristic
297C4—31no—
298C4—31no—
299C4—61no—
300C4—122no—
301C4—182no—
302C4—361no—
303C4—361no—
304C4D3×#22612yes—
305C3—41no—
306C3—81no—
307C3#8539311yescharacteristic, complemented
308C2—64no—
309C2#1059642511yescharacteristic
310C2C2×D3×#22611yescharacteristic
311C2#1059646011yescharacteristic
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)#41
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. C4×C3⋊#226C2semidirect
3. C3×#5391C2semidirect
4. D3×#859C2semidirect
5. #10593211C2semidirect
19. #5391D3semidirect
189. D12#226semidirect
307. C3#85393semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D122
C4×C3⋊#2262
C3×#53912
D3×#8592
#105932112
#853933
#811193
#638984