Finite Group Databasebeta

← #11921695

Subgroups — #11921695

Order 1152.

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

normal subgroups only

This group has 471 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
2#10593625C211yesmaximal, characteristic, complemented
3C3×C4×#226C211yesmaximal, characteristic, complemented
4D3×#859C212yesmaximal, complemented
5#10593210C212yesmaximal, complemented
6#85393—31nomaximal
7#79735—31nomaximal
8C2×D3×SL(2,𝔽3)C2212yes—
9C2×C3×#226C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11Dic3×D12—41nomaximal
12C3⋊#859C2212yes—
13C3×#859C2211yescharacteristic
14#5579—31no—
15Q8×D12D311yescharacteristic
16D4×SL(2,𝔽3)—31no—
17#5419—32no—
18C2×#859—32no—
19#5391—31no—
20#5391—61no—
21C4×#226D311yescharacteristic, complemented
22C3×C4×Q16—31no—
23#5186—32no—
24#5031—31no—
25#5021—31no—
26#4884—32no—
27#4813—32no—
28D3×#103—32no—
29C8×D12—31no—
30#4681—31no—
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×#226C2×D311yescharacteristic
43C2×C3×Q16—31no—
44C3×C4×Q8—31no—
45C3×C4×Q8C2×D311yescharacteristic
46#946—32no—
47C2×#216—32no—
48D4×Dic3—121no—
49C2×D3×C8—32no—
50#897—32no—
51#896—62no—
52D3×C4⋊C4—32no—
53C4×D12—31no—
54C4×D12—61no—
55#870—61no—
56#869—31no—
57#869—31no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)C2×D311yescharacteristic
60#859—32no—
61#859—62no—
62#859C2×D311yescharacteristic
63C3×#107—31no—
64C3×#103—31no—
65C3×#103—31no—
66C3×C4×C8—31no—
67#835—32no—
68(C2×D3)⋊C8—32no—
69#816—32no—
70#816—32no—
71#810—32no—
72#807—32no—
73C4⋊(C3⋊C8)—31no—
74C22×C3×D3—42no—
75C2×C32×C4—41no—
76C2×C3⋊Dic3—42no—
77C2×C3×Dic3—41no—
78C2×C3×Dic3—41no—
79C3×D12—44no—
80C3×SL(2,𝔽3)#4111yescharacteristic
81D3×Dic3—84no—
82#554—91no—
83D4×Q8—31no—
84#487—91no—
85#484—92no—
86#477—91no—
87#467—92no—
88#462—91no—
89#451—92no—
90C4×Q16—31no—
91C8×D4—91no—
92C2×#103—92no—
93C2×C3×Q8—31no—
94C2×C3×Q8C22×D311yescharacteristic
95C2×C3×D4—121no—
96C22×Dic3—122no—
97D3×Q8—22no—
98D3×Q8—62no—
99C2×D12S411yescharacteristic
100C2×C4×D3—32no—
101C2×C4×D3—62no—
102C2×Dic6—32no—
103C2×Dic6—61no—
104C2×SL(2,𝔽3)—21no—
105C2×SL(2,𝔽3)—62no—
106C2×SL(2,𝔽3)C22×D311yescharacteristic
107#226—21no—
108C3×Q16—61no—
109C3×Q16—61no—
110C2×C3×C8—31no—
111C2×C3×C8—31no—
112C3×C4⋊C4—31no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—61no—
115C3×C42—31no—
116C3×C42—61no—
117#217—122no—
118#216—122no—
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4—62no—
121C4⋊Dic3—31no—
122C4⋊Dic3—32no—
123C4⋊Dic3—61no—
124C4⋊Dic3—61no—
125C4⋊Dic3—121no—
126C4⋊Dic3—241no—
127Dic3⋊C4—62no—
128Dic3⋊C4—62no—
129C4×Dic3—41no—
130C2×C3⋊C8—32no—
131D3×C8—122no—
132C22×C32—41no—
133C2×C3×D3—44no—
134C2×C3×D3—82no—
135C2×C3×D3—82no—
136C32×C4—42no—
137C3⋊Dic3—82no—
138C3×Dic3—42no—
139C3×Dic3—81no—
140C22×Q8—32no—
141C2×Q16—31no—
142C2×Q16—92no—
143C22×C8—92no—
144C4⋊Q8—91no—
145C4⋊Q8—91no—
146#125—181no—
147C22⋊Q8—92no—
148C22⋊Q8—92no—
149C22⋊Q8—182no—
150C4×Q8—31no—
151C4×Q8D3211yescharacteristic
152C4×D4—91no—
153C4×D4—181no—
154C2×C4⋊C4—92no—
155#107—31no—
156#107—92no—
157C4⋊C8—91no—
158#103—31no—
159#103—31no—
160#103—92no—
161#103—92no—
162#103—92no—
163#103—92no—
164C22⋊C8—92no—
165C4×C8—31no—
166C23×C3—122no—
167C22×D3C2×S412yes—
168C3×Q8—31no—
169C3×Q8—61no—
170C3×Q8—61no—
171C3×Q8#23711yescharacteristic
172C3×D4—124no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—41no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—81no—
179C2×C3×C4C2×S411yescharacteristic
180C2×Dic3—32no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—62no—
184C2×Dic3—122no—
185C2×Dic3—242no—
186C2×Dic3—244no—
187D12#22614yescomplemented
188C4×D3—62no—
189C4×D3—62no—
190C4×D3—242no—
191Dic6—62no—
192Dic6—62no—
193Dic6—241no—
194SL(2,𝔽3)—21no—
195SL(2,𝔽3)#23511yescharacteristic
196C3×C8—61no—
197C3×C8—61no—
198C3⋊C8—62no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C3×D3—84no—
203C2×Q8—31no—
204C2×Q8—62no—
205C2×Q8—92no—
206C2×Q8—181no—
207C2×Q8—182no—
208C2×Q8C2×D3211yescharacteristic
209C2×D4—31no—
210C22×C4—92no—
211C22×C4—182no—
212Q16—61no—
213Q16—61no—
214Q16—362no—
215C2×C8—31no—
216C2×C8—31no—
217C2×C8—92no—
218C2×C8—362no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—61no—
222C4⋊C4—92no—
223C4⋊C4—91no—
224C4⋊C4—181no—
225C4⋊C4—182no—
226C4⋊C4—182no—
227C4⋊C4—181no—
228C22⋊C4—92no—
229C22⋊C4—182no—
230C42—31no—
231C42—61no—
232C22×C3—41no—
233C22×C3—81no—
234C22×C3—124no—
235C22×C3—242no—
236C22×C3—242no—
237C22×C3C22×S411yescharacteristic
238C2×D3—22no—
239C2×D3—22no—
240C2×D3C2×#22614yes—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—42no—
244C3×C4—61no—
245C3×C4—82no—
246C3×C4—121no—
247C3×C4—121no—
248C3×C4C2×#22612yes—
249Dic3—42no—
250Dic3—62no—
251Dic3—81no—
252Dic3—122no—
253Dic3—242no—
254Dic3—482no—
255C32—41noSylow
256C23—32no—
257Q8—31no—
258Q8—61no—
259Q8—61no—
260Q8—182no—
261Q8—182no—
262Q8—721no—
263Q8#379511yescharacteristic
264D4—34no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—92no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4—722no—
274C2×C4D3×S411yescharacteristic
275C8—61no—
276C8—61no—
277C8—182no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—81no—
282C2×C3—81no—
283C2×C3—81no—
284C2×C3—244no—
285C2×C3#591411yescharacteristic
286C2×C3C22×#22611yescharacteristic
287C2×C3#592611yescharacteristic
288D3—24no—
289C22—34no—
290C22—62no—
291C22—62no—
292C22C2×D3×S411yescharacteristic
293C4—31no—
294C4—31no—
295C4—61no—
296C4—121no—
297C4—121no—
298C4—182no—
299C4—362no—
300C4D3×#22612yes—
301C3—41no—
302C3—81no—
303C3#8539311yescharacteristic, complemented
304C2—64no—
305C2#1059642411yescharacteristic
306C2C2×D3×#22611yescharacteristic
307C2#1059645911yescharacteristic
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. #10593625C2semidirect
3. C3×C4×#226C2semidirect
4. D3×#859C2semidirect
5. #10593210C2semidirect
21. C4×#226D3semidirect
187. D12#226semidirect
303. 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
#105936252
C3×C4×#2262
D3×#8592
#105932102
#853933
#797353
Dic3×D124