Finite Group Databasebeta

← #11921673

Subgroups — #11921673

Order 1152.

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

normal subgroups only

This group has 579 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, complemented
2#10593620C211yesmaximal, characteristic, complemented
3C3×C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593217C212yesmaximal, complemented
5D3×#859C212yesmaximal, complemented
6#85397—31nomaximal
7#79678—31nomaximal
8C2×#3795—41nomaximal
9C2×D3×SL(2,𝔽3)C2212yes—
10C2×C3×GL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2212yes—
13C3×#859C2211yescharacteristic
14Q8×D12D311yescharacteristic, complemented
15#5551—31no—
16D4×SL(2,𝔽3)—31no—
17#5421—32no—
18C2×#859—32no—
19C4×GL(2,𝔽3)D311yescharacteristic, complemented
20#5386—31no—
21#5386—61no—
22C3×C4×SD16—31no—
23#5170—32no—
24#5020—31no—
25#5019—31no—
26#4862—32no—
27D3×#103—32no—
28#4778—32no—
29C8×D12—31no—
30#4681—31no—
31C2×#634—41no—
32C2×C3×D12—41no—
33C2×C3×C4×D3—41no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×#625—42no—
36C2×D3×Dic3—42no—
37#3795—48no—
38D3×SL(2,𝔽3)—22no—
39C3×GL(2,𝔽3)—21no—
40#3298—91noSylow
41C2×D3×Q8C2×D312yes—
42C2×#237—121no—
43C2×#235—61no—
44C22×SL(2,𝔽3)—32no—
45C2×GL(2,𝔽3)C2×D311yescharacteristic
46C2×C3×SD16—31no—
47C3×C4×Q8C2×D311yescharacteristic
48C3×C4×D4—31no—
49#938—32no—
50C2×#214—32no—
51#924—61no—
52C2×D3×C8—32no—
53#897—32no—
54D3×C4⋊C4—32no—
55#883—62no—
56C4×D12—31no—
57#869—31no—
58#869—31no—
59C4×SL(2,𝔽3)—21no—
60C4×SL(2,𝔽3)C2×D311yescharacteristic
61#859—32no—
62#859—62no—
63#859C2×D311yescharacteristic
64C3×#106—31no—
65C3×#103—31no—
66C3×#102—31no—
67C3×C4×C8—31no—
68#835—32no—
69(C2×D3)⋊C8—32no—
70#816—32no—
71#816—32no—
72#810—32no—
73#808—32no—
74C4⋊(C3⋊C8)—31no—
75C22×C3×D3—41no—
76C22×C3×D3—42no—
77C2×C32×C4—41no—
78C2×C3⋊Dic3—42no—
79#634—44no—
80C2×C3×Dic3—41no—
81C3×D12—44no—
82C3×C4×D3—44no—
83C3×SL(2,𝔽3)#4111yescharacteristic, complemented
84#625—48no—
85D3×Dic3—48no—
86D4×Q8—31no—
87#547—91no—
88#487—91no—
89#483—92no—
90#475—91no—
91#465—92no—
92#461—91no—
93#448—92no—
94C4×SD16—31no—
95C8×D4—91no—
96C2×#103—92no—
97C2×C3×Q8C22×D311yescharacteristic
98C2×C3×D4—31no—
99C2×C3×D4—121no—
100C22×C3×C4—61no—
101C2×#67—62no—
102C2×#67—122no—
103C22×Dic3—122no—
104D3×Q8—22no—
105D3×Q8—62no—
106#237—128no—
107#235—124no—
108C2×D12S411yescharacteristic
109C2×C4×D3—32no—
110C2×C4×D3—41no—
111C2×C4×D3—62no—
112C2×Dic6—32no—
113C2×Dic6—61no—
114C2×Dic6—121no—
115C2×Dic6—241no—
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)—62no—
118C2×SL(2,𝔽3)C22×D311yescharacteristic
119GL(2,𝔽3)—21no—
120C3×SD16—61no—
121C3×SD16—61no—
122C2×C3×C8—31no—
123C2×C3×C8—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—31no—
126C3×C22⋊C4—61no—
127C3×C42—31no—
128#217—62no—
129#214—122no—
130(C2×D3)⋊C4—32no—
131C4⋊Dic3—31no—
132C4⋊Dic3—32no—
133C4⋊Dic3—61no—
134C4⋊Dic3—61no—
135Dic3⋊C4—62no—
136C2×C3⋊C8—32no—
137D3×C8—122no—
138C22×C32—41no—
139C2×C3×D3—42no—
140C2×C3×D3—44no—
141C2×C3×D3—82no—
142C2×C3×D3—82no—
143C2×C3×D3—81no—
144C2×C3×D3—81no—
145C32×C4—42no—
146C3⋊Dic3—44no—
147C3×Dic3—42no—
148C2×#41—181no—
149C22×Q8—32no—
150C2×SD16—31no—
151C2×SD16—92no—
152C22×C8—92no—
153C4⋊Q8—91no—
154C4⋊Q8—91no—
155#123—182no—
156C22⋊Q8—92no—
157C22⋊Q8—181no—
158#121—92no—
159C4×Q8D3211yescharacteristic
160C4×D4—31no—
161C4×D4—91no—
162C2×C4⋊C4—92no—
163#106—31no—
164#106—92no—
165C4⋊C8—91no—
166#103—31no—
167#103—92no—
168#103—92no—
169#103—92no—
170#103—92no—
171#102—31no—
172C22⋊C8—92no—
173C4×C8—31no—
174C23×C3—61no—
175C23×C3—122no—
176C22×D3—41no—
177C22×D3C2×S412yes—
178C3×Q8—31no—
179C3×Q8#23711yescharacteristic, complemented
180C3×D4—61no—
181C3×D4—61no—
182C3×D4—124no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—41no—
186C2×C3×C4—61no—
187C2×C3×C4—81no—
188C2×C3×C4—122no—
189C2×C3×C4C2×S411yescharacteristic
190#67—128no—
191#67—124no—
192C2×Dic3—32no—
193C2×Dic3—41no—
194C2×Dic3—62no—
195C2×Dic3—122no—
196C2×Dic3—128no—
197C2×Dic3—242no—
198D12GL(2,𝔽3)14yescomplemented
199C4×D3—44no—
200C4×D3—62no—
201C4×D3—62no—
202C4×D3—124no—
203Dic6—62no—
204Dic6—62no—
205Dic6—124no—
206Dic6—122no—
207Dic6—244no—
208SL(2,𝔽3)—21no—
209SL(2,𝔽3)#23511yescharacteristic, complemented
210C3×C8—61no—
211C3×C8—61no—
212C3⋊C8—62no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—41no—
216C3×D3—84no—
217C3×D3—82no—
218#41—364no—
219C2×Q8—62no—
220C2×Q8—92no—
221C2×Q8—181no—
222C2×Q8—182no—
223C2×Q8C2×D3211yescharacteristic
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—182no—
227C22×C4—61no—
228C22×C4—92no—
229C22×C4—182no—
230SD16—61no—
231SD16—61no—
232SD16—362no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—92no—
236C2×C8—362no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—92no—
240C4⋊C4—91no—
241C4⋊C4—181no—
242C4⋊C4—181no—
243C4⋊C4—182no—
244C22⋊C4—61no—
245C22⋊C4—92no—
246C22⋊C4—182no—
247C42—31no—
248C22×C3—41no—
249C22×C3—81no—
250C22×C3—124no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3—242no—
255C22×C3—242no—
256C22×C3C22×S411yescharacteristic
257C2×D3—22no—
258C2×D3—22no—
259C2×D3—42no—
260C2×D3—81no—
261C2×D3—81no—
262C2×D3C2×GL(2,𝔽3)14yes—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—42no—
266C3×C4—61no—
267C3×C4—82no—
268C3×C4—121no—
269C3×C4C2×GL(2,𝔽3)12yes—
270Dic3—42no—
271Dic3—62no—
272Dic3—124no—
273Dic3—122no—
274Dic3—244no—
275C32—41noSylow
276C23—32no—
277C23—61no—
278Q8—31no—
279Q8—182no—
280Q8—182no—
281Q8—362no—
282Q8#379511yescharacteristic, complemented
283D4—34no—
284D4—61no—
285D4—61no—
286D4—364no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—92no—
291C2×C4—122no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4—364no—
296C2×C4D3×S411yescharacteristic
297C8—61no—
298C8—61no—
299C8—182no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—81no—
304C2×C3—81no—
305C2×C3—81no—
306C2×C3—244no—
307C2×C3—241no—
308C2×C3#591411yescharacteristic
309C2×C3C22×GL(2,𝔽3)11yescharacteristic
310C2×C3#592711yescharacteristic
311D3—24no—
312D3—82no—
313C22—34no—
314C22—62no—
315C22—62no—
316C22—121no—
317C22—121no—
318C22—121no—
319C22C2×D3×S411yescharacteristic
320C4—31no—
321C4—31no—
322C4—61no—
323C4—121no—
324C4—182no—
325C4—362no—
326C4D3×GL(2,𝔽3)12yes—
327C3—41no—
328C3—81no—
329C3#8539711yescharacteristic, complemented
330C2—64no—
331C2—241no—
332C2#1059642411yescharacteristic
333C2C2×D3×GL(2,𝔽3)11yescharacteristic
334C2#1059645211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
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
1. SL(2,𝔽3)×D12C2semidirect
2. #10593620C2semidirect
3. C3×C4×GL(2,𝔽3)C2semidirect
4. #10593217C2semidirect
5. D3×#859C2semidirect
14. Q8×D12D3semidirect
19. C4×GL(2,𝔽3)D3semidirect
83. C3×SL(2,𝔽3)#41semidirect
179. C3×Q8#237semidirect
198. D12GL(2,𝔽3)semidirect
209. SL(2,𝔽3)#235semidirect
282. Q8#3795semidirect
329. C3#85397semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D122
#105936202
C3×C4×GL(2,𝔽3)2
#105932172
D3×#8592
#853973
#796783
C2×#37954