Finite Group Databasebeta

← #11922385

Subgroups — #11922385

Order 1152.

593 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×C3×#986C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4Dic3×GL(2,𝔽3)C212yesmaximal, complemented
5#10593204C212yesmaximal, complemented
6#85366—31nomaximal
7#83930—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#986C414yescomplemented
10C2×C3×GL(2,𝔽3)C2212yes—
11#63898—41nomaximal
12Dic3×SL(2,𝔽3)C2212yescomplemented
13C3⋊#859C2211yescharacteristic, complemented
14C3⋊#859C2211yescharacteristic, complemented
15C2×#986D311yescharacteristic
16C2×C3×AGL(1,ℤ/8)—31no—
17#5827D311yescharacteristic, complemented
18#5798—31no—
19#520⋊C3—31no—
20#5424—31no—
21#5424—61no—
22#5394—32no—
23C4×GL(2,𝔽3)—32no—
24#5233—31no—
25C2×#832—31no—
26#5164—32no—
27Dic3×SD16—32no—
28#5152—32no—
29Dic3×D8—32no—
30#5122—31no—
31#5121—31no—
32C2×C3×D12—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×D3×Dic3—42no—
36C3×GL(2,𝔽3)C2×C414yescomplemented
37C4×C3⋊Dic3—41no—
38C3×C4⋊Dic3—41no—
39(C2×D3)⋊Dic3—42no—
40#2946—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42C22×C3×D4—31no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3C2×D311yescharacteristic
45#986Dic314yescomplemented
46C2×GL(2,𝔽3)C2×D312yes—
47C3×AGL(1,ℤ/8)—34no—
48C3×AGL(1,ℤ/8)—34no—
49C2×C3×SD16—32no—
50C2×C3×D8—32no—
51C2×C3×M16—31no—
52C2×#217—62no—
53Q8×Dic3C2×D312yes—
54D4×Dic3—32no—
55D4×Dic3—32no—
56D4×Dic3—32no—
57D4×Dic3—61no—
58D4×Dic3—61no—
59#926—31no—
60#926—31no—
61C2×C4⋊Dic3—31no—
62C22×C3⋊C8—31no—
63#893—121no—
64C4×SL(2,𝔽3)—32no—
65#859—31no—
66#859—31no—
67#859—61no—
68#859—61no—
69#835—31no—
70#835—31no—
71#832—31no—
72#832—32no—
73#832—31no—
74#832—32no—
75#818—32no—
76#817—32no—
77#815—32no—
78C8×Dic3—32no—
79C22×C3×D3—42no—
80C2×C32×C4—41no—
81C2×C3⋊Dic3—41no—
82C2×C3⋊Dic3—41no—
83C2×C3×Dic3—42no—
84C3×D12—44no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic
86D3×Dic3—84no—
87C2×AGL(1,ℤ/8)—31no—
88#520—31no—
89#518—91no—
90#442—92no—
91#440—92no—
92C4×SD16—92no—
93C4×D8—92no—
94#428—91no—
95#416—91no—
96C2×#102—91no—
97#405—91no—
98C24×C3—61no—
99C3×#41—32no—
100C3×#41C4×D312yescomplemented
101C2×C3×Q8C22×D311yescharacteristic
102C2×C3×D4—31no—
103C2×C3×D4—32no—
104C2×C3×D4—34no—
105C2×C3×D4—62no—
106C2×C3×D4—61no—
107C22×C3×C4—31no—
108C22×Dic3—61no—
109C22×Dic3—62no—
110C22×Dic3—62no—
111C2×D12—41no—
112C2×C4×D3—122no—
113#41⋊C3—22no—
114#41⋊C3C2×Dic312yescomplemented
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)C22×D311yescharacteristic
117GL(2,𝔽3)C2×Dic314yescomplemented
118C3×SD16—34no—
119C3×SD16—34no—
120C3×D8—38no—
121C3×M16—34no—
122C2×C3×C8—32no—
123C3×C4⋊C4—121no—
124#217—32no—
125#217—62no—
126#217—61no—
127#217—62no—
128(C2×D3)⋊C4—122no—
129C4⋊Dic3—31no—
130C4⋊Dic3—32no—
131C4⋊Dic3—31no—
132C4⋊Dic3—31no—
133C4⋊Dic3S411yescharacteristic
134C4×Dic3—32no—
135C4×Dic3—61no—
136C4×Dic3—121no—
137C4×Dic3—241no—
138C2×C3⋊C8—31no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—62no—
141C22×C32—41no—
142C2×C3×D3—44no—
143C2×C3×D3—82no—
144C2×C3×D3—82no—
145C32×C4—42no—
146C3⋊Dic3—42no—
147C3⋊Dic3—81no—
148C3×Dic3—82no—
149C2×#41D3211yescharacteristic
150C22×D4—31no—
151AGL(1,ℤ/8)—34no—
152AGL(1,ℤ/8)—34no—
153C2×SD16—32no—
154C2×D8—32no—
155C2×M16—31no—
156C22×C8—91no—
157C4×Q8—32no—
158C4×D4—92no—
159C4×D4—92no—
160C4×D4—92no—
161C4×D4—181no—
162C4×D4—181no—
163#117—91no—
164#117—91no—
165C2×C4⋊C4—91no—
166C2×C22⋊C4—182no—
167#107—92no—
168#106—92no—
169#103—91no—
170#103—91no—
171#102—92no—
172#102—92no—
173#102—91no—
174#102—91no—
175#97—92no—
176C4×C8—92no—
177C23×C3—31no—
178C23×C3—64no—
179C23×C3—62no—
180C23×C3—241no—
181C23×C3—241no—
182C22×D3—42no—
183C3×Q8—31no—
184C3×Q8C2×C4×D311yescharacteristic
185C3×D4—32no—
186C3×D4—34no—
187C3×D4—61no—
188C3×D4—62no—
189C3×D4—64no—
190C2×C3×C4—31no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—81no—
195C2×C3×C4—122no—
196C2×C3×C4C2×S411yescharacteristic
197C2×Dic3—32no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—121no—
201C2×Dic3—121no—
202C2×Dic3—241no—
203C2×Dic3—241no—
204C2×Dic3—242no—
205C2×Dic3—241no—
206C2×Dic3—242no—
207C2×Dic3C2×S412yes—
208D12—44no—
209C4×D3—244no—
210SL(2,𝔽3)—21no—
211SL(2,𝔽3)C22×Dic311yescharacteristic
212C3×C8—34no—
213C3⋊C8—61no—
214C3⋊C8—61no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C3×D3—84no—
219C24—61no—
220#41—32no—
221#41D3×Dic312yescomplemented
222C2×Q8C2×D3211yescharacteristic
223C2×D4—31no—
224C2×D4—32no—
225C2×D4—34no—
226C2×D4—62no—
227C2×D4—61no—
228C22×C4—31no—
229C22×C4—182no—
230C22×C4—181no—
231C22×C4—182no—
232SD16—34no—
233SD16—34no—
234D8—38no—
235M16—34no—
236C2×C8—32no—
237C2×C8—91no—
238C2×C8—91no—
239C2×C8—182no—
240C4⋊C4—31no—
241C4⋊C4—92no—
242C4⋊C4—91no—
243C4⋊C4—91no—
244C4⋊C4—91no—
245C22⋊C4—92no—
246C22⋊C4—182no—
247C22⋊C4—182no—
248C22⋊C4—181no—
249C42—92no—
250C42—181no—
251C22×C3—32no—
252C22×C3—41no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—64no—
256C22×C3—81no—
257C22×C3—122no—
258C22×C3—122no—
259C22×C3—244no—
260C22×C3C22×S411yescharacteristic
261C2×D3—44no—
262C2×D3—82no—
263C2×D3—82no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—42no—
267C3×C4—82no—
268C3×C4—242no—
269C3×C4C4×S412yes—
270Dic3—22no—
271Dic3—62no—
272Dic3—122no—
273Dic3—121no—
274Dic3—121no—
275Dic3—241no—
276Dic3—242no—
277Dic3—242no—
278C32—41noSylow
279C23—31no—
280C23—64no—
281C23—62no—
282C23—241no—
283C23—241no—
284Q8—31no—
285Q8C2×D3×Dic311yescharacteristic
286D4—32no—
287D4—34no—
288D4—61no—
289D4—62no—
290D4—64no—
291C2×C4—31no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—92no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—722no—
299C2×C4—721no—
300C2×C4—722no—
301C2×C4D3×S411yescharacteristic
302C8—34no—
303C8—181no—
304C8—181no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—62no—
309C2×C3—81no—
310C2×C3—81no—
311C2×C3—81no—
312C2×C3—124no—
313C2×C3C2×C4×S411yescharacteristic
314C2×C3#592511yescharacteristic
315C2×C3#592611yescharacteristic
316D3—84no—
317C22—32no—
318C22—61no—
319C22—61no—
320C22—64no—
321C22—122no—
322C22—122no—
323C22—244no—
324C22C2×D3×S411yescharacteristic
325C4—31no—
326C4—31no—
327C4—62no—
328C4—182no—
329C4—361no—
330C4—361no—
331C4Dic3×S412yes—
332C3—41no—
333C3—81no—
334C3#8536611yescharacteristic, complemented
335C2—62no—
336C2—124no—
337C2#1059644811yescharacteristic
338C2#1059646011yescharacteristic
339C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3×#986C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC2×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. #10593658C2semidirect
4. Dic3×GL(2,𝔽3)C2semidirect
5. #10593204C2semidirect
9. C3×#986C4semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
13. C3⋊#859C22semidirect
14. C3⋊#859C22semidirect
17. #5827D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. C3×GL(2,𝔽3)C2×C4semidirect
45. #986Dic3semidirect
100. C3×#41C4×D3semidirect
114. #41⋊C3C2×Dic3semidirect
117. GL(2,𝔽3)C2×Dic3semidirect
221. #41D3×Dic3semidirect
334. C3#85366semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9862
#5827⋊C32
#105936582
Dic3×GL(2,𝔽3)2
#105932042
#853663
#839303
#638984