Finite Group Databasebeta

← #11922381

Subgroups — #11922381

Order 1152.

613 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 613 conjugacy classes of subgroups — too many to draw. Only its 67 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×#64267C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3C3×#5424C211yesmaximal, characteristic, complemented
4#10593214C212yesmaximal, complemented
5#10593207C212yesmaximal, complemented
6#85366—31nomaximal
7#78820—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#64267C414yescomplemented
10C2×C3⋊GL(2,𝔽3)C2212yes—
11#63894—41nomaximal
12Dic3×SL(2,𝔽3)C2212yescomplemented
13C3×#859C2211yescharacteristic, complemented
14C3×#859C2211yescharacteristic, complemented
15C2×#986—31no—
16C2×#986—61no—
17#5827D311yescharacteristic, complemented
18C2×#949—31no—
19#5521—31no—
20#520⋊C3—31no—
21#5424D311yescharacteristic, complemented
22#5394—32no—
23C4×GL(2,𝔽3)—32no—
24C3×#416—31no—
25C2×#821—31no—
26#5101—31no—
27#4995—31no—
28#4817—32no—
29#4788—32no—
30#4785—32no—
31#4756—32no—
32C2×C4⋊(C3⋊D3)—41no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×#624—42no—
36C3⋊GL(2,𝔽3)C2×C414yescomplemented
37C3×C4⋊Dic3—41no—
38C3×C4×Dic3—41no—
39#3722—42no—
40#2946—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42C22×D12—31no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3C2×D311yescharacteristic
45#986—34no—
46#986—64no—
47C2×GL(2,𝔽3)—32no—
48C2×GL(2,𝔽3)—62no—
49C22×C3×C8—31no—
50C3×#117—31no—
51#949—34no—
52#949—34no—
53Q8×Dic3C2×D312yes—
54C2×#215—32no—
55D4×Dic3—32no—
56C2×#213—32no—
57#926—31no—
58C2×C4⋊Dic3—31no—
59C2×#208—31no—
60#893—32no—
61#893—32no—
62#893—121no—
63D3×C22⋊C4—62no—
64C4×D12—61no—
65C4×D12—61no—
66C4×SL(2,𝔽3)—32no—
67#859C2×D311yescharacteristic, complemented
68#859C2×D311yescharacteristic, complemented
69C3×#103—31no—
70C3×#103—31no—
71C3×#102—31no—
72C3×#102—31no—
73#821—32no—
74#821—32no—
75#815—32no—
76C8×Dic3—32no—
77#808—32no—
78#807—32no—
79C22×C3⋊D3—42no—
80C2×C32×C4—41no—
81C4⋊(C3⋊D3)—44no—
82C2×C3×Dic3—41no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—42no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic
86#624—84no—
87C2×AGL(1,ℤ/8)—91no—
88#520—31no—
89#518—91no—
90#442—92no—
91#440—92no—
92C4×SD16—92no—
93C4×D8—92no—
94#428—91no—
95#416—31no—
96C2×#102—91no—
97#405—91no—
98C23×D3—61no—
99C3×#41—32no—
100C3×#41C4×D312yescomplemented
101C2×C3×Q8C22×D311yescharacteristic
102C2×C3×D4—31no—
103C22×C3×C4—31no—
104C22×Dic3—61no—
105C2×D12—32no—
106C2×D12—34no—
107C2×D12—62no—
108C2×D12—61no—
109C2×D12—121no—
110C2×D12—241no—
111C2×C4×D3—62no—
112C2×C4×D3—62no—
113C2×C4×D3—122no—
114#41⋊C3—22no—
115#41⋊C3C4×D312yescomplemented
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)C22×D311yescharacteristic
118GL(2,𝔽3)—34no—
119GL(2,𝔽3)—64no—
120C2×C3×C8—31no—
121C2×C3×C8—31no—
122C2×C3×C8—62no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—121no—
126C3×C22⋊C4—61no—
127C3×C42—61no—
128#217—32no—
129#215—34no—
130#215—34no—
131#213—38no—
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4—62no—
134(C2×D3)⋊C4—122no—
135C4⋊Dic3—31no—
136C4⋊Dic3—32no—
137C4⋊Dic3S411yescharacteristic
138C4×Dic3—32no—
139C4×Dic3—41no—
140#208—34no—
141C2×C3⋊C8—32no—
142C22×C32—41no—
143C2×C3⋊D3—44no—
144C2×C3⋊D3—82no—
145C2×C3⋊D3—82no—
146C32×C4—42no—
147C3×Dic3—42no—
148C3×Dic3—82no—
149C3×Dic3—81no—
150C2×#41D3211yescharacteristic
151C22×D4—91no—
152AGL(1,ℤ/8)—94no—
153AGL(1,ℤ/8)—94no—
154C2×SD16—92no—
155C2×D8—92no—
156C2×M16—91no—
157C22×C8—31no—
158C4×Q8—32no—
159C4×D4—92no—
160C4×D4—92no—
161C4×D4—92no—
162C4×D4—181no—
163C4×D4—181no—
164#117—31no—
165#117—91no—
166C2×C4⋊C4—91no—
167C2×C22⋊C4—182no—
168#107—92no—
169#106—92no—
170#103—31no—
171#103—31no—
172#102—31no—
173#102—31no—
174#102—92no—
175#102—92no—
176#97—92no—
177C4×C8—92no—
178C23×C3—31no—
179C22×D3—62no—
180C22×D3—64no—
181C22×D3—122no—
182C22×D3—242no—
183C22×D3—241no—
184C22×D3—241no—
185C3×Q8—31no—
186C3×Q8C2×C4×D311yescharacteristic
187C3×D4—32no—
188C3×D4—61no—
189C2×C3×C4—31no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—41no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—81no—
196C2×C3×C4—122no—
197C2×C3×C4C2×S411yescharacteristic
198C2×Dic3—32no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—241no—
202C2×Dic3C2×S412yes—
203D12—34no—
204D12—62no—
205D12—64no—
206D12—124no—
207D12—244no—
208C4×D3—242no—
209C4×D3—244no—
210C4×D3—242no—
211SL(2,𝔽3)—21no—
212SL(2,𝔽3)C2×C4×D311yescharacteristic
213C3×C8—61no—
214C3×C8—61no—
215C3⋊C8—34no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C3⋊D3—84no—
220C24—181no—
221#41—32no—
222#41#62412yescomplemented
223C2×Q8C2×D3211yescharacteristic
224C2×D4—31no—
225C2×D4—92no—
226C2×D4—94no—
227C2×D4—182no—
228C2×D4—181no—
229C22×C4—31no—
230C22×C4—182no—
231C22×C4—182no—
232C22×C4—181no—
233SD16—94no—
234SD16—94no—
235D8—98no—
236M16—94no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—62no—
240C2×C8—92no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—31no—
244C4⋊C4—92no—
245C4⋊C4—91no—
246C22⋊C4—61no—
247C22⋊C4—92no—
248C22⋊C4—182no—
249C22⋊C4—182no—
250C42—61no—
251C42—92no—
252C22×C3—32no—
253C22×C3—41no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—81no—
257C22×C3C22×S411yescharacteristic
258C2×D3—64no—
259C2×D3—124no—
260C2×D3—122no—
261C2×D3—122no—
262C2×D3—244no—
263C2×D3—242no—
264C2×D3—242no—
265C2×D3—244no—
266C2×D3—482no—
267C2×D3—482no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—42no—
271C3×C4—82no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—242no—
275C3×C4C4×S412yes—
276Dic3—22no—
277Dic3—42no—
278Dic3—62no—
279Dic3—81no—
280C32—41noSylow
281C23—31no—
282C23—184no—
283C23—182no—
284C23—721no—
285C23—721no—
286Q8—31no—
287Q8C2×#62411yescharacteristic
288D4—32no—
289D4—61no—
290D4—94no—
291D4—182no—
292D4—184no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—92no—
300C2×C4—722no—
301C2×C4—722no—
302C2×C4—721no—
303C2×C4D3×S411yescharacteristic
304C8—61no—
305C8—61no—
306C8—94no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—62no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3—81no—
314C2×C3C2×C4×S411yescharacteristic
315C2×C3#592511yescharacteristic
316C2×C3#592611yescharacteristic
317D3—124no—
318D3—244no—
319D3—484no—
320C22—32no—
321C22—61no—
322C22—61no—
323C22—184no—
324C22—362no—
325C22—362no—
326C22—724no—
327C22C2×D3×S411yescharacteristic
328C4—31no—
329C4—31no—
330C4—62no—
331C4—121no—
332C4—121no—
333C4—182no—
334C4(C3⋊S4)⋊C412yes—
335C3—41no—
336C3—81no—
337C3#8536611yescharacteristic, complemented
338C2—62no—
339C2—364no—
340C2#1059644911yescharacteristic
341C2#1059645911yescharacteristic
342C2C2×(C3⋊S4)⋊C411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×#64267C2
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. C3×#5424C2semidirect
4. #10593214C2semidirect
5. #10593207C2semidirect
9. #64267C4semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
13. C3×#859C22semidirect
14. C3×#859C22semidirect
17. #5827D3semidirect
21. #5424D3semidirect
33. C3×#41⋊C3C2×C4semidirect
36. C3⋊GL(2,𝔽3)C2×C4semidirect
67. #859C2×D3semidirect
68. #859C2×D3semidirect
100. C3×#41C4×D3semidirect
115. #41⋊C3C4×D3semidirect
222. #41#624semidirect
337. C3#85366semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
#5827⋊C32
C3×#54242
#105932142
#105932072
#853663
#788203
#638944