Finite Group Databasebeta

← #11922293

Subgroups — #11922293

Order 1152.

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

normal subgroups only

This group has 533 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×#984C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic
3#10593658C211yesmaximal, characteristic
4#10593198C212yesmaximal, complemented
5Dic3×#226C212yesmaximal, complemented
6#85360—31nomaximal
7#83965—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C3×#984C414yescomplemented
10C2×C3×#226C2212yes—
11#63843—41nomaximal
12Dic3×SL(2,𝔽3)C2212yes—
13C3⋊#859C2211yescharacteristic
14C3⋊#859C2211yescharacteristic
15C2×#984D311yescharacteristic
16C2×C3×#137—31no—
17#5827D311yescharacteristic
18#5812—31no—
19#520⋊C3—31no—
20#5424—31no—
21#5424—61no—
22#5388—32no—
23C4×#226—32no—
24#5233—31no—
25C2×#835—31no—
26#5184—32no—
27Dic3×Q16—32no—
28#5164—32no—
29Dic3×SD16—32no—
30#5122—31no—
31#5121—31no—
32C2×C3×Dic6—41no—
33C3×#41⋊C3C2×C412yes—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C3×#226C2×C414yescomplemented
36C4×C3⋊Dic3—41no—
37C3×C4⋊Dic3—41no—
38Dic3⋊Dic3—42no—
39Dic32—42no—
40#2947—91noSylow
41C2×C3×#41C2×D311yescharacteristic
42C22×C3×Q8—31no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3C2×D311yescharacteristic
45#984Dic314yescomplemented
46C2×#226C2×D312yes—
47C3×#137—34no—
48C3×#137—34no—
49C2×C3×Q16—32no—
50C2×C3×SD16—32no—
51C2×C3×M16—31no—
52Q8×Dic3—32no—
53Q8×Dic3—32no—
54Q8×Dic3—61no—
55Q8×Dic3—61no—
56Q8×Dic3C2×D312yes—
57D4×Dic3—32no—
58#926—31no—
59#926—31no—
60#926—62no—
61C2×C4⋊Dic3—31no—
62C22×C3⋊C8—31no—
63#887—121no—
64C4×SL(2,𝔽3)—32no—
65#859—31no—
66#859—31no—
67#859—61no—
68#859—61no—
69#835—32no—
70#835—32no—
71#835—31no—
72#835—31no—
73#832—31no—
74#832—31no—
75#818—32no—
76#817—32no—
77#815—32no—
78C8×Dic3—32no—
79C2×C32×C4—41no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3—41no—
82C2×C3×Dic3—42no—
83C2×C3×Dic3—42no—
84C3×Dic6—44no—
85C3×SL(2,𝔽3)C22×C411yescharacteristic
86C2×#137—31no—
87#520—31no—
88#519—91no—
89#441—92no—
90#440—92no—
91C4×Q16—92no—
92C4×SD16—92no—
93#428—91no—
94#416—91no—
95C2×#103—91no—
96#405—91no—
97C3×#41—32no—
98C3×#41C4×D312yes—
99C2×C3×Q8—32no—
100C2×C3×Q8—34no—
101C2×C3×Q8—62no—
102C2×C3×Q8—61no—
103C2×C3×Q8C22×D311yescharacteristic
104C2×C3×D4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—61no—
107C22×Dic3—61no—
108C2×Dic6—41no—
109#41⋊C3—22no—
110#41⋊C3C2×Dic312yes—
111C2×SL(2,𝔽3)—21no—
112C2×SL(2,𝔽3)C22×D311yescharacteristic
113#226C2×Dic314yescomplemented
114C3×Q16—34no—
115C3×Q16—34no—
116C3×SD16—38no—
117C3×M16—34no—
118C2×C3×C8—32no—
119C3×C4⋊C4—121no—
120#217—32no—
121#217—61no—
122C4⋊Dic3—31no—
123C4⋊Dic3—32no—
124C4⋊Dic3—31no—
125C4⋊Dic3—31no—
126C4⋊Dic3—62no—
127C4⋊Dic3—62no—
128C4⋊Dic3S411yescharacteristic
129Dic3⋊C4—122no—
130C4×Dic3—32no—
131C4×Dic3—62no—
132C4×Dic3—61no—
133C4×Dic3—62no—
134C4×Dic3—122no—
135C4×Dic3—121no—
136C4×Dic3—241no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—31no—
139C2×C3⋊C8—62no—
140C22×C32—41no—
141C32×C4—42no—
142C3⋊Dic3—42no—
143C3⋊Dic3—81no—
144C3×Dic3—44no—
145C3×Dic3—82no—
146C2×#41D3211yescharacteristic
147C22×Q8—31no—
148#137—34no—
149#137—34no—
150C2×Q16—32no—
151C2×SD16—32no—
152C2×M16—31no—
153C22×C8—91no—
154C4×Q8—32no—
155C4×Q8—92no—
156C4×Q8—92no—
157C4×Q8—181no—
158C4×Q8—181no—
159C4×D4—92no—
160#117—91no—
161#117—91no—
162#117—182no—
163C2×C4⋊C4—91no—
164#107—92no—
165#106—92no—
166#103—92no—
167#103—92no—
168#103—91no—
169#103—91no—
170#102—91no—
171#102—91no—
172#97—92no—
173C4×C8—92no—
174C23×C3—31no—
175C3×Q8—31no—
176C3×Q8—34no—
177C3×Q8—62no—
178C3×Q8—64no—
179C3×Q8C2×C4×D311yescharacteristic
180C3×D4—32no—
181C3×D4—61no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—41no—
186C2×C3×C4—64no—
187C2×C3×C4—62no—
188C2×C3×C4—81no—
189C2×C3×C4—122no—
190C2×C3×C4C2×S411yescharacteristic
191C2×Dic3—32no—
192C2×Dic3—42no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—121no—
196C2×Dic3—121no—
197C2×Dic3—241no—
198C2×Dic3—241no—
199C2×Dic3—241no—
200C2×Dic3C2×S412yes—
201Dic6—44no—
202SL(2,𝔽3)—21no—
203SL(2,𝔽3)C22×Dic311yescharacteristic
204C3×C8—34no—
205C3⋊C8—61no—
206C3⋊C8—61no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210#41—32no—
211#41D3×Dic312yes—
212C2×Q8—32no—
213C2×Q8—34no—
214C2×Q8—62no—
215C2×Q8—61no—
216C2×Q8C2×D3211yescharacteristic
217C2×D4—31no—
218C22×C4—31no—
219C22×C4—61no—
220C22×C4—181no—
221Q16—34no—
222Q16—34no—
223SD16—38no—
224M16—34no—
225C2×C8—32no—
226C2×C8—91no—
227C2×C8—91no—
228C2×C8—182no—
229C4⋊C4—31no—
230C4⋊C4—92no—
231C4⋊C4—91no—
232C4⋊C4—91no—
233C4⋊C4—91no—
234C4⋊C4—182no—
235C4⋊C4—182no—
236C22⋊C4—92no—
237C22⋊C4—181no—
238C42—92no—
239C42—182no—
240C42—182no—
241C42—181no—
242C22×C3—32no—
243C22×C3—41no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—81no—
247C22×C3C22×S411yescharacteristic
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—42no—
251C3×C4—64no—
252C3×C4—82no—
253C3×C4—242no—
254C3×C4C4×S412yes—
255Dic3—22no—
256Dic3—44no—
257Dic3—62no—
258Dic3—122no—
259Dic3—121no—
260Dic3—121no—
261Dic3—241no—
262Dic3—242no—
263Dic3—242no—
264C32—41noSylow
265C23—31no—
266Q8—31no—
267Q8—34no—
268Q8—62no—
269Q8—64no—
270Q8C2×D3×Dic311yescharacteristic
271D4—32no—
272D4—61no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—32no—
277C2×C4—64no—
278C2×C4—62no—
279C2×C4—92no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—721no—
283C2×C4D3×S411yescharacteristic
284C8—34no—
285C8—181no—
286C8—181no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—62no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3C2×C4×S411yescharacteristic
295C2×C3#592411yescharacteristic
296C2×C3#592711yescharacteristic
297C22—32no—
298C22—61no—
299C22—61no—
300C22C2×D3×S411yescharacteristic
301C4—31no—
302C4—31no—
303C4—62no—
304C4—64no—
305C4—182no—
306C4—361no—
307C4—361no—
308C4Dic3×S412yes—
309C3—41no—
310C3—81no—
311C3#8536011yescharacteristic, complemented
312C2—62no—
313C2#1059645011yescharacteristic
314C2#1059645311yescharacteristic
315C2C2×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×#41⋊C3C22
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
4. #10593198C2semidirect
5. Dic3×#226C2semidirect
9. C3×#984C4semidirect
35. C3×#226C2×C4semidirect
45. #984Dic3semidirect
113. #226C2×Dic3semidirect
311. C3#85360semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9842
#5827⋊C32
#105936582
#105931982
Dic3×#2262
#853603
#839653
#638434