Finite Group Databasebeta

← #11922297

Subgroups — #11922297

Order 1152.

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

normal subgroups only

This group has 621 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
1C2×C3⋊#994C211yesmaximal, characteristic
2C2×C3×#984C211yesmaximal, characteristic, complemented
3#10593658C211yesmaximal, characteristic, complemented
4#10593194C214yesmaximal, complemented
5#85527—31nomaximal
6#83982—31nomaximal
7C2×C3×#41⋊C3C2211yescharacteristic
8C3⋊#994—22no—
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#984—22no—
11C2×C3×#226C2212yescomplemented
12#63873—41nomaximal
13C3⋊#859C2212yescomplemented
14C2×#1009D311yescharacteristic
15C2×#994—31no—
16C2×#984D311yescharacteristic, complemented
17C2×C3×#137—31no—
18#5816—31no—
19C22×#215—31no—
20#5425—34no—
21#5424—31no—
22#5424—61no—
23#5233—31no—
24#5187—34no—
25#5169—34no—
26#5131—31no—
27#5129—31no—
28C2×C3×Dic6—41no—
29C2×C3×Dic6—41no—
30C3×#41⋊C3D412yes—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊(#41⋊C3)—22no—
33C3×#226—22no—
34C4×C3⋊Dic3—41no—
35#3724—44no—
36#3009—91noSylow
37C2×C3×#41C2×D311yescharacteristic
38C22×C3×Q8—31no—
39#1009—22no—
40#1009—62no—
41C2×#239C2×D312yes—
42C2×D3×D4—32no—
43C2×#235—31no—
44C22×D12—31no—
45#994—62no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3—32no—
48C2×#41⋊C3C2×D311yescharacteristic
49#984—22no—
50C2×#226C2×D312yescomplemented
51C3×#137—62no—
52C3×#137—62no—
53C2×C3×Q16—32no—
54C2×C3×SD16—32no—
55C2×C3×M16—31no—
56#947—61no—
57#946—34no—
58#944—61no—
59#944—121no—
60C2×#215—34no—
61C2×#215—64no—
62#929—62no—
63#926—31no—
64C22×C3⋊C8—31no—
65#859—32no—
66#859—62no—
67#835—32no—
68#832—32no—
69#821—34no—
70(C2×D3)⋊C8—34no—
71C2×C32×C4—41no—
72C2×C3⋊Dic3—42no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3—42no—
75C3×Dic6—82no—
76C3×Dic6—82no—
77C3×SL(2,𝔽3)C2×D411yescharacteristic
78C2×H(5,𝔽2)—31no—
79C2×#137—31no—
80C22×SD16—91no—
81#536—91no—
82#452—94no—
83#450—94no—
84#418—91no—
85#416—91no—
86#408—91no—
87C23×D3—61no—
88C3×#41—32no—
89C3×#41#6712yes—
90C2×C3×Q8—32no—
91C2×C3×Q8—62no—
92C2×C3×Q8—62no—
93C2×C3×Q8—61no—
94C2×C3×Q8—121no—
95C2×C3×Q8C22×D311yescharacteristic
96C2×C3×D4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—61no—
99C2×#67—32no—
100#239—22no—
101#239—62no—
102D3×D4—64no—
103D3×D4—64no—
104#235—62no—
105#235—62no—
106C2×D12—32no—
107C2×D12—34no—
108C2×D12—31no—
109C2×D12—62no—
110C2×C4×D3—32no—
111C2×Dic6—41no—
112C2×Dic6S411yescharacteristic
113#41⋊C3—22no—
114#41⋊C3—62no—
115#41⋊C3#6712yes—
116C2×SL(2,𝔽3)—21no—
117C2×SL(2,𝔽3)C22×D311yescharacteristic
118#226—22no—
119C3×Q16—62no—
120C3×Q16—62no—
121C3×SD16—64no—
122C3×M16—62no—
123C2×C3×C8—32no—
124#217—61no—
125#215—68no—
126(C2×D3)⋊C4—64no—
127C4⋊Dic3—32no—
128Dic3⋊C4—64no—
129Dic3⋊C4—124no—
130C4×Dic3—61no—
131C4×Dic3—121no—
132C4×Dic3—241no—
133C2×C3⋊C8—32no—
134C2×C3⋊C8—62no—
135C22×C32—41no—
136C32×C4—42no—
137C3⋊Dic3—44no—
138C3×Dic3—82no—
139C3×Dic3—82no—
140H(5,𝔽2)—62no—
141H(5,𝔽2)—182no—
142C2×#41—32no—
143C2×#41—91no—
144C2×#41D3211yescharacteristic
145C22×Q8—31no—
146C22×D4—91no—
147C22×D4—92no—
148#137—62no—
149#137—62no—
150C2×Q16—32no—
151C2×SD16—32no—
152C2×SD16—94no—
153C2×SD16—184no—
154C2×M16—31no—
155C22×C8—91no—
156C4⋊Q8—181no—
157#124—181no—
158#123—182no—
159C22⋊Q8—94no—
160#117—91no—
161#103—92no—
162#102—94no—
163#102—92no—
164C22⋊C8—94no—
165C23×C3—31no—
166C22×D3—32no—
167C22×D3—64no—
168C22×D3—242no—
169C3×Q8—31no—
170C3×Q8—62no—
171C3×Q8—62no—
172C3×Q8—122no—
173C3×Q8—242no—
174C3×Q8C2×#6711yescharacteristic
175C3×D4—32no—
176C3×D4—61no—
177C2×C3×C4—31no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—41no—
181C2×C3×C4—62no—
182C2×C3×C4—81no—
183C2×C3×C4—122no—
184C2×C3×C4—122no—
185C2×C3×C4C2×S411yescharacteristic
186#67—64no—
187C2×Dic3—42no—
188C2×Dic3—62no—
189C2×Dic3—122no—
190C2×Dic3—242no—
191C2×Dic3C2×S412yes—
192D12—34no—
193D12—62no—
194D12—62no—
195C4×D3—62no—
196C4×D3—62no—
197Dic6—22no—
198Dic6—82no—
199SL(2,𝔽3)—21no—
200SL(2,𝔽3)C2×#6711yescharacteristic
201C3×C8—62no—
202C3⋊C8—62no—
203C2×C32—42no—
204C2×C32—41no—
205C24—181no—
206#41—32no—
207#41—62no—
208#41—182no—
209#41—182no—
210#41—182no—
211#41#62512yes—
212C2×Q8—31no—
213C2×Q8—32no—
214C2×Q8—62no—
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8C2×D3211yescharacteristic
218C2×D4—31no—
219C2×D4—92no—
220C2×D4—94no—
221C2×D4—92no—
222C2×D4—91no—
223C2×D4—182no—
224C2×D4—184no—
225C2×D4—184no—
226C22×C4—31no—
227C22×C4—61no—
228C22×C4—92no—
229Q16—62no—
230Q16—62no—
231SD16—64no—
232SD16—188no—
233M16—62no—
234C2×C8—32no—
235C2×C8—92no—
236C2×C8—182no—
237C4⋊C4—92no—
238C4⋊C4—184no—
239C22⋊C4—184no—
240C22⋊C4—181no—
241C42—181no—
242C22×C3—32no—
243C22×C3—41no—
244C22×C3—62no—
245C22×C3—81no—
246C22×C3C22×S411yescharacteristic
247C2×D3—64no—
248C2×D3—62no—
249C2×D3—244no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—42no—
253C3×C4—82no—
254C3×C4—122no—
255C3×C4—242no—
256C3×C4GL(2,ℤ/4)12yes—
257Dic3—22no—
258Dic3—82no—
259Dic3—124no—
260Dic3—122no—
261Dic3—244no—
262C32—41noSylow
263C23—31no—
264C23—92no—
265C23—184no—
266C23—722no—
267Q8—31no—
268Q8—62no—
269Q8—62no—
270Q8—62no—
271Q8—122no—
272Q8C2×#62511yescharacteristic
273D4—32no—
274D4—61no—
275D4—94no—
276D4—182no—
277D4—184no—
278D4—182no—
279C2×C4—31no—
280C2×C4—32no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—62no—
284C2×C4—122no—
285C2×C4—182no—
286C2×C4—182no—
287C2×C4—182no—
288C2×C4D3×S411yescharacteristic
289C8—62no—
290C8—182no—
291C2×C3—42no—
292C2×C3—41no—
293C2×C3—62no—
294C2×C3—82no—
295C2×C3—81no—
296C2×C3#592412yes—
297C2×C3C2×GL(2,ℤ/4)11yescharacteristic
298D3—122no—
299C22—32no—
300C22—62no—
301C22—184no—
302C22—182no—
303C22—724no—
304C22C2×D3×S411yescharacteristic
305C4—31no—
306C4—31no—
307C4—62no—
308C4—122no—
309C4—362no—
310C4#6420912yes—
311C3—41no—
312C3—81no—
313C3#8552711yescharacteristic, complemented
314C2—62no—
315C2—362no—
316C2#1059645012yes—
317C2C2×#6420911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3⋊#994C2
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×D4
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. C2×C3×#984C2semidirect
3. #10593658C2semidirect
4. #10593194C2semidirect
11. C2×C3×#226C22semidirect
13. C3⋊#859C22semidirect
16. C2×#984D3semidirect
50. C2×#226C2×D3semidirect
313. C3#85527semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C2×C3×#9842
#105936582
#105931942
#855273
#839823
#638734