Finite Group Databasebeta

← #11922360

Subgroups — #11922360

Order 1152.

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

normal subgroups only

This group has 421 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
1#5827⋊C3C211yesmaximal, characteristic
2#10593658C211yesmaximal, characteristic
3C3×#5426C211yesmaximal, characteristic
4#10593212C212yesmaximal, complemented
5#10593196C212yesmaximal, complemented
6#85361—31nomaximal
7#78873—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#63919—41nomaximal
10Dic3×SL(2,𝔽3)C2212yes—
11C3⋊#859C2211yescharacteristic
12C3⋊#859C2211yescharacteristic
13C3×#859C2212yes—
14#5827D311yescharacteristic
15#5500—31no—
16#520⋊C3—31no—
17#5426D311yescharacteristic
18#5424—31no—
19#5424—61no—
20#5389—32no—
21#5386—32no—
22C3×#417—31no—
23#5233—31no—
24#5121—31no—
25#5119—31no—
26C2×#808—31no—
27#4796—32no—
28#4791—32no—
29#4766—32no—
30#4761—32no—
31C3×#41⋊C3Q812yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C2×#627—42no—
34C4×C3⋊Dic3—41no—
35C3×C4⋊Dic3—41no—
36C3×C4⋊Dic3—41no—
37#3724—42no—
38#3084—91noSylow
39C2×C3×#41C2×D311yescharacteristic
40C2×#41⋊C3—21no—
41C2×#41⋊C3C2×D311yescharacteristic
42C2×C3×M16—31no—
43C2×C3×C4⋊C4—31no—
44Q8×Dic3C2×D312yes—
45D4×Dic3—32no—
46#926—31no—
47#926—31no—
48C2×C4⋊Dic3—31no—
49#924—62no—
50C22×C3⋊C8—31no—
51#890—32no—
52#890—61no—
53#888—32no—
54#888—61no—
55#888—121no—
56C4×SL(2,𝔽3)—32no—
57#859—31no—
58#859—31no—
59#859—61no—
60#859—61no—
61#859C2×D312yes—
62C3×#103—32no—
63C3×#102—32no—
64#835—31no—
65#835—31no—
66#832—31no—
67#832—31no—
68#818—32no—
69#817—32no—
70Dic3⋊C8—32no—
71Dic3⋊C8—32no—
72#808—32no—
73#808—32no—
74C2×C32×C4—41no—
75C2×C3⋊Dic3—41no—
76C2×C3⋊Dic3—41no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—42no—
79C3×SL(2,𝔽3)C2×Q811yescharacteristic
80#627—84no—
81#544—91no—
82#520—31no—
83#479—92no—
84#478—92no—
85#476—92no—
86#475—92no—
87#428—91no—
88C2×#106—91no—
89#424—91no—
90#417—31no—
91#416—91no—
92C3×#41—32no—
93C3×#41Dic612yescomplemented
94C2×C3×Q8C22×D311yescharacteristic
95C2×C3×D4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—61no—
98C22×Dic3—61no—
99C2×Dic6—62no—
100C2×Dic6—122no—
101#41⋊C3—22no—
102#41⋊C3Dic612yescomplemented
103C2×SL(2,𝔽3)—21no—
104C2×SL(2,𝔽3)C22×D311yescharacteristic
105C3×M16—62no—
106C2×C3×C8—32no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—61no—
109C3×C4⋊C4—121no—
110#217—32no—
111#217—61no—
112C4⋊Dic3—31no—
113C4⋊Dic3—32no—
114C4⋊Dic3—31no—
115C4⋊Dic3—31no—
116C4⋊Dic3—41no—
117C4⋊Dic3—62no—
118C4⋊Dic3S411yescharacteristic
119Dic3⋊C4—62no—
120Dic3⋊C4—62no—
121Dic3⋊C4—122no—
122C4×Dic3—32no—
123C4×Dic3—61no—
124C4×Dic3—121no—
125C4×Dic3—241no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—31no—
128C2×C3⋊C8—62no—
129C22×C32—41no—
130C32×C4—42no—
131C3⋊Dic3—42no—
132C3⋊Dic3—42no—
133C3×Dic3—82no—
134C3×Dic3—82no—
135C2×#41D3211yescharacteristic
136C2×M16—31no—
137C22×C8—91no—
138C4⋊Q8—92no—
139C4⋊Q8—181no—
140#125—92no—
141#125—181no—
142C22⋊Q8—182no—
143C4×Q8—32no—
144C4×D4—92no—
145#117—91no—
146#117—91no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—91no—
149#107—92no—
150#106—92no—
151#106—92no—
152#106—92no—
153C4⋊C8—92no—
154C4⋊C8—92no—
155#103—32no—
156#103—91no—
157#103—91no—
158#102—32no—
159#102—91no—
160#102—91no—
161C23×C3—31no—
162C3×Q8—31no—
163C3×Q8C2×Dic611yescharacteristic
164C3×D4—32no—
165C3×D4—61no—
166C2×C3×C4—31no—
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—41no—
170C2×C3×C4—62no—
171C2×C3×C4—81no—
172C2×C3×C4—122no—
173C2×C3×C4—241no—
174C2×C3×C4C2×S411yescharacteristic
175C2×Dic3—32no—
176C2×Dic3—42no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—121no—
180C2×Dic3—121no—
181C2×Dic3—241no—
182C2×Dic3—241no—
183C2×Dic3—241no—
184C2×Dic3C2×S412yes—
185Dic6—242no—
186Dic6—244no—
187SL(2,𝔽3)—21no—
188SL(2,𝔽3)C2×Dic611yescharacteristic
189C3×C8—62no—
190C3⋊C8—61no—
191C3⋊C8—61no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32—41no—
195#41—32no—
196#41#62712yescomplemented
197C2×Q8—182no—
198C2×Q8C2×D3211yescharacteristic
199C2×D4—31no—
200C22×C4—31no—
201C22×C4—61no—
202C22×C4—181no—
203M16—62no—
204C2×C8—32no—
205C2×C8—91no—
206C2×C8—91no—
207C2×C8—182no—
208C4⋊C4—31no—
209C4⋊C4—32no—
210C4⋊C4—61no—
211C4⋊C4—92no—
212C4⋊C4—91no—
213C4⋊C4—91no—
214C4⋊C4—91no—
215C4⋊C4—182no—
216C4⋊C4—182no—
217C4⋊C4—182no—
218C22⋊C4—92no—
219C22⋊C4—181no—
220C42—92no—
221C42—181no—
222C22×C3—32no—
223C22×C3—41no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—81no—
227C22×C3C22×S411yescharacteristic
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—42no—
231C3×C4—82no—
232C3×C4—122no—
233C3×C4—242no—
234C3×C4A4⋊Q812yes—
235Dic3—22no—
236Dic3—62no—
237Dic3—82no—
238Dic3—122no—
239Dic3—122no—
240Dic3—121no—
241Dic3—121no—
242Dic3—242no—
243Dic3—242no—
244C32—41noSylow
245C23—31no—
246Q8—31no—
247Q8—722no—
248Q8C2×#62711yescharacteristic
249D4—32no—
250D4—61no—
251C2×C4—31no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—62no—
256C2×C4—92no—
257C2×C4—181no—
258C2×C4—181no—
259C2×C4—241no—
260C2×C4—721no—
261C2×C4D3×S411yescharacteristic
262C8—62no—
263C8—181no—
264C8—181no—
265C2×C3—41no—
266C2×C3—41no—
267C2×C3—41no—
268C2×C3—62no—
269C2×C3—81no—
270C2×C3—81no—
271C2×C3—81no—
272C2×C3C2×A4⋊Q811yescharacteristic
273C2×C3#592411yescharacteristic
274C2×C3#592611yescharacteristic
275C22—32no—
276C22—61no—
277C22—61no—
278C22C2×D3×S411yescharacteristic
279C4—31no—
280C4—31no—
281C4—62no—
282C4—122no—
283C4—182no—
284C4—361no—
285C4—361no—
286C4#6420412yes—
287C3—41no—
288C3—81no—
289C3#8536111yescharacteristic, complemented
290C2—62no—
291C2#1059645011yescharacteristic
292C2#1059646011yescharacteristic
293C2C2×#6420411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×#41C2×D3
socleC22×C3C22×S4
prime radicalC2×C3×#41⋊C3C22
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×Q8
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
4. #10593212C2semidirect
5. #10593196C2semidirect
31. C3×#41⋊C3Q8semidirect
93. C3×#41Dic6semidirect
102. #41⋊C3Dic6semidirect
196. #41#627semidirect
289. C3#85361semidirect

Maximal subgroups

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

subgroupindex
#5827⋊C32
#105936582
C3×#54262
#105932122
#105931962
#853613
#788733
#639194