Finite Group Databasebeta

← #11922312

Subgroups — #11922312

Order 1152.

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

normal subgroups only

This group has 409 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#10593202C212yesmaximal, complemented
5#10593199C212yesmaximal, complemented
6#85363—31nomaximal
7#78872—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9#63872—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#5391—32no—
21#5389—32no—
22C3×#417—31no—
23#5233—31no—
24#5121—31no—
25#5119—31no—
26C2×#807—31no—
27#4799—32no—
28#4793—32no—
29#4763—32no—
30#4759—32no—
31C3×#41⋊C3Q812yescomplemented
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C4×C3⋊Dic3—41no—
34C3×C4⋊Dic3—41no—
35C3×C4⋊Dic3—41no—
36Dic3⋊Dic3—42no—
37Dic3⋊Dic3—42no—
38#3083—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#890—121no—
54#888—32no—
55#888—61no—
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#807—32no—
73#807—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#544—91no—
81#520—31no—
82#479—92no—
83#478—92no—
84#477—92no—
85#474—92no—
86#428—91no—
87C2×#107—91no—
88#424—91no—
89#417—31no—
90#416—91no—
91C3×#41—32no—
92C3×#41Dic612yes—
93C2×C3×Q8C22×D311yescharacteristic
94C2×C3×D4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—61no—
97C22×Dic3—61no—
98C2×Dic6—62no—
99#41⋊C3—22no—
100#41⋊C3Dic612yescomplemented
101C2×SL(2,𝔽3)—21no—
102C2×SL(2,𝔽3)C22×D311yescharacteristic
103C3×M16—62no—
104C2×C3×C8—32no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—61no—
107C3×C4⋊C4—121no—
108#217—32no—
109#217—61no—
110C4⋊Dic3—31no—
111C4⋊Dic3—32no—
112C4⋊Dic3—31no—
113C4⋊Dic3—31no—
114C4⋊Dic3—41no—
115C4⋊Dic3—62no—
116C4⋊Dic3—122no—
117C4⋊Dic3S411yescharacteristic
118Dic3⋊C4—62no—
119Dic3⋊C4—62no—
120Dic3⋊C4—122no—
121C4×Dic3—32no—
122C4×Dic3—61no—
123C4×Dic3—121no—
124C4×Dic3—241no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—62no—
128C22×C32—41no—
129C32×C4—42no—
130C3⋊Dic3—81no—
131C3⋊Dic3—81no—
132C3×Dic3—82no—
133C3×Dic3—82no—
134C2×#41D3211yescharacteristic
135C2×M16—31no—
136C22×C8—91no—
137C4⋊Q8—92no—
138C4⋊Q8—181no—
139#125—92no—
140#125—181no—
141C22⋊Q8—182no—
142C4×Q8—32no—
143C4×D4—92no—
144#117—91no—
145#117—91no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—91no—
148#107—92no—
149#107—92no—
150#107—92no—
151#106—92no—
152C4⋊C8—92no—
153C4⋊C8—92no—
154#103—32no—
155#103—91no—
156#103—91no—
157#102—32no—
158#102—91no—
159#102—91no—
160C23×C3—31no—
161C3×Q8—31no—
162C3×Q8C2×Dic611yescharacteristic
163C3×D4—32no—
164C3×D4—61no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—41no—
169C2×C3×C4—62no—
170C2×C3×C4—81no—
171C2×C3×C4—122no—
172C2×C3×C4—241no—
173C2×C3×C4C2×S411yescharacteristic
174C2×Dic3—32no—
175C2×Dic3—42no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—121no—
179C2×Dic3—121no—
180C2×Dic3—241no—
181C2×Dic3—241no—
182C2×Dic3—241no—
183C2×Dic3C2×S412yes—
184Dic6—242no—
185SL(2,𝔽3)—21no—
186SL(2,𝔽3)C2×Dic611yescharacteristic
187C3×C8—62no—
188C3⋊C8—61no—
189C3⋊C8—61no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193#41—32no—
194#41#62712yes—
195C2×Q8—182no—
196C2×Q8C2×D3211yescharacteristic
197C2×D4—31no—
198C22×C4—31no—
199C22×C4—61no—
200C22×C4—181no—
201M16—62no—
202C2×C8—32no—
203C2×C8—91no—
204C2×C8—91no—
205C2×C8—182no—
206C4⋊C4—31no—
207C4⋊C4—32no—
208C4⋊C4—61no—
209C4⋊C4—92no—
210C4⋊C4—91no—
211C4⋊C4—91no—
212C4⋊C4—91no—
213C4⋊C4—182no—
214C4⋊C4—182no—
215C4⋊C4—182no—
216C22⋊C4—92no—
217C22⋊C4—181no—
218C42—92no—
219C42—181no—
220C22×C3—32no—
221C22×C3—41no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—81no—
225C22×C3C22×S411yescharacteristic
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—122no—
231C3×C4—242no—
232C3×C4A4⋊Q812yes—
233Dic3—22no—
234Dic3—62no—
235Dic3—82no—
236Dic3—121no—
237Dic3—121no—
238Dic3—241no—
239Dic3—241no—
240Dic3—242no—
241Dic3—242no—
242C32—41noSylow
243C23—31no—
244Q8—31no—
245Q8—722no—
246Q8C2×#62711yescharacteristic
247D4—32no—
248D4—61no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—62no—
254C2×C4—92no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—241no—
258C2×C4—721no—
259C2×C4D3×S411yescharacteristic
260C8—62no—
261C8—181no—
262C8—181no—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—41no—
266C2×C3—62no—
267C2×C3—81no—
268C2×C3—81no—
269C2×C3—81no—
270C2×C3C2×A4⋊Q811yescharacteristic
271C2×C3#592511yescharacteristic
272C2×C3#592711yescharacteristic
273C22—32no—
274C22—61no—
275C22—61no—
276C22C2×D3×S411yescharacteristic
277C4—31no—
278C4—31no—
279C4—62no—
280C4—122no—
281C4—182no—
282C4—361no—
283C4—361no—
284C4#6420412yes—
285C3—41no—
286C3—81no—
287C3#8536311yescharacteristic, complemented
288C2—62no—
289C2#1059644811yescharacteristic
290C2#1059645311yescharacteristic
291C2C2×#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. #10593202C2semidirect
5. #10593199C2semidirect
31. C3×#41⋊C3Q8semidirect
100. #41⋊C3Dic6semidirect
287. C3#85363semidirect

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
#105932022
#105931992
#853633
#788723
#638724