Finite Group Databasebeta

← #11922867

Subgroups — #11922867

Order 1152.

524 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 85 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×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2#10593654C212yesmaximal
3#10593625C211yesmaximal, characteristic, complemented
4#10593623C212yesmaximal, complemented
5#10593620C211yesmaximal, characteristic, complemented
6#85318—33nomaximal
7#82431—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9#64133—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2212yes—
13C3⋊#859C2212yes—
14C2×C3×C4×Q8D311yescharacteristic
15#5470—31no—
16C2×C4×SL(2,𝔽3)D313yes—
17#5420—36no—
18#5391—33no—
19#5389—36no—
20#5386—33no—
21#5225—32no—
22#5022—32no—
23#5022—32no—
24#5021—31no—
25#5021—31no—
26#5020—31no—
27#5020—31no—
28#4964—32no—
29#4924—31no—
30C22×C32×C4—41no—
31C2×#634—41no—
32C2×C3×SL(2,𝔽3)Q812yescomplemented
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34#3757—42no—
35C4⋊(C3⋊Dic3)—41no—
36#3750—42no—
37#3078—91noSylow
38C22×C3×Q8C2×D311yescharacteristic
39C22×SL(2,𝔽3)C2×D313yes—
40C3×C4×Q8—31no—
41C3×C4×Q8—31no—
42C3×C4×Q8C2×D311yescharacteristic
43C3×C4×Q8C2×D311yescharacteristic
44C2×C3×C4⋊C4—31no—
45C2×C3×C42—31no—
46#926—32no—
47#924—61no—
48#924—61no—
49#924—123no—
50C2×#208—32no—
51#870—32no—
52#869—31no—
53#869—31no—
54C4×Dic6—61no—
55C4×Dic6—61no—
56C4×SL(2,𝔽3)C2×D313yes—
57C4×SL(2,𝔽3)C2×D313yes—
58#859—36no—
59#859—36no—
60#835—32no—
61#835—32no—
62#835—32no—
63#835—32no—
64#808—32no—
65#808—32no—
66#807—32no—
67#807—32no—
68C4⋊(C3⋊C8)—31no—
69C4⋊(C3⋊C8)—31no—
70C4⋊(C3⋊C8)—32no—
71C23×C32—41no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4—82no—
75C2×C3⋊Dic3—42no—
76C2×C3⋊Dic3—42no—
77#634—82no—
78C3×SL(2,𝔽3)C2×Q811yescharacteristic
79#533—91no—
80C2×C4×Q8C3⋊D311yescharacteristic
81#479—92no—
82#479—92no—
83#477—91no—
84#477—91no—
85#475—91no—
86#475—91no—
87#428—92no—
88#423—91no—
89#419—92no—
90C2×C3×Q8—31no—
91C2×C3×Q8—32no—
92C2×C3×Q8Dic612yescomplemented
93C2×C3×Q8C22×D311yescharacteristic
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—43no—
97C22×C3×C4S411yescharacteristic
98C2×Dic6—61no—
99C2×Dic6—61no—
100C2×Dic6—123no—
101C2×SL(2,𝔽3)Dic616yescomplemented
102C2×SL(2,𝔽3)C22×D313yes—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—31no—
105C3×C4⋊C4—61no—
106C3×C42—31no—
107C3×C42—31no—
108C3×C42—31no—
109C3×C42—31no—
110#217—62no—
111#217—126no—
112C4⋊Dic3—32no—
113C4⋊Dic3—32no—
114C4⋊Dic3—61no—
115C4⋊Dic3—61no—
116C4⋊Dic3—123no—
117Dic3⋊C4—62no—
118Dic3⋊C4—62no—
119Dic3⋊C4—126no—
120C4×Dic3—62no—
121#208—64no—
122C2×C3⋊C8—32no—
123C2×C3⋊C8—32no—
124C22×C32—42no—
125C22×C32—41no—
126C22×C32—81no—
127C22×C32—81no—
128C32×C4—42no—
129C32×C4—81no—
130C3⋊Dic3—82no—
131C3⋊Dic3—82no—
132C22×Q8C2×C3⋊D311yescharacteristic
133C2×M16—92no—
134C4⋊Q8—91no—
135C4⋊Q8—91no—
136#125—92no—
137C22⋊Q8—181no—
138C22⋊Q8—181no—
139C4×Q8—31no—
140C4×Q8—31no—
141C4×Q8—181no—
142C4×Q8—181no—
143C4×Q8C2×C3⋊D311yescharacteristic
144C4×Q8C2×C3⋊D311yescharacteristic
145#117—92no—
146C2×C4⋊C4—31no—
147C2×C42—31no—
148#107—92no—
149#107—92no—
150#106—92no—
151#106—92no—
152C4⋊C8—91no—
153C4⋊C8—91no—
154C4⋊C8—92no—
155#103—92no—
156#103—92no—
157#103—92no—
158#103—92no—
159C23×C3—43no—
160C23×C3C2×S411yescharacteristic
161C3×Q8—32no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8C2×Dic611yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—43no—
173C2×C3×C4—43no—
174C2×C3×C4—86no—
175C2×C3×C4—121no—
176C2×C3×C4C2×S411yescharacteristic
177C2×C3×C4C2×S411yescharacteristic
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—126no—
181C2×Dic3—126no—
182Dic6—122no—
183Dic6—122no—
184Dic6—246no—
185SL(2,𝔽3)C2×Dic613yes—
186C3⋊C8—62no—
187C3⋊C8—62no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—82no—
192C2×Q8—31no—
193C2×Q8—32no—
194C2×Q8—181no—
195C2×Q8—181no—
196C2×Q8#63412yescomplemented
197C2×Q8C22×C3⋊D311yescharacteristic
198C22×C4—31no—
199C22×C4—31no—
200C22×C4C3⋊S411yescharacteristic
201M16—184no—
202C2×C8—92no—
203C2×C8—92no—
204C4⋊C4—31no—
205C4⋊C4—31no—
206C4⋊C4—61no—
207C4⋊C4—92no—
208C4⋊C4—92no—
209C4⋊C4—182no—
210C4⋊C4—181no—
211C4⋊C4—182no—
212C4⋊C4—181no—
213C22⋊C4—182no—
214C42—31no—
215C42—31no—
216C42—31no—
217C42—31no—
218C42—182no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—46no—
222C22×C3—43no—
223C22×C3—83no—
224C22×C3—83no—
225C22×C3A4⋊Q812yes—
226C22×C3C22×S411yescharacteristic
227C3×C4—21no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—32no—
231C3×C4—46no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—83no—
235C3×C4#98312yes—
236Dic3—122no—
237Dic3—122no—
238Dic3—246no—
239Dic3—246no—
240C32—41noSylow
241C23C2×C3⋊S411yescharacteristic
242Q8—32no—
243Q8—31no—
244Q8—61no—
245Q8—362no—
246Q8—362no—
247Q8C2×#63411yescharacteristic
248C2×C4—22no—
249C2×C4—32no—
250C2×C4—32no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—121no—
256C2×C4—182no—
257C2×C4—182no—
258C2×C4C2×C3⋊S411yescharacteristic
259C2×C4C2×C3⋊S411yescharacteristic
260C8—182no—
261C8—182no—
262C2×C3—22no—
263C2×C3—43no—
264C2×C3—43no—
265C2×C3—43no—
266C2×C3—86no—
267C2×C3C2×A4⋊Q811yescharacteristic
268C2×C3C2×#98311yescharacteristic
269C2×C3#592311yescharacteristic
270C22—21no—
271C22—21no—
272C22#6425912yes—
273C22C22×C3⋊S411yescharacteristic
274C4—21no—
275C4—31no—
276C4—31no—
277C4—32no—
278C4—61no—
279C4—61no—
280C4—362no—
281C4—362no—
282C4#6426412yes—
283C3—43no—
284C3#8531811yescharacteristic, complemented
285C2—22no—
286C2C2×#6425911yescharacteristic
287C2C2×#6426411yescharacteristic
288C2#1059658611yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #10593625C2semidirect
4. #10593623C2semidirect
5. #10593620C2semidirect
32. C2×C3×SL(2,𝔽3)Q8semidirect
92. C2×C3×Q8Dic6semidirect
101. C2×SL(2,𝔽3)Dic6semidirect
196. C2×Q8#634semidirect
284. C3#85318semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#105936542
#105936252
#105936232
#105936202
#853183
#824313
#641334