Finite Group Databasebeta

← #11921105

Subgroups — #11921105

Order 1152.

518 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊GU(2,𝔽3)C212yesmaximal, complemented
3#73149—33nomaximal
4#72138—31nomaximal
5C2×C3×#41⋊C3—22no—
6C2×C3×#41⋊C3C411yescharacteristic, complemented
7C2×C3×#41⋊C3C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C411yescharacteristic, complemented
9C3⋊GU(2,𝔽3)—22no—
10#63657—41nomaximal
11C22×C3×#41D311yescharacteristic
12C22×#41⋊C3D313yes—
13C2×GU(2,𝔽3)—36no—
14C2×#837—32no—
15C2×#837—32no—
16C4×#217—31no—
17#5207—31no—
18C22×C32×C4—41no—
19C3×#41⋊C3—21no—
20C3×#41⋊C3D412yes—
21C2×C3×SL(2,𝔽3)—21no—
22C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
23C2×C3⋊(C3⋊C8)—42no—
24#1789—91noSylow
25C2×C3×#41—22no—
26C2×C3×#41—31no—
27C2×C3×#41—31no—
28C2×C3×#41—62no—
29C2×C3×#41Dic311yescharacteristic
30C2×C3×#41C2×D311yescharacteristic
31C22×C3×Q8Dic311yescharacteristic
32C22×C3×D4—31no—
33C23×C3×C4—31no—
34C2×#41⋊C3—26no—
35C2×#41⋊C3Dic313yescomplemented
36C2×#41⋊C3C2×D313yes—
37C22×SL(2,𝔽3)Dic313yescomplemented
38C2×#217—61no—
39C2×C4×Dic3—32no—
40C2×#208—32no—
41GU(2,𝔽3)—66no—
42#837—62no—
43#837—64no—
44#837—62no—
45C3⋊#95—61no—
46#830—61no—
47#830—61no—
48#830—123no—
49C23×C32—41no—
50C2×C32×C4—41no—
51C2×C32×C4—41no—
52C2×C32×C4—82no—
53C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
54C3⋊(C3⋊C8)—82no—
55C22×#41C3⋊D311yescharacteristic
56C2×#104—92no—
57C2×#104—92no—
58#407—91no—
59C4×C22⋊C4—91no—
60C24×C3—31no—
61C3×#41—21no—
62C3×#41—32no—
63C3×#41—34no—
64C3×#41—62no—
65C3×#41—62no—
66C3×#41—61no—
67C3×#41—62no—
68C3×#41#6712yes—
69C2×C3×Q8—21no—
70C2×C3×Q8—31no—
71C2×C3×Q8—61no—
72C2×C3×Q8C2×Dic311yescharacteristic
73C2×C3×D4—31no—
74C2×C3×D4—31no—
75C2×C3×D4—61no—
76C2×C3×D4—62no—
77C2×C3×D4—62no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4—31no—
83C22×C3×C4—43no—
84C22×C3×C4—62no—
85C22×C3×C4—62no—
86C22×C3×C4S411yescharacteristic
87C22×Dic3—62no—
88#41⋊C3—23no—
89#41⋊C3#6716yes—
90C2×SL(2,𝔽3)—23no—
91C2×SL(2,𝔽3)C2×Dic313yes—
92#217—64no—
93C4×Dic3—62no—
94C4×Dic3—62no—
95#208—62no—
96#208—62no—
97C2×C3⋊C8—62no—
98C2×C3⋊C8—126no—
99C22×C32—42no—
100C22×C32—41no—
101C22×C32—81no—
102C22×C32—81no—
103C32×C4—42no—
104C32×C4—81no—
105C2×#41—22no—
106C2×#41—31no—
107C2×#41—31no—
108C2×#41—62no—
109C2×#41C3⋊Dic311yescharacteristic
110C2×#41C2×C3⋊D311yescharacteristic
111C22×Q8C3⋊Dic311yescharacteristic
112C22×D4—31no—
113C23×C4—31no—
114C2×M16—92no—
115C2×C22⋊C4—181no—
116C2×C42—92no—
117#104—182no—
118#104—184no—
119#104—182no—
120C22⋊C8—181no—
121C22⋊C8—181no—
122#95—181no—
123C23×C3—31no—
124C23×C3—31no—
125C23×C3—43no—
126C23×C3—62no—
127C23×C3—62no—
128C23×C3—121no—
129C23×C3SL(2,ℤ/4)11yescharacteristic
130C3×Q8—31no—
131C3×Q8—32no—
132C3×Q8—61no—
133C3×Q8#21711yescharacteristic
134C3×D4—32no—
135C3×D4—32no—
136C3×D4—61no—
137C3×D4—61no—
138C3×D4—64no—
139C2×C3×C4—22no—
140C2×C3×C4—31no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—31no—
144C2×C3×C4—43no—
145C2×C3×C4—43no—
146C2×C3×C4—61no—
147C2×C3×C4—61no—
148C2×C3×C4—62no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—62no—
152C2×C3×C4—86no—
153C2×C3×C4SL(2,ℤ/4)11yescharacteristic
154C2×C3×C4C2×S411yescharacteristic
155C2×Dic3—64no—
156C2×Dic3—122no—
157C2×Dic3—122no—
158SL(2,𝔽3)#21713yescomplemented
159C3⋊C8—122no—
160C3⋊C8—246no—
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—82no—
165C24—31no—
166#41—21no—
167#41—32no—
168#41—34no—
169#41—62no—
170#41—62no—
171#41—61no—
172#41—62no—
173#41#63812yes—
174C2×Q8—21no—
175C2×Q8—31no—
176C2×Q8—61no—
177C2×Q8C2×C3⋊Dic311yescharacteristic
178C2×D4—31no—
179C2×D4—31no—
180C2×D4—61no—
181C2×D4—62no—
182C2×D4—62no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—62no—
189C22×C4—62no—
190C22×C4—182no—
191C22×C4C3⋊S411yescharacteristic
192M16—182no—
193M16—182no—
194C2×C8—182no—
195C22⋊C4—184no—
196C42—182no—
197C42—182no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—32no—
201C22×C3—46no—
202C22×C3—43no—
203C22×C3—61no—
204C22×C3—61no—
205C22×C3—61no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3—83no—
209C22×C3—83no—
210C22×C3—124no—
211C22×C3GU(2,𝔽3)12yescomplemented
212C22×C3C2×SL(2,ℤ/4)11yescharacteristic
213C3×C4—21no—
214C3×C4—31no—
215C3×C4—31no—
216C3×C4—46no—
217C3×C4—61no—
218C3×C4—83no—
219C3×C4GL(2,ℤ/4)12yes—
220Dic3—124no—
221C32—41noSylow
222C23—31no—
223C23—31no—
224C23—62no—
225C23—62no—
226C23—121no—
227C23C3⋊SL(2,ℤ/4)11yescharacteristic
228Q8—31no—
229Q8—32no—
230Q8—61no—
231Q8#375711yescharacteristic
232D4—32no—
233D4—32no—
234D4—61no—
235D4—61no—
236D4—64no—
237C2×C4—22no—
238C2×C4—31no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—31no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—62no—
248C2×C4—184no—
249C2×C4—362no—
250C2×C4—362no—
251C2×C4C3⋊SL(2,ℤ/4)11yescharacteristic
252C2×C4C2×C3⋊S411yescharacteristic
253C8—362no—
254C2×C3—22no—
255C2×C3—43no—
256C2×C3—43no—
257C2×C3—43no—
258C2×C3—62no—
259C2×C3—86no—
260C2×C3—121no—
261C2×C3C2×GU(2,𝔽3)11yescharacteristic
262C2×C3#542311yescharacteristic
263C2×C3#543211yescharacteristic
264C22—21no—
265C22—21no—
266C22—32no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22—61no—
271C22—61no—
272C22—124no—
273C22C3⋊GU(2,𝔽3)12yescomplemented
274C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
275C4—21no—
276C4—31no—
277C4—31no—
278C4—61no—
279C4—364no—
280C4#6426912yes—
281C3—43no—
282C3#7314911yescharacteristic, complemented
283C2—22no—
284C2—62no—
285C2—121no—
286C2C2×C3⋊GU(2,𝔽3)11yescharacteristic
287C2#1059365711yescharacteristic
288C2#1059366611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC22×C3×#41D3
socleC22×C3C2×SL(2,ℤ/4)
prime radicalC22×C3×#41⋊C3C2
cosocleC2×C3×#41⋊C3C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊GU(2,𝔽3)C2semidirect
6. C2×C3×#41⋊C3C4semidirect
8. C22×C3×SL(2,𝔽3)C4semidirect
35. C2×#41⋊C3Dic3semidirect
37. C22×SL(2,𝔽3)Dic3semidirect
53. C3×SL(2,𝔽3)C22⋊C4semidirect
158. SL(2,𝔽3)#217semidirect
211. C22×C3GU(2,𝔽3)semidirect
273. C22C3⋊GU(2,𝔽3)semidirect
282. C3#73149semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3⋊GU(2,𝔽3)2
#731493
#721383
#636574