Finite Group Databasebeta

← #11922942

Subgroups — #11922942

Order 1152.

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

normal subgroups only

This group has 586 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
1C3×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2#10593653C212yesmaximal, complemented
3C2×C3⋊#859C212yesmaximal, complemented
4#10593625C211yesmaximal, characteristic, complemented
5C4×C3⋊#226C211yesmaximal, characteristic, complemented
6#85393—33nomaximal
7#83551—31nomaximal
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C3⋊#226C2211yescharacteristic
10D4×C3⋊Dic3—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3⋊#859C2211yescharacteristic
13C3⋊#859C2212yes—
14C3×D4×Q8D311yescharacteristic
15#5543—31no—
16D4×SL(2,𝔽3)D313yes—
17#5419—36no—
18C2×#859—36no—
19#5391—33no—
20C4×#226—33no—
21#5228—32no—
22C2×#835—32no—
23#5090—31no—
24#5079—31no—
25#5051—32no—
26#5044—32no—
27C4×#216—31no—
28#5021—31no—
29D4×C3⋊C8—31no—
30C2×C32×D4—41no—
31C22×C3⋊Dic3—42no—
32C2×C3×SL(2,𝔽3)—22no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊#226—21no—
35#3757—42no—
36C4⋊(C3⋊Dic3)—41no—
37C4×C3⋊Dic3—41no—
38#3299—91noSylow
39C22×C3×Q8C2×D312yes—
40C22×SL(2,𝔽3)C2×D316yes—
41C2×#226—33no—
42C3×C4⋊Q8—31no—
43C3×C22⋊Q8—32no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×D4—31no—
46C2×#216—31no—
47C2×#216—32no—
48D4×Dic3—61no—
49D4×Dic3—123no—
50C2×C4⋊Dic3—32no—
51#924—32no—
52C22×C3⋊C8—32no—
53#890—61no—
54#878—62no—
55#869—31no—
56C4×Dic6—31no—
57C4×SL(2,𝔽3)C2×D313yes—
58#859—33no—
59#859—36no—
60#835—32no—
61#835—32no—
62#835—32no—
63#835—31no—
64#835—32no—
65#830—32no—
66#810—31no—
67#807—32no—
68#807—31no—
69C4⋊(C3⋊C8)—31no—
70C4×C3⋊C8—31no—
71C23×C32—42no—
72C32×D4—44no—
73C2×C32×C4—41no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—41no—
76C2×C3⋊Dic3—42no—
77C2×C3⋊Dic3—84no—
78C3×SL(2,𝔽3)#4111yescharacteristic
79#554—91no—
80D4×Q8C3⋊D311yescharacteristic
81#487—91no—
82#484—92no—
83#477—91no—
84#467—92no—
85#462—91no—
86#451—92no—
87C4×Q16—91no—
88C8×D4—91no—
89C2×#103—92no—
90C2×C3×Q8—22no—
91C2×C3×Q8—32no—
92C2×C3×Q8—62no—
93C2×C3×Q8C22×D311yescharacteristic
94C2×C3×D4—43no—
95C2×C3×D4S411yescharacteristic
96C22×C3×C4—32no—
97C22×Dic3—62no—
98C22×Dic3—126no—
99C2×Dic6—31no—
100C2×Dic6—61no—
101C2×SL(2,𝔽3)—26no—
102C2×SL(2,𝔽3)C22×D313yes—
103#226—63no—
104C3×C4⋊C4—31no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4—61no—
107C3×C22⋊C4—32no—
108C3×C42—31no—
109#217—62no—
110#217—126no—
111#216—61no—
112#216—61no—
113#216—122no—
114C4⋊Dic3—31no—
115C4⋊Dic3—32no—
116C4⋊Dic3—62no—
117C4⋊Dic3—61no—
118C4⋊Dic3—123no—
119Dic3⋊C4—61no—
120Dic3⋊C4—62no—
121C4×Dic3—61no—
122C4×Dic3—123no—
123C2×C3⋊C8—31no—
124C2×C3⋊C8—31no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—122no—
127C22×C32—44no—
128C22×C32—41no—
129C22×C32—82no—
130C22×C32—82no—
131C32×C4—42no—
132C3⋊Dic3—42no—
133C3⋊Dic3—82no—
134C3⋊Dic3—81no—
135C22×Q8C2×C3⋊D312yes—
136C2×Q16—91no—
137C2×Q16—92no—
138C22×C8—92no—
139C4⋊Q8—31no—
140C4⋊Q8—91no—
141#125—181no—
142C22⋊Q8—32no—
143C22⋊Q8—92no—
144C22⋊Q8—182no—
145C4×Q8—91no—
146C4×Q8C2×C3⋊D311yescharacteristic
147C4×D4—31no—
148C4×D4—181no—
149C2×C4⋊C4—92no—
150#107—91no—
151#107—92no—
152C4⋊C8—91no—
153#103—91no—
154#103—92no—
155#103—92no—
156#103—92no—
157#103—91no—
158#103—92no—
159C22⋊C8—92no—
160C4×C8—91no—
161C23×C3—46no—
162C23×C3C2×S412yes—
163C3×Q8—31no—
164C3×Q8—62no—
165C3×Q8—62no—
166C3×Q8#23711yescharacteristic
167C3×D4—412no—
168C3×D4#22614yescomplemented
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—43no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4C2×S411yescharacteristic
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—62no—
179C2×Dic3—123no—
180C2×Dic3—126no—
181C2×Dic3—123no—
182C2×Dic3—242no—
183C2×Dic3—2412no—
184Dic6—61no—
185Dic6—61no—
186Dic6—241no—
187SL(2,𝔽3)#23713yes—
188C3⋊C8—61no—
189C3⋊C8—62no—
190C3⋊C8—61no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32—84no—
195C2×Q8—22no—
196C2×Q8—32no—
197C2×Q8—62no—
198C2×Q8—91no—
199C2×Q8—181no—
200C2×Q8C22×C3⋊D311yescharacteristic
201C2×D4C3⋊S411yescharacteristic
202C22×C4—32no—
203C22×C4—182no—
204Q16—181no—
205Q16—181no—
206Q16—362no—
207C2×C8—91no—
208C2×C8—91no—
209C2×C8—92no—
210C2×C8—362no—
211C4⋊C4—31no—
212C4⋊C4—31no—
213C4⋊C4—61no—
214C4⋊C4—91no—
215C4⋊C4—92no—
216C4⋊C4—182no—
217C4⋊C4—181no—
218C4⋊C4—182no—
219C4⋊C4—181no—
220C22⋊C4—32no—
221C22⋊C4—182no—
222C42—31no—
223C42—181no—
224C22×C3—22no—
225C22×C3—22no—
226C22×C3—412no—
227C22×C3—43no—
228C22×C3—86no—
229C22×C3—86no—
230C22×C3C2×#22614yes—
231C22×C3C22×S411yescharacteristic
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—46no—
235C3×C4—62no—
236C3×C4—61no—
237C3×C4C2×#22612yes—
238Dic3—126no—
239Dic3—121no—
240Dic3—122no—
241Dic3—121no—
242Dic3—246no—
243Dic3—243no—
244C32—41noSylow
245C23C2×C3⋊S412yes—
246Q8—31no—
247Q8—62no—
248Q8—62no—
249Q8—181no—
250Q8—181no—
251Q8—721no—
252Q8#383011yescharacteristic
253D4C3⋊#22614yescomplemented
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—32no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—182no—
262C2×C4—722no—
263C2×C4C2×C3⋊S411yescharacteristic
264C8—181no—
265C8—182no—
266C8—181no—
267C2×C3—24no—
268C2×C3—43no—
269C2×C3—43no—
270C2×C3—43no—
271C2×C3—812no—
272C2×C3#591411yescharacteristic
273C2×C3C22×#22611yescharacteristic
274C2×C3#592611yescharacteristic
275C22—22no—
276C22—22no—
277C22C2×C3⋊#22614yes—
278C22C22×C3⋊S411yescharacteristic
279C4—31no—
280C4—31no—
281C4—62no—
282C4—61no—
283C4—361no—
284C4—361no—
285C4—362no—
286C4C2×C3⋊#22612yes—
287C3—43no—
288C3#8539311yescharacteristic, complemented
289C2—24no—
290C2#1059657711yescharacteristic
291C2C22×C3⋊#22611yescharacteristic
292C2#1059658911yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×D4×Q8D3
socleC22×C3C22×S4
prime radicalC3×D4×SL(2,𝔽3)C2
hypercenterC2×D4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #10593653C2semidirect
3. C2×C3⋊#859C2semidirect
4. #10593625C2semidirect
5. C4×C3⋊#226C2semidirect
168. C3×D4#226semidirect
253. D4C3⋊#226semidirect
288. C3#85393semidirect

Maximal subgroups

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

subgroupindex
C3×D4×SL(2,𝔽3)2
#105936532
C2×C3⋊#8592
#105936252
C4×C3⋊#2262
#853933
#835513
D4×C3⋊Dic34