Finite Group Databasebeta

← #11922871

Subgroups — #11922871

Order 1152.

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

normal subgroups only

This group has 540 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#10593653C212yesmaximal
3#10593625C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
5C4×C3⋊#226C212yesmaximal, complemented
6#85322—33nomaximal
7#82447—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊#226C2212yes—
10#64135—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2212yes—
14C2×C3×C4×Q8D311yescharacteristic
15#5470—31no—
16C2×C4×SL(2,𝔽3)D313yes—
17#5419—36no—
18#5391—33no—
19C4⋊#226—33no—
20C4×#226—36no—
21#5228—32no—
22#5031—31no—
23#5031—31no—
24C4×#216—32no—
25C4×#216—32no—
26#5021—31no—
27#5021—31no—
28#4970—32no—
29#4925—31no—
30C22×C32×C4—41no—
31C2×C3×SL(2,𝔽3)—21no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊#226—22no—
34#3757—42no—
35C4⋊(C3⋊Dic3)—41no—
36C4⋊(C3⋊Dic3)—41no—
37C4×C3⋊Dic3—42no—
38#3104—91noSylow
39C22×C3×Q8C2×D311yescharacteristic
40C22×SL(2,𝔽3)C2×D313yes—
41C2×#226—36no—
42C3×C4×Q8—31no—
43C3×C4×Q8—31no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×Q8C2×D311yescharacteristic
46C2×C3×C4⋊C4—31no—
47C2×C3×C42—31no—
48C2×#216—32no—
49C2×#216—32no—
50#926—61no—
51#926—123no—
52#924—32no—
53#924—61no—
54#870—61no—
55#869—31no—
56#869—31no—
57C4×Dic6—32no—
58C4×Dic6—61no—
59C4×SL(2,𝔽3)C2×D313yes—
60C4×SL(2,𝔽3)C2×D313yes—
61#859—36no—
62#835—32no—
63#835—32no—
64#830—32no—
65#810—32no—
66#810—32no—
67#807—32no—
68#807—32no—
69C4⋊(C3⋊C8)—31no—
70C4⋊(C3⋊C8)—31no—
71C4×C3⋊C8—32no—
72C23×C32—41no—
73C2×C32×C4—44no—
74C2×C32×C4—41no—
75C2×C32×C4—41no—
76C2×C3⋊Dic3—42no—
77C2×C3⋊Dic3—42no—
78C3×SL(2,𝔽3)#4111yescharacteristic
79#533—91no—
80C2×C4×Q8C3⋊D311yescharacteristic
81#484—92no—
82#477—91no—
83#477—91no—
84#462—91no—
85#462—91no—
86#451—92no—
87C4×Q16—92no—
88C4×Q16—92no—
89#431—91no—
90C2×C3×Q8—21no—
91C2×C3×Q8—31no—
92C2×C3×Q8—61no—
93C2×C3×Q8C22×D311yescharacteristic
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—43no—
97C22×C3×C4S411yescharacteristic
98C2×Dic6—32no—
99C2×Dic6—61no—
100C2×SL(2,𝔽3)—23no—
101C2×SL(2,𝔽3)C22×D313yes—
102#226—66no—
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—
112#216—62no—
113#216—64no—
114#216—62no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3—61no—
118C4⋊Dic3—61no—
119C4⋊Dic3—123no—
120C4⋊Dic3—123no—
121Dic3⋊C4—62no—
122Dic3⋊C4—62no—
123C4×Dic3—62no—
124C4×Dic3—126no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—32no—
127C22×C32—42no—
128C22×C32—41no—
129C22×C32—81no—
130C22×C32—81no—
131C32×C4—42no—
132C32×C4—42no—
133C3⋊Dic3—82no—
134C3⋊Dic3—82no—
135C22×Q8C2×C3⋊D311yescharacteristic
136C2×Q16—92no—
137C2×Q16—92no—
138C4⋊Q8—91no—
139C4⋊Q8—91no—
140#125—181no—
141C22⋊Q8—92no—
142C22⋊Q8—181no—
143C4×Q8—31no—
144C4×Q8—31no—
145C4×Q8—92no—
146C4×Q8—181no—
147C4×Q8C2×C3⋊D311yescharacteristic
148C4×Q8C2×C3⋊D311yescharacteristic
149#117—181no—
150C2×C4⋊C4—31no—
151C2×C42—31no—
152#107—92no—
153#107—92no—
154C4⋊C8—91no—
155C4⋊C8—91no—
156#103—92no—
157#103—92no—
158#103—92no—
159#103—92no—
160C22⋊C8—92no—
161C4×C8—92no—
162C23×C3—43no—
163C23×C3C2×S411yescharacteristic
164C3×Q8—31no—
165C3×Q8—32no—
166C3×Q8—61no—
167C3×Q8#23511yescharacteristic
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—412no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—121no—
178C2×C3×C4#22614yescomplemented
179C2×C3×C4C2×S411yescharacteristic
180C2×C3×C4C2×S411yescharacteristic
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—126no—
184C2×Dic3—126no—
185Dic6—62no—
186Dic6—62no—
187Dic6—122no—
188SL(2,𝔽3)#23513yes—
189C3⋊C8—62no—
190C3⋊C8—62no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32—82no—
195C2×Q8—21no—
196C2×Q8—31no—
197C2×Q8—61no—
198C2×Q8—92no—
199C2×Q8—181no—
200C2×Q8C22×C3⋊D311yescharacteristic
201C22×C4—31no—
202C22×C4—31no—
203C22×C4C3⋊S411yescharacteristic
204Q16—182no—
205Q16—184no—
206Q16—182no—
207C2×C8—92no—
208C2×C8—92no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—61no—
212C4⋊C4—92no—
213C4⋊C4—182no—
214C4⋊C4—182no—
215C4⋊C4—181no—
216C4⋊C4—181no—
217C4⋊C4—181no—
218C22⋊C4—182no—
219C42—31no—
220C42—31no—
221C42—31no—
222C42—31no—
223C42—182no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—46no—
227C22×C3—43no—
228C22×C3—83no—
229C22×C3—83no—
230C22×C3C2×#22612yes—
231C22×C3C22×S411yescharacteristic
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—46no—
235C3×C4—46no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4—61no—
239C3×C4C2×#22612yes—
240C3×C4C2×#22612yes—
241Dic3—122no—
242Dic3—122no—
243Dic3—246no—
244Dic3—246no—
245C32—41noSylow
246C23C2×C3⋊S411yescharacteristic
247Q8—31no—
248Q8—32no—
249Q8—61no—
250Q8—182no—
251Q8—182no—
252Q8—362no—
253Q8#382811yescharacteristic
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—121no—
261C2×C4—182no—
262C2×C4—182no—
263C2×C4C3⋊#22614yescomplemented
264C2×C4C2×C3⋊S411yescharacteristic
265C2×C4C2×C3⋊S411yescharacteristic
266C8—182no—
267C8—182no—
268C2×C3—22no—
269C2×C3—43no—
270C2×C3—43no—
271C2×C3—43no—
272C2×C3—86no—
273C2×C3#591211yescharacteristic
274C2×C3C22×#22611yescharacteristic
275C2×C3#592311yescharacteristic
276C22—21no—
277C22—21no—
278C22C2×C3⋊#22612yes—
279C22C22×C3⋊S411yescharacteristic
280C4—31no—
281C4—31no—
282C4—61no—
283C4—61no—
284C4—61no—
285C4—362no—
286C4—362no—
287C4C2×C3⋊#22612yes—
288C4C2×C3⋊#22612yes—
289C3—43no—
290C3#8532211yescharacteristic, complemented
291C2—22no—
292C2#1059657511yescharacteristic
293C2C22×C3⋊#22611yescharacteristic
294C2#1059658611yescharacteristic
subgroupG/H
centerC2×C4C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC22×C3C22×S4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)#41
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
3. #10593625C2semidirect
4. C4⋊(C3⋊#226)C2semidirect
5. C4×C3⋊#226C2semidirect
178. C2×C3×C4#226semidirect
263. C2×C4C3⋊#226semidirect
290. C3#85322semidirect

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
#105936532
#105936252
C4⋊(C3⋊#226)2
C4×C3⋊#2262
#853223
#824473
#641354