Finite Group Databasebeta

← #11922868

Subgroups — #11922868

Order 1152.

532 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 73 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
3C4⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
4#10593622C212yesmaximal, complemented
5#10593620C211yesmaximal, characteristic, complemented
6#85319—33nomaximal
7#82448—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊#226C2212yes—
10#64133—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—
18C4⋊#226—33no—
19#5388—36no—
20#5386—33no—
21#5228—32no—
22#5031—31no—
23#5031—31no—
24#5030—32no—
25#5030—32no—
26#5020—31no—
27#5020—31no—
28#4971—32no—
29#4926—31no—
30C22×C32×C4—41no—
31C2×#634—41no—
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊#226—22no—
35#3757—42no—
36C4⋊(C3⋊Dic3)—41no—
37#3750—42no—
38#3113—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#924—32no—
52#924—61no—
53#924—123no—
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#808—32no—
68#808—32no—
69C4⋊(C3⋊C8)—31no—
70C4⋊(C3⋊C8)—31no—
71#803—32no—
72C23×C32—41no—
73C2×C32×C4—41no—
74C2×C32×C4—41no—
75C2×C32×C4—82no—
76C2×C3⋊Dic3—42no—
77C2×C3⋊Dic3—42no—
78#634—82no—
79C3×SL(2,𝔽3)#4111yescharacteristic
80#533—91no—
81C2×C4×Q8C3⋊D311yescharacteristic
82#483—92no—
83#475—91no—
84#475—91no—
85#462—91no—
86#462—91no—
87#451—92no—
88#441—92no—
89#441—92no—
90#432—91no—
91C2×C3×Q8—21no—
92C2×C3×Q8—31no—
93C2×C3×Q8—61no—
94C2×C3×Q8C22×D311yescharacteristic
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—43no—
98C22×C3×C4S411yescharacteristic
99C2×Dic6—32no—
100C2×Dic6—61no—
101C2×Dic6—123no—
102C2×SL(2,𝔽3)—23no—
103C2×SL(2,𝔽3)C22×D313yes—
104#226—66no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—61no—
108C3×C42—31no—
109C3×C42—31no—
110C3×C42—31no—
111C3×C42—31no—
112#217—62no—
113#217—126no—
114#216—62no—
115#216—64no—
116#216—62no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119C4⋊Dic3—61no—
120C4⋊Dic3—61no—
121C4⋊Dic3—123no—
122Dic3⋊C4—62no—
123Dic3⋊C4—62no—
124Dic3⋊C4—126no—
125C4×Dic3—62no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—32no—
128C22×C32—42no—
129C22×C32—41no—
130C22×C32—81no—
131C22×C32—81no—
132C32×C4—42no—
133C32×C4—81no—
134C3⋊Dic3—82no—
135C3⋊Dic3—82no—
136C22×Q8C2×C3⋊D311yescharacteristic
137C2×Q16—92no—
138C2×Q16—92no—
139C4⋊Q8—91no—
140C4⋊Q8—91no—
141#125—181no—
142C22⋊Q8—92no—
143C22⋊Q8—181no—
144C4×Q8—31no—
145C4×Q8—31no—
146C4×Q8—92no—
147C4×Q8—181no—
148C4×Q8C2×C3⋊D311yescharacteristic
149C4×Q8C2×C3⋊D311yescharacteristic
150#117—181no—
151C2×C4⋊C4—31no—
152C2×C42—31no—
153#106—92no—
154#106—92no—
155C4⋊C8—91no—
156C4⋊C8—91no—
157#103—92no—
158#103—92no—
159#103—92no—
160#103—92no—
161C22⋊C8—92no—
162#97—92no—
163C23×C3—43no—
164C23×C3C2×S411yescharacteristic
165C3×Q8—31no—
166C3×Q8—32no—
167C3×Q8—61no—
168C3×Q8#23511yescharacteristic
169C2×C3×C4—22no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—43no—
175C2×C3×C4—43no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—86no—
180C2×C3×C4C2×S411yescharacteristic
181C2×C3×C4C2×S411yescharacteristic
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—126no—
185C2×Dic3—126no—
186Dic6—62no—
187Dic6—62no—
188Dic6—122no—
189Dic6—246no—
190SL(2,𝔽3)#23513yes—
191C3⋊C8—62no—
192C3⋊C8—62no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—82no—
197C2×Q8—21no—
198C2×Q8—31no—
199C2×Q8—61no—
200C2×Q8—92no—
201C2×Q8—181no—
202C2×Q8C22×C3⋊D311yescharacteristic
203C22×C4—31no—
204C22×C4—31no—
205C22×C4C3⋊S411yescharacteristic
206Q16—182no—
207Q16—184no—
208Q16—182no—
209C2×C8—92no—
210C2×C8—92no—
211C4⋊C4—31no—
212C4⋊C4—31no—
213C4⋊C4—61no—
214C4⋊C4—92no—
215C4⋊C4—181no—
216C4⋊C4—181no—
217C4⋊C4—182no—
218C4⋊C4—182no—
219C4⋊C4—181no—
220C22⋊C4—182no—
221C42—31no—
222C42—31no—
223C42—31no—
224C42—31no—
225C42—182no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—46no—
229C22×C3—43no—
230C22×C3—83no—
231C22×C3—83no—
232C22×C3#98412yes—
233C22×C3C22×S411yescharacteristic
234C3×C4—21no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—46no—
238C3×C4—61no—
239C3×C4—61no—
240C3×C4—61no—
241C3×C4—83no—
242C3×C4#98312yes—
243Dic3—122no—
244Dic3—122no—
245Dic3—246no—
246Dic3—246no—
247C32—41noSylow
248C23C2×C3⋊S411yescharacteristic
249Q8—31no—
250Q8—32no—
251Q8—61no—
252Q8—182no—
253Q8—182no—
254Q8—362no—
255Q8#382811yescharacteristic
256C2×C4—22no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—182no—
265C2×C4—182no—
266C2×C4C2×C3⋊S411yescharacteristic
267C2×C4C2×C3⋊S411yescharacteristic
268C8—182no—
269C8—182no—
270C2×C3—22no—
271C2×C3—43no—
272C2×C3—43no—
273C2×C3—43no—
274C2×C3—86no—
275C2×C3#591211yescharacteristic
276C2×C3C2×#98311yescharacteristic
277C2×C3C2×#98411yescharacteristic
278C22—21no—
279C22—21no—
280C22#6426512yes—
281C22C22×C3⋊S411yescharacteristic
282C4—21no—
283C4—31no—
284C4—31no—
285C4—61no—
286C4—61no—
287C4—61no—
288C4—362no—
289C4—362no—
290C4#6426412yes—
291C3—43no—
292C3#8531911yescharacteristic, complemented
293C2—22no—
294C2#1059657511yescharacteristic
295C2C2×#6426411yescharacteristic
296C2C2×#6426511yescharacteristic
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)#41
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. C4⋊(C3⋊#226)C2semidirect
4. #10593622C2semidirect
5. #10593620C2semidirect
292. C3#85319semidirect

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
C4⋊(C3⋊#226)2
#105936222
#105936202
#853193
#824483
#641334