Finite Group Databasebeta

← #11921635

Subgroups — #11921635

Order 1152.

463 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#5570⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic
3C3×C4×#226C211yesmaximal, characteristic, complemented
4#10593214C212yesmaximal, complemented
5#10593208C212yesmaximal, complemented
6#85420—31nomaximal
7#79701—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2212yes—
9C2×C3×#226C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63894—41nomaximal
12Dic3×SL(2,𝔽3)C2212yes—
13C3×#859C2211yescharacteristic
14#5585—31no—
15#5570D311yescharacteristic, complemented
16#557⋊C3—31no—
17#5395—32no—
18C4⋊GL(2,𝔽3)—31no—
19C4⋊GL(2,𝔽3)—61no—
20C4×GL(2,𝔽3)—32no—
21C4×#226D311yescharacteristic, complemented
22C3×C4×Q16—31no—
23#5185—32no—
24#5027—31no—
25#5020—31no—
26#4891—32no—
27#4800—32no—
28#4788—32no—
29#4694—31no—
30C8×Dic6—31no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C2×#624—42no—
34C3⋊GL(2,𝔽3)—22no—
35C3×#226—21no—
36C3×C4⋊Dic3—41no—
37C3×C4×Dic3—41no—
38#3722—42no—
39#3365—91noSylow
40C2×GL(2,𝔽3)—32no—
41C2×GL(2,𝔽3)—62no—
42C2×#226C2×D311yescharacteristic
43C2×C3×Q16—31no—
44C3×C4×Q8—31no—
45C3×C4×Q8C2×D311yescharacteristic
46#947—32no—
47Q8×Dic3C2×D312yes—
48C2×#215—32no—
49#894—62no—
50#893—32no—
51#893—121no—
52#890—32no—
53#875—61no—
54#874—31no—
55C4×D12—61no—
56#869—31no—
57C4×Dic6—31no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)—32no—
60C4×SL(2,𝔽3)C2×D311yescharacteristic
61#859C2×D311yescharacteristic
62C3×#107—31no—
63C3×#103—31no—
64C3×#103—31no—
65C3×C4×C8—31no—
66#835—32no—
67#821—32no—
68#821—32no—
69Dic3⋊C8—32no—
70C8×Dic3—32no—
71#809—32no—
72#808—32no—
73C4⋊(C3⋊C8)—31no—
74C22×C3⋊D3—42no—
75C2×C32×C4—41no—
76C4⋊(C3⋊D3)—44no—
77C2×C3×Dic3—41no—
78C2×C3×Dic3—41no—
79C2×C3×Dic3—42no—
80C3×SL(2,𝔽3)#4111yescharacteristic
81#624—84no—
82#559—91no—
83#557—31no—
84#488—92no—
85#486—91no—
86#478—92no—
87#475—91no—
88#464—92no—
89#460—91no—
90C8×Q8—91no—
91C4×Q16—31no—
92C4×SD16—92no—
93C2×C3×Q8—31no—
94C2×C3×Q8C22×D311yescharacteristic
95C2×D12—32no—
96C2×D12—61no—
97C2×D12—61no—
98C2×D12—121no—
99C2×D12—241no—
100C2×C4×D3—62no—
101C2×C4×D3—122no—
102C2×Dic6—31no—
103C2×SL(2,𝔽3)—21no—
104C2×SL(2,𝔽3)C22×D311yescharacteristic
105GL(2,𝔽3)—62no—
106GL(2,𝔽3)—122no—
107#226—21no—
108C3×Q16—61no—
109C3×Q16—61no—
110C2×C3×C8—31no—
111C2×C3×C8—31no—
112C3×C4⋊C4—31no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—121no—
116C3×C42—31no—
117C3×C42—61no—
118#215—62no—
119#215—62no—
120(C2×D3)⋊C4—62no—
121(C2×D3)⋊C4—62no—
122(C2×D3)⋊C4—62no—
123(C2×D3)⋊C4—122no—
124C4⋊Dic3—32no—
125C4⋊Dic3—61no—
126C4⋊Dic3S411yescharacteristic
127Dic3⋊C4—32no—
128C4×Dic3—32no—
129C4×Dic3—41no—
130C2×C3⋊C8—32no—
131C22×C32—41no—
132C2×C3⋊D3—44no—
133C2×C3⋊D3—82no—
134C2×C3⋊D3—82no—
135C32×C4—42no—
136C3×Dic3—42no—
137C3×Dic3—82no—
138C3×Dic3—81no—
139C2×Q16—31no—
140C2×SD16—92no—
141C4⋊Q8—91no—
142#127—91no—
143#125—92no—
144#124—92no—
145#124—181no—
146#123—182no—
147C4×Q8—31no—
148C4×Q8—32no—
149C4×Q8—91no—
150C4×Q8D3211yescharacteristic
151C4×D4—92no—
152C4×D4—181no—
153#107—31no—
154#106—92no—
155C4⋊C8—91no—
156C4⋊C8—92no—
157#103—31no—
158#103—31no—
159#103—92no—
160#102—92no—
161#102—92no—
162#102—92no—
163C4×C8—31no—
164C4×C8—92no—
165C22×D3—62no—
166C22×D3—122no—
167C22×D3—242no—
168C3×Q8—31no—
169C3×Q8—61no—
170C3×Q8—61no—
171C3×Q8#23911yescharacteristic
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—41no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—81no—
178C2×C3×C4—122no—
179C2×C3×C4C2×S411yescharacteristic
180C2×Dic3—32no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3C2×S412yes—
184D12—62no—
185D12—62no—
186D12—124no—
187D12—122no—
188D12—244no—
189D12—241no—
190C4×D3—242no—
191C4×D3—244no—
192Dic6—62no—
193SL(2,𝔽3)—21no—
194SL(2,𝔽3)#23511yescharacteristic
195C3×C8—61no—
196C3×C8—61no—
197C3⋊C8—62no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C3⋊D3—84no—
202C2×Q8—31no—
203C2×Q8—91no—
204C2×Q8C2×D3211yescharacteristic
205C2×D4—92no—
206C2×D4—181no—
207C2×D4—181no—
208C22×C4—182no—
209Q16—61no—
210Q16—61no—
211SD16—182no—
212SD16—182no—
213C2×C8—31no—
214C2×C8—31no—
215C2×C8—92no—
216C4⋊C4—31no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—61no—
220C4⋊C4—92no—
221C4⋊C4—92no—
222C4⋊C4—181no—
223C22⋊C4—182no—
224C22⋊C4—182no—
225C22⋊C4—182no—
226C42—31no—
227C42—61no—
228C42—92no—
229C22×C3—41no—
230C22×C3—81no—
231C22×C3C22×S411yescharacteristic
232C2×D3—124no—
233C2×D3—122no—
234C2×D3—122no—
235C2×D3—122no—
236C2×D3—242no—
237C2×D3—242no—
238C2×D3—244no—
239C2×D3—482no—
240C2×D3—482no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—42no—
244C3×C4—61no—
245C3×C4—82no—
246C3×C4—121no—
247C3×C4—121no—
248C3×C4—242no—
249C3×C4#98512yes—
250Dic3—22no—
251Dic3—42no—
252Dic3—62no—
253Dic3—81no—
254C32—41noSylow
255C23—182no—
256Q8—31no—
257Q8—61no—
258Q8—61no—
259Q8—182no—
260Q8#379911yescharacteristic
261D4—182no—
262D4—182no—
263D4—362no—
264D4—721no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—32no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—92no—
271C2×C4—722no—
272C2×C4D3×S411yescharacteristic
273C8—61no—
274C8—61no—
275C8—182no—
276C2×C3—41no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—81no—
280C2×C3—81no—
281C2×C3—81no—
282C2×C3#591911yescharacteristic
283C2×C3C2×#98511yescharacteristic
284C2×C3#592611yescharacteristic
285D3—244no—
286D3—242no—
287D3—484no—
288C22—362no—
289C22—362no—
290C22—362no—
291C22C2×D3×S411yescharacteristic
292C4—31no—
293C4—31no—
294C4—62no—
295C4—61no—
296C4—121no—
297C4—121no—
298C4—182no—
299C4#6419812yes—
300C3—41no—
301C3—81no—
302C3#8542011yescharacteristic, complemented
303C2—722no—
304C2#1059642111yescharacteristic
305C2C2×#6419811yescharacteristic
306C2#1059645911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
prime radicalC4⋊(C3⋊GL(2,𝔽3))C2
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #5570⋊C3C2semidirect
3. C3×C4×#226C2semidirect
4. #10593214C2semidirect
5. #10593208C2semidirect
15. #5570D3semidirect
21. C4×#226D3semidirect
302. C3#85420semidirect

Maximal subgroups

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

subgroupindex
#5570⋊C32
C4⋊(C3⋊GL(2,𝔽3))2
C3×C4×#2262
#105932142
#105932082
#854203
#797013
#638944