Finite Group Databasebeta

← #11921648

Subgroups — #11921648

Order 1152.

449 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 53 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×#5391C211yesmaximal, characteristic, complemented
4#10593214C212yesmaximal
5#10593196C212yesmaximal, complemented
6#85418—31nomaximal
7#81136—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9C3×C4×SL(2,𝔽3)C2211yescharacteristic
10#63900—41nomaximal
11Dic3×SL(2,𝔽3)C2212yes—
12C3⋊#859C2211yescharacteristic
13C3×#859C2212yes—
14#5730—31no—
15#5570D311yescharacteristic, complemented
16#557⋊C3—31no—
17C4×GL(2,𝔽3)—31no—
18C4×GL(2,𝔽3)—32no—
19C4×GL(2,𝔽3)—61no—
20#5391D311yescharacteristic, complemented
21#5389—32no—
22C3×#477—31no—
23#5080—31no—
24C4×#215—31no—
25#4891—32no—
26#4878—32no—
27#4846—31no—
28#4830—31no—
29#4796—32no—
30#4788—32no—
31C2×C4×C3⋊D3—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C2×#624—42no—
34C3⋊GL(2,𝔽3)Q812yescomplemented
35C3×C4⋊Dic3—41no—
36C3×C4⋊Dic3—41no—
37#3724—42no—
38#3381—91noSylow
39C2×GL(2,𝔽3)—31no—
40C2×GL(2,𝔽3)—61no—
41C3×C4⋊Q8—31no—
42C3×C4×Q8C2×D311yescharacteristic
43#946—62no—
44Q8×Dic3C2×D312yes—
45C2×#215—31no—
46#897—61no—
47#893—32no—
48D3×C4⋊C4—61no—
49D3×C4⋊C4—121no—
50#890—32no—
51#890—32no—
52C4×D12—31no—
53#869—31no—
54C4×Dic6—31no—
55C4×SL(2,𝔽3)—21no—
56C4×SL(2,𝔽3)—32no—
57C4×SL(2,𝔽3)C2×D311yescharacteristic
58#859—31no—
59#859—61no—
60#859C2×D312yes—
61C3×#107—32no—
62C3×C4⋊C8—31no—
63C3×#103—32no—
64#835—31no—
65#821—32no—
66#818—32no—
67Dic3⋊C8—32no—
68C8×Dic3—32no—
69#809—31no—
70#808—31no—
71#808—32no—
72#808—32no—
73C4×C3⋊C8—31no—
74C22×C3⋊D3—41no—
75C2×C32×C4—41no—
76C2×C3⋊Dic3—41no—
77C4×C3⋊D3—44no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—42no—
80C3×SL(2,𝔽3)C2×Q811yescharacteristic
81#624—84no—
82#557—31no—
83#554—91no—
84#499—92no—
85#498—91no—
86#479—92no—
87#478—92no—
88#477—31no—
89#474—91no—
90C8×Q8—91no—
91C4×SD16—92no—
92C4×SD16—91no—
93C2×C3×Q8—61no—
94C2×C3×Q8C22×D311yescharacteristic
95C2×D12—31no—
96C2×C4×D3—61no—
97C2×C4×D3—62no—
98C2×C4×D3—121no—
99C2×C4×D3—122no—
100C2×C4×D3—241no—
101C2×Dic6—31no—
102C2×SL(2,𝔽3)—21no—
103C2×SL(2,𝔽3)C22×D311yescharacteristic
104GL(2,𝔽3)—32no—
105GL(2,𝔽3)—62no—
106C2×C3×C8—32no—
107C3×C4⋊C4—31no—
108C3×C4⋊C4—32no—
109C3×C4⋊C4—61no—
110C3×C4⋊C4—121no—
111C3×C42—31no—
112#215—32no—
113#215—32no—
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4—61no—
116C4⋊Dic3—32no—
117C4⋊Dic3—31no—
118C4⋊Dic3—41no—
119C4⋊Dic3—61no—
120C4⋊Dic3—62no—
121C4⋊Dic3S411yescharacteristic
122Dic3⋊C4—32no—
123Dic3⋊C4—62no—
124Dic3⋊C4—122no—
125C4×Dic3—32no—
126C2×C3⋊C8—31no—
127C2×C3⋊C8—31no—
128C22×C32—41no—
129C2×C3⋊D3—42no—
130C2×C3⋊D3—42no—
131C2×C3⋊D3—42no—
132C32×C4—42no—
133C3⋊Dic3—42no—
134C3×Dic3—82no—
135C3×Dic3—82no—
136C2×SD16—91no—
137C4⋊Q8—31no—
138C4⋊Q8—91no—
139#125—92no—
140#125—92no—
141C22⋊Q8—181no—
142C22⋊Q8—182no—
143C4×Q8—32no—
144C4×Q8—91no—
145C4×Q8D3211yescharacteristic
146C4×D4—91no—
147C4×D4—92no—
148C2×C4⋊C4—181no—
149#107—32no—
150#107—92no—
151#106—92no—
152#106—91no—
153#106—92no—
154C4⋊C8—31no—
155C4⋊C8—92no—
156#103—32no—
157#103—91no—
158#102—91no—
159#102—92no—
160C4×C8—91no—
161C4×C8—92no—
162C22×D3—61no—
163C22×D3—121no—
164C22×D3—241no—
165C3×Q8—31no—
166C3×Q8—241no—
167C3×Q8D3×Q811yescharacteristic
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—41no—
171C2×C3×C4—62no—
172C2×C3×C4—81no—
173C2×C3×C4—122no—
174C2×C3×C4C2×S411yescharacteristic
175C2×Dic3—32no—
176C2×Dic3—42no—
177C2×Dic3—61no—
178C2×Dic3—121no—
179C2×Dic3—241no—
180C2×Dic3C2×S412yes—
181D12—32no—
182D12—61no—
183C4×D3—124no—
184C4×D3—122no—
185C4×D3—244no—
186C4×D3—242no—
187C4×D3—244no—
188Dic6—62no—
189SL(2,𝔽3)—21no—
190SL(2,𝔽3)D3×Q811yescharacteristic
191C3×C8—62no—
192C3⋊C8—32no—
193C3⋊C8—61no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C3⋊D3—44no—
198C2×Q8—61no—
199C2×Q8—91no—
200C2×Q8C2×D3211yescharacteristic
201C2×D4—91no—
202C22×C4—182no—
203C22×C4—181no—
204SD16—92no—
205SD16—92no—
206C2×C8—32no—
207C2×C8—91no—
208C2×C8—91no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—32no—
212C4⋊C4—61no—
213C4⋊C4—92no—
214C4⋊C4—92no—
215C4⋊C4—91no—
216C4⋊C4—182no—
217C4⋊C4—181no—
218C4⋊C4—182no—
219C22⋊C4—181no—
220C22⋊C4—182no—
221C42—31no—
222C42—92no—
223C22×C3—41no—
224C22×C3—81no—
225C22×C3C22×S411yescharacteristic
226C2×D3—62no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—121no—
231C2×D3—121no—
232C2×D3—242no—
233C2×D3—242no—
234C2×D3—242no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—42no—
238C3×C4—61no—
239C3×C4—82no—
240C3×C4—122no—
241C3×C4—242no—
242C3×C4#98512yes—
243Dic3—22no—
244Dic3—62no—
245Dic3—82no—
246Dic3—122no—
247Dic3—121no—
248Dic3—242no—
249C32—41noSylow
250C23—181no—
251Q8—31no—
252Q8—182no—
253Q8—241no—
254Q8#379711yescharacteristic
255D4—92no—
256D4—181no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—32no—
260C2×C4—62no—
261C2×C4—92no—
262C2×C4—181no—
263C2×C4—362no—
264C2×C4—722no—
265C2×C4D3×S411yescharacteristic
266C8—62no—
267C8—92no—
268C8—181no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—81no—
273C2×C3—81no—
274C2×C3—81no—
275C2×C3Q8×S411yescharacteristic
276C2×C3C2×#98511yescharacteristic
277C2×C3#592611yescharacteristic
278D3—124no—
279D3—122no—
280D3—244no—
281C22—182no—
282C22—361no—
283C22—361no—
284C22C2×D3×S411yescharacteristic
285C4—31no—
286C4—31no—
287C4—62no—
288C4—61no—
289C4—122no—
290C4—182no—
291C4—361no—
292C4#6419812yes—
293C3—41no—
294C3—81no—
295C3#8541811yescharacteristic, complemented
296C2—362no—
297C2(C3⋊S4)⋊Q811yescharacteristic
298C2C2×#6419811yescharacteristic
299C2#1059646011yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
prime radicalC2×C3⋊GL(2,𝔽3)C22
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)C2×Q8
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×#5391C2semidirect
5. #10593196C2semidirect
15. #5570D3semidirect
20. #5391D3semidirect
34. C3⋊GL(2,𝔽3)Q8semidirect
295. C3#85418semidirect

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×#53912
#105932142
#105931962
#854183
#811363
#639004