Finite Group Databasebeta

← #11921703

Subgroups — #11921703

Order 1152.

581 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 581 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#10593620C211yesmaximal, characteristic, complemented
3C3×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593217C212yesmaximal, complemented
5Dic3×GL(2,𝔽3)C212yesmaximal, complemented
6#85325—31nomaximal
7#80360—31nomaximal
8C2×#3795—41nomaximal
9C2×D3×SL(2,𝔽3)C2211yescharacteristic
10C2×C3×GL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13C3⋊#859C2212yes—
14#5716—31no—
15C4×D3×Q8D311yescharacteristic, complemented
16C2×C4×SL(2,𝔽3)—31no—
17#5421—32no—
18C4⋊GL(2,𝔽3)D311yescharacteristic, complemented
19C4×GL(2,𝔽3)—32no—
20#5386—31no—
21#5386—61no—
22C3×#460—31no—
23#5170—32no—
24Dic3×SD16—32no—
25#5075—31no—
26#5020—31no—
27#4851—31no—
28#4833—31no—
29#4777—32no—
30#4758—32no—
31C2×#634—41no—
32C2×C3×D12—41no—
33C2×C3×C4×D3—41no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×#625—42no—
36C2×D3×Dic3—42no—
37#3795—48no—
38D3×SL(2,𝔽3)—21no—
39C3×GL(2,𝔽3)—22no—
40#3103—91noSylow
41C2×D3×Q8C2×D311yescharacteristic
42C2×#237—61no—
43C2×#235—121no—
44C22×SL(2,𝔽3)—31no—
45C2×GL(2,𝔽3)C2×D312yes—
46C2×C3×SD16—32no—
47C3×#127—31no—
48C3×C4×Q8C2×D311yescharacteristic
49Q8×Dic3C2×D311yescharacteristic
50#938—32no—
51#938—61no—
52#936—61no—
53D4×Dic3—32no—
54D4×Dic3—61no—
55C2×#214—32no—
56D3×C4⋊C4—31no—
57#887—31no—
58C42×D3—31no—
59#869—31no—
60C4×Dic6—31no—
61C4×SL(2,𝔽3)—21no—
62C4×SL(2,𝔽3)—31no—
63C4×SL(2,𝔽3)C2×D311yescharacteristic
64#859—32no—
65#859—62no—
66C3×C4⋊C8—31no—
67C3×#102—32no—
68#835—32no—
69#832—32no—
70(C2×D3)⋊C8—32no—
71#817—32no—
72#816—32no—
73C8×Dic3—32no—
74#808—32no—
75C4⋊(C3⋊C8)—31no—
76C22×C3×D3—42no—
77C22×C3×D3—41no—
78C2×C32×C4—41no—
79C2×C3⋊Dic3—42no—
80#634—44no—
81C2×C3×Dic3—41no—
82C3×D12—44no—
83C3×C4×D3—44no—
84C3×SL(2,𝔽3)#4111yescharacteristic, complemented
85#625—48no—
86D3×Dic3—48no—
87#532—91no—
88C2×C4×Q8—31no—
89#481—92no—
90#475—91no—
91#475—91no—
92#460—31no—
93#460—91no—
94#448—92no—
95C4×SD16—92no—
96C4×SD16—92no—
97#431—91no—
98C2×C3×Q8C22×D311yescharacteristic
99C2×C3×D4—32no—
100C2×C3×D4—61no—
101C2×C3×D4—61no—
102C22×C3×C4—121no—
103C2×#67—62no—
104C2×#67—122no—
105C22×Dic3—62no—
106D3×Q8—21no—
107D3×Q8—61no—
108#237—124no—
109#235—128no—
110C2×D12—41no—
111C2×C4×D3—31no—
112C2×C4×D3—31no—
113C2×C4×D3—122no—
114C2×C4×D3S411yescharacteristic
115C2×Dic6—31no—
116C2×Dic6—61no—
117C2×Dic6—121no—
118C2×Dic6—241no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)—61no—
121C2×SL(2,𝔽3)C22×D311yescharacteristic
122GL(2,𝔽3)—22no—
123C3×SD16—62no—
124C3×SD16—62no—
125C2×C3×C8—32no—
126C3×C4⋊C4—31no—
127C3×C42—31no—
128#217—62no—
129#217—62no—
130#214—64no—
131C4⋊Dic3—31no—
132C4⋊Dic3—32no—
133C4⋊Dic3—61no—
134Dic3⋊C4—61no—
135C4×Dic3—31no—
136C4×Dic3—31no—
137C4×Dic3—31no—
138C2×C3⋊C8—32no—
139C22×C32—41no—
140C2×C3×D3—44no—
141C2×C3×D3—42no—
142C2×C3×D3—81no—
143C2×C3×D3—81no—
144C2×C3×D3—82no—
145C2×C3×D3—82no—
146C32×C4—42no—
147C3⋊Dic3—44no—
148C3×Dic3—42no—
149C2×#41—181no—
150C22×Q8—31no—
151C2×SD16—32no—
152C2×SD16—92no—
153C4⋊Q8—91no—
154#127—31no—
155#124—181no—
156#121—92no—
157#121—181no—
158C4×Q8—31no—
159C4×Q8—91no—
160C4×Q8—91no—
161C4×Q8D3211yescharacteristic
162C4×D4—92no—
163C4×D4—181no—
164C2×C4⋊C4—91no—
165C2×C42—91no—
166#106—92no—
167#106—92no—
168C4⋊C8—31no—
169C4⋊C8—91no—
170#103—92no—
171#103—92no—
172#102—32no—
173#102—92no—
174C22⋊C8—92no—
175C4×C8—92no—
176C23×C3—62no—
177C23×C3—121no—
178C22×D3—42no—
179C22×D3C2×S411yescharacteristic
180C3×Q8—31no—
181C3×Q8#23511yescharacteristic, complemented
182C3×D4—62no—
183C3×D4—62no—
184C3×D4—122no—
185C3×D4—122no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4—124no—
191C2×C3×C4—121no—
192C2×C3×C4C2×S411yescharacteristic
193#67—128no—
194#67—124no—
195C2×Dic3—31no—
196C2×Dic3—31no—
197C2×Dic3—62no—
198C2×Dic3—122no—
199C2×Dic3—124no—
200C2×Dic3—242no—
201C2×Dic3C2×S411yescharacteristic
202D12—44no—
203C4×D3—61no—
204C4×D3—61no—
205C4×D3—121no—
206C4×D3—128no—
207C4×D3GL(2,𝔽3)14yescomplemented
208Dic6—32no—
209Dic6—61no—
210Dic6—124no—
211Dic6—122no—
212Dic6—244no—
213SL(2,𝔽3)—21no—
214SL(2,𝔽3)#23711yescharacteristic, complemented
215C3×C8—62no—
216C3⋊C8—62no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—41no—
220C3×D3—82no—
221C3×D3—84no—
222#41—364no—
223C2×Q8—61no—
224C2×Q8—91no—
225C2×Q8—181no—
226C2×Q8—181no—
227C2×Q8C2×D3211yescharacteristic
228C2×D4—32no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—182no—
232C22×C4—31no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4—182no—
236SD16—62no—
237SD16—62no—
238SD16—184no—
239C2×C8—32no—
240C2×C8—92no—
241C4⋊C4—31no—
242C4⋊C4—91no—
243C4⋊C4—92no—
244C4⋊C4—181no—
245C4⋊C4—181no—
246C22⋊C4—182no—
247C22⋊C4—182no—
248C42—31no—
249C42—91no—
250C42—91no—
251C42—91no—
252C22×C3—41no—
253C22×C3—81no—
254C22×C3—122no—
255C22×C3—122no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3—241no—
259C22×C3—241no—
260C22×C3C22×S411yescharacteristic
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—44no—
264C2×D3—82no—
265C2×D3—82no—
266C2×D3C2×GL(2,𝔽3)12yes—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—42no—
270C3×C4—61no—
271C3×C4—82no—
272C3×C4—122no—
273C3×C4C2×GL(2,𝔽3)12yes—
274Dic3—61no—
275Dic3—61no—
276Dic3—124no—
277Dic3—122no—
278Dic3—244no—
279Dic3C2×GL(2,𝔽3)12yes—
280C32—41noSylow
281C23—31no—
282C23—62no—
283Q8—31no—
284Q8—92no—
285Q8—181no—
286Q8—362no—
287Q8#379511yescharacteristic, complemented
288D4—62no—
289D4—62no—
290D4—122no—
291D4—122no—
292D4—364no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—34no—
296C2×C4—31no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—181no—
300C2×C4—181no—
301C2×C4—182no—
302C2×C4—364no—
303C2×C4—361no—
304C2×C4D3×S411yescharacteristic
305C8—62no—
306C8—182no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—81no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3—242no—
314C2×C3—242no—
315C2×C3#591211yescharacteristic
316C2×C3C22×GL(2,𝔽3)11yescharacteristic
317C2×C3#592311yescharacteristic
318D3—22no—
319D3—84no—
320C22—32no—
321C22—61no—
322C22—61no—
323C22—122no—
324C22—122no—
325C22—122no—
326C22C2×D3×S411yescharacteristic
327C4—31no—
328C4—31no—
329C4—32no—
330C4—61no—
331C4—181no—
332C4—181no—
333C4—362no—
334C4D3×GL(2,𝔽3)12yes—
335C3—41no—
336C3—81no—
337C3#8532511yescharacteristic, complemented
338C2—62no—
339C2—242no—
340C2#1059642711yescharacteristic
341C2C2×D3×GL(2,𝔽3)11yescharacteristic
342C2#1059645511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
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. C4×D3×SL(2,𝔽3)C2semidirect
2. #10593620C2semidirect
3. C3×C4⋊GL(2,𝔽3)C2semidirect
4. #10593217C2semidirect
5. Dic3×GL(2,𝔽3)C2semidirect
15. C4×D3×Q8D3semidirect
18. C4⋊GL(2,𝔽3)D3semidirect
84. C3×SL(2,𝔽3)#41semidirect
181. C3×Q8#235semidirect
207. C4×D3GL(2,𝔽3)semidirect
214. SL(2,𝔽3)#237semidirect
287. Q8#3795semidirect
337. C3#85325semidirect

Maximal subgroups

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

subgroupindex
C4×D3×SL(2,𝔽3)2
#105936202
C3×C4⋊GL(2,𝔽3)2
#105932172
Dic3×GL(2,𝔽3)2
#853253
#803603
C2×#37954