Finite Group Databasebeta

← #83815⋊C3 #11921342

Subgroups — #83815⋊C3 #11921342

Order 1152.

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

normal subgroups only

This group has 386 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
1C2×#1010⋊C3C211yesmaximal, characteristic
2C3×#568⋊C3C211yesmaximal, characteristic, complemented
3#5819⋊C3C211yesmaximal, characteristic, complemented
4#83815C311yesmaximal, characteristic, complemented
5#2889⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#1010⋊C3—21no—
8#1010⋊C3—21no—
9C2×D3×SL(2,𝔽3)C412yes—
10C3×#131⋊C3—21no—
11C3⋊(#131⋊C3)—21no—
12C3×#823—41nomaximal
13C2×#1010C2×C311yescharacteristic
14C2×2−1+4⋊C3—31no—
15C3×#568C2×C311yescharacteristic, complemented
16#5819C2×C311yescharacteristic, complemented
17#568⋊C3—21no—
18#568⋊C3—31no—
19#568⋊C3D311yescharacteristic, complemented
20C2×#824—31no—
21C2×#824—31no—
22#5132—31no—
23#5132—31no—
24C2×C3×D12—41no—
25C3×#41⋊C3D411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28D3×SL(2,𝔽3)—22no—
29C32×M16—41no—
30C3×#208—41no—
31#2889—31noSylow
32C2×C3×#41C22×C311yescharacteristic
33#1010—21no—
34#1010—21no—
35#1010—61no—
36#1010—61no—
37C2×D3×Q8C3×C412yes—
38C2×#237—32no—
39C2×#235—31no—
40C22×Dic6—31no—
412−1+4⋊C3—61no—
422−1+4⋊C3—61no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3C2×D311yescharacteristic
45C22×SL(2,𝔽3)—32no—
46C3×#131—21no—
47C3×#131—61no—
48C2×C3×M16—31no—
49C2×C3×M16—31no—
50#948—21no—
51#948—61no—
52C2×#208—31no—
53C2×#208—31no—
54#131⋊C3—22no—
55#131⋊C3—21no—
56#131⋊C3—61no—
57C3×#100—121no—
58#824—32no—
59#824—31no—
60#824—31no—
61#823—31no—
62#823A411yescharacteristic
63C22×C3×D3—42no—
64C2×C32×C4—41no—
65C3×D12—81no—
66C3×D12—81no—
67C3×SL(2,𝔽3)C22⋊C411yescharacteristic
68C32×C8—81no—
69C3×C3⋊C8—81no—
70C2×2−1+4—31no—
71#568—31no—
72#568C3×D311yescharacteristic, complemented
73#413—91no—
74#413—91no—
75C2×#101—91no—
76C2×#101—91no—
77C3×#41—31no—
78C3×#41—31no—
79C3×#41C3×D411yescharacteristic
80C3×#41C3×D411yescharacteristic
81C2×C3×Q8C2×C3×C411yescharacteristic
82C2×C3×D4—31no—
83C2×C3×D4—121no—
84C22×C3×C4—31no—
85C2×#67—32no—
86C22×Dic3—32no—
87D3×Q8—22no—
88D3×Q8—62no—
89#237—64no—
90#237—62no—
91#237—62no—
92#235—61no—
93#235—61no—
94#235—61no—
95#235—61no—
96C2×D12C2×A411yescharacteristic
97C2×C4×D3—32no—
98C2×Dic6—31no—
99C2×Dic6—32no—
100C2×Dic6—61no—
101C2×Dic6—61no—
102C2×Dic6—61no—
103C2×Dic6—61no—
104#41⋊C3—21no—
105#41⋊C3—21no—
106#41⋊C3D1211yescharacteristic
107#41⋊C3#6711yescharacteristic
108C2×SL(2,𝔽3)—21no—
109C2×SL(2,𝔽3)—62no—
110C2×SL(2,𝔽3)C4×D311yescharacteristic
111C3×M16—31no—
112C3×M16—41no—
113C3×M16—61no—
114C3×M16—61no—
115C3×M16—81no—
116C3×M16—121no—
117C3×M16C2×A411yescharacteristic
118C2×C3×C8—61no—
119C2×C3×C8—61no—
120#208—31no—
121#208—61no—
122#208—61no—
123#208C2×A411yescharacteristic
124C2×C3⋊C8—61no—
125C2×C3⋊C8—61no—
126C22×C32—41no—
127C2×C3×D3—82no—
128C2×C3×D3—162no—
129C32×C4—41no—
130C32×C4—41no—
1312−1+4—61no—
1322−1+4—61no—
1332−1+4—181no—
1342−1+4—181no—
135C2×#41—91no—
136C2×#41—92no—
137C2×#41C2×C3×D311yescharacteristic
138C22×Q8—32no—
139C22×Q8—91no—
140#131—21no—
141#131—61no—
142#131—61no—
143#131—181no—
144C2×M16—31no—
145C2×M16—31no—
146C2×M16—91no—
147C2×M16—91no—
148#101—91no—
149#101—92no—
150#101—91no—
151#100—31no—
152#100—91no—
153C23×C3—31no—
154C23×C3—122no—
155C22×D3C4×A412yes—
156C3×Q8—31no—
157C3×Q8C3×C22⋊C411yescharacteristic
158C3×D4—31no—
159C3×D4—31no—
160C3×D4—32no—
161C3×D4—241no—
162C3×D4—241no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—41no—
169C2×C3×C4—81no—
170C2×C3×C4C22×A411yescharacteristic
171#67—62no—
172#67—62no—
173C2×Dic3—32no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176D12—21no—
177D12—21no—
178C4×D3—62no—
179C4×D3—62no—
180Dic6—61no—
181Dic6—61no—
182Dic6—62no—
183Dic6—62no—
184SL(2,𝔽3)—21no—
185SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
186C3×C8—21no—
187C3×C8—61no—
188C3×C8—81no—
189C3×C8—82no—
190C3×C8—241no—
191C3⋊C8—21no—
192C3⋊C8—61no—
193C2×C32—41no—
194C2×C32—81no—
195C3×D3—162no—
196#41—31no—
197#41—31no—
198#41—181no—
199#41—181no—
200#41—181no—
201#41—181no—
202#41—184no—
203#41—182no—
204#41—182no—
205#41C3×D1211yescharacteristic
206#41C3×#6711yescharacteristic
207C2×Q8—62no—
208C2×Q8—92no—
209C2×Q8—91no—
210C2×Q8—181no—
211C2×Q8—181no—
212C2×Q8—181no—
213C2×Q8—181no—
214C2×Q8—182no—
215C2×Q8C3×C4×D311yescharacteristic
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—92no—
219C22×C4—31no—
220C22×C4—92no—
221C22×C4—92no—
222M16—31no—
223M16—31no—
224M16—61no—
225M16—61no—
226M16—91no—
227M16—181no—
228M16—181no—
229M16D3×A411yescharacteristic
230C2×C8—61no—
231C2×C8—61no—
232C2×C8—181no—
233C2×C8—181no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—41no—
237C22×C3—81no—
238C22×C3—121no—
239C22×C3—242no—
240C22×C3—482no—
241C22×C3C2×C4×A411yescharacteristic
242C2×D3—22no—
243C2×D3—42no—
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4D4×A411yescharacteristic
251C3×C4D4×A411yescharacteristic
252Dic3—62no—
253C32—41noSylow
254C23—31no—
255C23—32no—
256Q8—31no—
257Q8—182no—
258Q8—182no—
259Q8—181no—
260Q8—181no—
261Q8C3×(C2×D3)⋊C411yescharacteristic
262D4—31no—
263D4—31no—
264D4—32no—
265D4—61no—
266D4—61no—
267D4—182no—
268D4—182no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—92no—
275C2×C4—182no—
276C2×C4—182no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4C2×D3×A411yescharacteristic
280C8—21no—
281C8—61no—
282C8—61no—
283C8—181no—
284C2×C3—21no—
285C2×C3—41no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3—161no—
291C2×C3—482no—
292C2×C3A4×C22⋊C411yescharacteristic
293D3—42no—
294C22—31no—
295C22—31no—
296C22—62no—
297C22—121no—
298C22—122no—
299C22C4×D3×A411yescharacteristic
300C4—31no—
301C4—31no—
302C4—182no—
303C4A4×D1211yescharacteristic
304C4A4×#6711yescharacteristic
305C3—41no—
306C3—81no—
307C3#2889⋊C311yescharacteristic, complemented
308C2—21no—
309C2—61no—
310C2—61no—
311C2—122no—
312C2A4×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC2A4×(C2×D3)⋊C4
commutatorC2×C3×Q8C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×#568C2×C3
socleC2×C3A4×C22⋊C4
prime radicalC2×#1010⋊C3C2
cosocleC2×C3×#41C22×C3
hypercenterM16D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×#568⋊C3C2semidirect
3. #5819⋊C3C2semidirect
4. #83815C3semidirect
15. C3×#568C2×C3semidirect
16. #5819C2×C3semidirect
19. #568⋊C3D3semidirect
72. #568C3×D3semidirect
307. C3#2889⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1010⋊C32
C3×#568⋊C32
#5819⋊C32
#838153
#2889⋊C33
C3×#8234