Finite Group Databasebeta

← C3×#85473 #11922757

Subgroups — C3×#85473 #11922757

Order 1152.

533 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#5926C212yesmaximal, complemented
2C3×#578⋊C3C211yesmaximal, characteristic, complemented
3C3×#5431C212yesmaximal, complemented
4C3×#5407C211yesmaximal, characteristic, complemented
5C3×#5404C211yesmaximal, characteristic, complemented
6#85473C311yesmaximal, characteristic, direct product factor
7C3×#3290—31nomaximal
8C3×2−1+4⋊C3C2212yescomplemented
9C3×#986C2212yescomplemented
10C3×#985C2211yescharacteristic
11C3×#983—22no—
12C2×C3×GL(2,𝔽3)—22no—
13C3×D3×D8—41nomaximal
14C3×#131⋊C3C2211yescharacteristic, complemented
15C3×GU(2,𝔽3)C2211yescharacteristic, complemented
16#5926C2×C312yescomplemented
17C3×#578D311yescharacteristic, complemented
18C3×#577—32no—
19C3×#576—31no—
20#578⋊C3—21no—
21#578⋊C3C2×C311yescharacteristic, complemented
22#5431C2×C312yescomplemented
23#5407C2×C311yescharacteristic, complemented
24#5404C2×C311yescharacteristic, complemented
25C3×#495—31no—
26C3×#473—31no—
27C3×#472—31no—
28C3×#471—32no—
29C3×#455—32no—
30C3×#454—32no—
31C3×#443—31no—
32C3×#443—31no—
33C3×D3×D4—42no—
34C3×#41⋊C3C2311yescharacteristic
35C2×C3×SL(2,𝔽3)—22no—
36C3×GL(2,𝔽3)—22no—
37C3×GL(2,𝔽3)D411yescharacteristic
38C3×#226D411yescharacteristic
39C32×D8—41no—
40C3×#213—42no—
41C3×D24—41no—
42C3×D3×C8—41no—
43#3290—31noSylow
44C3×2−1+4C2×D312yescomplemented
45C3×H(5,𝔽2)—32no—
462−1+4⋊C3—22no—
472−1+4⋊C3C22×C312yescomplemented
48#986C22×C312yescomplemented
49#985C22×C311yescharacteristic
50#983—22no—
51C2×GL(2,𝔽3)—22no—
52C3×#137—32no—
53C3×#137—32no—
54C3×#137—62no—
55C3×AGL(1,ℤ/8)—32no—
56C3×AGL(1,ℤ/8)—32no—
57C3×AGL(1,ℤ/8)—62no—
58C3×AGL(1,ℤ/8)—62no—
59C3×#135—31no—
60C3×#135—31no—
61C3×#135—62no—
62C3×#135—61no—
63C2×C3×Q16—31no—
64C2×C3×SD16—32no—
65C2×C3×SD16—62no—
66C2×C3×D8—31no—
67C2×C3×D8—61no—
68C3×#131—31no—
69C3×#131—32no—
70C3×#131C2×D311yescharacteristic, complemented
71C3×#124—32no—
72D3×D8—41no—
73#131⋊C3—21no—
74#131⋊C3C22×C311yescharacteristic, complemented
75GU(2,𝔽3)C22×C311yescharacteristic, complemented
76C3×#108—31no—
77C3×#108—31no—
78C3×#104—32no—
79C3×#104—31no—
80C3×#104—31no—
81C3×#101—32no—
82C3×#100—32no—
83C3×C4×C8—31no—
84C22×C3×D3—82no—
85C32×D4—42no—
86C3×#67—82no—
87C3×D12—42no—
88C3×C4×D3—41no—
89C3×SL(2,𝔽3)C2×D411yescharacteristic
90C32×C8—41no—
91C3×C3⋊C8—41no—
92#578C3×D311yescharacteristic, complemented
93#577—32no—
94#576—31no—
95#495—31no—
96#473—31no—
97#472—31no—
98#471—32no—
99#455—32no—
100#454—32no—
101#443—31no—
102#443—31no—
103C3×#41—31no—
104C3×#41—32no—
105C3×#41—62no—
106C3×#41—62no—
107C3×#41—62no—
108C3×#41C22×D311yescharacteristic
109C2×C3×Q8—22no—
110C2×C3×Q8—32no—
111C2×C3×D4—32no—
112C2×C3×D4—62no—
113C2×C3×D4—62no—
114C2×C3×D4—122no—
115D3×D4—42no—
116#41⋊C3—21no—
117#41⋊C3C23×C311yescharacteristic
118C2×SL(2,𝔽3)—22no—
119C2×SL(2,𝔽3)—42no—
120GL(2,𝔽3)—22no—
121GL(2,𝔽3)C3×D411yescharacteristic
122#226C3×D411yescharacteristic
123C3×Q16—31no—
124C3×Q16—31no—
125C3×Q16—61no—
126C3×Q16—62no—
127C3×SD16—31no—
128C3×SD16—31no—
129C3×SD16—32no—
130C3×SD16—62no—
131C3×SD16—62no—
132C3×SD16—62no—
133C3×SD16—62no—
134C3×SD16—62no—
135C3×D8—31no—
136C3×D8—41no—
137C3×D8—61no—
138C3×D8—62no—
139C3×D8—62no—
140C3×D8—61no—
141C3×D8—81no—
142C3×D8S411yescharacteristic
143C3×M16—31no—
144C3×M16—32no—
145C3×M16—31no—
146C3×M16—61no—
147C3×M16—62no—
148C2×C3×C8—31no—
149C2×C3×C8—31no—
150C2×C3×C8—62no—
151C2×C3×C8—61no—
152C3×C22⋊C4—122no—
153C3×C42—31no—
154#213—42no—
155D24—41no—
156D3×C8—41no—
157C22×C32—82no—
158C2×C3×D3—41no—
159C2×C3×D3—82no—
160C2×C3×D3—164no—
161C32×C4—41no—
162C3×Dic3—41no—
1632−1+4C2×C3×D312yescomplemented
164H(5,𝔽2)—32no—
165#137—32no—
166#137—32no—
167#137—62no—
168AGL(1,ℤ/8)—32no—
169AGL(1,ℤ/8)—32no—
170AGL(1,ℤ/8)—62no—
171AGL(1,ℤ/8)—62no—
172#135—31no—
173#135—31no—
174#135—62no—
175#135—61no—
176C2×Q16—31no—
177C2×SD16—32no—
178C2×SD16—62no—
179C2×D8—31no—
180C2×D8—61no—
181#131—31no—
182#131—32no—
183#131C2×C3×D311yescharacteristic, complemented
184#124—32no—
185#108—31no—
186#108—31no—
187#104—32no—
188#104—31no—
189#104—31no—
190#101—32no—
191#100—32no—
192C4×C8—31no—
193C23×C3—62no—
194C23×C3—122no—
195C22×D3—82no—
196C3×Q8—31no—
197C3×Q8—32no—
198C3×Q8—62no—
199C3×Q8D3×D411yescharacteristic
200C3×D4—31no—
201C3×D4—31no—
202C3×D4—42no—
203C3×D4—62no—
204C3×D4—62no—
205C3×D4—61no—
206C3×D4—62no—
207C3×D4—82no—
208C3×D4—122no—
209C3×D4—122no—
210C3×D4C2×S412yes—
211C2×C3×C4—31no—
212C2×C3×C4—62no—
213C2×C3×C4—61no—
214C2×C3×C4—62no—
215C2×C3×C4—61no—
216C2×C3×C4—122no—
217#67—82no—
218D12—42no—
219C4×D3—41no—
220SL(2,𝔽3)—21no—
221SL(2,𝔽3)C2×C3×D411yescharacteristic
222C3×C8—31no—
223C3×C8—31no—
224C3×C8—31no—
225C3×C8—41no—
226C3×C8—62no—
227C3×C8—61no—
228C3×C8—81no—
229C3×C8C2×S411yescharacteristic
230C3⋊C8—41no—
231C2×C32—41no—
232C2×C32—162no—
233C3×D3—42no—
234C3×D3—162no—
235#41—31no—
236#41—32no—
237#41—62no—
238#41—62no—
239#41—62no—
240#41C22×C3×D311yescharacteristic
241C2×Q8—22no—
242C2×Q8—32no—
243C2×D4—32no—
244C2×D4—62no—
245C2×D4—62no—
246C2×D4—122no—
247Q16—31no—
248Q16—31no—
249Q16—61no—
250Q16—62no—
251SD16—31no—
252SD16—31no—
253SD16—32no—
254SD16—62no—
255SD16—62no—
256SD16—62no—
257SD16—62no—
258SD16—62no—
259D8—31no—
260D8—61no—
261D8—62no—
262D8—62no—
263D8—61no—
264D8C3×S411yescharacteristic
265M16—31no—
266M16—32no—
267M16—31no—
268M16—61no—
269M16—62no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—62no—
273C2×C8—61no—
274C22⋊C4—122no—
275C42—31no—
276C22×C3—22no—
277C22×C3—31no—
278C22×C3—61no—
279C22×C3—82no—
280C22×C3—122no—
281C22×C3—162no—
282C22×C3—242no—
283C22×C3—482no—
284C2×D3—41no—
285C2×D3—82no—
286C2×D3—164no—
287C3×C4—31no—
288C3×C4—41no—
289C3×C4—62no—
290C3×C4—61no—
291C3×C4—81no—
292C3×C4—121no—
293C3×C4C22×S411yescharacteristic
294Dic3—41no—
295C32—41noSylow
296C23—62no—
297C23—122no—
298Q8—31no—
299Q8—32no—
300Q8—62no—
301Q8C3×D3×D411yescharacteristic
302D4—31no—
303D4—31no—
304D4—62no—
305D4—62no—
306D4—61no—
307D4—62no—
308D4—122no—
309D4—122no—
310D4C2×C3×S412yes—
311C2×C4—31no—
312C2×C4—62no—
313C2×C4—61no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—122no—
317C8—31no—
318C8—31no—
319C8—31no—
320C8—62no—
321C8—61no—
322C8C2×C3×S411yescharacteristic
323C2×C3—42no—
324C2×C3—41no—
325C2×C3—61no—
326C2×C3—81no—
327C2×C3—121no—
328C2×C3—162no—
329C2×C3—242no—
330C2×C3—322no—
331C2×C3D4×S411yescharacteristic
332D3—42no—
333D3—162no—
334C22—22no—
335C22—31no—
336C22—61no—
337C22—122no—
338C22—242no—
339C22—482no—
340C4—31no—
341C4—62no—
342C4—61no—
343C4—121no—
344C4C22×C3×S411yescharacteristic
345C3—41no—
346C3—81no—
347C3#8547311yescharacteristic, direct product factor
348C2—42no—
349C2—61no—
350C2—121no—
351C2—242no—
352C2C3×D4×S411yescharacteristic
subgroupG/H
center, socleC2×C3D4×S4
commutator, cosocle#41⋊C3C23×C3
FrattiniC4C22×C3×S4
FittingC3×#578D3
hypercenterC3×D8S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8

Complements

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

NH ≅ G/Ntype
1. C3×#5926C2semidirect
2. C3×#578⋊C3C2semidirect
3. C3×#5431C2semidirect
4. C3×#5407C2semidirect
5. C3×#5404C2semidirect
6. #85473C3direct (N × H)
8. C3×2−1+4⋊C3C22semidirect
9. C3×#986C22semidirect
14. C3×#131⋊C3C22semidirect
15. C3×GU(2,𝔽3)C22semidirect
16. #5926C2×C3semidirect
17. C3×#578D3semidirect
21. #578⋊C3C2×C3semidirect
22. #5431C2×C3semidirect
23. #5407C2×C3semidirect
24. #5404C2×C3semidirect
44. C3×2−1+4C2×D3semidirect
47. 2−1+4⋊C3C22×C3semidirect
48. #986C22×C3semidirect
70. C3×#131C2×D3semidirect
74. #131⋊C3C22×C3semidirect
75. GU(2,𝔽3)C22×C3semidirect
92. #578C3×D3semidirect
163. 2−1+4C2×C3×D3semidirect
183. #131C2×C3×D3semidirect
347. C3#85473direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#59262
C3×#578⋊C32
C3×#54312
C3×#54072
C3×#54042
#854733
C3×#32903
C3×D3×D84