Finite Group Databasebeta

← #10592761⋊C3 #13142926

Subgroups — #10592761⋊C3 #13142926

Order 1728.

587 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 90 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#11722627C211yesmaximal, characteristic, complemented
2H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
3#64141⋊C3C212yesmaximal, complemented
4#63663⋊C3C211yesmaximal, characteristic, complemented
5#63632⋊C3C212yesmaximal, complemented
6#10592761C311yesmaximal, characteristic, complemented
7C3×#5062—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10#88722—22no—
11Q8×(C3⋊D3)⋊C3—22no—
12#3828⋊C3—22no—
13#3828⋊C3C2212yes—
14C22⋊C4×H(3,𝔽3)—22no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#3753⋊C3—22no—
17#3754⋊C3—22no—
18#3748⋊C3C2212yes—
19#64364C2×C311yescharacteristic, complemented
20C3×#1007—31no—
21C32×#124—21no—
22C32×#124D311yescharacteristic, complemented
23C32×#124C2×C311yescharacteristic, complemented
24#64141C2×C312yescomplemented
25C3×#932—32no—
26#63663C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#836—31no—
29C3×#805—32no—
30C23×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—22no—
32D4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)D412yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36#638⋊C3—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)D412yes—
39C4×(C3⋊D3)⋊C3—24no—
40C4×(C3⋊D3)⋊C3—22no—
41#634⋊C3—22no—
42#634⋊C3D412yes—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C3×#454—91no—
45#5062—31no—
46#5062—91no—
47C2×C32×Q8—21no—
48C2×C32×Q8C2×D311yescharacteristic
49C2×C32×Q8C22×C311yescharacteristic
50C2×C32×D4—21no—
51C2×C32×D4C2×D311yescharacteristic
52C2×C32×D4C22×C311yescharacteristic
53#3832—22no—
54Q8×C3⋊D3—22no—
55#3828—22no—
56#3828C22×C312yes—
57C3×#239—62no—
58C3×D3×Q8—62no—
59C3×#235—32no—
60C3×#235—62no—
61C32×C22⋊C4—24no—
62C32×C22⋊C4—22no—
63C32×C22⋊C4—22no—
64C32×C42—21no—
65C32×C42C2×D311yescharacteristic
66C32×C42C22×C311yescharacteristic
67#3754—22no—
68#3753—22no—
69#3748C22×C312yes—
70C3×#214—62no—
71C3×#213—62no—
72C3×#208—32no—
73C22×H(3,𝔽3)—42no—
74C22×H(3,𝔽3)—41no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C2×(C3⋊D3)⋊C3—22no—
77C4×H(3,𝔽3)—22no—
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×D412yes—
80(C3⋊Dic3)⋊C3—22no—
81C3×2−1+4—91no—
82#1007—31no—
83#1007—91no—
84C3×AGL(1,ℤ/8)—92no—
85C3×#124—31no—
86C3×#124—31no—
87C3×#124—61no—
88C3×#124C3×D311yescharacteristic
89#932—32no—
90#932—92no—
91C3×#104—92no—
92C3×#101—91no—
93#836—31no—
94#836—91no—
95#805—32no—
96#805—92no—
97C23×C32—22no—
98C23×C32—21no—
99C23×C32—21no—
100C32×Q8—22no—
101C32×Q8—22no—
102C32×Q8—42no—
103C32×D4—22no—
104C32×D4—22no—
105C32×D4—42no—
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—22no—
110C2×C32×C4—21no—
111C2×C32×C4#6712yes—
112C2×C32×C4C3×D412yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C23×C311yescharacteristic
115#638—22no—
116C4⋊(C3⋊D3)—22no—
117C4⋊(C3⋊D3)C3×D412yes—
118C4×C3⋊D3—24no—
119C4×C3⋊D3—22no—
120#634—22no—
121#634C3×D412yes—
122C3×#67—62no—
123C3×D12—32no—
124C3×D12—62no—
125C3×C4×D3—64no—
126C3×C4×D3—62no—
127C3×Dic6—32no—
128C3×Dic6—62no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—62no—
131#454—91noSylow
132C2×H(3,𝔽3)—21no—
133C2×H(3,𝔽3)—81no—
134C2×H(3,𝔽3)#12011yescharacteristic
135(C3⋊D3)⋊C3—42no—
136C3×#41—92no—
137C3×#41—182no—
138C3×#41—182no—
139C2×C3×Q8—31no—
140C2×C3×Q8—31no—
141C2×C3×Q8—61no—
142C2×C3×Q8—182no—
143C2×C3×Q8C2×C3×D311yescharacteristic
144C2×C3×D4—31no—
145C2×C3×D4—31no—
146C2×C3×D4—61no—
147C2×C3×D4C2×C3×D311yescharacteristic
148#239—62no—
149#239—182no—
150D3×Q8—62no—
151D3×Q8—182no—
152#235—32no—
153#235—62no—
154#235—92no—
155#235—182no—
156C3×SD16—182no—
157C3×D8—182no—
158C3×M16—92no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—62no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—64no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42C2×C3×D311yescharacteristic
167#214—62no—
168#214—182no—
169#213—62no—
170#213—182no—
171#208—32no—
172#208—92no—
173C22×C32—21no—
174C22×C32—42no—
175C22×C32—44no—
176C22×C32—42no—
177C22×C32—42no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32C2×#6711yescharacteristic
181C22×C32C2×C3×D411yescharacteristic
182C2×C3⋊D3—22no—
183C2×C3×D3—62no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—42no—
188C32×C4—42no—
189C32×C4—41no—
190C32×C4—41no—
191C32×C4D3×D412yes—
192C32×C4C2×C3×D412yes—
193C3⋊Dic3—22no—
194C3×Dic3—62no—
1952−1+4—91no—
196AGL(1,ℤ/8)—92no—
197#124(C3⋊D3)⋊C311yescharacteristic, complemented
198#104—92no—
199#101—91no—
200H(3,𝔽3)#45411yescharacteristic, complemented, Sylow
201C23×C3—21no—
202C23×C3—61no—
203C23×C3—61no—
204C23×C3—62no—
205C3×Q8—22no—
206C3×Q8—62no—
207C3×Q8—62no—
208C3×Q8—92no—
209C3×Q8—122no—
210C3×Q8—182no—
211C3×D4—22no—
212C3×D4—62no—
213C3×D4—62no—
214C3×D4—92no—
215C3×D4—122no—
216C3×D4—182no—
217C3×D4—182no—
218C2×C3×C4—21no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—61no—
228C2×C3×C4—182no—
229C2×C3×C4—184no—
230C2×C3×C4C3×#6712yes—
231C2×C3×C4C22×C3×D311yescharacteristic
232#67—62no—
233#67—182no—
234D12—32no—
235D12—62no—
236D12—92no—
237D12—182no—
238C4×D3—64no—
239C4×D3—62no—
240C4×D3—182no—
241C4×D3—184no—
242Dic6—32no—
243Dic6—62no—
244Dic6—92no—
245Dic6—182no—
246C3×C8—182no—
247C3⋊C8—62no—
248C3⋊C8—182no—
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—82no—
254C2×C32—81no—
255C2×C32—81no—
256C2×C32#93711yescharacteristic
257C2×C32C3×#12011yescharacteristic
258C3⋊D3—42no—
259C3×D3—122no—
260#41—92no—
261#41—182no—
262#41—182no—
263C2×Q8—182no—
264C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
265C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
266SD16—182no—
267D8—182no—
268M16—92no—
269C22⋊C4—22no—
270C42C2×(C3⋊D3)⋊C311yescharacteristic
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—42no—
274C22×C3—41no—
275C22×C3—61no—
276C22×C3—121no—
277C22×C3—121no—
278C22×C3—122no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3—124no—
282C22×C3—182no—
283C22×C3C2×C3×#6711yescharacteristic
284C2×D3—62no—
285C2×D3—182no—
286C3×C4—22no—
287C3×C4—32no—
288C3×C4—32no—
289C3×C4—41no—
290C3×C4—62no—
291C3×C4—62no—
292C3×C4—62no—
293C3×C4—122no—
294C3×C4—121no—
295C3×C4—121no—
296C3×C4—122no—
297C3×C4—182no—
298C3×C4C3×D3×D412yes—
299Dic3—62no—
300Dic3—182no—
301C32—21no—
302C32#506211yescharacteristic, complemented
303C32C3×#45411yescharacteristic, complemented
304C23—21no—
305Q8—22no—
306Q8—92no—
307Q8—182no—
308D4—22no—
309D4—92no—
310D4—182no—
311D4—182no—
312C2×C4—21no—
313C2×C4—184no—
314C2×C4—182no—
315C2×C4#638⋊C312yes—
316C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
317C8—182no—
318C2×C3—21no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—62no—
324C2×C3—61no—
325C2×C3—81no—
326C2×C3—241no—
327C2×C3—241no—
328C2×C3—242no—
329C2×C3—362no—
330C2×C3C3×#93711yescharacteristic
331D3—122no—
332D3—362no—
333C22—42no—
334C22—41no—
335C22—182no—
336C22C2×#638⋊C311yescharacteristic
337C4—22no—
338C4—41no—
339C4—182no—
340C4D4×(C3⋊D3)⋊C312yes—
341C3—31no—
342C3—31no—
343C3—61no—
344C3C3×#506211yescharacteristic
345C2—21no—
346C2—81no—
347C2—362no—
348C2#64146⋊C311yescharacteristic
subgroupG/H
centerC2#64146⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC2×C3C3×#937
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #11722627C2semidirect
2. H(3,𝔽3)×#124C2semidirect
3. #64141⋊C3C2semidirect
4. #63663⋊C3C2semidirect
5. #63632⋊C3C2semidirect
6. #10592761C3semidirect
19. #64364C2×C3semidirect
22. C32×#124D3semidirect
23. C32×#124C2×C3semidirect
24. #64141C2×C3semidirect
26. #63663C2×C3semidirect
27. #63632C2×C3semidirect
197. #124(C3⋊D3)⋊C3semidirect
200. H(3,𝔽3)#454semidirect
302. C32#5062semidirect
303. C32C3×#454semidirect

Maximal subgroups

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

subgroupindex
#117226272
H(3,𝔽3)×#1242
#64141⋊C32
#63663⋊C32
#63632⋊C32
#105927613
C3×#50623