Finite Group Databasebeta

← #10592754⋊C3 #13142921

Subgroups — #10592754⋊C3 #13142921

Order 1728.

426 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 92 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3#63662⋊C3C212yesmaximal, complemented
4#63659⋊C3C212yesmaximal
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10592754C311yesmaximal, characteristic, complemented
7C3×#5055—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#634⋊C3—21no—
11C22⋊C4×H(3,𝔽3)—21no—
12C22⋊C4×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×#124—21no—
18C32×#124D311yescharacteristic
19C32×#124C2×C311yescharacteristic
20#64078C2×C311yescharacteristic, complemented
21C3×#869—31no—
22#63662C2×C312yescomplemented
23#63659C2×C312yes—
24#63630C2×C311yescharacteristic, complemented
25C3×#835—32no—
26C3×#832—32no—
27C3×#803—31no—
28C23×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)—22no—
30D4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3—22no—
36#634⋊C3—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38C3×#489—91no—
39#5055—31no—
40#5055—91no—
41C2×C32×Q8—21no—
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×Q8C22×C311yescharacteristic
44C2×C32×D4—21no—
45C2×C32×D4C2×D311yescharacteristic
46C2×C32×D4C22×C311yescharacteristic
47C2×#634—21no—
48C2×C3×Dic6—61no—
49C32×C22⋊C4—22no—
50C32×C22⋊C4—22no—
51C32×C22⋊C4—21no—
52C32×C22⋊C4—21no—
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42—21no—
56C32×C42C2×D311yescharacteristic
57C32×C42C22×C311yescharacteristic
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C312yes—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—61no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)—41no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)—41no—
71C4×H(3,𝔽3)#4112yes—
72(C3⋊Dic3)⋊C3—42no—
73C3×C4⋊Q8—91no—
74C3×#124—31no—
75C3×#124—31no—
76C3×#124—61no—
77C3×#124C3×D311yescharacteristic
78#869—31no—
79#869—91no—
80C3×#103—92no—
81C3×#102—92no—
82C3×#97—91no—
83#835—32no—
84#835—92no—
85#832—32no—
86#832—92no—
87#803—31no—
88#803—91no—
89C23×C32—22no—
90C23×C32—21no—
91C23×C32—21no—
92C32×Q8—22no—
93C32×Q8—22no—
94C32×Q8—42no—
95C32×D4—22no—
96C32×D4—22no—
97C32×D4—42no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4#6711yescharacteristic
105C2×C32×C4#6711yescharacteristic
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C3×D411yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C32×C4C23×C311yescharacteristic
110C2×C3⋊Dic3—22no—
111#634—42no—
112C2×C3×Dic3—62no—
113C3×Dic6—122no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116#489—91noSylow
117C2×H(3,𝔽3)—81no—
118C2×H(3,𝔽3)#12411yescharacteristic
119C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
120C2×H(3,𝔽3)#13711yescharacteristic
121C2×C3×Q8—31no—
122C2×C3×Q8—31no—
123C2×C3×Q8—61no—
124C2×C3×Q8—181no—
125C2×C3×Q8C2×C3×D311yescharacteristic
126C2×C3×D4—31no—
127C2×C3×D4—31no—
128C2×C3×D4—61no—
129C2×C3×D4C2×C3×D311yescharacteristic
130C2×Dic6—61no—
131C2×Dic6—181no—
132C2×C3×C8—92no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—181no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—21no—
137C3×C22⋊C4—61no—
138C3×C22⋊C4—61no—
139C3×C22⋊C4—61no—
140C3×C22⋊C4—61no—
141C3×C22⋊C4—62no—
142C3×C22⋊C4—62no—
143C3×C42—31no—
144C3×C42—31no—
145C3×C42—61no—
146C3×C42C2×C3×D311yescharacteristic
147C4⋊Dic3—32no—
148C4⋊Dic3—61no—
149C4⋊Dic3—92no—
150C4⋊Dic3—181no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153C22×C32—21no—
154C22×C32—42no—
155C22×C32—42no—
156C22×C32—42no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32C2×#6711yescharacteristic
164C22×C32C2×C3×D411yescharacteristic
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—21no—
169C32×C4—22no—
170C32×C4—41no—
171C32×C4—41no—
172C32×C4—42no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4#23712yes—
176C32×C4C3×#4112yes—
177C3⋊Dic3—42no—
178C3×Dic3—122no—
179C4⋊Q8—91no—
180#124(C3⋊D3)⋊C311yescharacteristic
181#103—92no—
182#102—92no—
183#97—91no—
184H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
185C23×C3—21no—
186C23×C3—61no—
187C23×C3—61no—
188C23×C3—62no—
189C3×Q8—22no—
190C3×Q8—62no—
191C3×Q8—62no—
192C3×Q8—122no—
193C3×Q8—362no—
194C3×D4—22no—
195C3×D4—62no—
196C3×D4—62no—
197C3×D4—122no—
198C2×C3×C4—21no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—182no—
212C2×C3×C4C3×#6711yescharacteristic
213C2×C3×C4C3×#6711yescharacteristic
214C2×C3×C4C22×C3×D311yescharacteristic
215C2×Dic3—62no—
216C2×Dic3—182no—
217Dic6—122no—
218Dic6—362no—
219C3×C8—182no—
220C3⋊C8—62no—
221C3⋊C8—182no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—82no—
226C2×C32—81no—
227C2×C32—81no—
228C2×C32#93611yescharacteristic
229C2×C32#94911yescharacteristic
230C2×C32#95111yescharacteristic
231C2×C32C3×#12411yescharacteristic
232C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
233C2×C32C3×#13711yescharacteristic
234C2×Q8—181no—
235C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
236C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
237C2×C8—92no—
238C4⋊C4—92no—
239C4⋊C4—181no—
240C22⋊C4—21no—
241C22⋊C4—21no—
242C42C2×(C3⋊D3)⋊C311yescharacteristic
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—61no—
249C22×C3—121no—
250C22×C3—121no—
251C22×C3—121no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3—121no—
255C22×C3—122no—
256C22×C3—122no—
257C22×C3—122no—
258C22×C3C2×C3×#6711yescharacteristic
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—32no—
262C3×C4—32no—
263C3×C4—41no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—62no—
269C3×C4—121no—
270C3×C4—121no—
271C3×C4—121no—
272C3×C4—121no—
273C3×C4—122no—
274C3×C4—362no—
275C3×C4C3×#23712yes—
276Dic3—122no—
277Dic3—362no—
278C32—21no—
279C32#505511yescharacteristic, complemented
280C32C3×#48911yescharacteristic, complemented
281C23—21no—
282Q8—22no—
283Q8—362no—
284D4—22no—
285C2×C4—21no—
286C2×C4—182no—
287C2×C4#638⋊C311yescharacteristic
288C2×C4#638⋊C311yescharacteristic
289C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
290C8—182no—
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—81no—
301C2×C3—241no—
302C2×C3—241no—
303C2×C3—242no—
304C2×C3C3×#93611yescharacteristic
305C2×C3C3×#94911yescharacteristic
306C2×C3C3×#95111yescharacteristic
307C22—41no—
308C22—41no—
309C22—41no—
310C22C2×#638⋊C311yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—362no—
315C4#3830⋊C312yes—
316C3—31no—
317C3—31no—
318C3—61no—
319C3C3×#505511yescharacteristic
320C2—81no—
321C2#64145⋊C311yescharacteristic
322C2#64160⋊C311yescharacteristic
323C2#64158⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC22×C3C2×C3×#67
prime radicalC2×D4×H(3,𝔽3)C22
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #64078⋊C3C2semidirect
3. #63662⋊C3C2semidirect
5. #63630⋊C3C2semidirect
6. #10592754C3semidirect
20. #64078C2×C3semidirect
22. #63662C2×C3semidirect
24. #63630C2×C3semidirect
184. H(3,𝔽3)#489semidirect
279. C32#5055semidirect
280. C32C3×#489semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#64078⋊C32
#63662⋊C32
#63659⋊C32
#63630⋊C32
#105927543
C3×#50553