Finite Group Databasebeta

← #10592759⋊C3 #13142919

Subgroups — #10592759⋊C3 #13142919

Order 1728.

510 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 100 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
2C2×#3756⋊C3C212yesmaximal, complemented
3C2×#3754⋊C3C212yesmaximal
4#64078⋊C3C211yesmaximal, characteristic, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10592759C311yesmaximal, characteristic, complemented
7C3×#5060—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#634⋊C3C2212yes—
11C22⋊C4×H(3,𝔽3)—22no—
12C42×H(3,𝔽3)C2211yescharacteristic
13#3756⋊C3—24no—
14#3754⋊C3—24no—
15C4⋊((C3⋊Dic3)⋊C3)—22no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×#124—21no—
18C32×#124D311yescharacteristic
19C32×#124C2×C311yescharacteristic
20C2×#3756C2×C312yescomplemented
21C2×#3754C2×C312yes—
22#64078C2×C311yescharacteristic, complemented
23C2×C3×#216—32no—
24C2×C3×#214—32no—
25C3×#869—31no—
26#63630C2×C311yescharacteristic, complemented
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)D412yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37(C3⋊(C3⋊C8))⋊C3D414yes—
38C3×#497—91no—
39#5060—31no—
40#5060—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×#634C22×C312yes—
48C2×C3×Dic6—32no—
49C32×C22⋊C4—24no—
50C32×C22⋊C4—22no—
51C32×C22⋊C4—22no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C22×C311yescharacteristic
55#3756—24no—
56#3754—24no—
57C4⋊(C3⋊Dic3)—22no—
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C3×#216—64no—
60C3×#214—64no—
61C3×C4⋊Dic3—62no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)—42no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)—22no—
67C4×H(3,𝔽3)—41no—
68C4×H(3,𝔽3)C2×D412yes—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×Q16—92no—
71C2×C3×SD16—92no—
72C3×C4⋊Q8—91no—
73C3×#124—31no—
74C3×#124—31no—
75C3×#124—61no—
76C3×#124C3×D311yescharacteristic
77C2×#216—32no—
78C2×#216—92no—
79C2×#214—32no—
80C2×#214—92no—
81#869—31no—
82#869—91no—
83C3×#97—91no—
84#803—31no—
85#803—91no—
86C23×C32—22no—
87C23×C32—21no—
88C23×C32—21no—
89C32×Q8—22no—
90C32×Q8—22no—
91C32×Q8—42no—
92C32×D4—22no—
93C32×D4—22no—
94C32×D4—42no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4#6712yes—
101C2×C32×C4C3×D412yes—
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C23×C311yescharacteristic
104C2×C3⋊Dic3—22no—
105#634—22no—
106#634—22no—
107C2×C3×Dic3—62no—
108C3×Dic6—62no—
109C3×Dic6—62no—
110C3⋊(C3⋊C8)C3×D414yes—
111C3×C3⋊C8—34no—
112#497—91noSylow
113C2×H(3,𝔽3)—81no—
114C2×H(3,𝔽3)#12711yescharacteristic
115C2×H(3,𝔽3)#13712yes—
116C2×C3×Q8—31no—
117C2×C3×Q8—31no—
118C2×C3×Q8—61no—
119C2×C3×Q8—92no—
120C2×C3×Q8C2×C3×D311yescharacteristic
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—61no—
124C2×C3×D4C2×C3×D311yescharacteristic
125C2×Dic6—32no—
126C2×Dic6—92no—
127C3×Q16—184no—
128C3×SD16—184no—
129C2×C3×C8—92no—
130C3×C4⋊C4—182no—
131C3×C22⋊C4—22no—
132C3×C22⋊C4—62no—
133C3×C22⋊C4—62no—
134C3×C22⋊C4—64no—
135C3×C42—31no—
136C3×C42—31no—
137C3×C42—61no—
138C3×C42C2×C3×D311yescharacteristic
139#216—64no—
140#216—184no—
141#214—64no—
142#214—184no—
143C4⋊Dic3—62no—
144C4⋊Dic3—182no—
145C2×C3⋊C8—32no—
146C2×C3⋊C8—92no—
147C22×C32—21no—
148C22×C32—42no—
149C22×C32—44no—
150C22×C32—41no—
151C22×C32—42no—
152C22×C32—41no—
153C22×C32—42no—
154C22×C32C2×#6711yescharacteristic
155C22×C32C2×C3×D411yescharacteristic
156C32×C4—22no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—42no—
160C32×C4—42no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4D3×D412yes—
164C32×C4C2×C3×D412yes—
165C3⋊Dic3—42no—
166C3×Dic3—122no—
167C2×Q16—92no—
168C2×SD16—92no—
169C4⋊Q8—91no—
170#124(C3⋊D3)⋊C311yescharacteristic
171#97—91no—
172H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
173C23×C3—21no—
174C23×C3—61no—
175C23×C3—61no—
176C23×C3—62no—
177C3×Q8—22no—
178C3×Q8—62no—
179C3×Q8—62no—
180C3×Q8—122no—
181C3×Q8—182no—
182C3×Q8—182no—
183C3×D4—22no—
184C3×D4—62no—
185C3×D4—62no—
186C3×D4—122no—
187C2×C3×C4—21no—
188C2×C3×C4—32no—
189C2×C3×C4—32no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—61no—
197C2×C3×C4—182no—
198C2×C3×C4C3×#6712yes—
199C2×C3×C4C22×C3×D311yescharacteristic
200C2×Dic3—62no—
201C2×Dic3—182no—
202Dic6—62no—
203Dic6—62no—
204Dic6—182no—
205Dic6—182no—
206C3×C8—94no—
207C3⋊C8—34no—
208C3⋊C8—94no—
209C2×C32—21no—
210C2×C32—22no—
211C2×C32—82no—
212C2×C32—81no—
213C2×C32—81no—
214C2×C32#94011yescharacteristic
215C2×C32#95112yes—
216C2×C32C3×#12711yescharacteristic
217C2×C32C3×#13712yes—
218C2×Q8—92no—
219C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
220C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
221Q16—184no—
222SD16—184no—
223C2×C8—92no—
224C4⋊C4—182no—
225C22⋊C4—22no—
226C42C2×(C3⋊D3)⋊C311yescharacteristic
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—41no—
230C22×C3—42no—
231C22×C3—61no—
232C22×C3—121no—
233C22×C3—121no—
234C22×C3—122no—
235C22×C3—122no—
236C22×C3—122no—
237C22×C3—124no—
238C22×C3C2×C3×#6711yescharacteristic
239C3×C4—22no—
240C3×C4—32no—
241C3×C4—32no—
242C3×C4—41no—
243C3×C4—62no—
244C3×C4—62no—
245C3×C4—62no—
246C3×C4—122no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—122no—
250C3×C4—362no—
251C3×C4C3×D3×D412yes—
252Dic3—122no—
253Dic3—362no—
254C32—21no—
255C32#506011yescharacteristic, complemented
256C32C3×#49711yescharacteristic, complemented
257C23—21no—
258Q8—22no—
259Q8—182no—
260Q8—182no—
261D4—22no—
262C2×C4—21no—
263C2×C4—182no—
264C2×C4#638⋊C312yes—
265C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
266C8—94no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—32no—
270C2×C3—32no—
271C2×C3—61no—
272C2×C3—62no—
273C2×C3—81no—
274C2×C3—241no—
275C2×C3—241no—
276C2×C3—242no—
277C2×C3C3×#94011yescharacteristic
278C2×C3C3×#95112yes—
279C22—41no—
280C22—42no—
281C22C2×#638⋊C311yescharacteristic
282C4—22no—
283C4—41no—
284C4—362no—
285C4D4×(C3⋊D3)⋊C312yes—
286C3—31no—
287C3—31no—
288C3—61no—
289C3C3×#506011yescharacteristic
290C2—81no—
291C2#64149⋊C311yescharacteristic
292C2#64160⋊C312yes—
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×#497
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. C2×#3756⋊C3C2semidirect
4. #64078⋊C3C2semidirect
5. #63630⋊C3C2semidirect
6. #10592759C3semidirect
20. C2×#3756C2×C3semidirect
22. #64078C2×C3semidirect
26. #63630C2×C3semidirect
172. H(3,𝔽3)#497semidirect
255. C32#5060semidirect
256. C32C3×#497semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#3756⋊C32
C2×#3754⋊C32
#64078⋊C32
#63630⋊C32
#105927593
C3×#50603