Finite Group Databasebeta

← #13139288

Subgroups — #13139288

Order 1728.

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

normal subgroups only

This group has 311 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
1#64128⋊C3C211yesmaximal, characteristic
2#11720851C211yesmaximal, characteristic, complemented
3#11720570C211yesmaximal, characteristic, complemented
4#63656⋊C3C211yesmaximal, characteristic, complemented
5#11720255C211yesmaximal, characteristic, complemented
6#11720249C211yesmaximal, characteristic, complemented
7#11720247C211yesmaximal, characteristic, complemented
8#10591044—31nomaximal
9#10591042—31nomaximal
10#88722—21no—
11((C3⋊D3)⋊C3)⋊Q8—21no—
12Q16×H(3,𝔽3)C2211yescharacteristic
13#88530C2211yescharacteristic
14#3755⋊C3—21no—
15#3745⋊C3C2211yescharacteristic
16C8×(C3⋊D3)⋊C3C2211yescharacteristic
17#3746⋊C3C2211yescharacteristic
18#88439—21no—
19#88386C2211yescharacteristic
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
21#64128D311yescharacteristic
22C3×#919—31no—
23#63802—31no—
24D3×Dic12—31no—
25#63656D311yescharacteristic, complemented
26C3×#829—31no—
27C3×#829—31no—
28C3×#829—31no—
29C3×#829—61no—
30#63549—31no—
31#63549—31no—
32#63548—31no—
33#63547—31no—
34#63544—31no—
35#63543—31no—
36Q8×H(3,𝔽3)—21no—
37#6229—21no—
38C4⋊((C3⋊D3)⋊C3)—21no—
39C4⋊((C3⋊D3)⋊C3)—41no—
40C4×(C3⋊D3)⋊C3—41no—
41C4×(C3⋊D3)⋊C3D411yescharacteristic
42#634⋊C3—21no—
43((C3⋊D3)⋊C3)⋊C4—41no—
44#6196—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
47#5195—91no—
48#4918—91no—
49#4918—91no—
50#3832—21no—
51C3×#239—61no—
52#3797—61no—
53D3×Dic6—61no—
54C32×Q16—21no—
55C32×Q16C2×D311yescharacteristic
56C32×Q16C2×D311yescharacteristic
57#3755—21no—
58#3746C2×D311yescharacteristic
59#3745C2×D311yescharacteristic
60C8×C3⋊D3C2×D311yescharacteristic
61C3×#215—61no—
62C3×Dic12—31no—
63C3×Dic12—31no—
64C3×Dic12—31no—
65C3×Dic12—61no—
66C3×D24—31no—
67C3×D3×C8—31no—
68#3719—61no—
69#3718—61no—
70C3⋊(C3⋊C16)C2×D311yescharacteristic
71C3×C3⋊C16—31no—
72C3×C3⋊C16—31no—
73C3×C3⋊C16—31no—
74C3×C3⋊C16—61no—
75C2×(C3⋊D3)⋊C3—41no—
76C2×(C3⋊D3)⋊C3D811yescharacteristic
77C4×H(3,𝔽3)—41no—
78C4×H(3,𝔽3)C2×D411yescharacteristic
79#1084—41no—
80(C3⋊Dic3)⋊C3—41no—
81(C3⋊Dic3)⋊C3D811yescharacteristic
82C3×#135—91no—
83#919—31no—
84#919—91no—
85D3×Q16—91no—
86D3×Q16—91no—
87C2×Dic12—91no—
88C3×Q32—91no—
89C3×Q32—91no—
90#829—31no—
91#829—31no—
92#829—31no—
93#829—61no—
94#829—91no—
95#828—91no—
96#828—91no—
97#827—91no—
98#827—91no—
99#812—91no—
100#800—91no—
101#800—91no—
102#798—91no—
103#798—91no—
104C32×Q8—21no—
105C32×Q8—21no—
106C32×Q8—41no—
107C4⋊(C3⋊D3)—21no—
108C4⋊(C3⋊D3)—41no—
109C4×C3⋊D3—41no—
110C4×C3⋊D3#6711yescharacteristic
111#634—21no—
112C3×D12—61no—
113C3×D12—121no—
114C3×C4×D3—31no—
115C3×C4×D3—121no—
116C3×Dic6—61no—
117C3×Dic6—61no—
118C3×Dic6—61no—
119C3×Dic6—121no—
120#627—121no—
121#627—121no—
122#624—121no—
123D3×Dic3—121no—
124C32×C8—21no—
125C32×C8C22×D311yescharacteristic
126C32×C8C22×D311yescharacteristic
127C3⋊(C3⋊C8)#6711yescharacteristic
128C3×C3⋊C8—31no—
129#510—271noSylow
130C2×H(3,𝔽3)C2×D811yescharacteristic
131(C3⋊D3)⋊C3—21no—
132(C3⋊D3)⋊C3—81no—
133C3×#41—181no—
134#239—61no—
135#239—181no—
136D3×Q8—181no—
137D3×Q8—181no—
138C2×Dic6—181no—
139C3×Q16—31no—
140C3×Q16—31no—
141C3×Q16—61no—
142C3×Q16—91no—
143C3×Q16—91no—
144C3×Q16D3211yescharacteristic
145C3×SD16—181no—
146C3×D8—91no—
147C2×C3×C8—91no—
148#216—181no—
149#216—181no—
150#215—61no—
151#215—181no—
152Dic12—31no—
153Dic12—31no—
154Dic12—31no—
155Dic12—61no—
156Dic12—91no—
157Dic12—181no—
158D24—31no—
159D24—91no—
160D3×C8—31no—
161D3×C8—91no—
162C3×C16—91no—
163C3×C16—91no—
164C3⋊C16—31no—
165C3⋊C16—31no—
166C3⋊C16—31no—
167C3⋊C16—61no—
168C3⋊C16—91no—
169C2×C3⋊D3—41no—
170C2×C3⋊D3#21311yescharacteristic
171C2×C3×D3—31no—
172C2×C3×D3—121no—
173C32×C4—21no—
174C32×C4—41no—
175C32×C4—41no—
176C32×C4—81no—
177C32×C4D3×D411yescharacteristic
178C32×C4C2×#6711yescharacteristic
179C3⋊Dic3—41no—
180C3⋊Dic3#21311yescharacteristic
181C3×Dic3—31no—
182C3×Dic3—121no—
183C3×Dic3—121no—
184C3×Dic3—121no—
185C3×Dic3—241no—
186#135—91no—
187C2×Q16—271no—
188Q32—91no—
189Q32—91no—
190SD32—271no—
191SD32—271no—
192M32—271no—
193H(3,𝔽3)#51011yescharacteristic, complemented, Sylow
194C3×Q8—21no—
195C3×Q8—61no—
196C3×Q8—61no—
197C3×Q8—121no—
198C3×Q8—181no—
199C3×Q8—181no—
200C3×D4—181no—
201C3×D4—361no—
202C2×C3×C4—91no—
203C2×C3×C4—361no—
204C2×Dic3—361no—
205D12—61no—
206D12—121no—
207D12—181no—
208D12—361no—
209C4×D3—31no—
210C4×D3—91no—
211C4×D3—121no—
212C4×D3—361no—
213C4×D3—361no—
214C4×D3—361no—
215Dic6—61no—
216Dic6—61no—
217Dic6—61no—
218Dic6—121no—
219Dic6—181no—
220Dic6—361no—
221Dic6—361no—
222Dic6—361no—
223C3×C8—31no—
224C3×C8—31no—
225C3×C8—61no—
226C3×C8—91no—
227C3×C8C2×D3211yescharacteristic
228C3⋊C8—31no—
229C3⋊C8—91no—
230C2×C32—21no—
231C2×C32D3×D811yescharacteristic
232C2×C32C2×#21311yescharacteristic
233C3⋊D3—21no—
234C3⋊D3—81no—
235C3×D3—61no—
236C3×D3—241no—
237#41—181no—
238C2×Q8—541no—
239Q16—91no—
240Q16—91no—
241Q16—541no—
242Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic
243SD16—181no—
244D8—91no—
245C2×C8—91no—
246C16—91no—
247C16—91no—
248C22×C3—91no—
249C22×C3—361no—
250C2×D3—31no—
251C2×D3—91no—
252C2×D3—121no—
253C2×D3—361no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—41no—
257C3×C4—61no—
258C3×C4—91no—
259C3×C4—121no—
260C3×C4—121no—
261C3×C4—241no—
262C3×C4—361no—
263C3×C4—361no—
264C3×C4D3×#6711yescharacteristic
265Dic3—31no—
266Dic3—91no—
267Dic3—121no—
268Dic3—121no—
269Dic3—121no—
270Dic3—241no—
271Dic3—361no—
272C32—21no—
273C32#491811yescharacteristic, complemented
274C32#519511yescharacteristic, complemented
275Q8—21no—
276Q8—181no—
277Q8—181no—
278Q8—1081no—
279D4—181no—
280D4—361no—
281C2×C4—91no—
282C2×C4—361no—
283C2×C4—1081no—
284C8—91no—
285C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—61no—
289C2×C3—181no—
290C2×C3—721no—
291C2×C3D3×#21311yescharacteristic
292D3—61no—
293D3—181no—
294D3—241no—
295D3—721no—
296C22—91no—
297C22—361no—
298C4—41no—
299C4—91no—
300C4—361no—
301C4—361no—
302C4#8867611yescharacteristic
303C3—31no—
304C3—31no—
305C3—61no—
306C3#1059104211yescharacteristic
307C2—181no—
308C2—721no—
309C2#1172119411yescharacteristic
subgroupG/H
centerC2#11721194
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3D3×#213
prime radical#64128⋊C3C2
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#510
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
2. #11720851C2semidirect
3. #11720570C2semidirect
4. #63656⋊C3C2semidirect
5. #11720255C2semidirect
6. #11720249C2semidirect
7. #11720247C2semidirect
25. #63656D3semidirect
193. H(3,𝔽3)#510semidirect
273. C32#4918semidirect
274. C32#5195semidirect

Maximal subgroups

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

subgroupindex
#64128⋊C32
#117208512
#117205702
#63656⋊C32
#117202552
#117202492
#117202472
#105910443
#105910423