Finite Group Databasebeta

← #13139290

Subgroups — #13139290

Order 1728.

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

normal subgroups only

This group has 318 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, complemented
2#11720855C211yesmaximal, characteristic, complemented
3#11720570C211yesmaximal, characteristic, complemented
4#63655⋊C3C211yesmaximal, characteristic, complemented
5#11720255C211yesmaximal, characteristic, complemented
6#11720248C211yesmaximal, characteristic, complemented
7((C3⋊D3)⋊C3)⋊C16C211yesmaximal, characteristic, complemented
8#10591049—31nomaximal
9#10591048—31nomaximal
10#88722—21no—
11#88648—21no—
12Q16×H(3,𝔽3)C2211yescharacteristic
13#88530C2211yescharacteristic
14#3755⋊C3—21no—
15#3745⋊C3C2211yescharacteristic
16#3745⋊C3C2211yescharacteristic
17C8×(C3⋊D3)⋊C3C2211yescharacteristic
18#88438—21no—
19#88386C2211yescharacteristic
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
21#64128D311yescharacteristic, complemented
22C3×#919—31no—
23#63810—31no—
24#63808—31no—
25#63655D311yescharacteristic, complemented
26C3×#829—31no—
27C3×#829—31no—
28C3×#829—61no—
29C3×#828—31no—
30#63549—31no—
31#63547—31no—
32#63546—31no—
33#63546—31no—
34#63542—31no—
35D3×C3⋊C16—31no—
36Q8×H(3,𝔽3)—21no—
37#6229—21no—
38C4⋊((C3⋊D3)⋊C3)—21no—
39C4⋊((C3⋊D3)⋊C3)—21no—
40C4⋊((C3⋊D3)⋊C3)—41no—
41C4×(C3⋊D3)⋊C3—41no—
42C4×(C3⋊D3)⋊C3D411yescharacteristic
43#6198—41no—
44((C3⋊D3)⋊C3)⋊C4—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
47#5194—91no—
48#4919—91no—
49#4916—91no—
50#3832—21no—
51C3×#239—61no—
52#3799—61no—
53#3796—61no—
54C32×Q16—21no—
55C32×Q16C2×D311yescharacteristic
56C32×Q16C2×D311yescharacteristic
57#3755—21no—
58#3745C2×D311yescharacteristic
59#3745C2×D311yescharacteristic
60C8×C3⋊D3C2×D311yescharacteristic
61C3×#215—61no—
62C3×Dic12—31no—
63C3×Dic12—31no—
64C3×Dic12—61no—
65C3×D24—31no—
66C3×D24—31no—
67C3×D3×C8—31no—
68#3717—61no—
69#3715—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)⋊C3—41no—
77C2×(C3⋊D3)⋊C3D811yescharacteristic
78C4×H(3,𝔽3)—41no—
79C4×H(3,𝔽3)C2×D411yescharacteristic
80#1084—41no—
81(C3⋊Dic3)⋊C3D811yescharacteristic
82C3×#135—91no—
83#919—31no—
84#919—91no—
85#919—91no—
86#912—91no—
87#904—91no—
88C3×Q32—91no—
89C3×SD32—91no—
90#829—31no—
91#829—31no—
92#829—61no—
93#828—31no—
94#828—91no—
95#828—91no—
96#827—91no—
97#826—91no—
98#826—91no—
99C2×C3⋊C16—91no—
100#800—91no—
101D48—91no—
102D3×C16—91no—
103D3×C16—91no—
104C32×Q8—21no—
105C32×Q8—21no—
106C32×Q8—41no—
107C4⋊(C3⋊D3)—21no—
108C4⋊(C3⋊D3)—21no—
109C4⋊(C3⋊D3)—41no—
110C4×C3⋊D3—41no—
111C4×C3⋊D3#6711yescharacteristic
112C3×D12—61no—
113C3×D12—61no—
114C3×D12—121no—
115C3×C4×D3—31no—
116C3×C4×D3—121no—
117C3×Dic6—61no—
118C3×Dic6—61no—
119C3×Dic6—121no—
120#626—121no—
121#626—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#508—271noSylow
130C2×H(3,𝔽3)C2×D811yescharacteristic
131(C3⋊D3)⋊C3—21no—
132(C3⋊D3)⋊C3—81no—
133(C3⋊D3)⋊C3—81no—
134C3×#41—181no—
135#239—61no—
136#239—181no—
137#239—181no—
138#237—181no—
139#235—181no—
140C3×Q16—31no—
141C3×Q16—31no—
142C3×Q16—61no—
143C3×Q16—91no—
144C3×Q16D3211yescharacteristic
145C3×SD16—181no—
146C3×D8—91no—
147C3×D8—91no—
148C2×C3×C8—91no—
149#215—61no—
150#215—181no—
151#215—181no—
152#214—181no—
153Dic12—31no—
154Dic12—31no—
155Dic12—61no—
156D24—31no—
157D24—31no—
158D24—91no—
159D24—91no—
160#204—181no—
161D3×C8—31no—
162D3×C8—91no—
163C3×C16—91no—
164C3×C16—91no—
165C3⋊C16—31no—
166C3⋊C16—31no—
167C3⋊C16—31no—
168C3⋊C16—61no—
169C3⋊C16—91no—
170C2×C3⋊D3—41no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3#21311yescharacteristic
173C2×C3×D3—31no—
174C2×C3×D3—121no—
175C2×C3×D3—121no—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—81no—
180C32×C4D3×D411yescharacteristic
181C32×C4C2×#6711yescharacteristic
182C3⋊Dic3#21311yescharacteristic
183C3×Dic3—31no—
184C3×Dic3—121no—
185C3×Dic3—121no—
186C3×Dic3—241no—
187#135—91no—
188#135—271no—
189Q32—91no—
190SD32—91no—
191SD32—271no—
192D16—271no—
193C2×C16—271no—
194H(3,𝔽3)#50811yescharacteristic, complemented, Sylow
195C3×Q8—21no—
196C3×Q8—61no—
197C3×Q8—61no—
198C3×Q8—121no—
199C3×Q8—181no—
200C3×D4—181no—
201C3×D4—181no—
202C3×D4—361no—
203C2×C3×C4—91no—
204C2×C3×C4—361no—
205#67—361no—
206#67—361no—
207C2×Dic3—361no—
208D12—61no—
209D12—61no—
210D12—121no—
211D12—181no—
212D12—181no—
213D12—361no—
214D12—361no—
215C4×D3—31no—
216C4×D3—91no—
217C4×D3—121no—
218C4×D3—361no—
219C4×D3—361no—
220C4×D3—361no—
221Dic6—61no—
222Dic6—61no—
223Dic6—121no—
224C3×C8—31no—
225C3×C8—31no—
226C3×C8—61no—
227C3×C8—91no—
228C3×C8C2×D3211yescharacteristic
229C3⋊C8—31no—
230C3⋊C8—91no—
231C2×C32—21no—
232C2×C32D3×D811yescharacteristic
233C2×C32C2×#21311yescharacteristic
234C3⋊D3—21no—
235C3⋊D3—81no—
236C3⋊D3—81no—
237C3×D3—61no—
238C3×D3—241no—
239C3×D3—241no—
240#41—181no—
241#41—541no—
242Q16—91no—
243Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic
244SD16—181no—
245SD16—541no—
246D8—91no—
247D8—91no—
248C2×C8—91no—
249C16—91no—
250C16—91no—
251C22×C3—91no—
252C22×C3—361no—
253C22×C3—361no—
254C2×D3—31no—
255C2×D3—91no—
256C2×D3—121no—
257C2×D3—121no—
258C2×D3—361no—
259C2×D3—361no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—41no—
263C3×C4—61no—
264C3×C4—91no—
265C3×C4—121no—
266C3×C4—121no—
267C3×C4—241no—
268C3×C4—361no—
269C3×C4D3×#6711yescharacteristic
270Dic3—31no—
271Dic3—91no—
272Dic3—121no—
273Dic3—121no—
274Dic3—241no—
275C32—21no—
276C32#491611yescharacteristic, complemented
277C32#519411yescharacteristic, complemented
278Q8—21no—
279Q8—181no—
280D4—181no—
281D4—181no—
282D4—361no—
283D4—1081no—
284C2×C4—91no—
285C2×C4—361no—
286C2×C4—1081no—
287C8—91no—
288C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C2×C3—31no—
290C2×C3—31no—
291C2×C3—61no—
292C2×C3—181no—
293C2×C3—721no—
294C2×C3—721no—
295C2×C3D3×#21311yescharacteristic
296D3—61no—
297D3—181no—
298D3—241no—
299D3—241no—
300D3—721no—
301D3—721no—
302C22—91no—
303C22—361no—
304C22—361no—
305C4—41no—
306C4—91no—
307C4—361no—
308C4#8867611yescharacteristic
309C3—31no—
310C3—31no—
311C3—61no—
312C3#1059104611yescharacteristic
313C2—181no—
314C2—721no—
315C2—721no—
316C2#1172119411yescharacteristic
subgroupG/H
centerC2#11721194
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3D3×#213
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#508
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
1. #64128⋊C3C2semidirect
2. #11720855C2semidirect
3. #11720570C2semidirect
4. #63655⋊C3C2semidirect
5. #11720255C2semidirect
6. #11720248C2semidirect
7. ((C3⋊D3)⋊C3)⋊C16C2semidirect
21. #64128D3semidirect
25. #63655D3semidirect
194. H(3,𝔽3)#508semidirect
276. C32#4916semidirect
277. C32#5194semidirect

Maximal subgroups

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

subgroupindex
#64128⋊C32
#117208552
#117205702
#63655⋊C32
#117202552
#117202482
((C3⋊D3)⋊C3)⋊C162
#105910493
#105910483