Finite Group Databasebeta

← #13150600

Subgroups — #13150600

Order 1728.

1067 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#88740C211yesmaximal, characteristic
2C2×#3748⋊C3C212yesmaximal, complemented
3#11721065C214yesmaximal, complemented
4#11721064C218yesmaximal, complemented
5#10594776—32nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7C2×#6235C412yes—
8#88740C414yes—
9#88740C2214yes—
10C2×C4⋊#272C411yescharacteristic
11C2×C4×#272C2212yes—
12C2×#6229C411yescharacteristic
13#3748⋊C3C2214yes—
14#3748⋊C3C2214yescomplemented
15C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
16#88435C2218yescomplemented
17#88434C2218yescomplemented
18C2×C3×#235—32no—
19C2×C3×#235—61no—
20C2×#3748D312yescomplemented
21C2×C3×#208—32no—
22#63815—38no—
23#63814—38no—
24D3×#208—38no—
25C23×H(3,𝔽3)C2×C411yescharacteristic
26C22×#272C2×C412yes—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×C4×H(3,𝔽3)C2312yes—
31#6235C2×C418yes—
32C2×#1084C2×C412yes—
33C4⋊#272C2×C414yes—
34C4×#272C2×C414yes—
35C4×#272C2314yes—
36#6229C2×C414yes—
37(C3⋊(C3⋊C8))⋊C3C2318yes—
38#5819—91no—
39#5745—92no—
40C22×C32×C4—21no—
41C22×C32×C4C2×D312yes—
42C2×C3×#67—34no—
43C2×C3×#67—62no—
44C3×#235—34no—
45C3×#235—34no—
46C3×#235—34no—
47C3×#235—34no—
48C3×#235—64no—
49C3×#235—64no—
50C2×C3×D12—32no—
51C2×C3×D12—61no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—62no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—61no—
56#3748C2×D314yes—
57#3748C2×D314yescomplemented
58C2×C3⋊(C3⋊C8)C2×D314yescomplemented
59C3×#208—34no—
60C3×#208—34no—
61C2×C3×C3⋊C8—34no—
62#3711—316no—
63D3×C3⋊C8—316no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)C22×C411yescharacteristic
66C22×H(3,𝔽3)C22×C412yes—
67C2×#272—42no—
68C2×#272C22×C414yes—
69C4×H(3,𝔽3)C22×C412yes—
70C4×H(3,𝔽3)C22×C411yescharacteristic
71C4×H(3,𝔽3)C2411yescharacteristic
72#1084C22×C414yes—
73C2×C3×#41—91no—
74C2×#235—32no—
75C2×#235—61no—
76C2×C3×M16—92no—
77#948—98no—
78C2×#208—32no—
79C2×#208—92no—
80C2×#208—94no—
81#907—98no—
82D3×M16—98no—
83#901—98no—
84C23×C32—21no—
85C23×C32C4×D312yes—
86C22×C3×D3—34no—
87C22×C3×D3—62no—
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4—22no—
92C2×C32×C4C4×D312yes—
93C2×C32×C4C4×D312yes—
94C2×C32×C4C4×D312yes—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C22×D312yes—
97C2×C32×C4C22×D312yes—
98C3×#67—38no—
99C3×#67—38no—
100C3×#67—68no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—62no—
103C3×D12—34no—
104C3×D12—34no—
105C3×D12—64no—
106C3×C4×D3—38no—
107C3×C4×D3—38no—
108C3×C4×D3—64no—
109C3×C4×D3—64no—
110C3×Dic6—34no—
111C3×Dic6—34no—
112C3×Dic6—64no—
113C3⋊(C3⋊C8)C22×D318yes—
114C3×C3⋊C8—38no—
115#568—271noSylow
116C2×H(3,𝔽3)—21no—
117C2×H(3,𝔽3)—22no—
118C2×H(3,𝔽3)C23×C411yescharacteristic
119#272—24no—
120C3×#41—94no—
121C3×#41—94no—
122C2×C3×Q8—91no—
123C2×C3×D4—92no—
124C2×C3×D4—91no—
125C22×C3×C4—32no—
126C22×C3×C4—61no—
127C22×C3×C4—92no—
128C22×C3×C4D3211yescharacteristic
129C2×#67—34no—
130C2×#67—62no—
131#235—34no—
132#235—34no—
133#235—34no—
134#235—34no—
135#235—64no—
136#235—64no—
137C2×D12—32no—
138C2×D12—61no—
139C2×C4×D3—34no—
140C2×C4×D3—62no—
141C2×Dic6—32no—
142C2×Dic6—61no—
143C3×M16—94no—
144C3×M16—94no—
145C2×C3×C8—94no—
146#208—34no—
147#208—34no—
148#208—94no—
149#208—94no—
150#208—98no—
151C2×C3⋊C8—34no—
152C2×C3⋊C8—94no—
153C2×C3⋊C8—98no—
154ΓL(1,𝔽25)—916no—
155D3×C8—916no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—42no—
159C22×C32—42no—
160C22×C32C2×C4×D312yes—
161C22×C32C2×C4×D312yes—
162C22×C32C2×C4×D312yes—
163C2×C3×D3—38no—
164C2×C3×D3—64no—
165C2×C3×D3—124no—
166C2×C3×D3—124no—
167C32×C4—21no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4C2×C4×D312yes—
171C32×C4C2×C4×D312yes—
172C32×C4C2×C4×D312yes—
173C32×C4C23×D312yes—
174C3×Dic3—38no—
175C3×Dic3—64no—
176C2×#41—91no—
177#131—278no—
178C2×M16—92no—
179C2×M16—274no—
180H(3,𝔽3)#56811yescharacteristic, complemented, Sylow
181C23×C3—32no—
182C23×C3—61no—
183C23×C3—92no—
184C23×C3#62411yescharacteristic
185C22×D3—34no—
186C22×D3—62no—
187C3×Q8—94no—
188C3×D4—98no—
189C3×D4—94no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—61no—
197C2×C3×C4—62no—
198C2×C3×C4—61no—
199C2×C3×C4—62no—
200C2×C3×C4—92no—
201C2×C3×C4—94no—
202C2×C3×C4—94no—
203C2×C3×C4#62411yescharacteristic
204C2×C3×C4#62412yes—
205C2×C3×C4C2×D3212yes—
206C2×C3×C4C2×D3211yescharacteristic
207#67—38no—
208#67—38no—
209#67—68no—
210C2×Dic3—34no—
211C2×Dic3—62no—
212D12—34no—
213D12—34no—
214D12—64no—
215C4×D3—38no—
216C4×D3—38no—
217C4×D3—64no—
218C4×D3—64no—
219Dic6—34no—
220Dic6—34no—
221Dic6—64no—
222C3×C8—98no—
223C3⋊C8—38no—
224C3⋊C8—98no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—42no—
231C2×C32C22×C4×D312yes—
232C3×D3—68no—
233C3×D3—124no—
234#41—94no—
235#41—94no—
236C2×Q8—91no—
237C2×D4—92no—
238C2×D4—91no—
239C22×C4—92no—
240C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
241M16—94no—
242M16—94no—
243M16—278no—
244C2×C8—94no—
245C2×C8—278no—
246C22×C3—32no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—41no—
250C22×C3—61no—
251C22×C3—62no—
252C22×C3—94no—
253C22×C3—122no—
254C22×C3—122no—
255C22×C3—362no—
256C22×C3C2×#62411yescharacteristic
257C22×C3C2×#62412yes—
258C2×D3—38no—
259C2×D3—64no—
260C2×D3—124no—
261C2×D3—124no—
262C3×C4—32no—
263C3×C4—32no—
264C3×C4—32no—
265C3×C4—32no—
266C3×C4—61no—
267C3×C4—62no—
268C3×C4—61no—
269C3×C4—94no—
270C3×C4C2×#62411yescharacteristic
271C3×C4C2×#62412yes—
272C3×C4C22×D3211yescharacteristic
273Dic3—38no—
274Dic3—64no—
275C32—21no—
276C32#574512yescomplemented
277C23—92no—
278C23#619911yescharacteristic
279Q8—94no—
280D4—98no—
281D4—94no—
282C2×C4—92no—
283C2×C4—94no—
284C2×C4—94no—
285C2×C4#619911yescharacteristic
286C2×C4#619912yes—
287C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
288C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C8—98no—
290C2×C3—21no—
291C2×C3—22no—
292C2×C3—32no—
293C2×C3—62no—
294C2×C3—62no—
295C2×C3—62no—
296C2×C3—62no—
297C2×C3—61no—
298C2×C3—122no—
299C2×C3—184no—
300C2×C3C22×#62411yescharacteristic
301D3—68no—
302D3—124no—
303C22—41no—
304C22—94no—
305C22—362no—
306C22C2×#619911yescharacteristic
307C22C2×#619912yes—
308C4—94no—
309C4C2×#619911yescharacteristic
310C4C2×#619912yes—
311C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
312C3—32no—
313C3—61no—
314C3#1059478011yescharacteristic
315C2—21no—
316C2—22no—
317C2—184no—
318C2C22×#619911yescharacteristic
subgroupG/H
centerC4C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC2×C3C22×#624
prime radicalC2×#88740C2
cosocleC4×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#568
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3748⋊C3C2semidirect
3. #11721065C2semidirect
4. #11721064C2semidirect
14. #3748⋊C3C22semidirect
15. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
16. #88435C22semidirect
17. #88434C22semidirect
20. C2×#3748D3semidirect
57. #3748C2×D3semidirect
58. C2×C3⋊(C3⋊C8)C2×D3semidirect
180. H(3,𝔽3)#568semidirect
276. C32#5745semidirect

Maximal subgroups

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

subgroupindex
C2×#887402
C2×#3748⋊C32
#117210652
#117210642
#105947763