Finite Group Databasebeta

← #13140053

Subgroups — #13140053

Order 1728.

901 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×C4×#272C211yesmaximal, characteristic
2C2×#88435C212yesmaximal, complemented
3#63657⋊C3C212yesmaximal, complemented
4#11720323C212yesmaximal, complemented
5#10591154—32nomaximal
6C23×#272C411yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C411yescharacteristic
9C2×C4×#272—24no—
10C2×C4×#272C411yescharacteristic
11C2×C4×#272C411yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
15#88435—24no—
16C22×C3×C4×D3—32no—
17C22×C3×C4×D3—61no—
18C2×#3711—34no—
19#63657D312yescomplemented
20C3×#830—32no—
21(C2×D3)⋊(C3⋊C8)—34no—
22C23×H(3,𝔽3)C2×C411yescharacteristic
23C22×#272—22no—
24C22×#272—22no—
25C22×#272—22no—
26C22×#272C2×C411yescharacteristic
27C22×#272C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31C2×#1084—22no—
32C2×#1084C2×C411yescharacteristic
33C2×#1084C2×C411yescharacteristic
34C4×#272—22no—
35C4×#272—22no—
36C4×#272—22no—
37C4×#272D414yes—
38(C3⋊(C3⋊C8))⋊C3—24no—
39#5207—91no—
40#4726—92no—
41C23×C3×D3—32no—
42C23×C3×D3—61no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D312yes—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—61no—
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—32no—
49C2×C3×C4×D3—32no—
50C2×C3×C4×D3—32no—
51C2×C3×C4×D3—68no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—68no—
57C2×C3⋊(C3⋊C8)C2×D314yescomplemented
58C2×C3×C3⋊C8—34no—
59#3711—68no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)M1612yescomplemented
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×#272—28no—
65C2×#272—21no—
66C2×#272—21no—
67C2×#272—21no—
68C2×#272—21no—
69C2×#272C22⋊C412yes—
70C2×#272M1612yescomplemented
71C4×H(3,𝔽3)—21no—
72C4×H(3,𝔽3)C2×D412yes—
73#1084—21no—
74#1084C22⋊C412yes—
75C23×C3×C4—91no—
76C22×C4×D3—32no—
77C22×C4×D3—61no—
78C2×#208—92no—
79C2×ΓL(1,𝔽25)—94no—
80C3×C22⋊C8—92no—
81#830—32no—
82#830—92no—
83#830—92no—
84(C2×D3)⋊C8—94no—
85C23×C32—21no—
86C23×C32C4×D312yes—
87C22×C3×D3—32no—
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C22×C3×D3—61no—
91C22×C3×D3—64no—
92C22×C3×D3—64no—
93C22×C3×D3—64no—
94C22×C3×D3—64no—
95C22×C3×D3—64no—
96C22×C3×D3—64no—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4C4×D312yes—
102C2×C32×C4C22×D312yes—
103C2×C3×Dic3—32no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C3×C4×D3—38no—
110C3×C4×D3—64no—
111C3×C4×D3—64no—
112C3×C4×D3—64no—
113C3×C4×D3—64no—
114C3×C4×D3—64no—
115C3×C4×D3—64no—
116C3×C4×D3—64no—
117C3⋊(C3⋊C8)—24no—
118C3×C3⋊C8—64no—
119#407—271noSylow
120C2×H(3,𝔽3)—22no—
121C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
122C2×H(3,𝔽3)C2×M1611yescharacteristic
123C2×H(3,𝔽3)C2×M1611yescharacteristic
124#272—22no—
125#272—22no—
126C24×C3—91no—
127C23×D3—32no—
128C23×D3—61no—
129C22×C3×C4—32no—
130C22×C3×C4—61no—
131C22×C3×C4—91no—
132C22×C3×C4—91no—
133C22×C3×C4—91no—
134C22×C3×C4—91no—
135C22×C3×C4—91no—
136C22×C3×C4—184no—
137C22×C3×C4D3211yescharacteristic
138C22×Dic3—32no—
139C22×Dic3—61no—
140C2×C4×D3—32no—
141C2×C4×D3—32no—
142C2×C4×D3—32no—
143C2×C4×D3—32no—
144C2×C4×D3—61no—
145C2×C4×D3—61no—
146C2×C4×D3—61no—
147C2×C4×D3—61no—
148C2×C4×D3—68no—
149C2×C4×D3—68no—
150C2×C3×C8—94no—
151#208—184no—
152C2×C3⋊C8—34no—
153C2×C3⋊C8—94no—
154ΓL(1,𝔽25)—188no—
155C22×C32—22no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32ΓL(1,𝔽25)14yescomplemented
162C22×C32C2×C4×D312yes—
163C2×C3×D3—34no—
164C2×C3×D3—34no—
165C2×C3×D3—62no—
166C2×C3×D3—62no—
167C2×C3×D3—62no—
168C2×C3×D3—616no—
169C2×C3×D3—616no—
170C2×C3×D3—62no—
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4D3×D414yes—
181C3×Dic3—34no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C23×C4—91no—
186C2×M16—272no—
187C22⋊C8—92no—
188C22⋊C8—272no—
189H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
190C23×C3—32no—
191C23×C3—61no—
192C23×C3—91no—
193C23×C3—91no—
194C23×C3—182no—
195C23×C3—182no—
196C23×C3—182no—
197C23×C3#62411yescharacteristic
198C22×D3—32no—
199C22×D3—32no—
200C22×D3—61no—
201C22×D3—61no—
202C22×D3—64no—
203C22×D3—64no—
204C22×D3—64no—
205C22×D3—64no—
206C22×D3—64no—
207C22×D3—64no—
208C2×C3×C4—22no—
209C2×C3×C4—32no—
210C2×C3×C4—32no—
211C2×C3×C4—64no—
212C2×C3×C4—64no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—91no—
216C2×C3×C4—94no—
217C2×C3×C4—91no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×C3×C4#62411yescharacteristic
223C2×C3×C4C2×D3211yescharacteristic
224C2×Dic3—32no—
225C2×Dic3—32no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228C2×Dic3—64no—
229C2×Dic3—64no—
230C4×D3—38no—
231C4×D3—64no—
232C4×D3—64no—
233C4×D3—64no—
234C4×D3—64no—
235C4×D3—64no—
236C4×D3—64no—
237C4×D3—64no—
238C3×C8—184no—
239C3⋊C8—64no—
240C3⋊C8—184no—
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32D3×C22⋊C412yes—
247C2×C32C2×ΓL(1,𝔽25)12yes—
248C2×C32D3×M1612yes—
249C3×D3—64no—
250C3×D3—64no—
251C3×D3—64no—
252C3×D3—64no—
253C24—91no—
254C22×C4—91no—
255C22×C4—91no—
256C22×C4—91no—
257C22×C4—91no—
258C22×C4—91no—
259C22×C4—184no—
260C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
261M16—544no—
262C2×C8—94no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—32no—
266C22×C3—34no—
267C22×C3—62no—
268C22×C3—62no—
269C22×C3—62no—
270C22×C3—62no—
271C22×C3—61no—
272C22×C3—62no—
273C22×C3—92no—
274C22×C3—92no—
275C22×C3—188no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3—181no—
280C22×C3#371212yes—
281C22×C3C2×#62411yescharacteristic
282C2×D3—34no—
283C2×D3—34no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—62no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—616no—
292C2×D3—616no—
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—62no—
296C3×C4—21no—
297C3×C4—34no—
298C3×C4—62no—
299C3×C4—62no—
300C3×C4—62no—
301C3×C4—92no—
302C3×C4—181no—
303C3×C4#381112yes—
304Dic3—34no—
305Dic3—62no—
306Dic3—62no—
307Dic3—62no—
308C32—21no—
309C32#472612yescomplemented
310C23—91no—
311C23—91no—
312C23—182no—
313C23—182no—
314C23—182no—
315C23#619911yescharacteristic
316C2×C4—22no—
317C2×C4—91no—
318C2×C4—94no—
319C2×C4—91no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4#619911yescharacteristic
325C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C8—184no—
327C2×C3—22no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—32no—
331C2×C3—64no—
332C2×C3—64no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—182no—
337C2×C3—182no—
338C2×C3#6381811yescharacteristic
339C2×C3C2×#371211yescharacteristic
340C2×C3#6397411yescharacteristic
341D3—64no—
342D3—64no—
343D3—64no—
344D3—64no—
345C22—21no—
346C22—21no—
347C22—92no—
348C22—92no—
349C22—188no—
350C22—181no—
351C22—181no—
352C22—181no—
353C22—181no—
354C22#8843512yescomplemented
355C22C2×#619911yescharacteristic
356C4—21no—
357C4—92no—
358C4—181no—
359C4#8867712yes—
360C3—32no—
361C3—61no—
362C3#1059115911yescharacteristic
363C2—22no—
364C2—182no—
365C2—182no—
366C2#1172106511yescharacteristic
367C2C2×#8843511yescharacteristic
368C2#1172132011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×#272C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88435C2semidirect
3. #63657⋊C3C2semidirect
4. #11720323C2semidirect
14. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
19. #63657D3semidirect
57. C2×C3⋊(C3⋊C8)C2×D3semidirect
62. C22×H(3,𝔽3)M16semidirect
70. C2×#272M16semidirect
161. C22×C32ΓL(1,𝔽25)semidirect
189. H(3,𝔽3)#407semidirect
309. C32#4726semidirect
354. C22#88435semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×#884352
#63657⋊C32
#117203232
#105911543