Finite Group Databasebeta

← #13140253

Subgroups — #13140253

Order 1728.

1391 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1391 conjugacy classes of subgroups — too many to draw. Only its 195 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
2C22×#6199C212yesmaximal
3#63658⋊C3C212yesmaximal, complemented
4#11720334C212yesmaximal, complemented
5D3×C3⋊#95—32nomaximal
6C23×#272C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C2×C4×#272—24no—
10C2×C4×#272C414yescomplemented
11C22×(C3⋊Dic3)⋊C3C2214yes—
12C2×#6199—24no—
13C2×#6199—24no—
14C2×#6199C418yescomplemented
15C22×C3×C4×D3—32no—
16C22×C3×C4×D3—61no—
17C22×D3×Dic3—34no—
18#63658D312yescomplemented
19C3×C3⋊#95—32no—
20#63579—34no—
21C23×H(3,𝔽3)C2311yescharacteristic
22C22×#272C2×C414yes—
23C22×#272C2×C412yes—
24C22×#272D412yes—
25C22×#272D414yes—
26C22×#272Q812yes—
27C2×C4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)C2×C412yescomplemented
29C2×#1084—22no—
30C2×#1084C2×C412yescomplemented
31C4×#272—28no—
32C2×(C3⋊Dic3)⋊C3—24no—
33C2×(C3⋊Dic3)⋊C3—24no—
34C2×(C3⋊Dic3)⋊C3C2×C418yescomplemented
35#6199—216no—
36C2×C3⋊#95—91no—
37D3×#95—92no—
38C23×C3×D3—32no—
39C23×C3×D3—61no—
40C22×C32×C4—21no—
41C22×C32×C4C2×D312yes—
42C22×C3⋊Dic3C2×D314yes—
43C22×C3×Dic3—34no—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—61no—
46C2×C3×C4×D3—38no—
47C2×C3×C4×D3—68no—
48C2×C3×C4×D3—64no—
49C2×C3×C4×D3—68no—
50C2×D3×Dic3—316no—
51C2×D3×Dic3—68no—
52C2×D3×Dic3—68no—
53C22×H(3,𝔽3)C22×C412yes—
54C22×H(3,𝔽3)C22×C411yescharacteristic
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C22×H(3,𝔽3)C2×D412yes—
57C22×H(3,𝔽3)C2×Q811yescharacteristic
58C2×#272C4214yes—
59C2×#272C22⋊C418yes—
60C2×#272C22⋊C414yes—
61C2×#272C4⋊C418yes—
62C2×#272C4⋊C414yes—
63C4×H(3,𝔽3)—22no—
64#1084—22no—
65(C3⋊Dic3)⋊C3—28no—
66C23×C3×C4—91no—
67C23×Dic3—92no—
68C22×C4×D3—32no—
69C22×C4×D3—61no—
70C22×C4×D3—94no—
71C3×#95—92no—
72C3⋊#95—32no—
73C3⋊#95—92no—
74C3⋊#95—92no—
75C3⋊#95—94no—
76C23×C32—21no—
77C23×C32C22×D312yes—
78C22×C3×D3—38no—
79C22×C3×D3—34no—
80C22×C3×D3—34no—
81C22×C3×D3—38no—
82C22×C3×D3—34no—
83C22×C3×D3—64no—
84C22×C3×D3—62no—
85C22×C3×D3—62no—
86C22×C3×D3—64no—
87C22×C3×D3—62no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4—22no—
91C2×C32×C4C4×D314yescomplemented
92C2×C3⋊Dic3—24no—
93C2×C3⋊Dic3—24no—
94C2×C3⋊Dic3C4×D318yescomplemented
95C2×C3×Dic3—38no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C3×C4×D3—616no—
103C3×C4×D3—616no—
104D3×Dic3—632no—
105C2×#95—271noSylow
106C2×H(3,𝔽3)C2×C4211yescharacteristic
107C2×H(3,𝔽3)C2×C22⋊C412yes—
108C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
109C2×H(3,𝔽3)C2×C4⋊C412yes—
110C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
111#272#9518yescomplemented
112C24×C3—91no—
113C23×D3—32no—
114C23×D3—61no—
115C22×C3×C4—32no—
116C22×C3×C4—61no—
117C22×C3×C4—94no—
118C22×C3×C4—91no—
119C22×C3×C4—94no—
120C22×C3×C4—184no—
121C22×C3×C4D3211yescharacteristic
122C22×Dic3—32no—
123C22×Dic3—34no—
124C22×Dic3—61no—
125C22×Dic3—94no—
126C22×Dic3—98no—
127C22×Dic3—184no—
128C22×Dic3—184no—
129C2×C4×D3—38no—
130C2×C4×D3—64no—
131C2×C4×D3—68no—
132C2×C4×D3—68no—
133C2×C4×D3—916no—
134C2×C4×D3—188no—
135C2×C4×D3—188no—
136C22×C32—21no—
137C22×C32—22no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—22no—
141C22×C32C2×C4×D314yes—
142C22×C32C2×C4×D312yes—
143C22×C32D3×D412yes—
144C22×C32D3×D414yes—
145C22×C32D3×Q812yes—
146C2×C3×D3—38no—
147C2×C3×D3—316no—
148C2×C3×D3—38no—
149C2×C3×D3—316no—
150C2×C3×D3—38no—
151C2×C3×D3—64no—
152C2×C3×D3—64no—
153C2×C3×D3—68no—
154C2×C3×D3—68no—
155C2×C3×D3—64no—
156C32×C4—24no—
157C32×C4—24no—
158C3⋊Dic3—28no—
159C3×Dic3—64no—
160C3×Dic3—68no—
161C3×Dic3—64no—
162C23×C4—91no—
163C23×C4—272no—
164#95—92no—
165#95—272no—
166H(3,𝔽3)C2×#9511yescharacteristic, complemented, Sylow
167C23×C3—32no—
168C23×C3—61no—
169C23×C3—92no—
170C23×C3—94no—
171C23×C3—92no—
172C23×C3—92no—
173C23×C3—94no—
174C23×C3C2×D3211yescharacteristic
175C22×D3—34no—
176C22×D3—38no—
177C22×D3—38no—
178C22×D3—34no—
179C22×D3—34no—
180C22×D3—62no—
181C22×D3—64no—
182C22×D3—62no—
183C22×D3—64no—
184C22×D3—62no—
185C2×C3×C4—22no—
186C2×C3×C4—34no—
187C2×C3×C4—64no—
188C2×C3×C4—64no—
189C2×C3×C4—62no—
190C2×C3×C4—98no—
191C2×C3×C4—92no—
192C2×C3×C4—188no—
193C2×C3×C4—184no—
194C2×C3×C4—184no—
195C2×C3×C4—182no—
196C2×C3×C4#62412yes—
197C2×Dic3—34no—
198C2×Dic3—38no—
199C2×Dic3—62no—
200C2×Dic3—64no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—98no—
205C2×Dic3—184no—
206C2×Dic3—184no—
207C2×Dic3—1816no—
208C4×D3—616no—
209C4×D3—616no—
210C4×D3—1832no—
211C2×C32—21no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32C42×D312yes—
217C2×C32D3×C22⋊C414yes—
218C2×C32D3×C22⋊C412yes—
219C2×C32D3×C4⋊C414yes—
220C2×C32D3×C4⋊C412yes—
221C3×D3—316no—
222C3×D3—68no—
223C24—91no—
224C22×C4—94no—
225C22×C4—91no—
226C22×C4—94no—
227C22×C4—184no—
228C22×C4—278no—
229C22×C4—544no—
230C22×C4—544no—
231C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
232C22×C3—32no—
233C22×C3—34no—
234C22×C3—32no—
235C22×C3—32no—
236C22×C3—34no—
237C22×C3—61no—
238C22×C3—62no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3—62no—
242C22×C3—94no—
243C22×C3—98no—
244C22×C3—94no—
245C22×C3—98no—
246C22×C3—94no—
247C22×C3#379711yescharacteristic
248C22×C3C4×D3212yes—
249C22×C3#380211yescharacteristic
250C22×C3C2×#62411yescharacteristic
251C22×C3#381112yes—
252C2×D3—38no—
253C2×D3—316no—
254C2×D3—38no—
255C2×D3—316no—
256C2×D3—38no—
257C2×D3—64no—
258C2×D3—64no—
259C2×D3—68no—
260C2×D3—64no—
261C2×D3—68no—
262C3×C4—22no—
263C3×C4—64no—
264C3×C4—64no—
265C3×C4—188no—
266C3×C4—182no—
267Dic3—64no—
268Dic3—64no—
269Dic3—68no—
270Dic3—188no—
271C32—21no—
272C32D3×#9512yescomplemented
273C23—92no—
274C23—94no—
275C23—92no—
276C23—92no—
277C23—94no—
278C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
279C2×C4—22no—
280C2×C4—98no—
281C2×C4—92no—
282C2×C4—188no—
283C2×C4—184no—
284C2×C4—182no—
285C2×C4—184no—
286C2×C4—5416no—
287C2×C4#619912yescomplemented
288C2×C3—32no—
289C2×C3—32no—
290C2×C3—34no—
291C2×C3—32no—
292C2×C3—34no—
293C2×C3—61no—
294C2×C3—61no—
295C2×C3—62no—
296C2×C3—61no—
297C2×C3—62no—
298C2×C3—98no—
299C2×C3C4×#62411yescharacteristic
300C2×C3#6388312yes—
301C2×C3#6390011yescharacteristic
302C2×C3#6392112yes—
303C2×C3#6397411yescharacteristic
304D3—316no—
305D3—68no—
306C22—94no—
307C22—94no—
308C22—98no—
309C22—94no—
310C22—98no—
311C22#8865111yescharacteristic
312C22C4×((C3⋊D3)⋊C3)⋊C212yes—
313C22#8865511yescharacteristic
314C22C2×#619911yescharacteristic
315C22#8867712yes—
316C4—22no—
317C4—182no—
318C4—188no—
319C3—32no—
320C3—61no—
321C3#1059137011yescharacteristic
322C2—98no—
323C2C4×#619911yescharacteristic
324C2#1172114712yes—
325C2#1172115211yescharacteristic
326C2#1172116112yes—
327C2#1172132011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#95
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #63658⋊C3C2semidirect
4. #11720334C2semidirect
10. C2×C4×#272C4semidirect
14. C2×#6199C4semidirect
18. #63658D3semidirect
28. C2×C4×H(3,𝔽3)C2×C4semidirect
30. C2×#1084C2×C4semidirect
34. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
91. C2×C32×C4C4×D3semidirect
94. C2×C3⋊Dic3C4×D3semidirect
111. #272#95semidirect
166. H(3,𝔽3)C2×#95semidirect
272. C32D3×#95semidirect
287. C2×C4#6199semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×#61992
#63658⋊C32
#117203342
D3×C3⋊#953