Finite Group Databasebeta

← #13140220

Subgroups — #13140220

Order 1728.

1055 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 171 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×C4⋊((C3⋊Dic3)⋊C3)C212yesmaximal, complemented
3C2×#88451C212yesmaximal
4#11720334C212yesmaximal, complemented
5#10591313—32nomaximal
6C23×#272C2211yescharacteristic
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C2×C4×#272—24no—
10C2×C4×#272C414yescomplemented
11C22×(C3⋊Dic3)⋊C3C2214yes—
12C4⋊((C3⋊Dic3)⋊C3)—24no—
13#88451—24no—
14C22×C3×C4×D3—32no—
15C22×C3×C4×D3—61no—
16C2×C4⋊(C3⋊Dic3)D312yescomplemented
17C2×C3×C4⋊Dic3—32no—
18C2×(C2×D3)⋊Dic3—34no—
19#63579—34no—
20C23×H(3,𝔽3)C2311yescharacteristic
21C22×#272—22no—
22C22×#272—22no—
23C22×#272C2×C412yes—
24C22×#272D412yes—
25C22×#272Q812yes—
26C2×C4×H(3,𝔽3)C2×C412yes—
27C2×C4×H(3,𝔽3)D414yes—
28C2×#1084—22no—
29C2×#1084C2×C412yes—
30C4×#272—28no—
31C2×(C3⋊Dic3)⋊C3—24no—
32C2×(C3⋊Dic3)⋊C3—24no—
33C2×(C3⋊Dic3)⋊C3—24no—
34#5212—91no—
35#4987—92no—
36C23×C3×D3—32no—
37C23×C3×D3—61no—
38C22×C32×C4—21no—
39C22×C32×C4C2×D312yes—
40C22×C3⋊Dic3C2×D314yes—
41C22×C3×Dic3—32no—
42C22×C3×Dic3—34no—
43C22×C3×Dic3—61no—
44C2×C3×C4×D3—38no—
45C2×C3×C4×D3—64no—
46C2×C3×C4×D3—68no—
47C2×C3×C4×D3—64no—
48C2×C3×C4×D3—64no—
49C4⋊(C3⋊Dic3)—24no—
50C3×C4⋊Dic3—64no—
51(C2×D3)⋊Dic3—68no—
52C22×H(3,𝔽3)C22×C411yescharacteristic
53C22×H(3,𝔽3)C2×D412yes—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C22×H(3,𝔽3)C2×Q811yescharacteristic
56C22×H(3,𝔽3)#4112yes—
57C2×#272—22no—
58C2×#272—22no—
59C2×#272—24no—
60C2×#272—24no—
61C2×#272C4⋊C414yescomplemented
62C4×H(3,𝔽3)C22⋊C414yescomplemented
63#1084—22no—
64(C3⋊Dic3)⋊C3—44no—
65C23×C3×C4—91no—
66C22×C4×D3—32no—
67C22×C4×D3—61no—
68C2×C3×C4⋊C4—92no—
69C2×#217—92no—
70C2×(C2×D3)⋊C4—94no—
71C2×C4⋊Dic3—32no—
72C2×C4⋊Dic3—92no—
73C3⋊#95—92no—
74C3⋊#95—94no—
75C23×C32—21no—
76C23×C32C22×D312yes—
77C22×C3×D3—34no—
78C22×C3×D3—34no—
79C22×C3×D3—34no—
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C22×C3×D3—62no—
83C22×C3×D3—64no—
84C22×C3×D3—64no—
85C22×C3×D3—62no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—62no—
89C2×C32×C4—22no—
90C2×C32×C4—24no—
91C2×C32×C4C4×D314yes—
92C2×C32×C4D1214yes—
93C2×C32×C4#6714yes—
94C2×C3⋊Dic3—24no—
95C2×C3⋊Dic3—24no—
96C2×C3⋊Dic3—24no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C3×C4×D3—616no—
106C3×C4×D3—616no—
107#380—271noSylow
108C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
109C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
110C2×H(3,𝔽3)#11711yescharacteristic
111C2×H(3,𝔽3)#12112yes—
112C2×H(3,𝔽3)C22⋊Q812yes—
113#272—24no—
114C24×C3—91no—
115C23×D3—32no—
116C23×D3—61no—
117C22×C3×C4—32no—
118C22×C3×C4—61no—
119C22×C3×C4—91no—
120C22×C3×C4—94no—
121C22×C3×C4—94no—
122C22×C3×C4—184no—
123C22×C3×C4D3211yescharacteristic
124C22×Dic3—32no—
125C22×Dic3—34no—
126C22×Dic3—61no—
127C22×Dic3—94no—
128C2×C4×D3—38no—
129C2×C4×D3—64no—
130C2×C4×D3—64no—
131C2×C4×D3—64no—
132C2×C4×D3—68no—
133C3×C4⋊C4—184no—
134#217—184no—
135(C2×D3)⋊C4—188no—
136C4⋊Dic3—64no—
137C4⋊Dic3—184no—
138C22×C32—22no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—22no—
143C22×C32C2×C4×D312yes—
144C22×C32C2×D1212yes—
145C22×C32D3×D412yes—
146C22×C32#23712yes—
147C22×C32D3×Q812yes—
148C22×C32#23912yes—
149C22×C32C2×#6712yes—
150C2×C3×D3—38no—
151C2×C3×D3—64no—
152C2×C3×D3—64no—
153C2×C3×D3—64no—
154C2×C3×D3—68no—
155C2×C3×D3—68no—
156C2×C3×D3—64no—
157C2×C3×D3—64no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—64no—
161C2×C3×D3—64no—
162C32×C4—24no—
163C32×C4(C2×D3)⋊C418yescomplemented
164C3⋊Dic3—44no—
165C3×Dic3—64no—
166C3×Dic3—64no—
167C3×Dic3—124no—
168C23×C4—91no—
169C2×C4⋊C4—92no—
170C2×C22⋊C4—272no—
171#95—272no—
172H(3,𝔽3)#38011yescharacteristic, complemented, Sylow
173C23×C3—32no—
174C23×C3—61no—
175C23×C3—92no—
176C23×C3—92no—
177C23×C3—92no—
178C23×C3—182no—
179C23×C3—182no—
180C23×C3C2×D3211yescharacteristic
181C22×D3—34no—
182C22×D3—34no—
183C22×D3—34no—
184C22×D3—62no—
185C22×D3—62no—
186C22×D3—62no—
187C22×D3—62no—
188C22×D3—62no—
189C22×D3—62no—
190C22×D3—62no—
191C22×D3—64no—
192C22×D3—64no—
193C2×C3×C4—34no—
194C2×C3×C4—34no—
195C2×C3×C4—34no—
196C2×C3×C4—62no—
197C2×C3×C4—64no—
198C2×C3×C4—92no—
199C2×C3×C4—182no—
200C2×C3×C4—188no—
201C2×C3×C4—184no—
202C2×C3×C4—184no—
203C2×C3×C4—184no—
204C2×C3×C4#62412yes—
205C2×C3×C4#62614yes—
206C2×Dic3—34no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—62no—
210C2×Dic3—64no—
211C2×Dic3—64no—
212C2×Dic3—64no—
213C2×Dic3—64no—
214C2×Dic3—184no—
215C2×Dic3—184no—
216C2×Dic3—184no—
217C4×D3—616no—
218C4×D3—616no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—22no—
223C2×C32—21no—
224C2×C32D3×C4⋊C412yes—
225C2×C32#89212yes—
226C2×C32#89512yes—
227C2×C32#89712yes—
228C2×C32C2×(C2×D3)⋊C412yes—
229C2×C32#93812yes—
230C2×C32#94612yes—
231C3×D3—68no—
232C3×D3—68no—
233C24—91no—
234C22×C4—91no—
235C22×C4—94no—
236C22×C4—94no—
237C22×C4—184no—
238C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
239C4⋊C4—184no—
240C22⋊C4—544no—
241C22×C3—32no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—32no—
245C22×C3—32no—
246C22×C3—32no—
247C22×C3—32no—
248C22×C3—62no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—62no—
253C22×C3—94no—
254C22×C3—182no—
255C22×C3—184no—
256C22×C3—184no—
257C22×C3—182no—
258C22×C3#379612yes—
259C22×C3#379711yescharacteristic
260C22×C3#380211yescharacteristic
261C22×C3C2×#62411yescharacteristic
262C22×C3C2×#62612yes—
263C2×D3—38no—
264C2×D3—64no—
265C2×D3—64no—
266C2×D3—64no—
267C2×D3—64no—
268C2×D3—64no—
269C2×D3—64no—
270C2×D3—64no—
271C2×D3—64no—
272C2×D3—64no—
273C2×D3—68no—
274C2×D3—68no—
275C3×C4—38no—
276C3×C4—64no—
277C3×C4—182no—
278C3×C4—364no—
279C3×C4#372214yes—
280Dic3—64no—
281Dic3—64no—
282Dic3—124no—
283Dic3—364no—
284C32—21no—
285C32#498712yescomplemented
286C23—92no—
287C23—92no—
288C23—92no—
289C23—182no—
290C23—182no—
291C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C2×C4—92no—
293C2×C4—182no—
294C2×C4—188no—
295C2×C4—184no—
296C2×C4—184no—
297C2×C4—184no—
298C2×C4#619912yes—
299C2×C4#620014yes—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—32no—
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—32no—
307C2×C3—61no—
308C2×C3—61no—
309C2×C3—62no—
310C2×C3—61no—
311C2×C3—62no—
312C2×C3—184no—
313C2×C3#6384911yescharacteristic
314C2×C3#6387112yes—
315C2×C3#6390011yescharacteristic
316C2×C3#6391612yes—
317C2×C3C2×#372211yescharacteristic
318D3—68no—
319D3—68no—
320C22—94no—
321C22—182no—
322C22—182no—
323C22—184no—
324C22—184no—
325C22#8864912yes—
326C22#8865111yescharacteristic
327C22#8865511yescharacteristic
328C22C2×#619911yescharacteristic
329C22C2×#620012yes—
330C4—182no—
331C4—364no—
332C4#8845114yescomplemented
333C3—32no—
334C3—61no—
335C3#1059132511yescharacteristic
336C2—184no—
337C2#1172108911yescharacteristic
338C2#1172109412yes—
339C2#1172115211yescharacteristic
340C2#1172115912yes—
341C2C2×#8845111yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
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)#380
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #11720334C2semidirect
10. C2×C4×#272C4semidirect
16. C2×C4⋊(C3⋊Dic3)D3semidirect
61. C2×#272C4⋊C4semidirect
62. C4×H(3,𝔽3)C22⋊C4semidirect
163. C32×C4(C2×D3)⋊C4semidirect
172. H(3,𝔽3)#380semidirect
285. C32#4987semidirect
332. C4#88451semidirect

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#884512
#117203342
#105913133