Finite Group Databasebeta

← #13139039

Subgroups — #13139039

Order 1728.

629 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88650C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic, complemented
3C2×#88435C211yesmaximal, characteristic, complemented
4#63647⋊C3C212yesmaximal, complemented
5#11720317C212yesmaximal, complemented
6#10590953—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C4×(C3⋊D3)⋊C3C2212yes—
9C2×#6200C412yes—
10C2×#6198C411yescharacteristic
11C2×#6196C411yescharacteristic
12#88650—22no—
13#88650—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C8×#272—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88435—22no—
19C2×#3798—32no—
20C2×C3×D3×C8—32no—
21C2×C3×D3×C8—61no—
22C2×#3711—32no—
23#63647D312yescomplemented
24C3×(C2×D3)⋊C8—32no—
25#63557—32no—
26(C2×D3)⋊(C3⋊C8)—32no—
27C22×#272C2×C411yescharacteristic
28C22×(C3⋊D3)⋊C3C2×C412yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084C2×C411yescharacteristic
31C4×#272D412yes—
32C4×#272D412yes—
33C2×(C3⋊Dic3)⋊C3C2×C412yes—
34C4×(C3⋊D3)⋊C3—22no—
35C4×(C3⋊D3)⋊C3—22no—
36#6200—24no—
37#6198—22no—
38#6196—22no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5110—91no—
43#4727—92no—
44C2×C4×C3⋊D3C2×D312yes—
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—61no—
48C2×#627—32no—
49C2×#626—32no—
50C2×#626—32no—
51C2×#625—32no—
52#3798—64no—
53#3798—64no—
54C2×C32×C8—21no—
55C2×C32×C8C2×D312yes—
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C2×C3×C3⋊C8—32no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—61no—
60C3×D3×C8—64no—
61C3×D3×C8—64no—
62#3711—64no—
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×#272—21no—
65C2×#272—21no—
66C2×#272C22⋊C412yes—
67C2×(C3⋊D3)⋊C3—22no—
68C2×(C3⋊D3)⋊C3—22no—
69C2×(C3⋊D3)⋊C3—22no—
70C4×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)C2×D411yescharacteristic
72#1084C22⋊C412yes—
73(C3⋊Dic3)⋊C3—22no—
74C2×#235—91no—
75C2×#235—92no—
76C22×C3×C8—91no—
77C2×#208—91no—
78C2×ΓL(1,𝔽25)—92no—
79C2×D3×C8—32no—
80C2×D3×C8—61no—
81C3×C22⋊C8—92no—
82#830—92no—
83(C2×D3)⋊C8—32no—
84(C2×D3)⋊C8—92no—
85(C2×D3)⋊C8—92no—
86(C2×D3)⋊C8—92no—
87C22×C3⋊D3C4×D312yes—
88C22×C3×D3—32no—
89C22×C3×D3—32no—
90C22×C3×D3—61no—
91C2×C32×C4—21no—
92C2×C32×C4C22×D312yes—
93C2×C3⋊Dic3C4×D312yes—
94C4×C3⋊D3—22no—
95C4×C3⋊D3—22no—
96C2×C3×Dic3—32no—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—61no—
99C3×C4×D3—34no—
100C3×C4×D3—34no—
101C3×C4×D3—62no—
102C3×C4×D3—62no—
103C3×C4×D3—62no—
104C3×C4×D3—62no—
105#627—64no—
106#626—64no—
107#626—64no—
108#625—64no—
109C32×C8—22no—
110C32×C8—22no—
111C3⋊(C3⋊C8)—22no—
112C3×C3⋊C8—62no—
113C3×C3⋊C8—62no—
114C3×C3⋊C8—62no—
115#408—271noSylow
116C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
117C2×H(3,𝔽3)#13111yescharacteristic
118C2×H(3,𝔽3)#13111yescharacteristic
119#272—22no—
120(C3⋊D3)⋊C3—42no—
121C22×C3×C4—92no—
122C22×C3×C4—91no—
123C2×#67—92no—
124C2×#67—92no—
125C2×#67—92no—
126#235—184no—
127#235—184no—
128#235—182no—
129#235—182no—
130C2×D12—91no—
131C2×D12—92no—
132C2×C4×D3—32no—
133C2×C4×D3—32no—
134C2×C4×D3—61no—
135C2×C4×D3—92no—
136C2×Dic6—91no—
137C2×Dic6—92no—
138C2×C3×C8—32no—
139C2×C3×C8—61no—
140C2×C3×C8—92no—
141C2×C3×C8—91no—
142C2×C3×C8—182no—
143C2×C3×C8D3211yescharacteristic
144#208—182no—
145C2×C3⋊C8—32no—
146C2×C3⋊C8—32no—
147C2×C3⋊C8—61no—
148C2×C3⋊C8—92no—
149ΓL(1,𝔽25)—184no—
150D3×C8—64no—
151D3×C8—64no—
152C22×C32—21no—
153C22×C32C2×C4×D312yes—
154C2×C3⋊D3—22no—
155C2×C3⋊D3—22no—
156C2×C3⋊D3—22no—
157C2×C3×D3—34no—
158C2×C3×D3—62no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4D3×D412yes—
169C32×C4D3×D412yes—
170C3⋊Dic3—22no—
171C3×Dic3—34no—
172C3×Dic3—62no—
173C3×Dic3—62no—
174C2×#41—271no—
175C2×M16—271no—
176C22×C8—91no—
177C22⋊C8—92no—
178C22⋊C8—272no—
179H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
180C23×C3—92no—
181C23×C3—91no—
182C22×D3—32no—
183C22×D3—32no—
184C22×D3—61no—
185C22×D3—92no—
186C2×C3×C4—32no—
187C2×C3×C4—61no—
188C2×C3×C4—91no—
189C2×C3×C4—92no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4C2×D3211yescharacteristic
195#67—184no—
196#67—184no—
197#67—184no—
198C2×Dic3—32no—
199C2×Dic3—32no—
200C2×Dic3—61no—
201C2×Dic3—92no—
202D12—182no—
203D12—184no—
204C4×D3—34no—
205C4×D3—34no—
206C4×D3—62no—
207C4×D3—62no—
208C4×D3—62no—
209C4×D3—62no—
210C4×D3—182no—
211C4×D3—182no—
212Dic6—184no—
213Dic6—182no—
214C3×C8—21no—
215C3×C8—62no—
216C3×C8—62no—
217C3×C8—182no—
218C3×C8—181no—
219C3⋊C8—62no—
220C3⋊C8—62no—
221C3⋊C8—62no—
222C3⋊C8—182no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32D3×C22⋊C412yes—
227C2×C32#90112yes—
228C2×C32#90712yes—
229C3⋊D3—42no—
230C3×D3—64no—
231C3×D3—64no—
232C3×D3—122no—
233#41—542no—
234#41—542no—
235C2×Q8—271no—
236C2×D4—271no—
237C2×D4—272no—
238C22×C4—91no—
239C22×C4—92no—
240M16—542no—
241C2×C8—92no—
242C2×C8—91no—
243C2×C8—182no—
244C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
245C22×C3—32no—
246C22×C3—61no—
247C22×C3—92no—
248C22×C3—182no—
249C22×C3—182no—
250C22×C3—182no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3C4×D3211yescharacteristic
254C2×D3—34no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—182no—
264C2×D3—182no—
265C2×D3—182no—
266C3×C4—32no—
267C3×C4—32no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—92no—
271C3×C4—182no—
272C3×C4#380211yescharacteristic
273C3×C4#381111yescharacteristic
274Dic3—34no—
275Dic3—62no—
276Dic3—62no—
277Dic3—182no—
278C32—21no—
279C32#472712yescomplemented
280C23—91no—
281C23—92no—
282Q8—542no—
283D4—542no—
284D4—544no—
285C2×C4—91no—
286C2×C4—92no—
287C2×C4—92no—
288C2×C4—92no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C8—21no—
293C8—182no—
294C8—181no—
295C2×C3—32no—
296C2×C3—32no—
297C2×C3—32no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—182no—
302C2×C3—362no—
303C2×C3#6380311yescharacteristic
304C2×C3#6380511yescharacteristic
305C2×C3#6392111yescharacteristic
306D3—64no—
307D3—64no—
308D3—122no—
309D3—362no—
310C22—92no—
311C22—181no—
312C22—181no—
313C22—182no—
314C22—182no—
315C22—182no—
316C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
317C4—92no—
318C4—182no—
319C4#8865511yescharacteristic
320C4#8867711yescharacteristic
321C3—32no—
322C3—61no—
323C3#1059095811yescharacteristic
324C2—182no—
325C2—362no—
326C2#1172084011yescharacteristic
327C2#1172084211yescharacteristic
328C2#1172116111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC2×#88650C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#408
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C8×#272C2semidirect
3. C2×#88435C2semidirect
4. #63647⋊C3C2semidirect
5. #11720317C2semidirect
23. #63647D3semidirect
179. H(3,𝔽3)#408semidirect
279. C32#4727semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C2×C8×#2722
C2×#884352
#63647⋊C32
#117203172
#105909533