Finite Group Databasebeta

← #13139040

Subgroups — #13139040

Order 1728.

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

normal subgroups only

This group has 621 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×#88527C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4#63647⋊C3C212yesmaximal, complemented
5#11720317C212yesmaximal, complemented
6#10590955—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
16#88527—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18#88434—22no—
19C2×#3798—32no—
20C2×C3×ΓL(1,𝔽25)—32no—
21C2×C3×ΓL(1,𝔽25)—61no—
22C2×D3×C3⋊C8—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#5129—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×ΓL(1,𝔽25)—64no—
61C3×ΓL(1,𝔽25)—64no—
62D3×C3⋊C8—64no—
63C22×H(3,𝔽3)C22×C411yescharacteristic
64C2×#272—22no—
65C2×#272C22⋊C412yes—
66C2×(C3⋊D3)⋊C3—22no—
67C2×(C3⋊D3)⋊C3—22no—
68C2×(C3⋊D3)⋊C3—22no—
69C4×H(3,𝔽3)C2×D411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084C22⋊C412yes—
72(C3⋊Dic3)⋊C3—22no—
73C2×#235—91no—
74C2×#235—92no—
75C2×C3×M16—91no—
76C22×C3⋊C8—91no—
77C2×ΓL(1,𝔽25)—32no—
78C2×ΓL(1,𝔽25)—61no—
79C2×D3×C8—92no—
80C3×C22⋊C8—92no—
81#830—92no—
82(C2×D3)⋊C8—32no—
83(C2×D3)⋊C8—92no—
84(C2×D3)⋊C8—92no—
85(C2×D3)⋊C8—92no—
86C22×C3⋊D3C4×D312yes—
87C22×C3×D3—32no—
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C2×C32×C4—21no—
91C2×C32×C4C22×D312yes—
92C2×C3⋊Dic3C4×D312yes—
93C4×C3⋊D3—22no—
94C4×C3⋊D3—22no—
95C2×C3×Dic3—32no—
96C2×C3×Dic3—32no—
97C2×C3×Dic3—61no—
98C3×C4×D3—34no—
99C3×C4×D3—34no—
100C3×C4×D3—62no—
101C3×C4×D3—62no—
102C3×C4×D3—62no—
103C3×C4×D3—62no—
104#627—64no—
105#626—64no—
106#626—64no—
107#625—64no—
108C32×C8—22no—
109C32×C8—22no—
110C3⋊(C3⋊C8)—22no—
111C3×C3⋊C8—62no—
112C3×C3⋊C8—62no—
113C3×C3⋊C8—62no—
114#408—271noSylow
115C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
116C2×H(3,𝔽3)#13112yes—
117#272—22no—
118(C3⋊D3)⋊C3—42no—
119C22×C3×C4—92no—
120C22×C3×C4—91no—
121C2×#67—92no—
122C2×#67—92no—
123C2×#67—92no—
124#235—184no—
125#235—184no—
126#235—182no—
127#235—182no—
128C2×D12—91no—
129C2×D12—92no—
130C2×C4×D3—32no—
131C2×C4×D3—32no—
132C2×C4×D3—61no—
133C2×C4×D3—92no—
134C2×Dic6—91no—
135C2×Dic6—92no—
136C3×M16—182no—
137C2×C3×C8—32no—
138C2×C3×C8—61no—
139C2×C3×C8—92no—
140C2×C3×C8—91no—
141C2×C3×C8D3211yescharacteristic
142C2×C3⋊C8—32no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8—92no—
146C2×C3⋊C8—182no—
147ΓL(1,𝔽25)—64no—
148ΓL(1,𝔽25)—64no—
149D3×C8—184no—
150C22×C32—21no—
151C22×C32C2×C4×D312yes—
152C2×C3⋊D3—22no—
153C2×C3⋊D3—22no—
154C2×C3⋊D3—22no—
155C2×C3×D3—34no—
156C2×C3×D3—62no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—122no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4D3×D412yes—
166C32×C4D3×D412yes—
167C3⋊Dic3—22no—
168C3×Dic3—34no—
169C3×Dic3—62no—
170C3×Dic3—62no—
171C2×#41—271no—
172C2×M16—91no—
173C22×C8—271no—
174C22⋊C8—92no—
175C22⋊C8—272no—
176H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
177C23×C3—92no—
178C23×C3—91no—
179C22×D3—32no—
180C22×D3—32no—
181C22×D3—61no—
182C22×D3—92no—
183C2×C3×C4—32no—
184C2×C3×C4—61no—
185C2×C3×C4—91no—
186C2×C3×C4—92no—
187C2×C3×C4—92no—
188C2×C3×C4—92no—
189C2×C3×C4—182no—
190C2×C3×C4—182no—
191C2×C3×C4C2×D3211yescharacteristic
192#67—184no—
193#67—184no—
194#67—184no—
195C2×Dic3—32no—
196C2×Dic3—32no—
197C2×Dic3—61no—
198C2×Dic3—92no—
199D12—182no—
200D12—184no—
201C4×D3—34no—
202C4×D3—34no—
203C4×D3—62no—
204C4×D3—62no—
205C4×D3—62no—
206C4×D3—62no—
207C4×D3—182no—
208C4×D3—182no—
209Dic6—184no—
210Dic6—182no—
211C3×C8—21no—
212C3×C8—62no—
213C3×C8—62no—
214C3×C8—182no—
215C3×C8—181no—
216C3⋊C8—62no—
217C3⋊C8—62no—
218C3⋊C8—62no—
219C3⋊C8—182no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32D3×C22⋊C412yes—
223C2×C32#90112yes—
224C2×C32#90712yes—
225C3⋊D3—42no—
226C3×D3—64no—
227C3×D3—122no—
228C3×D3—122no—
229#41—542no—
230#41—542no—
231C2×Q8—271no—
232C2×D4—271no—
233C2×D4—272no—
234C22×C4—91no—
235C22×C4—92no—
236M16—182no—
237C2×C8—92no—
238C2×C8—91no—
239C2×C8—542no—
240C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C22×C3—32no—
242C22×C3—61no—
243C22×C3—92no—
244C22×C3—182no—
245C22×C3—182no—
246C22×C3—182no—
247C22×C3—182no—
248C22×C3C4×D3211yescharacteristic
249C2×D3—34no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—62no—
253C2×D3—62no—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—122no—
257C2×D3—182no—
258C2×D3—182no—
259C2×D3—182no—
260C3×C4—32no—
261C3×C4—32no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—92no—
265C3×C4—182no—
266C3×C4#380211yescharacteristic
267C3×C4#381111yescharacteristic
268Dic3—34no—
269Dic3—62no—
270Dic3—62no—
271Dic3—182no—
272C32—21no—
273C32#472712yescomplemented
274C23—91no—
275C23—92no—
276Q8—542no—
277D4—542no—
278D4—544no—
279C2×C4—91no—
280C2×C4—92no—
281C2×C4—92no—
282C2×C4—92no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C8—21no—
287C8—182no—
288C8—181no—
289C2×C3—32no—
290C2×C3—32no—
291C2×C3—32no—
292C2×C3—62no—
293C2×C3—61no—
294C2×C3—182no—
295C2×C3—362no—
296C2×C3#6380412yes—
297C2×C3#6392111yescharacteristic
298D3—64no—
299D3—122no—
300D3—122no—
301D3—362no—
302C22—92no—
303C22—182no—
304C22—182no—
305C22—182no—
306C22—182no—
307C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C4—92no—
309C4—182no—
310C4#8865511yescharacteristic
311C4#8867711yescharacteristic
312C3—32no—
313C3—61no—
314C3#1059095911yescharacteristic
315C2—182no—
316C2—362no—
317C2#1172084112yes—
318C2#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×#88527C2semidirect
3. C2×#88434C2semidirect
4. #63647⋊C3C2semidirect
5. #11720317C2semidirect
23. #63647D3semidirect
176. H(3,𝔽3)#408semidirect
273. 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×#885272
C2×#884342
#63647⋊C32
#117203172
#105909553