Finite Group Databasebeta

← #64148⋊C3 #11721641

Subgroups — #64148⋊C3 #11721641

Order 864.

383 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 81 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×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic, complemented
3C2×#638⋊C3C211yesmaximal, characteristic, complemented
4C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#3757⋊C3C211yesmaximal, characteristic, complemented
6#3752⋊C3C211yesmaximal, characteristic, complemented
7#3750⋊C3C211yesmaximal, characteristic, complemented
8#64148C311yesmaximal, characteristic, complemented
9C3×#939—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C23×H(3,𝔽3)C2211yescharacteristic
12C22×(C3⋊D3)⋊C3C2211yescharacteristic
13D4×H(3,𝔽3)—22no—
14C2×C4×H(3,𝔽3)C2211yescharacteristic
15#638⋊C3—22no—
16#638⋊C3—22no—
17C2×(C3⋊Dic3)⋊C3—22no—
18C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×C32×D4—21no—
22C2×C32×D4D311yescharacteristic, complemented
23C2×C32×D4C2×C311yescharacteristic, complemented
24C2×#638C2×C311yescharacteristic, complemented
25C2×#638C2×C311yescharacteristic, complemented
26C22×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×C3×#67—31no—
28C2×C3×#67—31no—
29C22×C3×Dic3—31no—
30#3757C2×C311yescharacteristic, complemented
31#3752C2×C311yescharacteristic, complemented
32#3750C2×C311yescharacteristic, complemented
33C3×#217—31no—
34C3×(C2×D3)⋊C4—31no—
35C3×Dic3⋊C4—31no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—21no—
38C22×H(3,𝔽3)—21no—
39C22×H(3,𝔽3)—21no—
40C22×H(3,𝔽3)—21no—
41C22×H(3,𝔽3)D412yescomplemented
42C22×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊D3)⋊C3—21no—
44C2×(C3⋊D3)⋊C3—21no—
45C2×(C3⋊D3)⋊C3—21no—
46C4×H(3,𝔽3)—21no—
47(C3⋊Dic3)⋊C3—21no—
48(C3⋊Dic3)⋊C3—21no—
49(C3⋊Dic3)⋊C3D412yescomplemented
50C3×#121—91no—
51#939—31no—
52#939—91no—
53C23×C32—21no—
54C23×C32—21no—
55C23×C32C2×D311yescharacteristic
56C23×C32C2×D311yescharacteristic
57C23×C32C22×C311yescharacteristic
58C23×C32C22×C311yescharacteristic
59C22×C3⋊D3C22×C311yescharacteristic
60C22×C3×D3—31no—
61C32×D4—24no—
62C32×D4—22no—
63C32×D4—22no—
64C2×C32×C4—21no—
65C2×C32×C4C2×D311yescharacteristic
66C2×C32×C4C22×C311yescharacteristic
67#638—22no—
68#638—22no—
69C2×C3⋊Dic3—22no—
70C2×C3⋊Dic3C22×C311yescharacteristic
71C2×C3⋊Dic3C22×C311yescharacteristic
72C2×C3⋊Dic3C22×C311yescharacteristic
73C3×#67—62no—
74C3×#67—62no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—31no—
78C2×C3×Dic3—62no—
79C2×H(3,𝔽3)—22no—
80C2×H(3,𝔽3)—41no—
81C2×H(3,𝔽3)C2×D411yescharacteristic
82C2×H(3,𝔽3)C2×D411yescharacteristic
83C2×H(3,𝔽3)#4111yescharacteristic
84(C3⋊D3)⋊C3—41no—
85C2×C3×D4—31no—
86C2×C3×D4—31no—
87C2×C3×D4—61no—
88C2×C3×D4—91no—
89C2×C3×D4—91no—
90C2×C3×D4C3×D311yescharacteristic
91C22×C3×C4—91no—
92C2×#67—31no—
93C2×#67—31no—
94C2×#67—91no—
95C2×#67—91no—
96C22×Dic3—31no—
97C22×Dic3—91no—
98C3×C4⋊C4—91no—
99C3×C22⋊C4—91no—
100C3×C22⋊C4—91no—
101#217—31no—
102#217—91no—
103(C2×D3)⋊C4—31no—
104(C2×D3)⋊C4—91no—
105Dic3⋊C4—31no—
106Dic3⋊C4—91no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—22no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—22no—
119C22×C32—21no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32#6712yescomplemented
125C22×C32C3×D412yescomplemented
126C22×C32C22×D311yescharacteristic
127C22×C32C23×C311yescharacteristic
128C2×C3⋊D3—21no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C2×C3×D3—61no—
132C2×C3×D3—61no—
133C2×C3×D3—61no—
134C32×C4—22no—
135C32×C4—21no—
136C32×C4—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3—21no—
139C3⋊Dic3C3×D412yescomplemented
140C3×Dic3—32no—
141C3×Dic3—61no—
142C3×Dic3—61no—
143#121—91noSylow
144H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
145C23×C3—31no—
146C23×C3—31no—
147C23×C3—31no—
148C23×C3—31no—
149C23×C3—61no—
150C23×C3—61no—
151C23×C3—91no—
152C23×C3C2×C3×D311yescharacteristic
153C23×C3C2×C3×D311yescharacteristic
154C22×D3—31no—
155C22×D3—91no—
156C3×D4—22no—
157C3×D4—62no—
158C3×D4—62no—
159C3×D4—64no—
160C3×D4—182no—
161C3×D4—182no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—61no—
165C2×C3×C4—91no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—182no—
169C2×C3×C4C2×C3×D311yescharacteristic
170#67—62no—
171#67—62no—
172#67—182no—
173#67—182no—
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—62no—
178C2×Dic3—91no—
179C2×Dic3—91no—
180C2×Dic3—91no—
181C2×Dic3—182no—
182C2×C32—22no—
183C2×C32—22no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32D3×D411yescharacteristic
192C2×C32#23711yescharacteristic
193C2×C32C2×#6711yescharacteristic
194C2×C32C2×C3×D411yescharacteristic
195C2×C32C2×C3×D411yescharacteristic
196C2×C32C3×#4111yescharacteristic
197C3⋊D3—41no—
198C3×D3—121no—
199C2×D4—91no—
200C2×D4—91no—
201C2×D4(C3⋊D3)⋊C311yescharacteristic, complemented
202C22×C4—91no—
203C4⋊C4—91no—
204C22⋊C4—91no—
205C22⋊C4—91no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—32no—
212C22×C3—32no—
213C22×C3—31no—
214C22×C3—31no—
215C22×C3—61no—
216C22×C3—61no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—61no—
220C22×C3—61no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3—61no—
224C22×C3—61no—
225C22×C3—62no—
226C22×C3—62no—
227C22×C3—61no—
228C22×C3—121no—
229C22×C3—121no—
230C22×C3—121no—
231C22×C3—121no—
232C22×C3—181no—
233C22×C3—181no—
234C22×C3—181no—
235C22×C3C3×#6712yes—
236C22×C3C22×C3×D311yescharacteristic
237C2×D3—61no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—181no—
241C2×D3—181no—
242C2×D3—181no—
243C3×C4—21no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—62no—
247C3×C4—92no—
248C3×C4—181no—
249C3×C4—181no—
250Dic3—32no—
251Dic3—61no—
252Dic3—61no—
253Dic3—92no—
254Dic3—181no—
255Dic3—181no—
256C32—21no—
257C32#93911yescharacteristic, complemented
258C32C3×#12111yescharacteristic, complemented
259C23—91no—
260C23C2×(C3⋊D3)⋊C311yescharacteristic
261C23C2×(C3⋊D3)⋊C311yescharacteristic
262D4—22no—
263D4—182no—
264D4—182no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—182no—
269C2×C4C2×(C3⋊D3)⋊C311yescharacteristic
270C2×C3—22no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—41no—
278C2×C3—62no—
279C2×C3—62no—
280C2×C3—61no—
281C2×C3—61no—
282C2×C3—61no—
283C2×C3—122no—
284C2×C3—121no—
285C2×C3—121no—
286C2×C3—122no—
287C2×C3—361no—
288C2×C3C3×D3×D411yescharacteristic
289C2×C3C3×#23711yescharacteristic
290C2×C3C2×C3×#6711yescharacteristic
291D3—121no—
292D3—361no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—181no—
299C22—181no—
300C22—181no—
301C22#638⋊C312yescomplemented
302C22C22×(C3⋊D3)⋊C311yescharacteristic
303C4—21no—
304C4—92no—
305C4—181no—
306C4—181no—
307C3—31no—
308C3—31no—
309C3—61no—
310C3C3×#93911yescharacteristic
311C2—22no—
312C2—41no—
313C2—361no—
314C2D4×(C3⋊D3)⋊C311yescharacteristic
315C2#3830⋊C311yescharacteristic
316C2C2×#638⋊C311yescharacteristic
subgroupG/H
centerC22C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C23×C3
Frattini, socleC22×C3C22×C3×D3
FittingC2×D4×H(3,𝔽3)C2
hypercenterC2×D4(C3⋊D3)⋊C3
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×H(3,𝔽3)C2semidirect
2. C2×#638⋊C3C2semidirect
3. C2×#638⋊C3C2semidirect
4. C22×(C3⋊Dic3)⋊C3C2semidirect
5. #3757⋊C3C2semidirect
6. #3752⋊C3C2semidirect
7. #3750⋊C3C2semidirect
8. #64148C3semidirect
22. C2×C32×D4D3semidirect
23. C2×C32×D4C2×C3semidirect
24. C2×#638C2×C3semidirect
25. C2×#638C2×C3semidirect
26. C22×C3⋊Dic3C2×C3semidirect
30. #3757C2×C3semidirect
31. #3752C2×C3semidirect
32. #3750C2×C3semidirect
41. C22×H(3,𝔽3)D4semidirect
49. (C3⋊Dic3)⋊C3D4semidirect
124. C22×C32#67semidirect
125. C22×C32C3×D4semidirect
139. C3⋊Dic3C3×D4semidirect
144. H(3,𝔽3)#121semidirect
201. C2×D4(C3⋊D3)⋊C3semidirect
257. C32#939semidirect
258. C32C3×#121semidirect
301. C22#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D4×H(3,𝔽3)2
C2×#638⋊C32
C22×(C3⋊Dic3)⋊C32
#3757⋊C32
#3752⋊C32
#3750⋊C32
#641483
C3×#9393