Finite Group Databasebeta

← #11721157

Subgroups — #11721157

Order 864.

383 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 56 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×C4⋊#272C211yesmaximal, characteristic, complemented
2C2×#6200C211yesmaximal, characteristic, complemented
3C2×#6200C211yesmaximal, characteristic, complemented
4C2×#6199C211yesmaximal, characteristic, complemented
5#3752⋊C3C211yesmaximal, characteristic, complemented
6#3750⋊C3C211yesmaximal, characteristic, complemented
7#88451C211yesmaximal, characteristic, complemented
8#63886—31nomaximal
9#63885—31nomaximal
10C22×#272C2211yescharacteristic
11C22×#272C2211yescharacteristic
12C22×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#272—22no—
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6200—22no—
19#6200—22no—
20#6199—22no—
21C2×C3×D12—31no—
22C2×C3×D12—31no—
23C2×C3×D12—61no—
24C2×#626—31no—
25C2×#626—31no—
26C2×#625—31no—
27C2×#625—31no—
28C2×D3×Dic3—31no—
29C2×D3×Dic3—31no—
30#3752D311yescharacteristic, complemented
31#3750D311yescharacteristic, complemented
32C3×(C2×D3)⋊C4—31no—
33C3×Dic3⋊C4—31no—
34(C2×D3)⋊Dic3—31no—
35(C2×D3)⋊Dic3—31no—
36C22×H(3,𝔽3)C2311yescharacteristic
37C2×#272—21no—
38C2×#272—21no—
39C2×#272—21no—
40C2×#272—21no—
41C2×#272—21no—
42C2×#272D412yescomplemented
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
50#939—91no—
51#895—91no—
52#884—91no—
53C22×C3⋊D3C2×D311yescharacteristic
54C22×C3×D3—31no—
55C22×C3×D3—31no—
56C22×C3×D3—31no—
57C22×C3×D3—31no—
58C22×C3×D3—31no—
59C22×C3×D3—61no—
60C22×C3×D3—61no—
61C2×C32×C4—21no—
62C2×C32×C4C2×D311yescharacteristic
63C2×C32×C4C2×D311yescharacteristic
64C2×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊Dic3C2×D311yescharacteristic
67C2×C3×Dic3—31no—
68C2×C3×Dic3—31no—
69C2×C3×Dic3—31no—
70C3×D12—62no—
71C3×D12—62no—
72C3×D12—64no—
73#626—62no—
74#626—62no—
75#625—62no—
76#625—62no—
77D3×Dic3—62no—
78D3×Dic3—62no—
79C2×H(3,𝔽3)C2×D411yescharacteristic
80C2×H(3,𝔽3)C2×D411yescharacteristic
81C2×H(3,𝔽3)#4111yescharacteristic
82#272—22no—
83#272—41no—
84(C3⋊D3)⋊C3—41no—
85C2×C3×D4—91no—
86C2×#67—91no—
87C2×#67—91no—
88C2×#67—91no—
89C2×#67—91no—
90C22×Dic3—91no—
91C2×D12—31no—
92C2×D12—31no—
93C2×D12—61no—
94C2×D12—91no—
95C2×D12—91no—
96C2×C4×D3—91no—
97C2×C4×D3—91no—
98C3×C4⋊C4—91no—
99C3×C22⋊C4—91no—
100#217—91no—
101(C2×D3)⋊C4—31no—
102(C2×D3)⋊C4—91no—
103(C2×D3)⋊C4—91no—
104(C2×D3)⋊C4—181no—
105Dic3⋊C4—31no—
106Dic3⋊C4—91no—
107C22×C32—21no—
108C22×C32C22×D311yescharacteristic
109C22×C32C22×D311yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C2×C3×D3—32no—
114C2×C3×D3—32no—
115C2×C3×D3—62no—
116C2×C3×D3—61no—
117C2×C3×D3—61no—
118C2×C3×D3—61no—
119C2×C3×D3—61no—
120C2×C3×D3—62no—
121C2×C3×D3—61no—
122C2×C3×D3—61no—
123C2×C3×D3—61no—
124C2×C3×D3—61no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—61no—
129C2×C3×D3—61no—
130C2×C3×D3—121no—
131C2×C3×D3—121no—
132C2×C3×D3—121no—
133C2×C3×D3—121no—
134C32×C4—21no—
135C32×C4—22no—
136C32×C4—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3—21no—
139C3⋊Dic3D1212yescomplemented
140C3×Dic3—32no—
141C3×Dic3—61no—
142C3×Dic3—61no—
143#121—271noSylow
144H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C23×C3—91no—
147C23×C3—91no—
148C22×D3—31no—
149C22×D3—31no—
150C22×D3—31no—
151C22×D3—31no—
152C22×D3—31no—
153C22×D3—61no—
154C22×D3—61no—
155C22×D3—181no—
156C3×D4—182no—
157C2×C3×C4—31no—
158C2×C3×C4—61no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—121no—
163C2×C3×C4D3211yescharacteristic
164#67—182no—
165#67—182no—
166#67—182no—
167#67—182no—
168C2×Dic3—31no—
169C2×Dic3—31no—
170C2×Dic3—31no—
171C2×Dic3—91no—
172C2×Dic3—91no—
173C2×Dic3—181no—
174C2×Dic3—182no—
175D12—62no—
176D12—62no—
177D12—64no—
178D12—182no—
179D12—182no—
180C4×D3—182no—
181C4×D3—182no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32C2×D1211yescharacteristic
186C2×C32#23511yescharacteristic
187C2×C32D3×D411yescharacteristic
188C2×C32D3×D411yescharacteristic
189C2×C32D3×D411yescharacteristic
190C2×C32#23911yescharacteristic
191C3⋊D3—41no—
192C3×D3—62no—
193C3×D3—62no—
194C3×D3—122no—
195C3×D3—122no—
196C3×D3—121no—
197C3×D3—121no—
198C3×D3—121no—
199C2×D4—91no—
200C2×D4—271no—
201C2×D4—271no—
202C22×C4—271no—
203C4⋊C4—91no—
204C22⋊C4—91no—
205C22⋊C4—271no—
206C22×C3—31no—
207C22×C3—61no—
208C22×C3—92no—
209C22×C3—121no—
210C22×C3—181no—
211C22×C3—181no—
212C22×C3—181no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3—181no—
216C22×C3—181no—
217C22×C3—181no—
218C22×C3C2×D3211yescharacteristic
219C2×D3—32no—
220C2×D3—32no—
221C2×D3—62no—
222C2×D3—61no—
223C2×D3—61no—
224C2×D3—61no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—61no—
228C2×D3—61no—
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—62no—
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—121no—
237C2×D3—121no—
238C2×D3—121no—
239C2×D3—121no—
240C2×D3—181no—
241C2×D3—361no—
242C2×D3—361no—
243C3×C4—21no—
244C3×C4—61no—
245C3×C4—92no—
246C3×C4—121no—
247C3×C4—122no—
248C3×C4—181no—
249C3×C4—181no—
250Dic3—32no—
251Dic3—61no—
252Dic3—61no—
253Dic3—92no—
254Dic3—181no—
255Dic3—361no—
256C32—21no—
257C32#88411yescharacteristic, complemented
258C32#89511yescharacteristic, complemented
259C23—91no—
260C23—91no—
261C23—91no—
262D4—182no—
263D4—542no—
264D4—542no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—542no—
269C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—61no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—121no—
277C2×C3—121no—
278C2×C3—121no—
279C2×C3—182no—
280C2×C3—361no—
281C2×C3—361no—
282C2×C3#379911yescharacteristic
283C2×C3D3×D1211yescharacteristic
284C2×C3#381111yescharacteristic
285D3—62no—
286D3—62no—
287D3—122no—
288D3—121no—
289D3—121no—
290D3—122no—
291D3—121no—
292D3—361no—
293C22—92no—
294C22—181no—
295C22—181no—
296C22—181no—
297C22—181no—
298C22—181no—
299C22—181no—
300C22—181no—
301C22—181no—
302C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C4—21no—
304C4—92no—
305C4—181no—
306C4—181no—
307C3—31no—
308C3—61no—
309C3—121no—
310C3#6391011yescharacteristic
311C2—182no—
312C2—361no—
313C2—361no—
314C2#8865211yescharacteristic
315C2#8865411yescharacteristic
316C2#8867711yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#272C2semidirect
2. C2×#6200C2semidirect
3. C2×#6200C2semidirect
4. C2×#6199C2semidirect
5. #3752⋊C3C2semidirect
6. #3750⋊C3C2semidirect
7. #88451C2semidirect
30. #3752D3semidirect
31. #3750D3semidirect
42. C2×#272D4semidirect
49. (C3⋊Dic3)⋊C3D4semidirect
139. C3⋊Dic3D12semidirect
144. H(3,𝔽3)#121semidirect
257. C32#884semidirect
258. C32#895semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#2722
C2×#62002
C2×#61992
#3752⋊C32
#3750⋊C32
#884512
#638863
#638853