Finite Group Databasebeta

← #11721322

Subgroups — #11721322

Order 864.

383 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 64 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×#6235C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#6200C211yesmaximal, characteristic, complemented
5#88451C211yesmaximal, characteristic, complemented
6#88450C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8#63980—31nomaximal
9#63978—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C22×#272C2211yescharacteristic
12C22×(C3⋊D3)⋊C3C2211yescharacteristic
13#6235—22no—
14C2×#1084C2211yescharacteristic
15#638⋊C3—22no—
16C2×(C3⋊Dic3)⋊C3—22no—
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
20#6200—22no—
21C2×#638D311yescharacteristic, complemented
22C22×C3⋊Dic3D311yescharacteristic, complemented
23C2×C3×#67—31no—
24C2×C3×#67—31no—
25C2×C3×#67—31no—
26C2×C3×#67—61no—
27C22×C3×Dic3—31no—
28C2×#626—31no—
29C2×#625—31no—
30#3724—31no—
31Dic3⋊Dic3—31no—
32#3722—31no—
33(C2×D3)⋊Dic3—31no—
34(C2×D3)⋊Dic3—31no—
35(C2×D3)⋊Dic3—31no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—21no—
38C22×H(3,𝔽3)D412yescomplemented
39C22×H(3,𝔽3)C2311yescharacteristic
40C2×#272—21no—
41C2×#272—21no—
42C2×#272—21no—
43C2×(C3⋊D3)⋊C3—21no—
44C2×(C3⋊D3)⋊C3—21no—
45C2×(C3⋊D3)⋊C3—21no—
46#1084—21no—
47(C3⋊Dic3)⋊C3—21no—
48(C3⋊Dic3)⋊C3—21no—
49(C3⋊Dic3)⋊C3D412yescomplemented
50#939—91no—
51#939—91no—
52#930—91no—
53C23×C32—21no—
54C23×C32C2×D311yescharacteristic
55C23×C32C2×D311yescharacteristic
56C22×C3⋊D3C2×D311yescharacteristic
57C22×C3×D3—31no—
58C22×C3×D3—31no—
59C22×C3×D3—31no—
60C22×C3×D3—61no—
61#638—22no—
62C2×C3⋊Dic3—22no—
63C2×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊Dic3C2×D311yescharacteristic
66C3×#67—64no—
67C3×#67—62no—
68C3×#67—62no—
69C3×#67—62no—
70C2×C3×Dic3—31no—
71C2×C3×Dic3—31no—
72C2×C3×Dic3—31no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—62no—
77#626—62no—
78#625—62no—
79C2×H(3,𝔽3)—22no—
80C2×H(3,𝔽3)C2×D411yescharacteristic
81C2×H(3,𝔽3)C2×D411yescharacteristic
82C2×H(3,𝔽3)#4111yescharacteristic
83#272—41no—
84(C3⋊D3)⋊C3—41no—
85C2×C3×D4—91no—
86C2×C3×D4—91no—
87C22×C3×C4—91no—
88C2×#67—31no—
89C2×#67—31no—
90C2×#67—31no—
91C2×#67—61no—
92C2×#67—91no—
93C2×#67—91no—
94C2×#67—91no—
95C22×Dic3—31no—
96C22×Dic3—91no—
97C2×D12—91no—
98#217—91no—
99#217—91no—
100(C2×D3)⋊C4—91no—
101(C2×D3)⋊C4—91no—
102(C2×D3)⋊C4—91no—
103(C2×D3)⋊C4—91no—
104C4⋊Dic3—91no—
105Dic3⋊C4—91no—
106Dic3⋊C4—91no—
107C22×C32—21no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—22no—
112C22×C32—21no—
113C22×C32—41no—
114C22×C32—41no—
115C22×C32D1212yescomplemented
116C22×C32#6712yescomplemented
117C22×C32C22×D311yescharacteristic
118C22×C32C22×D311yescharacteristic
119C2×C3⋊D3—21no—
120C2×C3⋊D3—21no—
121C2×C3⋊D3—21no—
122C2×C3×D3—62no—
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—61no—
131C2×C3×D3—61no—
132C2×C3×D3—121no—
133C2×C3×D3—121no—
134C3⋊Dic3—21no—
135C3⋊Dic3—21no—
136C3⋊Dic3#6712yescomplemented
137C3×Dic3—32no—
138C3×Dic3—62no—
139C3×Dic3—61no—
140C3×Dic3—61no—
141C3×Dic3—61no—
142C3×Dic3—61no—
143#121—271noSylow
144H(3,𝔽3)#12111yescharacteristic, complemented, Sylow
145C23×C3—31no—
146C23×C3—31no—
147C23×C3—61no—
148C23×C3—91no—
149C23×C3—91no—
150C23×C3D3211yescharacteristic
151C22×D3—31no—
152C22×D3—31no—
153C22×D3—31no—
154C22×D3—61no—
155C22×D3—91no—
156C3×D4—182no—
157C3×D4—182no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—182no—
163#67—62no—
164#67—62no—
165#67—64no—
166#67—62no—
167#67—182no—
168#67—182no—
169#67—182no—
170C2×Dic3—31no—
171C2×Dic3—31no—
172C2×Dic3—31no—
173C2×Dic3—31no—
174C2×Dic3—31no—
175C2×Dic3—61no—
176C2×Dic3—62no—
177C2×Dic3—91no—
178C2×Dic3—91no—
179C2×Dic3—91no—
180C2×Dic3—182no—
181D12—182no—
182C2×C32—22no—
183C2×C32—22no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—42no—
188C2×C32C2×D1211yescharacteristic
189C2×C32#23511yescharacteristic
190C2×C32D3×D411yescharacteristic
191C2×C32#23711yescharacteristic
192C2×C32C2×#6711yescharacteristic
193C2×C32C2×#6711yescharacteristic
194C3⋊D3—41no—
195C3×D3—122no—
196C3×D3—121no—
197C3×D3—121no—
198C3×D3—121no—
199C2×D4—91no—
200C2×D4—91no—
201C2×D4—271no—
202C22×C4—91no—
203C4⋊C4—271no—
204C22⋊C4—271no—
205C22⋊C4—271no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—32no—
209C22×C3—32no—
210C22×C3—31no—
211C22×C3—31no—
212C22×C3—61no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—61no—
216C22×C3—62no—
217C22×C3—61no—
218C22×C3—121no—
219C22×C3—121no—
220C22×C3—181no—
221C22×C3—181no—
222C22×C3—181no—
223C22×C3—181no—
224C22×C3—181no—
225C22×C3—181no—
226C22×C3#62612yes—
227C22×C3C2×D3211yescharacteristic
228C2×D3—61no—
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—61no—
234C2×D3—62no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—121no—
239C2×D3—121no—
240C2×D3—181no—
241C2×D3—181no—
242C2×D3—181no—
243C3×C4—92no—
244C3×C4—181no—
245C3×C4—181no—
246C3×C4—181no—
247Dic3—32no—
248Dic3—61no—
249Dic3—61no—
250Dic3—62no—
251Dic3—61no—
252Dic3—61no—
253Dic3—92no—
254Dic3—181no—
255Dic3—181no—
256C32—21no—
257C32#93011yescharacteristic, complemented
258C32#93911yescharacteristic, complemented
259C23—91no—
260C23—91no—
261C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
262D4—182no—
263D4—182no—
264D4—542no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—182no—
270C2×C3—22no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—62no—
278C2×C3—62no—
279C2×C3—61no—
280C2×C3—61no—
281C2×C3—61no—
282C2×C3—122no—
283C2×C3—361no—
284C2×C3—361no—
285C2×C3#380411yescharacteristic
286C2×C3C2×#62611yescharacteristic
287C2×C3D3×#6711yescharacteristic
288D3—121no—
289D3—121no—
290D3—122no—
291D3—121no—
292D3—361no—
293C22—21no—
294C22—21no—
295C22—181no—
296C22—181no—
297C22—181no—
298C22—181no—
299C22—181no—
300C22—181no—
301C22#620012yescomplemented
302C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C4—92no—
304C4—181no—
305C4—181no—
306C4—181no—
307C3—31no—
308C3—31no—
309C3—61no—
310C3#6397811yescharacteristic
311C2—22no—
312C2—361no—
313C2—361no—
314C2#8867111yescharacteristic
315C2C2×#620011yescharacteristic
316C2#8867611yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#6235C2semidirect
2. C2×#638⋊C3C2semidirect
3. C22×(C3⋊Dic3)⋊C3C2semidirect
4. C2×#6200C2semidirect
5. #88451C2semidirect
6. #88450C2semidirect
7. #88448C2semidirect
21. C2×#638D3semidirect
22. C22×C3⋊Dic3D3semidirect
38. C22×H(3,𝔽3)D4semidirect
49. (C3⋊Dic3)⋊C3D4semidirect
115. C22×C32D12semidirect
116. C22×C32#67semidirect
136. C3⋊Dic3#67semidirect
144. H(3,𝔽3)#121semidirect
257. C32#930semidirect
258. C32#939semidirect
301. C22#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#62352
C2×#638⋊C32
C22×(C3⋊Dic3)⋊C32
C2×#62002
#884512
#884502
#884482
#639803
#639783