Finite Group Databasebeta

← #82891

Subgroups — #82891

Order 384.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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×C3×#123C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic
3C2×#929C211yesmaximal, characteristic, complemented
4C22×C4×Dic3C211yesmaximal, characteristic, complemented
5#5220C211yesmaximal, characteristic, complemented
6#4990C212yesmaximal, complemented
7#4976C212yesmaximal, complemented
8#4958C211yesmaximal, characteristic, complemented
9#4957C212yesmaximal, complemented
10#4945C212yesmaximal, complemented
11#4942C211yesmaximal, characteristic, complemented
12#2854—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C3×#123—24no—
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2212yes—
26#939—24no—
27#929—24no—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2211yescharacteristic
34C2×C4×Dic3—24no—
35C2×C4×Dic3C2212yes—
36C2×C4×Dic3C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#123—31no—
41C2×#123D311yescharacteristic, complemented
42C2×#121—31no—
43C22×C42—31no—
44#390—32no—
45#389—32no—
46#388—32no—
47#388—32no—
48#386—31no—
49#386—31no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4D414yescomplemented
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2312yes—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—24no—
68C22×Dic3—22no—
69C22×Dic3—24no—
70C22×Dic3—24no—
71C22×Dic3D414yes—
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C3×C4⋊C4—24no—
78C3×C22⋊C4—22no—
79C3×C22⋊C4—24no—
80#217—22no—
81#217—22no—
82#217—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—24no—
86Dic3⋊C4—24no—
87Dic3⋊C4—22no—
88Dic3⋊C4—24no—
89C4×Dic3—24no—
90C4×Dic3—24no—
91C22×D4—31no—
92C22×D4—31no—
93C22×D4C2×D311yescharacteristic
94C23×C4—31no—
95C23×C4—31no—
96C23×C4C2×D311yescharacteristic
97#123—24no—
98#123—64no—
99#121—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4C2×D312yes—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4C2×D311yescharacteristic
111C2×C22⋊C4C2×D312yes—
112C2×C42—32no—
113C2×C42—32no—
114C2×C42—64no—
115#95—32no—
116#95—32no—
117#95—32no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—21no—
125C23×C3C2×D412yes—
126C23×C3C2×D412yes—
127C23×C3#4112yes—
128C23×C3C2411yescharacteristic
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—22no—
132C22×D3—21no—
133C22×D3—22no—
134C3×D4—44no—
135C2×C3×C4—28no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4C2×D414yes—
145#67—44no—
146#67—44no—
147C2×Dic3—28no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—28no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3C2×D412yes—
159C2×Dic3C2×D412yes—
160C2×Dic3#4114yes—
161C2×Dic3#4114yes—
162C24—31no—
163C24C22×D311yescharacteristic
164C24C22×D311yescharacteristic
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—62no—
169C2×D4—64no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—62no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—31no—
176C22×C4—34no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—32no—
180C22×C4—31no—
181C22×C4—64no—
182C22×C4—64no—
183C22×C4—62no—
184C22×C4#6714yescomplemented
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D312yes—
188C4⋊C4—24no—
189C4⋊C4—62no—
190C4⋊C4—64no—
191C4⋊C4—64no—
192C22⋊C4—22no—
193C22⋊C4—24no—
194C22⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—62no—
197C22⋊C4—64no—
198C22⋊C4—62no—
199C22⋊C4—64no—
200C42—64no—
201C42—64no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—24no—
205C22×C3—24no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3#12114yes—
212C22×C3C22×D411yescharacteristic
213C22×C3C22×D411yescharacteristic
214C22×C3C2×#4111yescharacteristic
215C22×C3C2×#4112yes—
216C22×C3C2×#4112yes—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—42no—
221C2×D3—42no—
222C3×C4—24no—
223C3×C4—41no—
224C3×C4—42no—
225Dic3—22no—
226Dic3—24no—
227Dic3—41no—
228Dic3—42no—
229Dic3#12314yescomplemented
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—61no—
238C23—61no—
239C23—62no—
240C23—62no—
241C23—61no—
242C23D3×D412yes—
243C23#23712yes—
244C23C2×#6712yes—
245C23C23×D311yescharacteristic
246D4—44no—
247D4—124no—
248D4—124no—
249C2×C4—28no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—32no—
259C2×C4—34no—
260C2×C4—32no—
261C2×C4—34no—
262C2×C4—68no—
263C2×C4—68no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4C2×#6714yes—
274C2×C3—24no—
275C2×C3—81no—
276C2×C3C2×#12111yescharacteristic
277C2×C3C2×#12311yescharacteristic
278C2×C3#52511yescharacteristic
279C2×C3#52912yes—
280C2×C3#53212yes—
281D3—81no—
282C22—24no—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—122no—
292C22—121no—
293C22—121no—
294C22—121no—
295C22—122no—
296C22#93914yes—
297C22C2×D3×D411yescharacteristic
298C22C2×#23711yescharacteristic
299C22D3×#4112yes—
300C22D3×#4112yes—
301C22C22×#6711yescharacteristic
302C4—24no—
303C4—34no—
304C4—41no—
305C4—42no—
306C4—62no—
307C4—64no—
308C4—121no—
309C4—122no—
310C3#285411yescharacteristic, complemented, Sylow
311C2—24no—
312C2—81no—
313C2—241no—
314C2#564912yes—
315C2D3×#12311yescharacteristic
316C2#565611yescharacteristic
317C2C2×#93911yescharacteristic
318C2#583212yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radicalC2×#939C2
hypercenterC2×#123D3
hypocenterC3#2854
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
3. C2×#929C2semidirect
4. C22×C4×Dic3C2semidirect
5. #5220C2semidirect
6. #4990C2semidirect
7. #4976C2semidirect
8. #4958C2semidirect
9. #4957C2semidirect
10. #4945C2semidirect
11. #4942C2semidirect
41. C2×#123D3semidirect
59. C22×C3×C4D4semidirect
184. C22×C4#67semidirect
229. Dic3#123semidirect
310. C3#2854semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9392
C2×#9292
C22×C4×Dic32
#52202
#49902
#49762
#49582
#49572
#49452
#49422
#28543