Finite Group Databasebeta

← #74872

Subgroups — #74872

Order 384.

688 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 165 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×#939C211yesmaximal, characteristic
2C2×#935C211yesmaximal, characteristic, complemented
3C22×C4×Dic3C211yesmaximal, characteristic, complemented
4C2×#886C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4957C212yesmaximal, complemented
8#4951C212yesmaximal, complemented
9#4942C211yesmaximal, characteristic, complemented
10#4669C212yesmaximal, complemented
11#4647C212yesmaximal, complemented
12#2854—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24#939—24no—
25#935—24no—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×C4⋊Dic3C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×C4×Dic3—24no—
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2212yes—
36#886—24no—
37C3×#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#123—31no—
41C2×#123—31no—
42C2×#121—31no—
43C22×C42—31no—
44#390—32no—
45#389—32no—
46#388—32no—
47#388—32no—
48#386—31no—
49#386D311yescharacteristic, complemented
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—24no—
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C2×#67—24no—
67C22×Dic3—22no—
68C22×Dic3—22no—
69C22×Dic3—24no—
70C22×Dic3D414yes—
71C22×Dic3D414yescomplemented
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C3×C22⋊C4—22no—
78C3×C22⋊C4—24no—
79#217—22no—
80#217—24no—
81#217—22no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—22no—
85C4⋊Dic3—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—64no—
98#123—64no—
99#121—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—32no—
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#95C2×D312yes—
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—21no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4#4114yes—
145#67—44no—
146#67—44no—
147C2×Dic3—28no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—28no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3C2×D412yes—
159C2×Dic3C2×D414yes—
160C2×Dic3C2×D412yes—
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—24no—
175C22×C4—31no—
176C22×C4—34no—
177C22×C4—34no—
178C22×C4—31no—
179C22×C4—32no—
180C22×C4—32no—
181C22×C4—31no—
182C22×C4—64no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4C22×D312yes—
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D312yes—
188C4⋊C4—62no—
189C4⋊C4—64no—
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—24no—
226Dic3—22no—
227Dic3—41no—
228Dic3—42no—
229Dic3#12314yescomplemented
230C23—22no—
231C23—22no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—61no—
238C23—61no—
239C23—61no—
240C23—62no—
241C23—62no—
242C23#23512yes—
243C23D3×D412yes—
244C23D3×D412yes—
245C23C23×D311yescharacteristic
246D4—44no—
247D4—124no—
248D4—124no—
249C2×C4—28no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
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—62no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4#23714yes—
274C2×C3—24no—
275C2×C3—81no—
276C2×C3C2×#12111yescharacteristic
277C2×C3C2×#12311yescharacteristic
278C2×C3#52511yescharacteristic
279C2×C3#52912yes—
280C2×C3#53212yes—
281D3—81no—
282C22—22no—
283C22—24no—
284C22—22no—
285C22—24no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22—121no—
292C22—121no—
293C22—122no—
294C22—121no—
295C22—122no—
296C22#88414yes—
297C22C2×#23511yescharacteristic
298C22C2×D3×D411yescharacteristic
299C22C2×D3×D411yescharacteristic
300C22C2×#23712yes—
301C22D3×#4112yes—
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—
314C2C2×#88411yescharacteristic
315C2#553812yes—
316C2#558611yescharacteristic
317C2#559612yes—
318C2D3×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC2×#939C2
hypercenter#386D3
hypocenterC3#2854
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. C22×C4×Dic3C2semidirect
4. C2×#886C2semidirect
5. C3×#386C2semidirect
6. #5213C2semidirect
7. #4957C2semidirect
8. #4951C2semidirect
9. #4942C2semidirect
10. #4669C2semidirect
11. #4647C2semidirect
49. #386D3semidirect
71. C22×Dic3D4semidirect
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×#9392
C2×#9352
C22×C4×Dic32
C2×#8862
C3×#3862
#52132
#49572
#49512
#49422
#46692
#46472
#28543