Finite Group Databasebeta

← #82170

Subgroups — #82170

Order 384.

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

normal subgroups only

This group has 610 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
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic
3C2×#929C211yesmaximal, characteristic
4C2×#926C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic
6#5211C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8#4994C212yesmaximal, complemented
9#4990C212yesmaximal, complemented
10#4982C212yesmaximal, complemented
11#4976C212yesmaximal, complemented
12#2413—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C2×C3×C4⋊C4—24no—
18C2×C3×C4⋊C4C2212yes—
19C2×C3×C4⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23#929—24no—
24#929—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29#926—24no—
30C2×C4⋊Dic3C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2211yescharacteristic
35C2×C4×Dic3C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#123—31no—
41C2×#123—31no—
42C2×#117—31no—
43C22×C4⋊C4D311yescharacteristic, complemented
44#398—32no—
45#396—32no—
46#389—32no—
47#388—32no—
48#386—31no—
49#385—31no—
50#381—31no—
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—22no—
56C22×C3×C4—24no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4D414yes—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2312yes—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—22no—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C3×C4⋊C4—24no—
74C3×C4⋊C4—24no—
75#217—22no—
76#217—22no—
77#217D414yes—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82C4⋊Dic3—24no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85Dic3⋊C4—24no—
86Dic3⋊C4—24no—
87C4×Dic3—24no—
88C22×D4—31no—
89C23×C4C2×D311yescharacteristic
90C23×C4C2×D311yescharacteristic
91C23×C4C2×D311yescharacteristic
92#123—64no—
93#123—64no—
94#117—64no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4C2×D312yes—
102C2×C4⋊C4C2×D312yes—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C42—32no—
111#95—32no—
112#95—32no—
113#95—32no—
114#95—32no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3C2×D412yes—
118C23×C3C2×D412yes—
119C23×C3#4112yes—
120C23×C3C2411yescharacteristic
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—21no—
125C22×D3—22no—
126C2×C3×C4—28no—
127C2×C3×C4—28no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—44no—
138C2×C3×C4C2×D412yes—
139C2×C3×C4C2×D412yes—
140C2×C3×C4#4114yes—
141#67—44no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3C2×D414yes—
155C24—31no—
156C24C22×D311yescharacteristic
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—62no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—32no—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4#6714yes—
173C22×C4C22×D312yes—
174C22×C4C22×D312yes—
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—64no—
180C4⋊C4—64no—
181C4⋊C4—64no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C22⋊C4—34no—
185C22⋊C4—64no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—64no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C42—64no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3#12114yes—
197C22×C3C22×D411yescharacteristic
198C22×C3C22×D411yescharacteristic
199C22×C3C2×#4112yes—
200C22×C3C2×#4111yescharacteristic
201C22×C32−1+412yes—
202C2×D3—41no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—42no—
206C2×D3—42no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—24no—
210C3×C4—42no—
211Dic3—24no—
212Dic3—41no—
213Dic3—41no—
214Dic3—41no—
215Dic3—42no—
216C23—22no—
217C23—22no—
218C23—62no—
219C23—61no—
220C23—61no—
221C23—62no—
222C23—61no—
223C23D3×D412yes—
224C23#23712yes—
225C23C2×#6712yes—
226C23C23×D311yescharacteristic
227D4—124no—
228C2×C4—28no—
229C2×C4—28no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—34no—
240C2×C4—44no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4#23514yes—
254C2×C4C2×#6712yes—
255C2×C4C2×#6712yes—
256C2×C3—24no—
257C2×C3C2×#12111yescharacteristic
258C2×C3#52511yescharacteristic
259C2×C3#53611yescharacteristic
260C2×C3#54712yes—
261C2×C3#55212yes—
262D3—81no—
263C22—24no—
264C22—24no—
265C22—22no—
266C22—22no—
267C22—122no—
268C22—122no—
269C22—121no—
270C22—121no—
271C22—121no—
272C22#93914yes—
273C22C2×#23512yes—
274C22C2×D3×D411yescharacteristic
275C22C2×#23711yescharacteristic
276C22#100712yes—
277C22C22×#6711yescharacteristic
278C4—24no—
279C4—22no—
280C4—22no—
281C4—42no—
282C4—64no—
283C4—122no—
284C4—121no—
285C4—121no—
286C4—121no—
287C3#241311yescharacteristic, complemented, Sylow
288C2—24no—
289C2—241no—
290C2#550712yes—
291C2#551312yes—
292C2#579111yescharacteristic
293C2C2×#93911yescharacteristic
294C2#581611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC22×#67C22
hypercenterC22×C4⋊C4D3
hypocenterC3#2413
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
4. C2×#926C2semidirect
6. #5211C2semidirect
7. #5210C2semidirect
8. #4994C2semidirect
9. #4990C2semidirect
10. #4982C2semidirect
11. #4976C2semidirect
43. C22×C4⋊C4D3semidirect
287. C3#2413semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×#9292
C2×#9262
#52132
#52112
#52102
#49942
#49902
#49822
#49762
#24133