Finite Group Databasebeta

← #80337

Subgroups — #80337

Order 384.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#5716C211yesmaximal, characteristic, complemented
2#5557C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5158C212yesmaximal, complemented
5#5155C212yesmaximal, complemented
6#5075C211yesmaximal, characteristic, complemented
7#5019C211yesmaximal, characteristic, complemented
8#4849C211yesmaximal, characteristic, complemented
9#4833C211yesmaximal, characteristic, complemented
10#4783C212yesmaximal, complemented
11#4768C212yesmaximal, complemented
12#3212—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#938—21no—
18#938—21no—
19#938C2211yescharacteristic
20#936C2211yescharacteristic
21#936C2212yes—
22D4×Dic3—21no—
23D4×Dic3—21no—
24C2×#214C2212yes—
25C2×#214C2212yes—
26C4×#67—21no—
27#926—21no—
28C2×Dic12C2212yes—
29#897C2211yescharacteristic
30#890C2211yescharacteristic
31#883—21no—
32#879—21no—
33C42×D3C2211yescharacteristic
34#869C2211yescharacteristic
35C4×Dic6C2211yescharacteristic
36C3×C4⋊C8C2211yescharacteristic
37C3×#102C2212yes—
38#832C2212yes—
39(C2×D3)⋊C8C2212yes—
40#816C2212yes—
41C8×Dic3C2212yes—
42#810C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#529—31no—
46#495—32no—
47#488—32no—
48#462—31no—
49#461—31no—
50#460—31no—
51#459D311yescharacteristic, complemented
52#452—32no—
53#449—32no—
54#431—31no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C2×#67—21no—
61C2×#67—22no—
62C22×Dic3—22no—
63#237—42no—
64#237—42no—
65C2×C4×D3D411yescharacteristic
66C2×C4×D3D411yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×Dic6C2312yes—
69C2×Dic6C2311yescharacteristic
70C3×D8—42no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74C3×C42C2311yescharacteristic
75#217—21no—
76#217—21no—
77#217—21no—
78#217—22no—
79#217—22no—
80#214—42no—
81#214—42no—
82(C2×D3)⋊C4—21no—
83C4⋊Dic3—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3C2311yescharacteristic
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90C4×Dic3—21no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95Dic12—42no—
96C2×#41—32no—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16—32no—
100C2×D8C2×D312yes—
101C4⋊Q8—31no—
102#127C2×D311yescharacteristic
103#126—61no—
104#125—31no—
105#124—31no—
106#124—32no—
107#123—61no—
108C22⋊Q8—31no—
109#121—31no—
110#121—61no—
111#121—61no—
112C4×Q8—31no—
113C4×D4—61no—
114C4×D4—61no—
115C4×D4—61no—
116C4×D4C2×D311yescharacteristic
117#117—61no—
118C2×C42—31no—
119C4⋊C8—31no—
120C4⋊C8C2×D311yescharacteristic
121#103—32no—
122#103—32no—
123#102—32no—
124#102C2×D312yes—
125C22⋊C8—32no—
126C4×C8—32no—
127C23×C3—21no—
128C23×C3—22no—
129C22×D3C2×D411yescharacteristic
130C3×D4—21no—
131C3×D4—22no—
132C3×D4—21no—
133C3×D4—22no—
134C3×D4—42no—
135C3×D4—42no—
136C2×C3×C4—21no—
137C2×C3×C4—42no—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141#67—44no—
142#67—42no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—44no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C4×D3—22no—
151C4×D3—21no—
152C4×D3—21no—
153C4×D3—41no—
154Dic6—21no—
155Dic6—22no—
156Dic6—21no—
157Dic6—22no—
158C3×C8—22no—
159C3⋊C8—22no—
160#41—122no—
161#41—122no—
162C2×Q8—32no—
163C2×Q8—31no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—61no—
167C2×D4—62no—
168C2×D4C22×D312yes—
169C2×D4C22×D311yescharacteristic
170C22×C4—21no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—62no—
175Q16—122no—
176SD16—122no—
177SD16—122no—
178D8—42no—
179C2×C8—32no—
180C2×C8C22×D312yes—
181C4⋊C4—31no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4C22×D311yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—61no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—61no—
196C42—31no—
197C42—31no—
198C42—31no—
199C42—61no—
200C42C22×D311yescharacteristic
201C22×C3—42no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—42no—
205C22×C3—41no—
206C22×C3—42no—
207C22×C3C22×D411yescharacteristic
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—21no—
211C3×C4—21no—
212C3×C4—41no—
213C3×C4#13512yes—
214C3×C4H(5,𝔽2)11yescharacteristic
215C3×C4H(5,𝔽2)11yescharacteristic
216Dic3—21no—
217Dic3—21no—
218Dic3—21no—
219Dic3—41no—
220Dic3—42no—
221Dic3—41no—
222C23—21no—
223C23—22no—
224C23—31no—
225Q8—61no—
226Q8—62no—
227Q8—61no—
228Q8—62no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—42no—
235D4—124no—
236D4—122no—
237C2×C4—21no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—42no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—62no—
248C2×C4—121no—
249C2×C4—124no—
250C2×C4D3×D411yescharacteristic
251C2×C4D3×D411yescharacteristic
252C2×C4C23×D311yescharacteristic
253C8—22no—
254C8—62no—
255C2×C3—82no—
256C2×C3—81no—
257C2×C3#53511yescharacteristic
258C2×C3C2×#13511yescharacteristic
259C2×C3#57511yescharacteristic
260D3—41no—
261C22—41no—
262C22—42no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22C2×D3×D411yescharacteristic
271C4—21no—
272C4—41no—
273C4—61no—
274C4—61no—
275C4—61no—
276C4—121no—
277C4—122no—
278C4—121no—
279C4#91212yes—
280C4#100411yescharacteristic
281C4#100911yescharacteristic
282C3#321211yescharacteristic, complemented, Sylow
283C2—81no—
284C2—82no—
285C2—121no—
286C2#560711yescharacteristic
287C2C2×#91211yescharacteristic
288C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3212
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. #5557C2semidirect
3. C3×#459C2semidirect
4. #5158C2semidirect
5. #5155C2semidirect
6. #5075C2semidirect
7. #5019C2semidirect
8. #4849C2semidirect
9. #4833C2semidirect
10. #4783C2semidirect
11. #4768C2semidirect
51. #459D3semidirect
282. C3#3212semidirect

Maximal subgroups

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

subgroupindex
#57162
#55572
C3×#4592
#51582
#51552
#50752
#50192
#48492
#48332
#47832
#47682
#32123