Finite Group Databasebeta

← #80345

Subgroups — #80345

Order 384.

430 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 115 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#5548C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C212yesmaximal, complemented
5#5150C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5008C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4782C212yesmaximal, complemented
11#4761C212yesmaximal, complemented
12#3253—31nomaximal, Sylow
13C2×#237—21no—
14C2×#235—21no—
15C2×C3×D8C2212yes—
16C3×#127C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#940C2211yescharacteristic
19#938—21no—
20#938C2211yescharacteristic
21#938C2212yes—
22#936—21no—
23D4×Dic3—21no—
24D4×Dic3C2212yes—
25C2×#213C2212yes—
26C4×#67—21no—
27C2×ΓL(1,𝔽25)C2212yes—
28#895C2211yescharacteristic
29#888C2211yescharacteristic
30#885—21no—
31#884—21no—
32C4×D12C2211yescharacteristic
33C42×D3C2211yescharacteristic
34#869C2211yescharacteristic
35C3×C4⋊C8C2211yescharacteristic
36C3×#102C2212yes—
37#832C2212yes—
38#832C2212yes—
39#821C2212yes—
40#817C2212yes—
41Dic3⋊C8C2212yes—
42#808C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#532—31no—
46#476—31no—
47#476—31no—
48#476—32no—
49#469—32no—
50#469—32no—
51#459—32no—
52#459—31no—
53#459D311yescharacteristic, complemented
54#423—31no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2311yescharacteristic
58C2×C3×D4C2312yes—
59C22×C3×C4—21no—
60C2×#67—21no—
61C2×#67—22no—
62C2×#67—21no—
63C2×#67—21no—
64C22×Dic3—22no—
65#237—44no—
66#235—44no—
67C2×D12—21no—
68C2×D12C2311yescharacteristic
69C2×C4×D3—21no—
70C2×C4×D3D411yescharacteristic
71C2×C4×D3D411yescharacteristic
72C2×C4×D3C2311yescharacteristic
73C2×Dic6—21no—
74C2×Dic6—21no—
75C3×D8—42no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4C2311yescharacteristic
78C3×C22⋊C4—21no—
79C3×C42C2311yescharacteristic
80#217—21no—
81#217—22no—
82#217—22no—
83#213—42no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3C2312yes—
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90C4×Dic3D411yescharacteristic
91C4×Dic3D411yescharacteristic
92C4×Dic3C2311yescharacteristic
93C2×C3⋊C8C2312yes—
94ΓL(1,𝔽25)—42no—
95C2×#41—61no—
96C2×#41—61no—
97C2×D8—32no—
98C2×D8C2×D312yes—
99C2×M16—32no—
100C4⋊Q8—31no—
101C4⋊Q8—31no—
102#127—31no—
103#127C2×D311yescharacteristic
104#124—61no—
105#124—61no—
106#121—31no—
107#121—32no—
108#121—31no—
109#121—61no—
110#121—61no—
111C4×D4—32no—
112C4×D4—31no—
113C4×D4—61no—
114C4×D4—61no—
115C4×D4C2×D311yescharacteristic
116C2×C42—31no—
117#106—32no—
118#106—32no—
119C4⋊C8—32no—
120C4⋊C8—31no—
121C4⋊C8C2×D311yescharacteristic
122#102—32no—
123#102—32no—
124#102—32no—
125#102C2×D312yes—
126C23×C3—22no—
127C23×C3—21no—
128C22×D3—21no—
129C22×D3C2×D411yescharacteristic
130C3×D4—22no—
131C3×D4—21no—
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—42no—
142#67—42no—
143#67—44no—
144#67—42no—
145C2×Dic3—21no—
146C2×Dic3—22no—
147C2×Dic3—44no—
148C2×Dic3C2×D411yescharacteristic
149C2×Dic3C2×D411yescharacteristic
150C2×Dic3C2×D411yescharacteristic
151D12—21no—
152D12—21no—
153D12—42no—
154C4×D3—22no—
155C4×D3—21no—
156C4×D3—21no—
157C4×D3—22no—
158C4×D3—42no—
159Dic6—42no—
160Dic6—42no—
161C3×C8—22no—
162C3⋊C8—22no—
163#41—124no—
164#41—124no—
165C2×Q8—61no—
166C2×Q8—61no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—31no—
170C2×D4—61no—
171C2×D4—61no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—62no—
175C2×D4C22×D311yescharacteristic
176C2×D4C22×D312yes—
177C22×C4—21no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—62no—
182C22×C4—61no—
183D8—42no—
184D8—122no—
185M16—122no—
186C2×C8—32no—
187C2×C8C22×D312yes—
188C4⋊C4—31no—
189C4⋊C4—32no—
190C4⋊C4—61no—
191C4⋊C4—61no—
192C4⋊C4C22×D311yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—61no—
195C22⋊C4—62no—
196C22⋊C4—61no—
197C22⋊C4—62no—
198C22⋊C4—61no—
199C42—31no—
200C42—31no—
201C42—31no—
202C42C22×D311yescharacteristic
203C22×C3—42no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—42no—
209C22×C3C22×D411yescharacteristic
210C2×D3—21no—
211C2×D3—21no—
212C2×D3—21no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—41no—
216C3×C4—21no—
217C3×C4—41no—
218C3×C4AGL(1,ℤ/8)12yes—
219C3×C4H(5,𝔽2)11yescharacteristic
220C3×C42−1+411yescharacteristic
221Dic3—21no—
222Dic3—21no—
223Dic3—41no—
224Dic3—42no—
225Dic3AGL(1,ℤ/8)12yes—
226C23—21no—
227C23—22no—
228C23—31no—
229C23—61no—
230Q8—122no—
231Q8—122no—
232D4—21no—
233D4—22no—
234D4—21no—
235D4—22no—
236D4—42no—
237D4—42no—
238D4—61no—
239D4—61no—
240D4—122no—
241D4—122no—
242D4—122no—
243D4—122no—
244D4—124no—
245C2×C4—21no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—42no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—124no—
257C2×C4—122no—
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—22no—
262C8—62no—
263C2×C3—82no—
264C2×C3—81no—
265C2×C3#53711yescharacteristic
266C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
267C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
268D3—41no—
269D3—81no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—61no—
277C22—61no—
278C22—61no—
279C22—121no—
280C22—121no—
281C22—121no—
282C22C2×D3×D411yescharacteristic
283C4—21no—
284C4—32no—
285C4—41no—
286C4—61no—
287C4—61no—
288C4—121no—
289C4—122no—
290C4#91112yes—
291C4#100411yescharacteristic
292C4#101011yescharacteristic
293C3#325311yescharacteristic, complemented, Sylow
294C2—81no—
295C2—82no—
296C2—121no—
297C2—241no—
298C2#560811yescharacteristic
299C2C2×#91111yescharacteristic
300C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3253
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. #5548C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5150C2semidirect
6. #5072C2semidirect
7. #5008C2semidirect
8. #4842C2semidirect
9. #4837C2semidirect
10. #4782C2semidirect
11. #4761C2semidirect
53. #459D3semidirect
293. C3#3253semidirect

Maximal subgroups

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

subgroupindex
#57162
#55482
C3×#4592
#51592
#51502
#50722
#50082
#48422
#48372
#47822
#47612
#32533