Finite Group Databasebeta

← #80344

Subgroups — #80344

Order 384.

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

normal subgroups only

This group has 392 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#5159C212yesmaximal, complemented
5#5153C212yesmaximal, complemented
6#5075C211yesmaximal, characteristic, complemented
7#5007C211yesmaximal, characteristic, complemented
8#4851C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4780C212yesmaximal, complemented
11#4763C212yesmaximal, complemented
12#3251—31nomaximal, Sylow
13C2×#237—21no—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#938—21no—
18#938C2211yescharacteristic
19#938C2212yes—
20#936—21no—
21#936C2211yescharacteristic
22D4×Dic3—21no—
23D4×Dic3C2212yes—
24C2×#214C2212yes—
25C4×#67—21no—
26#926—21no—
27C2×ΓL(1,𝔽25)C2212yes—
28#897C2211yescharacteristic
29#890C2211yescharacteristic
30#883—21no—
31#879—21no—
32C42×D3C2211yescharacteristic
33#869C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35C3×C4⋊C8C2211yescharacteristic
36C3×#102C2212yes—
37#832C2212yes—
38#832C2212yes—
39#817C2212yes—
40#816C2212yes—
41Dic3⋊C8C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#529—31no—
46#478—32no—
47#475—31no—
48#474—31no—
49#469—32no—
50#468—32no—
51#463—32no—
52#460—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—
62C22×Dic3—22no—
63#237—44no—
64C2×C4×D3D411yescharacteristic
65C2×C4×D3D411yescharacteristic
66C2×C4×D3C2311yescharacteristic
67C2×Dic6—21no—
68C2×Dic6C2311yescharacteristic
69C3×D8—42no—
70C2×C3×C8C2312yes—
71C3×C4⋊C4C2311yescharacteristic
72C3×C22⋊C4—21no—
73C3×C42C2311yescharacteristic
74#217—21no—
75#217—21no—
76#217—21no—
77#217—22no—
78#217—22no—
79#214—42no—
80(C2×D3)⋊C4—21no—
81C4⋊Dic3—21no—
82C4⋊Dic3—21no—
83C4⋊Dic3—21no—
84C4⋊Dic3C2312yes—
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88C4×Dic3—21no—
89C4×Dic3D411yescharacteristic
90C4×Dic3D411yescharacteristic
91C4×Dic3C2311yescharacteristic
92C2×C3⋊C8C2312yes—
93ΓL(1,𝔽25)—42no—
94C2×#41—61no—
95C2×SD16—32no—
96C2×D8C2×D312yes—
97C2×M16—32no—
98C4⋊Q8—31no—
99#127C2×D311yescharacteristic
100#126—61no—
101#125—31no—
102#124—31no—
103#124—61no—
104#123—61no—
105C22⋊Q8—31no—
106#121—31no—
107#121—32no—
108#121—61no—
109C4×Q8—31no—
110C4×D4—32no—
111C4×D4—61no—
112C4×D4—61no—
113C4×D4C2×D311yescharacteristic
114#117—61no—
115C2×C42—31no—
116#107—32no—
117#106—32no—
118C4⋊C8—32no—
119C4⋊C8—31no—
120C4⋊C8C2×D311yescharacteristic
121#103—32no—
122#102—32no—
123#102—32no—
124#102C2×D312yes—
125C23×C3—21no—
126C23×C3—22no—
127C22×D3C2×D411yescharacteristic
128C3×D4—22no—
129C3×D4—21no—
130C3×D4—21no—
131C3×D4—22no—
132C3×D4—42no—
133C3×D4—42no—
134C2×C3×C4—21no—
135C2×C3×C4—42no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—44no—
140#67—42no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—44no—
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148C4×D3—22no—
149C4×D3—21no—
150C4×D3—21no—
151C4×D3—41no—
152Dic6—21no—
153Dic6—21no—
154Dic6—42no—
155C3×C8—22no—
156C3⋊C8—22no—
157#41—124no—
158C2×Q8—31no—
159C2×Q8—61no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—61no—
163C2×D4—62no—
164C2×D4C22×D311yescharacteristic
165C2×D4C22×D312yes—
166C22×C4—21no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—62no—
171SD16—122no—
172D8—42no—
173M16—122no—
174C2×C8—32no—
175C2×C8C22×D312yes—
176C4⋊C4—31no—
177C4⋊C4—32no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4C22×D311yescharacteristic
184C22⋊C4—21no—
185C22⋊C4—61no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C22⋊C4—61no—
191C42—31no—
192C42—31no—
193C42—31no—
194C42—61no—
195C42C22×D311yescharacteristic
196C22×C3—42no—
197C22×C3—41no—
198C22×C3—42no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—42no—
202C22×C3C22×D411yescharacteristic
203C2×D3—21no—
204C2×D3—21no—
205C2×D3—21no—
206C3×C4—21no—
207C3×C4—41no—
208C3×C4AGL(1,ℤ/8)12yes—
209C3×C4H(5,𝔽2)11yescharacteristic
210C3×C42−1+411yescharacteristic
211Dic3—21no—
212Dic3—21no—
213Dic3—21no—
214Dic3—41no—
215Dic3—42no—
216Dic3—41no—
217C23—21no—
218C23—22no—
219C23—31no—
220Q8—61no—
221Q8—61no—
222Q8—122no—
223D4—21no—
224D4—22no—
225D4—21no—
226D4—22no—
227D4—42no—
228D4—42no—
229D4—124no—
230D4—122no—
231C2×C4—21no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—42no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—121no—
243C2×C4—124no—
244C2×C4D3×D411yescharacteristic
245C2×C4D3×D411yescharacteristic
246C2×C4C23×D311yescharacteristic
247C8—22no—
248C8—62no—
249C2×C3—82no—
250C2×C3—81no—
251C2×C3#53711yescharacteristic
252C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
253C2×C3#57511yescharacteristic
254D3—41no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—42no—
261C22—61no—
262C22—61no—
263C22—61no—
264C22C2×D3×D411yescharacteristic
265C4—21no—
266C4—41no—
267C4—61no—
268C4—61no—
269C4—61no—
270C4—122no—
271C4—121no—
272C4—121no—
273C4#91112yes—
274C4#100411yescharacteristic
275C4#101011yescharacteristic
276C3#325111yescharacteristic, complemented, Sylow
277C2—81no—
278C2—82no—
279C2—121no—
280C2#560811yescharacteristic
281C2C2×#91111yescharacteristic
282C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3251
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. #5159C2semidirect
5. #5153C2semidirect
6. #5075C2semidirect
7. #5007C2semidirect
8. #4851C2semidirect
9. #4837C2semidirect
10. #4780C2semidirect
11. #4763C2semidirect
53. #459D3semidirect
276. C3#3251semidirect

Maximal subgroups

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

subgroupindex
#57162
#55572
C3×#4592
#51592
#51532
#50752
#50072
#48512
#48372
#47802
#47632
#32513