Finite Group Databasebeta

← #69031

Subgroups — #69031

Order 384.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#5748C211yesmaximal, characteristic
2C2×#909C211yesmaximal, characteristic, complemented
3#5526C211yesmaximal, characteristic, complemented
4#5139C211yesmaximal, characteristic, complemented
5#5120C211yesmaximal, characteristic, complemented
6#5006C211yesmaximal, characteristic, complemented
7C3×#344C211yesmaximal, characteristic, complemented
8#2009—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C2×C3×M16C2211yescharacteristic
14C3×#117C2211yescharacteristic
15#926C2211yescharacteristic
16#909—22no—
17#909—21no—
18#909—21no—
19#909—22no—
20#908—21no—
21#908—21no—
22C2×Dic12—21no—
23#904—22no—
24C2×#204—21no—
25#897—22no—
26#886—22no—
27#875—21no—
28#869—21no—
29#825—22no—
30#816—22no—
31#806—22no—
32#575—31no—
33C2×#137—31no—
34#536—31no—
35#421—31no—
36#419—31no—
37#410—31no—
38#344D311yescharacteristic, complemented
39C22×C3×C4C2311yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C22×Dic3—21no—
43#235—21no—
44#235—21no—
45#235—21no—
46#235—21no—
47#235—21no—
48#235—21no—
49#235—21no—
50#235—21no—
51C2×D12D411yescharacteristic
52C2×D12D411yescharacteristic
53C2×C4×D3—21no—
54C2×C4×D3—21no—
55C2×Dic6—21no—
56C2×Dic6—21no—
57C2×Dic6—21no—
58C2×Dic6—21no—
59C2×Dic6—21no—
60C2×Dic6D411yescharacteristic
61C2×Dic6D412yes—
62C2×Dic6D411yescharacteristic
63C3×M16—21no—
64C3×M16—21no—
65C3×M16—21no—
66C3×M16D412yes—
67C2×C3×C8—21no—
68C2×C3×C8D412yes—
69C3×C4⋊C4—21no—
70C3×C22⋊C4D412yes—
71C3×C42—21no—
72#217—21no—
73(C2×D3)⋊C4—24no—
74C4⋊Dic3—24no—
75C4⋊Dic3—21no—
76C4×Dic3—21no—
77Dic12—22no—
78Dic12—24no—
79D24—22no—
80#204—24no—
81#204—22no—
82#204—22no—
83C2×#41—31no—
84C2×#41—31no—
85C22×Q8—31no—
86#137—62no—
87#137—61no—
88#137—61no—
89#137—62no—
90AGL(1,ℤ/8)—61no—
91AGL(1,ℤ/8)—61no—
92#135—62no—
93C2×Q16—61no—
94C2×SD16—61no—
95C2×M16C2×D311yescharacteristic
96C2×M16C2×D311yescharacteristic
97C4⋊Q8—61no—
98#124—61no—
99#123—62no—
100C22⋊Q8—62no—
101#117—31no—
102#117C2×D311yescharacteristic
103#104—62no—
104#103—62no—
105#99—62no—
106C23×C3C2×D411yescharacteristic
107C22×D3—21no—
108C22×D3—21no—
109C2×C3×C4—22no—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4#4111yescharacteristic
116#67—41no—
117#67—41no—
118#67—41no—
119#67—41no—
120C2×Dic3—21no—
121C2×Dic3—22no—
122C2×Dic3—21no—
123C2×Dic3—41no—
124C2×Dic3—41no—
125C2×Dic3—41no—
126D12—21no—
127D12—21no—
128D12—21no—
129D12—21no—
130C4×D3—41no—
131C4×D3—41no—
132C4×D3—41no—
133C4×D3—41no—
134Dic6—22no—
135Dic6—21no—
136Dic6—21no—
137Dic6—21no—
138Dic6—22no—
139Dic6—21no—
140Dic6—41no—
141Dic6—41no—
142C3×C8—22no—
143C3×C8—22no—
144#41—61no—
145#41—61no—
146#41—61no—
147#41—61no—
148#41—61no—
149#41—61no—
150#41—61no—
151#41—61no—
152C2×Q8—32no—
153C2×Q8—31no—
154C2×Q8—31no—
155C2×Q8—61no—
156C2×Q8—61no—
157C2×Q8—61no—
158C2×Q8—61no—
159C2×Q8—61no—
160C2×D4—31no—
161C2×D4—31no—
162C2×D4—61no—
163C2×D4—61no—
164C22×C4—61no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4C22×D311yescharacteristic
168Q16—64no—
169Q16—62no—
170SD16—64no—
171SD16—62no—
172SD16—62no—
173D8—62no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16D1212yes—
178C2×C8—21no—
179C2×C8D1212yes—
180C4⋊C4—21no—
181C4⋊C4—64no—
182C4⋊C4—61no—
183C22⋊C4—64no—
184C22⋊C4—61no—
185C22⋊C4D1212yes—
186C42—21no—
187C42—61no—
188C22×C3—41no—
189C22×C3#12011yescharacteristic
190C22×C3#12111yescharacteristic
191C22×C3#12711yescharacteristic
192C2×D3—42no—
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—81no—
196C3×C4—42no—
197C3×C4#12011yescharacteristic
198C3×C4#12011yescharacteristic
199C3×C4#12111yescharacteristic
200C3×C4#12111yescharacteristic
201Dic3—42no—
202Dic3—41no—
203Dic3—41no—
204Dic3—81no—
205C23—61no—
206C23—61no—
207C23C2×D1211yescharacteristic
208Q8—62no—
209Q8—61no—
210Q8—61no—
211Q8—61no—
212Q8—62no—
213Q8—61no—
214Q8—121no—
215Q8—121no—
216D4—61no—
217D4—61no—
218D4—61no—
219D4—61no—
220D4—121no—
221D4—121no—
222D4—121no—
223D4—121no—
224C2×C4—22no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—121no—
229C2×C4—121no—
230C2×C4—121no—
231C2×C4—121no—
232C2×C4—121no—
233C2×C4—121no—
234C2×C4—121no—
235C2×C4C2×D1211yescharacteristic
236C2×C4C2×D1211yescharacteristic
237C2×C4D3×D411yescharacteristic
238C2×C4D3×D411yescharacteristic
239C2×C4D3×D411yescharacteristic
240C2×C4#23911yescharacteristic
241C8—22no—
242C8—22no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3#39211yescharacteristic
247D3—81no—
248D3—81no—
249C22—41no—
250C22—122no—
251C22—121no—
252C22—121no—
253C22—241no—
254C22#87411yescharacteristic
255C22#88211yescharacteristic
256C22#89511yescharacteristic
257C4—42no—
258C4—121no—
259C4—121no—
260C4—122no—
261C4—241no—
262C4#88211yescharacteristic
263C4#88211yescharacteristic
264C4#89511yescharacteristic
265C4#89511yescharacteristic
266C3#200911yescharacteristic, complemented, Sylow
267C2—21no—
268C2—21no—
269C2—21no—
270C2—241no—
271C2—241no—
272C2#467111yescharacteristic
subgroupG/H
centerC2#4671
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#344C2
socleC2×C3#392
prime radical#5748C2
hypercenter#344D3
hypocenterC3#2009
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#909C2semidirect
3. #5526C2semidirect
4. #5139C2semidirect
5. #5120C2semidirect
6. #5006C2semidirect
7. C3×#344C2semidirect
38. #344D3semidirect
266. C3#2009semidirect

Maximal subgroups

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

subgroupindex
#57482
C2×#9092
#55262
#51392
#51202
#50062
C3×#3442
#20093