Finite Group Databasebeta

← #70412

Subgroups — #70412

Order 384.

348 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 69 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#5825C211yesmaximal, characteristic, complemented
2#5822C211yesmaximal, characteristic
3C2×#909C211yesmaximal, characteristic, complemented
4C3×#421C211yesmaximal, characteristic, complemented
5#5006C211yesmaximal, characteristic, complemented
6#5004C211yesmaximal, characteristic, complemented
7#4559C211yesmaximal, characteristic, complemented
8#2009—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×#117C2211yescharacteristic
14#951—21no—
15#950—22no—
16#949—21no—
17#944—21no—
18#942—21no—
19#936—21no—
20#935—22no—
21#932—21no—
22#926C2211yescharacteristic
23#924—22no—
24C2×#208C2211yescharacteristic
25#909—22no—
26#909—22no—
27C2×Dic12—21no—
28C2×#204—21no—
29C3×#104—22no—
30#810—22no—
31#806—22no—
32#575—31no—
33C2×#137—31no—
34#536—31no—
35#421D311yescharacteristic, complemented
36#419—31no—
37#410—31no—
38#344—31no—
39C3×#41—21no—
40C3×#41—21no—
41C3×#41—21no—
42C3×#41—21no—
43C2×C3×Q8D411yescharacteristic
44C2×C3×D4—21no—
45C2×C3×D4D411yescharacteristic
46C22×C3×C4—21no—
47C22×C3×C4C2311yescharacteristic
48C2×#67—21no—
49C22×Dic3—21no—
50#235—21no—
51#235—21no—
52#235—21no—
53#235—21no—
54C2×D12D411yescharacteristic
55C2×C4×D3—21no—
56C2×Dic6—21no—
57C2×Dic6—21no—
58C2×Dic6—21no—
59C2×Dic6—21no—
60C2×Dic6—21no—
61C2×Dic6D411yescharacteristic
62C2×Dic6D412yes—
63C3×M16—21no—
64C3×M16D412yes—
65C2×C3×C8—21no—
66C3×C4⋊C4—21no—
67C3×C22⋊C4—21no—
68C3×C42—21no—
69#217—24no—
70#217D412yes—
71#216—22no—
72#215—22no—
73#214—22no—
74#213—22no—
75C4⋊Dic3—21no—
76Dic3⋊C4—24no—
77C4×Dic3—21no—
78#208—21no—
79#208—21no—
80C2×C3⋊C8D412yes—
81Dic12—24no—
82#204—24no—
83C2×#41—31no—
84C2×#41C2×D311yescharacteristic
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×M16—31no—
96C2×M16C2×D311yescharacteristic
97C4⋊Q8—61no—
98#124—61no—
99#123—62no—
100C22⋊Q8—62no—
101#117—31no—
102#117C2×D311yescharacteristic
103#104—22no—
104#103—62no—
105#99—62no—
106C23×C3—21no—
107C23×C3C2×D411yescharacteristic
108C22×D3—21no—
109C3×Q8—21no—
110C3×Q8—21no—
111C3×D4—21no—
112C3×D4—21no—
113C3×D4—41no—
114C3×D4—41no—
115C2×C3×C4—21no—
116C2×C3×C4—41no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C2×C3×C4C2×D411yescharacteristic
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4#4111yescharacteristic
125#67—41no—
126#67—41no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—41no—
131C2×Dic3—41no—
132D12—21no—
133D12—21no—
134C4×D3—41no—
135C4×D3—41no—
136Dic6—22no—
137Dic6—21no—
138Dic6—21no—
139Dic6—22no—
140Dic6—41no—
141Dic6—41no—
142C3×C8—22no—
143C3⋊C8—22no—
144#41—21no—
145#41—21no—
146#41—21no—
147#41—21no—
148#41—61no—
149#41—61no—
150#41—61no—
151#41—61no—
152C2×Q8—32no—
153C2×Q8—31no—
154C2×Q8—61no—
155C2×Q8—61no—
156C2×Q8—61no—
157C2×Q8—61no—
158C2×Q8—61no—
159C2×Q8D1211yescharacteristic
160C2×D4—21no—
161C2×D4—31no—
162C2×D4—61no—
163C2×D4D1211yescharacteristic
164C22×C4—21no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4C22×D311yescharacteristic
168Q16—64no—
169Q16—62no—
170SD16—64no—
171SD16—62no—
172SD16—62no—
173D8—62no—
174M16—21no—
175M16—61no—
176M16—61no—
177M16#6712yes—
178C2×C8—21no—
179C2×C8—32no—
180C4⋊C4—21no—
181C4⋊C4—64no—
182C4⋊C4—61no—
183C22⋊C4—21no—
184C22⋊C4—32no—
185C22⋊C4—64no—
186C42—21no—
187C42—61no—
188C22×C3—41no—
189C22×C3—42no—
190C22×C3—41no—
191C22×C3#12011yescharacteristic
192C22×C3#12111yescharacteristic
193C22×C3#12711yescharacteristic
194C2×D3—41no—
195C2×D3—81no—
196C3×C4—41no—
197C3×C4—81no—
198C3×C4#12011yescharacteristic
199C3×C4#12011yescharacteristic
200C3×C4#12111yescharacteristic
201C3×C4#12111yescharacteristic
202Dic3—42no—
203Dic3—42no—
204Dic3—41no—
205C23—21no—
206C23—61no—
207C23D3×D411yescharacteristic
208Q8—21no—
209Q8—21no—
210Q8—61no—
211Q8—62no—
212Q8—62no—
213Q8—61no—
214Q8—121no—
215Q8—121no—
216D4—21no—
217D4—21no—
218D4—41no—
219D4—41no—
220D4—61no—
221D4—61no—
222D4—121no—
223D4—121no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—121no—
232C2×C4—121no—
233C2×C4—121no—
234C2×C4—121no—
235C2×C4C2×D1211yescharacteristic
236C2×C4#23511yescharacteristic
237C2×C4D3×D411yescharacteristic
238C2×C4D3×D411yescharacteristic
239C2×C4D3×D411yescharacteristic
240C2×C4C2×#6711yescharacteristic
241C8—22no—
242C8—62no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—81no—
247C2×C3#39211yescharacteristic
248D3—81no—
249C22—41no—
250C22—42no—
251C22—41no—
252C22—121no—
253C22—241no—
254C22#88211yescharacteristic
255C22#88411yescharacteristic
256C22#94011yescharacteristic
257C4—41no—
258C4—81no—
259C4—122no—
260C4—121no—
261C4—122no—
262C4#88211yescharacteristic
263C4#88411yescharacteristic
264C4#93011yescharacteristic
265C4#93711yescharacteristic
266C3#200911yescharacteristic, complemented, Sylow
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—241no—
272C2#496011yescharacteristic
subgroupG/H
centerC2#4960
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#392
prime radical#5822C2
hypercenter#421D3
hypocenterC3#2009
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #5825C2semidirect
3. C2×#909C2semidirect
4. C3×#421C2semidirect
5. #5006C2semidirect
6. #5004C2semidirect
7. #4559C2semidirect
35. #421D3semidirect
266. C3#2009semidirect

Maximal subgroups

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

subgroupindex
#58252
#58222
C2×#9092
C3×#4212
#50062
#50042
#45592
#20093