Finite Group Databasebeta

← #70393

Subgroups — #70393

Order 384.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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#5833C211yesmaximal, characteristic
2C2×#951C211yesmaximal, characteristic, complemented
3C2×#909C211yesmaximal, characteristic, complemented
4C2×C3×#104C211yesmaximal, characteristic, complemented
5#4929C211yesmaximal, characteristic, complemented
6C2×#805C211yesmaximal, characteristic, complemented
7#4554C211yesmaximal, characteristic, complemented
8#2007—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C2×C3×C42C2211yescharacteristic
14#951—22no—
15#951—22no—
16#946—21no—
17C2×#216—21no—
18#939—22no—
19#938—21no—
20C2×#214—21no—
21#930—21no—
22C2×C4⋊Dic3C2211yescharacteristic
23#924—21no—
24C2×#208C2211yescharacteristic
25#909—22no—
26#909—22no—
27C2×Dic12—21no—
28C2×#204—21no—
29C3×#104—22no—
30C3⋊#95—22no—
31#805—22no—
32C2×#137—31no—
33C2×#137—31no—
34#537—31no—
35C2×#104—31no—
36C2×#104D311yescharacteristic, complemented
37#391—31no—
38#343—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×C4—21no—
48C22×C3×C4C2311yescharacteristic
49C2×#67—21no—
50C2×#67—22no—
51C22×Dic3—21no—
52C22×Dic3D412yes—
53#235—21no—
54#235—21no—
55#235—21no—
56#235—21no—
57C2×D12D411yescharacteristic
58C2×C4×D3—21no—
59C2×Dic6—21no—
60C2×Dic6—21no—
61C2×Dic6—21no—
62C2×Dic6—21no—
63C2×Dic6—21no—
64C2×Dic6D411yescharacteristic
65C2×Dic6D412yescomplemented
66C3×M16—21no—
67C3×M16D412yescomplemented
68C2×C3×C8—21no—
69C3×C42—21no—
70C3×C42—21no—
71#217—22no—
72#216—24no—
73#214—24no—
74(C2×D3)⋊C4—22no—
75C4⋊Dic3—21no—
76C4⋊Dic3—21no—
77Dic3⋊C4—22no—
78#208—21no—
79#208D412yescomplemented
80C2×C3⋊C8—21no—
81Dic12—24no—
82#204—24no—
83C2×#41—31no—
84C2×#41C2×D311yescharacteristic
85C22×Q8—31no—
86#137—62no—
87#137—62no—
88#137—62no—
89#137—62no—
90C2×Q16—61no—
91C2×Q16—61no—
92C2×SD16—61no—
93C2×SD16—61no—
94C2×M16—31no—
95C2×M16C2×D311yescharacteristic
96C22⋊Q8—61no—
97C22⋊Q8—61no—
98#121—61no—
99#121—62no—
100#121—61no—
101C2×C4⋊C4—31no—
102C2×C42C2×D311yescharacteristic
103#104—22no—
104#104—62no—
105#95—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—22no—
116C2×C3×C4—21no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4#4111yescharacteristic
127#67—41no—
128#67—41no—
129#67—82no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—41no—
134C2×Dic3—41no—
135C2×Dic3—42no—
136D12—21no—
137D12—21no—
138C4×D3—41no—
139C4×D3—41no—
140Dic6—22no—
141Dic6—21no—
142Dic6—22no—
143Dic6—21no—
144Dic6—41no—
145Dic6—41no—
146C3×C8—22no—
147C3⋊C8—22no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152#41—61no—
153#41—61no—
154#41—61no—
155#41—61no—
156C2×Q8—31no—
157C2×Q8—32no—
158C2×Q8—61no—
159C2×Q8—61no—
160C2×Q8—61no—
161C2×Q8—61no—
162C2×Q8—61no—
163C2×Q8D1211yescharacteristic
164C2×D4—21no—
165C2×D4—31no—
166C2×D4—61no—
167C2×D4—62no—
168C2×D4D1211yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—32no—
172C22×C4—61no—
173C22×C4—61no—
174C22×C4C22×D311yescharacteristic
175Q16—64no—
176Q16—64no—
177SD16—64no—
178SD16—64no—
179M16—21no—
180M16—32no—
181M16—61no—
182M16#6712yescomplemented
183C2×C8—21no—
184C2×C8—61no—
185C4⋊C4—62no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C42—21no—
191C42—21no—
192C22×C3—21no—
193C22×C3—21no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3#12011yescharacteristic
198C22×C3#12111yescharacteristic
199C22×C3#12711yescharacteristic
200C2×D3—41no—
201C2×D3—41no—
202C2×D3—41no—
203C3×C4—42no—
204C3×C4—41no—
205C3×C4#12011yescharacteristic
206C3×C4#12011yescharacteristic
207C3×C4#12111yescharacteristic
208C3×C4#12111yescharacteristic
209Dic3—42no—
210Dic3—41no—
211Dic3—42no—
212C23—21no—
213C23—61no—
214C23D3×D411yescharacteristic
215Q8—21no—
216Q8—21no—
217Q8—62no—
218Q8—61no—
219Q8—62no—
220Q8—61no—
221Q8—121no—
222Q8—121no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—61no—
228D4—61no—
229D4—121no—
230D4—121no—
231D4—242no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—62no—
239C2×C4—61no—
240C2×C4—62no—
241C2×C4—121no—
242C2×C4—121no—
243C2×C4—121no—
244C2×C4—121no—
245C2×C4—122no—
246C2×C4C2×D1211yescharacteristic
247C2×C4#23511yescharacteristic
248C2×C4D3×D411yescharacteristic
249C2×C4D3×D411yescharacteristic
250C2×C4D3×D411yescharacteristic
251C2×C4C2×#6711yescharacteristic
252C8—22no—
253C8—62no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—81no—
257C2×C3#39211yescharacteristic
258C2×C3#45511yescharacteristic
259C2×C3#45611yescharacteristic
260D3—81no—
261C22—21no—
262C22—21no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—121no—
267C22—121no—
268C22—121no—
269C22#88211yescharacteristic
270C22#88411yescharacteristic
271C22#94011yescharacteristic
272C4—42no—
273C4—41no—
274C4—121no—
275C4—122no—
276C4—122no—
277C4#88211yescharacteristic
278C4#88411yescharacteristic
279C4#93011yescharacteristic
280C4#93711yescharacteristic
281C3#200711yescharacteristic, complemented, Sylow
282C2—21no—
283C2—21no—
284C2—81no—
285C2—241no—
286C2#482511yescharacteristic
287C2#482611yescharacteristic
288C2#496011yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×#104C2
socleC22×C3#120
prime radical#5833C2
hypercenterC2×#104D3
hypocenterC3#2007
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#951C2semidirect
3. C2×#909C2semidirect
4. C2×C3×#104C2semidirect
5. #4929C2semidirect
6. C2×#805C2semidirect
7. #4554C2semidirect
36. C2×#104D3semidirect
65. C2×Dic6D4semidirect
67. C3×M16D4semidirect
79. #208D4semidirect
182. M16#67semidirect
281. C3#2007semidirect

Maximal subgroups

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

subgroupindex
#58332
C2×#9512
C2×#9092
C2×C3×#1042
#49292
C2×#8052
#45542
#20073