Finite Group Databasebeta

← #70416

Subgroups — #70416

Order 384.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#5822C211yesmaximal, characteristic
2#5745C211yesmaximal, characteristic, complemented
3C3×#421C211yesmaximal, characteristic, complemented
4#5234C211yesmaximal, characteristic, complemented
5#5002C211yesmaximal, characteristic, complemented
6#4558C211yesmaximal, characteristic, complemented
7#4533C211yesmaximal, characteristic, complemented
8#1881—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×C3×M16C2211yescharacteristic
12C3×#117C2211yescharacteristic
13#951—21no—
14#950—22no—
15#949—21no—
16#942C412yescomplemented
17#932C412yescomplemented
18C2×#208C2211yescharacteristic
19C2×#208C2211yescharacteristic
20C22×C3⋊C8C2211yescharacteristic
21#907—22no—
22D3×M16—22no—
23#901—22no—
24C3×#104—22no—
25#830—22no—
26#810—21no—
27#809—21no—
28#575—31no—
29#568—31no—
30#421D311yescharacteristic, complemented
31#416—31no—
32#408—31no—
33#344—31no—
34#341—31no—
35C3×#41—21no—
36C3×#41—21no—
37C3×#41—21no—
38C3×#41—21no—
39C2×C3×Q8C2×C411yescharacteristic
40C2×C3×D4—21no—
41C2×C3×D4C2×C411yescharacteristic
42C22×C3×C4—21no—
43C22×C3×C4C2311yescharacteristic
44C2×#67—21no—
45#235—21no—
46#235—21no—
47#235C2×C412yes—
48#235D412yes—
49C2×D12D411yescharacteristic
50C2×C4×D3—21no—
51C2×Dic6D411yescharacteristic
52C3×M16—21no—
53C3×M16D412yes—
54C2×C3×C8—21no—
55C3×C4⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C42—21no—
58#216—22no—
59#215—22no—
60#214—22no—
61#213—22no—
62#208—21no—
63#208—21no—
64#208—21no—
65#208C2×C412yes—
66C2×C3⋊C8—21no—
67C2×C3⋊C8—21no—
68C2×C3⋊C8—21no—
69C2×C3⋊C8—21no—
70C2×C3⋊C8D412yes—
71ΓL(1,𝔽25)—24no—
72D3×C8—24no—
73C2×#41—31no—
74C2×#41C2×D311yescharacteristic
75#137—32no—
76#137—61no—
77AGL(1,ℤ/8)—32no—
78AGL(1,ℤ/8)—61no—
79#135—62no—
80#131—62no—
81#131—62no—
82C2×M16—31no—
83C2×M16—31no—
84C2×M16—62no—
85C2×M16C2×D311yescharacteristic
86C22×C8—31no—
87#117C2×D311yescharacteristic
88#104—22no—
89#103—61no—
90#102—61no—
91C22⋊C8—62no—
92C23×C3—21no—
93C23×C3#4111yescharacteristic
94C22×D3—21no—
95C3×Q8—21no—
96C3×Q8—21no—
97C3×D4—21no—
98C3×D4—21no—
99C3×D4—41no—
100C3×D4—41no—
101C2×C3×C4—21no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4C22×C411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4#4111yescharacteristic
111#67—22no—
112#67—22no—
113C2×Dic3—21no—
114D12—21no—
115D12C22⋊C412yes—
116C4×D3—22no—
117C4×D3—22no—
118Dic6—21no—
119Dic6C22⋊C412yes—
120C3×C8—22no—
121C3⋊C8—22no—
122C3⋊C8—22no—
123C3⋊C8—41no—
124#41—21no—
125#41—21no—
126#41—21no—
127#41—21no—
128#41—32no—
129#41—32no—
130#41—61no—
131#41—61no—
132C2×Q8—31no—
133C2×Q8C4×D311yescharacteristic
134C2×D4—21no—
135C2×D4—31no—
136C2×D4—61no—
137C2×D4C4×D311yescharacteristic
138C22×C4—21no—
139C22×C4—61no—
140C22×C4C22×D311yescharacteristic
141Q16—62no—
142SD16—62no—
143SD16—62no—
144D8—62no—
145M16—21no—
146M16—32no—
147M16—64no—
148M16—61no—
149M16—61no—
150M16—61no—
151M16#6712yes—
152C2×C8—21no—
153C2×C8—32no—
154C2×C8—64no—
155C2×C8—61no—
156C2×C8—61no—
157C2×C8—61no—
158C2×C8—61no—
159C4⋊C4—21no—
160C22⋊C4—21no—
161C42—21no—
162C22×C3—41no—
163C22×C3—42no—
164C22×C3—41no—
165C22×C3C4×D411yescharacteristic
166C22×C3#12111yescharacteristic
167C22×C3#12311yescharacteristic
168C2×D3—22no—
169C2×D3—81no—
170C3×C4—41no—
171C3×C4—81no—
172C3×C4C2×C22⋊C411yescharacteristic
173C3×C4C4×D411yescharacteristic
174C3×C4#12011yescharacteristic
175C3×C4#12111yescharacteristic
176Dic3—22no—
177C23—21no—
178C23—61no—
179C23#23711yescharacteristic
180Q8—21no—
181Q8—21no—
182Q8—32no—
183Q8—61no—
184D4—21no—
185D4—21no—
186D4—32no—
187D4—41no—
188D4—41no—
189D4—62no—
190D4—62no—
191D4—61no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—61no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4C2×C4×D311yescharacteristic
200C2×C4#23511yescharacteristic
201C2×C4D3×D411yescharacteristic
202C2×C4D3×D411yescharacteristic
203C2×C4D3×D411yescharacteristic
204C2×C4C2×#6711yescharacteristic
205C8—22no—
206C8—62no—
207C8—62no—
208C8—121no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—21no—
212C2×C3—81no—
213C2×C3#38611yescharacteristic
214D3—42no—
215C22—41no—
216C22—42no—
217C22—41no—
218C22—62no—
219C22—241no—
220C22#67⋊C411yescharacteristic
221C22#88311yescharacteristic
222C22#93811yescharacteristic
223C4—41no—
224C4—62no—
225C4—81no—
226C4D3×C22⋊C411yescharacteristic
227C4#88411yescharacteristic
228C4C4×#6711yescharacteristic
229C4#93711yescharacteristic
230C3#188111yescharacteristic, complemented, Sylow
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—122no—
236C2#495611yescharacteristic
subgroupG/H
centerC4#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#386
prime radical#5822C2
cosocleC22×C3×C4C23
hypercenter#421D3
hypocenterC3#1881
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #5745C2semidirect
3. C3×#421C2semidirect
4. #5234C2semidirect
5. #5002C2semidirect
6. #4558C2semidirect
7. #4533C2semidirect
16. #942C4semidirect
17. #932C4semidirect
30. #421D3semidirect
230. C3#1881semidirect

Maximal subgroups

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

subgroupindex
#58222
#57452
C3×#4212
#52342
#50022
#45582
#45332
#18813