Finite Group Databasebeta

← #83309

Subgroups — #83309

Order 384.

396 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C3×#532C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic, complemented
3#5239C212yesmaximal, complemented
4#5238C212yesmaximal, complemented
5#5086C211yesmaximal, characteristic, complemented
6#5082C211yesmaximal, characteristic, complemented
7#5074C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5057C212yesmaximal, complemented
10#5054C212yesmaximal, complemented
11#4926C211yesmaximal, characteristic, complemented
12#3219—31nomaximal, Sylow
13C2×C3×#41C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#124C2212yes—
17C3×#121—21no—
18C3×#121—21no—
19C3×C4×D4—21no—
20C3×C4×D4—21no—
21C2×C3×C42C2211yescharacteristic
22C2×#216C2212yes—
23C2×#215C2212yes—
24C2×#214C2212yes—
25C2×#213C2212yes—
26#930C2211yescharacteristic
27#929—21no—
28C4×#67—21no—
29#924C2211yescharacteristic
30#876—21no—
31#875C2211yescharacteristic
32C4×D12C2211yescharacteristic
33#872—21no—
34#870C2211yescharacteristic
35C4×Dic6C2211yescharacteristic
36#835C2212yes—
37#835C2212yes—
38#832C2212yes—
39#832C2212yes—
40#830C2212yes—
41C4⋊(C3⋊C8)C2212yes—
42#803C2211yescharacteristic
43#803C2211yescharacteristic
44#532D311yescharacteristic, complemented
45#529—31no—
46#497—31no—
47#496—31no—
48#491—31no—
49#490—31no—
50#464—32no—
51#463—32no—
52#452—32no—
53#449—32no—
54#432—31no—
55C3×#41—42no—
56C3×#41—42no—
57C2×C3×Q8C2312yes—
58C2×C3×D4—22no—
59C2×C3×D4—21no—
60C2×C3×D4—21no—
61C2×C3×D4C2312yes—
62C22×C3×C4—22no—
63C22×C3×C4D411yescharacteristic
64C22×C3×C4D411yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×#67—21no—
67C2×D12C2311yescharacteristic
68C2×C4×D3—21no—
69C2×Dic6C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C42D411yescharacteristic
75C3×C42D411yescharacteristic
76C3×C42C2311yescharacteristic
77C3×C42C2311yescharacteristic
78#217—21no—
79#216—42no—
80#215—42no—
81#214—42no—
82#213—42no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—21no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3C2311yescharacteristic
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91Dic3⋊C4—21no—
92Dic3⋊C4—21no—
93C4×Dic3—21no—
94C2×C3⋊C8C2312yes—
95C2×C3⋊C8C2312yes—
96C2×#41C2×D312yes—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16—32no—
100C2×D8—32no—
101C4⋊Q8C2×D311yescharacteristic
102#127C2×D311yescharacteristic
103#126—61no—
104#125—31no—
105#124—31no—
106#124C2×D312yes—
107#123—61no—
108C22⋊Q8—31no—
109#121—21no—
110#121—21no—
111#121—31no—
112C4×Q8—31no—
113C4×D4—21no—
114C4×D4—21no—
115C4×D4—31no—
116C4×D4—61no—
117#117—61no—
118C2×C42C2×D311yescharacteristic
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102—32no—
123#102—32no—
124C22⋊C8—32no—
125#97—31no—
126#97—31no—
127C23×C3—22no—
128C23×C3C2×D411yescharacteristic
129C22×D3—21no—
130C3×Q8—22no—
131C3×Q8—22no—
132C3×D4—22no—
133C3×D4—22no—
134C3×D4—42no—
135C3×D4—42no—
136C3×D4—44no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—41no—
140C2×C3×C4—41no—
141C2×C3×C4—44no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2411yescharacteristic
148#67—42no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152D12—22no—
153C4×D3—42no—
154Dic6—22no—
155C3⋊C8—22no—
156C3⋊C8—22no—
157#41—42no—
158#41—42no—
159C2×Q8—31no—
160C2×Q8C22×D312yes—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—31no—
165C2×D4—61no—
166C2×D4C22×D312yes—
167C22×C4—22no—
168C22×C4—61no—
169C22×C4#6711yescharacteristic
170C22×C4#6711yescharacteristic
171C22×C4C22×D311yescharacteristic
172Q16—122no—
173SD16—122no—
174SD16—122no—
175D8—122no—
176C2×C8—32no—
177C2×C8—32no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—31no—
181C4⋊C4—31no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C42—61no—
194C42#6711yescharacteristic
195C42#6711yescharacteristic
196C42C22×D311yescharacteristic
197C42C22×D311yescharacteristic
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3C22×D411yescharacteristic
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—41no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—42no—
214C3×C4H(5,𝔽2)12yes—
215Dic3—41no—
216Dic3—41no—
217Dic3—41no—
218C23—22no—
219C23—61no—
220C23C2×#6711yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—62no—
224D4—22no—
225D4—22no—
226D4—42no—
227D4—42no—
228D4—44no—
229D4—62no—
230D4—122no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—122no—
240C2×C4C2×#6711yescharacteristic
241C2×C4C2×#6711yescharacteristic
242C2×C4C2×#6711yescharacteristic
243C2×C4C2×#6711yescharacteristic
244C2×C4C2×#6711yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—62no—
247C8—62no—
248C2×C3—41no—
249C2×C3—82no—
250C2×C3#53511yescharacteristic
251C2×C3#57511yescharacteristic
252C2×C3#57511yescharacteristic
253D3—81no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—121no—
261C22—121no—
262C22—121no—
263C22C22×#6711yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—121no—
271C4—121no—
272C4—121no—
273C4#100412yes—
274C3#321911yescharacteristic, complemented, Sylow
275C2—41no—
276C2—82no—
277C2—241no—
278C2#580511yescharacteristic
279C2#582211yescharacteristic
280C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#532C2
socleC22×C3C22×D4
hypercenter#532D3
hypocenterC3#3219
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5479C2semidirect
3. #5239C2semidirect
4. #5238C2semidirect
5. #5086C2semidirect
6. #5082C2semidirect
7. #5074C2semidirect
8. #5071C2semidirect
9. #5057C2semidirect
10. #5054C2semidirect
11. #4926C2semidirect
44. #532D3semidirect
274. C3#3219semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#54792
#52392
#52382
#50862
#50822
#50742
#50712
#50572
#50542
#49262
#32193