Finite Group Databasebeta

← #84117

Subgroups — #84117

Order 384.

700 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×C3×2−1+4C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic, complemented
3C2×#951C211yesmaximal, characteristic, complemented
4C22×#216C211yesmaximal, characteristic, complemented
5#5239C214yesmaximal, complemented
6#5234C211yesmaximal, characteristic, complemented
7#5233C211yesmaximal, characteristic, complemented
8#5228C214yesmaximal, complemented
9#5225C211yesmaximal, characteristic, complemented
10#3011—31nomaximal, Sylow
11C3×2−1+4—24no—
12C3×2−1+4—24no—
13C2×C3×#41—22no—
14C2×C3×#41—21no—
15C2×C3×#41C2212yes—
16C2×C3×#41C2211yescharacteristic
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8—21no—
19C22×C3×Q8C2212yes—
20C22×C3×Q8C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22#951—24no—
23#944—21no—
24C2×#216—24no—
25C2×#216C2212yes—
26C2×#216C2214yes—
27#936—21no—
28#935—22no—
29C2×#214C2212yes—
30#926C2211yescharacteristic
31#924C2214yes—
32C2×#208C2211yescharacteristic
33C22×C3⋊C8C2211yescharacteristic
34#835C2214yes—
35#835C2212yes—
36#832C2212yes—
37#830C2214yes—
38C2×2−1+4D311yescharacteristic
39C2×#137—31no—
40C22×Q16—31no—
41#536—31no—
42#452—34no—
43#451—34no—
44#419—31no—
45#416—31no—
46#408—31no—
47C3×#41—216no—
48C3×#41—28no—
49C3×#41—24no—
50C3×#41—24no—
51C3×#41—22no—
52C3×#41—22no—
53C3×#41—22no—
54C3×#41D414yes—
55C2×C3×Q8—28no—
56C2×C3×Q8—24no—
57C2×C3×Q8—24no—
58C2×C3×Q8—22no—
59C2×C3×Q8—21no—
60C2×C3×Q8—22no—
61C2×C3×Q8D414yes—
62C2×C3×Q8D411yescharacteristic
63C2×C3×Q8C2311yescharacteristic
64C2×C3×Q8C2312yes—
65C2×C3×D4—22no—
66C2×C3×D4—21no—
67C2×C3×D4D412yes—
68C2×C3×D4D411yescharacteristic
69C2×C3×D4C2311yescharacteristic
70C22×C3×C4—22no—
71C22×C3×C4—21no—
72C22×C3×C4—22no—
73C22×C3×C4—21no—
74C22×C3×C4C2312yes—
75C22×C3×C4C2311yescharacteristic
76C22×Dic3—21no—
77C2×Dic6—22no—
78C2×Dic6—22no—
79C2×Dic6—21no—
80C2×Dic6C2312yes—
81#217—24no—
82#217—21no—
83#216—28no—
84#216—24no—
85#214—24no—
86C4⋊Dic3C2312yes—
87Dic3⋊C4—24no—
88C4×Dic3—21no—
89#208—22no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
932−1+4—24no—
942−1+4—24no—
95C2×#41—22no—
96C2×#41—21no—
97C2×#41C2×D312yes—
98C2×#41C2×D311yescharacteristic
99C2×#41C2×D311yescharacteristic
100C22×Q8—21no—
101C22×Q8—31no—
102C22×Q8C2×D312yes—
103C22×Q8C2×D311yescharacteristic
104#137—64no—
105C2×Q16—34no—
106C2×Q16—32no—
107C2×Q16—64no—
108C2×SD16—32no—
109C2×M16—31no—
110C22×C8—31no—
111C4⋊Q8—61no—
112#124—61no—
113#123—62no—
114C22⋊Q8—34no—
115#117—31no—
116#103—32no—
117#103—34no—
118#102—32no—
119C22⋊C8—34no—
120C23×C3—21no—
121C23×C3C2×D412yes—
122C23×C3C2×D411yescharacteristic
123C3×Q8—28no—
124C3×Q8—24no—
125C3×Q8—22no—
126C3×Q8—21no—
127C3×Q8—22no—
128C3×Q8C2×D412yes—
129C3×Q8C2×D414yes—
130C3×D4—28no—
131C3×D4—24no—
132C3×D4—24no—
133C3×D4—22no—
134C3×D4—21no—
135C3×D4C2×D412yes—
136C2×C3×C4—28no—
137C2×C3×C4—28no—
138C2×C3×C4—24no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—24no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4C2×D412yes—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D412yes—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2411yescharacteristic
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—42no—
152Dic6—22no—
153Dic6—22no—
154Dic6—42no—
155C3⋊C8—22no—
156C3⋊C8—22no—
157#41—216no—
158#41—28no—
159#41—24no—
160#41—24no—
161#41—22no—
162#41—22no—
163#41—22no—
164#41#6714yes—
165C2×Q8—28no—
166C2×Q8—24no—
167C2×Q8—24no—
168C2×Q8—22no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—32no—
172C2×Q8—61no—
173C2×Q8—62no—
174C2×Q8—62no—
175C2×Q8#6714yes—
176C2×Q8#6711yescharacteristic
177C2×Q8C22×D311yescharacteristic
178C2×Q8C22×D312yes—
179C2×D4—22no—
180C2×D4—21no—
181C2×D4#6712yes—
182C2×D4#6711yescharacteristic
183C2×D4C22×D311yescharacteristic
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—61no—
189C22×C4C22×D312yes—
190C22×C4C22×D311yescharacteristic
191Q16—68no—
192Q16—64no—
193SD16—64no—
194M16—62no—
195C2×C8—32no—
196C2×C8—32no—
197C2×C8—62no—
198C4⋊C4—32no—
199C4⋊C4—64no—
200C22⋊C4—61no—
201C22⋊C4—64no—
202C42—61no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—42no—
208C22×C3#12012yes—
209C22×C3C22×D411yescharacteristic
210C3×C4—24no—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4#12012yes—
214C3×C4C22×D411yescharacteristic
215C3×C4C22×D411yescharacteristic
216Dic3—42no—
217Dic3—42no—
218C23—21no—
219C23C2×#6712yes—
220C23C2×#6711yescharacteristic
221Q8—28no—
222Q8—24no—
223Q8—22no—
224Q8—21no—
225Q8—22no—
226Q8—62no—
227Q8—62no—
228Q8—122no—
229Q8C2×#6712yes—
230Q8C2×#6714yes—
231D4—28no—
232D4—24no—
233D4—24no—
234D4—22no—
235D4—21no—
236D4C2×#6712yes—
237C2×C4—28no—
238C2×C4—28no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—122no—
248C2×C4C2×#6712yes—
249C2×C4C2×#6711yescharacteristic
250C2×C4C2×#6712yes—
251C2×C4C2×#6712yes—
252C2×C4C23×D311yescharacteristic
253C8—62no—
254C8—62no—
255C2×C3—22no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3C2×#12011yescharacteristic
259C2×C3#57812yes—
260C22—22no—
261C22—24no—
262C22—22no—
263C22—22no—
264C22—42no—
265C22#95312yes—
266C22C22×#6711yescharacteristic
267C4—24no—
268C4—22no—
269C4—22no—
270C4—122no—
271C4—122no—
272C4#95312yes—
273C4C22×#6711yescharacteristic
274C4C22×#6711yescharacteristic
275C3#301111yescharacteristic, complemented, Sylow
276C2—22no—
277C2—42no—
278C2—42no—
279C2#583812yes—
280C2C2×#95311yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC2×C3×2−1+4C2
socleC22×C3C22×D4
hypercenterC2×2−1+4D3
hypocenterC3#3011
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. C2×#951C2semidirect
4. C22×#216C2semidirect
5. #5239C2semidirect
6. #5234C2semidirect
7. #5233C2semidirect
8. #5228C2semidirect
9. #5225C2semidirect
275. C3#3011semidirect

Maximal subgroups

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

subgroupindex
C2×C3×2−1+42
#58252
C2×#9512
C22×#2162
#52392
#52342
#52332
#52282
#52252
#30113