Finite Group Databasebeta

← C4⋊(B(4,𝔽2)⋊C3) #11709607

Subgroups — C4⋊(B(4,𝔽2)⋊C3) #11709607

Order 768.

442 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C4×#997C211yesmaximal, characteristic, complemented
2C2×B(4,𝔽2)⋊C3C212yesmaximal, complemented
3#32926C311yesmaximal, characteristic, complemented, Sylow
4C2×#997C2211yescharacteristic
5C2×D4×A4—41nomaximal
6B(4,𝔽2)⋊C3—22no—
7C4×H(5,𝔽2)C2×C311yescharacteristic, complemented
8C2×B(4,𝔽2)C2×C312yescomplemented
9#2408—31no—
10#1889—32no—
11#1799—31no—
12C23×A4—42no—
13#997D411yescharacteristic, complemented
14D4×A4—44no—
15C2×C4×A4—41no—
16C2×C4×A4—81no—
17C4×SL(2,𝔽3)—81no—
18C2×H(5,𝔽2)C22×C311yescharacteristic
19C23×D4A411yescharacteristic, complemented
20C2×#121—61no—
21C2×#120—32no—
22#520—61no—
23#518—31no—
24#518—61no—
25C4×#41—61no—
26C2×C4×D4—31no—
27C2×C4×D4—61no—
28B(4,𝔽2)—22no—
29B(4,𝔽2)—62no—
30C2×#99—32no—
31#394—61no—
32#392—61no—
33#390—61no—
34#380—31no—
35#379—32no—
36#342—32no—
37C22×A4—44no—
38C22×A4—41no—
39C22×A4—81no—
40C22×A4—82no—
41C22×A4—82no—
42C2×C3×D4—161no—
43C2×SL(2,𝔽3)—81no—
44C4×A4—42no—
45C4×A4—82no—
46C25C2×A412yes—
47H(5,𝔽2)—31no—
48H(5,𝔽2)—61no—
49H(5,𝔽2)—61no—
50H(5,𝔽2)C3×D411yescharacteristic, complemented
51C2×#41—61no—
52C22×D4—31no—
53C22×D4—44no—
54C22×D4—61no—
55C22×D4—61no—
56C22×D4—61no—
57C22×D4—62no—
58C23×C4—21no—
59C23×C4—31no—
60C23×C4C2×A411yescharacteristic
61#121—124no—
62#120—62no—
63#120—62no—
64#120—62no—
65#120—62no—
66C4×Q8—21no—
67C4×Q8—61no—
68C4×D4—62no—
69C4×D4—62no—
70C4×D4—62no—
71C4×D4—62no—
72C4×D4—62no—
73C4×D4—61no—
74C4×D4—61no—
75#117—61no—
76#117—62no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—61no—
79C2×C4⋊C4—61no—
80C2×C22⋊C4—31no—
81C2×C22⋊C4—31no—
82C2×C22⋊C4—32no—
83C2×C22⋊C4—61no—
84C2×C22⋊C4—61no—
85C2×C22⋊C4—62no—
86C2×C22⋊C4—62no—
87C2×C22⋊C4—62no—
88C2×C42—31no—
89C2×C42—61no—
90#99—62no—
91#99—62no—
92#95—61no—
93C23×C3—162no—
94C2×A4—41no—
95C2×A4—41no—
96C2×A4—41no—
97C2×A4—81no—
98C2×A4—81no—
99C2×A4—81no—
100C2×A4—84no—
101C3×D4—164no—
102C2×C3×C4—161no—
103SL(2,𝔽3)—81no—
104C24—21no—
105C24—22no—
106C24—31no—
107C24—44no—
108C24—62no—
109C24—62no—
110C24—82no—
111C24C22×A411yescharacteristic
112#41—61no—
113#41—62no—
114#41—61no—
115#41—61no—
116#41—62no—
117#41—61no—
118C2×Q8—21no—
119C2×D4—31no—
120C2×D4—31no—
121C2×D4—31no—
122C2×D4—31no—
123C2×D4—41no—
124C2×D4—64no—
125C2×D4—61no—
126C2×D4—62no—
127C2×D4—62no—
128C2×D4—62no—
129C2×D4—61no—
130C2×D4—61no—
131C2×D4—62no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—121no—
135C2×D4—122no—
136C2×D4—124no—
137C2×D4—122no—
138C2×D4—122no—
139C2×D4—124no—
140C2×D4—242no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—42no—
147C22×C4—61no—
148C22×C4—61no—
149C22×C4—61no—
150C22×C4—61no—
151C22×C4—61no—
152C22×C4—62no—
153C22×C4—82no—
154C22×C4—122no—
155C4⋊C4—62no—
156C4⋊C4—61no—
157C4⋊C4—61no—
158C4⋊C4—122no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C22⋊C4—62no—
164C22⋊C4—122no—
165C22⋊C4—122no—
166C22⋊C4—124no—
167C22⋊C4—242no—
168C42—61no—
169C42—62no—
170C42—61no—
171C22×C3—164no—
172C22×C3—161no—
173C22×C3—322no—
174C22×C3—322no—
175A4—41no—
176A4—81no—
177C3×C4—162no—
178C23—21no—
179C23—31no—
180C23—31no—
181C23—31no—
182C23—31no—
183C23—42no—
184C23—41no—
185C23—41no—
186C23—61no—
187C23—64no—
188C23—61no—
189C23—61no—
190C23—84no—
191C23—81no—
192C23—81no—
193C23—122no—
194C23—122no—
195C23—122no—
196C23—122no—
197C23—122no—
198C23—121no—
199C23—121no—
200C23—244no—
201C23D4×A411yescharacteristic
202Q8—21no—
203Q8—61no—
204D4—44no—
205D4—62no—
206D4—62no—
207D4—62no—
208D4—62no—
209D4—61no—
210D4—61no—
211D4—62no—
212D4—124no—
213D4—244no—
214D4—244no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—61no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—121no—
227C2×C4—122no—
228C2×C4—121no—
229C2×C4—121no—
230C2×C4—122no—
231C2×C4—122no—
232C2×C4—122no—
233C2×C4—241no—
234C2×C4—241no—
235C2×C4—241no—
236C2×C4#120⋊C311yescharacteristic
237C2×C3—161no—
238C2×C3—161no—
239C2×C3—161no—
240C2×C3—324no—
241C22—31no—
242C22—31no—
243C22—41no—
244C22—44no—
245C22—61no—
246C22—61no—
247C22—61no—
248C22—61no—
249C22—82no—
250C22—82no—
251C22—81no—
252C22—121no—
253C22—124no—
254C22—124no—
255C22—121no—
256C22—121no—
257C22—121no—
258C22—121no—
259C22—242no—
260C22—242no—
261C22—241no—
262C22—241no—
263C22C2×#120⋊C311yescharacteristic
264C4—61no—
265C4—61no—
266C4—61no—
267C4—121no—
268C4—121no—
269C4—242no—
270C4B(4,𝔽2)⋊C312yescomplemented
271C3—161noSylow
272C2—61no—
273C2—61no—
274C2—84no—
275C2—121no—
276C2—121no—
277C2C4⋊(#120⋊C3)11yescharacteristic
278C2C2×B(4,𝔽2)⋊C311yescharacteristic
279C2#3027⋊C311yescharacteristic
subgroupG/H
center, socleC22C2×#120⋊C3
commutator, cosocleC2×H(5,𝔽2)C22×C3
FrattiniC24C22×A4
Fitting#32926C3
hypercenterC2×C4#120⋊C3
hypocenterH(5,𝔽2)C3×D4
LCSG ▷ C2×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4×#997C2semidirect
2. C2×B(4,𝔽2)⋊C3C2semidirect
3. #32926C3semidirect
7. C4×H(5,𝔽2)C2×C3semidirect
8. C2×B(4,𝔽2)C2×C3semidirect
13. #997D4semidirect
19. C23×D4A4semidirect
50. H(5,𝔽2)C3×D4semidirect
270. C4B(4,𝔽2)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×#9972
C2×B(4,𝔽2)⋊C32
#329263
C2×D4×A44