Finite Group Databasebeta

← C4⋊C4×GL(2,𝔽3) #11713019

Subgroups — C4⋊C4×GL(2,𝔽3) #11713019

Order 768.

695 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 87 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×C4×GL(2,𝔽3)C212yesmaximal
2C2×C4×GL(2,𝔽3)C211yesmaximal, characteristic
3C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4⋊#859C211yesmaximal, characteristic, complemented
5#73059C212yesmaximal, complemented
6C4⋊C4×SD16—31nomaximal, Sylow
7C22×GL(2,𝔽3)C2211yescharacteristic
8C2×D3×C4⋊C4—41nomaximal
9C2×C4×SL(2,𝔽3)C2212yes—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C2×#859C2211yescharacteristic
13C4×GL(2,𝔽3)—24no—
14C4×GL(2,𝔽3)C414yescomplemented
15C2×C4×SD16—31no—
16C2×C4×SD16—32no—
17Q8×C4⋊C4D311yescharacteristic, complemented
18D4×C4⋊C4—31no—
19#1941—32no—
20#1865—32no—
21#1864—32no—
22#1844—31no—
23#1803—31no—
24#1802—31no—
25C8×C4⋊C4—31no—
26C22×C4×D3—41no—
27C22×C4×D3—42no—
28C22×SL(2,𝔽3)C2311yescharacteristic
29C2×GL(2,𝔽3)C2×C412yes—
30C2×GL(2,𝔽3)D412yes—
31C2×GL(2,𝔽3)Q812yes—
32C2×C3×C4⋊C4—41no—
33C2×C4⋊Dic3—41no—
34C2×Dic3⋊C4—42no—
35D3×C4⋊C4—48no—
36C4×SL(2,𝔽3)—22no—
37C4×SL(2,𝔽3)C2×C412yescomplemented
38#859—22no—
39#859C2×C412yescomplemented
40C22×SD16—31no—
41C2×C4×Q8C2×D312yes—
42C2×C4×Q8C2×D311yescharacteristic
43C2×C4×D4—31no—
44C2×C4×D4—32no—
45C22×C4⋊C4—61no—
46C4×SD16—34no—
47C4×SD16—34no—
48C4×SD16—64no—
49C4×SD16—64no—
50C2×#106—31no—
51C2×#106—32no—
52C2×#106—32no—
53C2×C4⋊C8—31no—
54C2×#103—31no—
55C2×#103—32no—
56C2×#102—31no—
57C2×#102—32no—
58C2×C4×C8—31no—
59C2×C4×C8—32no—
60#389—32no—
61#389—32no—
62#385—62no—
63#384—31no—
64#384—31no—
65#380—61no—
66C4×C4⋊C4—31no—
67#340—32no—
68#340—32no—
69#336—32no—
70C23×D3—41no—
71C22×C3×C4—41no—
72C22×C3×C4—42no—
73C22×Dic3—41no—
74C22×Dic3—42no—
75C2×C4×D3—44no—
76C2×C4×D3—44no—
77C2×C4×D3—44no—
78C2×C4×D3—48no—
79C2×C4×D3—84no—
80C2×C4×D3—84no—
81C2×SL(2,𝔽3)C22×C411yescharacteristic
82C2×SL(2,𝔽3)C2×D411yescharacteristic
83C2×SL(2,𝔽3)C2×Q811yescharacteristic
84GL(2,𝔽3)C4⋊C414yesdirect product factor
85C3×C4⋊C4—44no—
86C4⋊Dic3—44no—
87Dic3⋊C4—48no—
88C22×Q8C22×D311yescharacteristic
89C22×D4—31no—
90C23×C4—61no—
91C23×C4—62no—
92C2×SD16—32no—
93C2×SD16—32no—
94C2×SD16—32no—
95C2×SD16—32no—
96C2×SD16—32no—
97C2×SD16—32no—
98C22×C8—31no—
99C22×C8—31no—
100C22×C8—32no—
101C4×Q8—22no—
102C4×Q8—32no—
103C4×Q8—62no—
104C4×Q8C4×D312yescomplemented
105C4×D4—34no—
106C4×D4—62no—
107C4×D4—64no—
108C4×D4—122no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—124no—
119C2×C4⋊C4S411yescharacteristic
120C2×C22⋊C4—61no—
121C2×C22⋊C4—62no—
122C2×C42—31no—
123C2×C42—32no—
124#106—34no—
125#106—64no—
126#106—122no—
127C4⋊C8—62no—
128#103—32no—
129#103—32no—
130#103—62no—
131#103—62no—
132#102—34no—
133#102—64no—
134C4×C8—34no—
135C4×C8—64no—
136#95—32no—
137#95—62no—
138C23×C3—41no—
139C22×D3—42no—
140C22×D3—42no—
141C22×D3—42no—
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—42no—
145C22×D3—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—44no—
149C2×C3×C4—42no—
150C2×C3×C4—82no—
151C2×C3×C4—82no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—44no—
156C2×Dic3—82no—
157C2×Dic3—82no—
158C4×D3—416no—
159C4×D3—816no—
160SL(2,𝔽3)C2×C4⋊C411yescharacteristic, complemented
161C24—61no—
162C2×Q8—31no—
163C2×Q8—31no—
164C2×Q8—31no—
165C2×Q8C2×C4×D311yescharacteristic
166C2×Q8D3×D411yescharacteristic
167C2×Q8D3×Q811yescharacteristic
168C2×D4—32no—
169C2×D4—32no—
170C2×D4—32no—
171C2×D4—61no—
172C2×D4—61no—
173C2×D4—61no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—32no—
177C22×C4—61no—
178C22×C4—64no—
179C22×C4—62no—
180C22×C4—124no—
181C22×C4—122no—
182C22×C4—122no—
183C22×C4—124no—
184C22×C4—242no—
185C22×C4C2×S412yes—
186C22×C4C2×S411yescharacteristic
187SD16—34no—
188SD16—34no—
189SD16—34no—
190SD16—34no—
191C2×C8—32no—
192C2×C8—32no—
193C2×C8—32no—
194C2×C8—32no—
195C2×C8—61no—
196C2×C8—61no—
197C2×C8—62no—
198C2×C8—62no—
199C2×C8—62no—
200C4⋊C4—32no—
201C4⋊C4—32no—
202C4⋊C4—61no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C4⋊C4—61no—
206C4⋊C4—62no—
207C4⋊C4—62no—
208C4⋊C4—62no—
209C4⋊C4—124no—
210C4⋊C4—122no—
211C4⋊C4—241no—
212C4⋊C4—241no—
213C4⋊C4GL(2,𝔽3)14yesdirect product factor
214C22⋊C4—64no—
215C22⋊C4—124no—
216C42—32no—
217C42—32no—
218C42—62no—
219C42—62no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C2×D3—44no—
228C2×D3—44no—
229C2×D3—44no—
230C2×D3—44no—
231C2×D3—44no—
232C2×D3—44no—
233C2×D3—44no—
234C3×C4—44no—
235C3×C4—84no—
236Dic3—44no—
237Dic3—84no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—121no—
242C23—121no—
243C23—121no—
244C23—121no—
245C23C22×S411yescharacteristic
246Q8—31no—
247Q8—31no—
248Q8—31no—
249Q8—61no—
250Q8D3×C4⋊C411yescharacteristic, complemented
251D4—34no—
252D4—62no—
253D4—62no—
254D4—62no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—128no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—248no—
277C2×C4C4×S412yes—
278C2×C4C2×GL(2,𝔽3)14yes—
279C2×C4C2×GL(2,𝔽3)12yes—
280C2×C4#98512yes—
281C8—34no—
282C8—62no—
283C8—122no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291D3—48no—
292C22—64no—
293C22—122no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22C2×C4×S411yescharacteristic
300C22D4×S411yescharacteristic
301C22C22×GL(2,𝔽3)11yescharacteristic
302C22Q8×S411yescharacteristic
303C22C2×#98511yescharacteristic
304C22#592411yescharacteristic
305C22#592611yescharacteristic
306C4—24no—
307C4—31no—
308C4—31no—
309C4—31no—
310C4—31no—
311C4—62no—
312C4—122no—
313C4—122no—
314C4—242no—
315C4C4×GL(2,𝔽3)14yescomplemented
316C3—41noSylow
317C2—124no—
318C2C4⋊C4×S411yescharacteristic
319C2C2×C4×GL(2,𝔽3)11yescharacteristic
320C2#8536411yescharacteristic
321C2D4×GL(2,𝔽3)11yescharacteristic
322C2#8540711yescharacteristic
323C2#8541811yescharacteristic
324C2Q8×GL(2,𝔽3)11yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingQ8×C4⋊C4D3
prime radicalC22×GL(2,𝔽3)C22
cosocleC22×SL(2,𝔽3)C23
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×SL(2,𝔽3)C2semidirect
4. C4⋊#859C2semidirect
5. #73059C2semidirect
14. C4×GL(2,𝔽3)C4semidirect
17. Q8×C4⋊C4D3semidirect
37. C4×SL(2,𝔽3)C2×C4semidirect
39. #859C2×C4semidirect
84. GL(2,𝔽3)C4⋊C4direct (N × H)
104. C4×Q8C4×D3semidirect
160. SL(2,𝔽3)C2×C4⋊C4semidirect
213. C4⋊C4GL(2,𝔽3)direct (N × H)
250. Q8D3×C4⋊C4semidirect
315. C4C4×GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C4×GL(2,𝔽3)2
C4⋊C4×SL(2,𝔽3)2
C4⋊#8592
#730592
C4⋊C4×SD163
C2×D3×C4⋊C44