Finite Group Databasebeta

← #11714027

Subgroups — #11714027

Order 768.

464 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C4×2−1+4⋊C3C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic, complemented
3#73155C212yesmaximal
4#73075C212yesmaximal, complemented
5C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
6#32970—31nomaximal, Sylow
7C2×2−1+4⋊C3C2211yescharacteristic
8#520⋊C3C2211yescharacteristic
9#519⋊C3—21no—
10C4×#41⋊C3C2211yescharacteristic
11C2×C4×SL(2,𝔽3)—21no—
12#5426C2212yes—
13C2×GU(2,𝔽3)C2212yes—
14#5391—21no—
15#5386—21no—
16#5007—41nomaximal
17C4×2−1+4D311yescharacteristic
18#3079—31no—
19#3079—31no—
20#2568—31no—
21#1860—32no—
22#1860—32no—
23#1857—32no—
24#1806—31no—
25#1806—31no—
26#1801—31no—
27#1793—32no—
282−1+4⋊C3—21no—
292−1+4⋊C3Q812yescomplemented
30C2×#41⋊C3C2311yescharacteristic
31C22×SL(2,𝔽3)—21no—
32C3×C4×D4—41no—
33#869—41no—
34C4×SL(2,𝔽3)—21no—
35C4×SL(2,𝔽3)D411yescharacteristic
36C4×SL(2,𝔽3)D411yescharacteristic
37GU(2,𝔽3)—22no—
38#859—22no—
39#832—42no—
40#807—42no—
41C4⋊(C3⋊C8)—41no—
42C2×2−1+4C2×D311yescharacteristic
43C2×C4⋊Q8—31no—
44#520—31no—
45#520—61no—
46#520C2×D311yescharacteristic
47#519—21no—
48#519—31no—
49C4×#41—31no—
50C4×#41—61no—
51C4×#41C2×D311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—31no—
54#477—61no—
55#477—61no—
56#476—61no—
57#476—61no—
58#475—61no—
59#475—61no—
60#474—61no—
61#474—61no—
62#428—32no—
63#428—32no—
64#423—31no—
65#423—31no—
66C2×#104—32no—
67C2×#104—32no—
68#417—32no—
69#417—32no—
70C2×#101—32no—
71#400—62no—
72#391—32no—
73#391—62no—
74#382—32no—
75C4×C4⋊C4—31no—
76#343—32no—
77#339—32no—
78#339—32no—
79C2×C3×D4—41no—
80C22×C3×C4—81no—
81C2×Dic6—81no—
82#41⋊C3C2×Q811yescharacteristic
83#41⋊C3#4111yescharacteristic
84C2×SL(2,𝔽3)—22no—
85C2×SL(2,𝔽3)C2×D411yescharacteristic
86C3×C4⋊C4—41no—
87C3×C22⋊C4—81no—
88C3×C42—41no—
89C4⋊Dic3—42no—
90C4⋊Dic3—81no—
91C2×C3⋊C8—42no—
922−1+4—21no—
932−1+4—32no—
942−1+4—61no—
952−1+4Dic612yescomplemented
96C2×#41—31no—
97C2×#41—61no—
98C2×#41C22×D311yescharacteristic
99C22×Q8—21no—
100C22×Q8—31no—
101C22×Q8—61no—
102C2×M16—32no—
103C2×M16—32no—
104C4⋊Q8—61no—
105C4⋊Q8—61no—
106C4⋊Q8—61no—
107C4⋊Q8—61no—
108C4×Q8—21no—
109C4×Q8—31no—
110C4×Q8—31no—
111C4×Q8—61no—
112C4×Q8—61no—
113C4×Q8—61no—
114C4×Q8—61no—
115C4×Q8—61no—
116C4×Q8#6711yescharacteristic
117C4×Q8#6711yescharacteristic
118C4×D4—31no—
119C4×D4—31no—
120C4×D4—31no—
121C4×D4—61no—
122C4×D4—61no—
123C4×D4—62no—
124C4×D4—61no—
125C4×D4S411yescharacteristic
126#117—31no—
127#117—31no—
128#117—61no—
129#117—62no—
130#117—61no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—32no—
133C2×C4⋊C4—32no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C42—31no—
138C2×C42—32no—
139C2×C42—32no—
140C2×C42—61no—
141#107—62no—
142#107—62no—
143#106—62no—
144#106—62no—
145C4⋊C8—61no—
146C4⋊C8—61no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149#104—62no—
150#104—64no—
151#104—62no—
152#103—62no—
153#103—62no—
154#102—62no—
155#102—62no—
156#101—64no—
157#95—61no—
158#95—64no—
159#95—64no—
160C23×C3—81no—
161C3×D4—42no—
162C3×D4—81no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—81no—
167C2×C3×C4—162no—
168C2×Dic3—82no—
169Dic6—162no—
170SL(2,𝔽3)C22⋊Q811yescharacteristic
171C3⋊C8—82no—
172#41—31no—
173#41—31no—
174#41—32no—
175#41—32no—
176#41—62no—
177#41—61no—
178#41—61no—
179#41—64no—
180#41—62no—
181#41C2×Dic611yescharacteristic
182#41#23511yescharacteristic
183C2×Q8—22no—
184C2×Q8—32no—
185C2×Q8—32no—
186C2×Q8—31no—
187C2×Q8—61no—
188C2×Q8—61no—
189C2×Q8—61no—
190C2×Q8—62no—
191C2×Q8—121no—
192C2×Q8—124no—
193C2×Q8—121no—
194C2×Q8C2×#6711yescharacteristic
195C2×D4—31no—
196C2×D4—61no—
197C2×D4C2×S411yescharacteristic
198C22×C4—21no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—32no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—62no—
207C22×C4—62no—
208M16—62no—
209M16—62no—
210M16—62no—
211M16—62no—
212C2×C8—62no—
213C2×C8—62no—
214C4⋊C4—31no—
215C4⋊C4—31no—
216C4⋊C4—31no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4—62no—
220C4⋊C4—62no—
221C4⋊C4—61no—
222C4⋊C4—62no—
223C4⋊C4—121no—
224C4⋊C4—122no—
225C4⋊C4—121no—
226C4⋊C4C2×S411yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—61no—
229C22⋊C4—61no—
230C22⋊C4—61no—
231C42—31no—
232C42—31no—
233C42—31no—
234C42—61no—
235C42—61no—
236C42—62no—
237C42—62no—
238C42—62no—
239C42—61no—
240C42C2×S411yescharacteristic
241C22×C3—41no—
242C22×C3—82no—
243C22×C3—161no—
244C22×C3—161no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—42no—
248C3×C4—81no—
249C3×C4—161no—
250Dic3—162no—
251C23—21no—
252C23—31no—
253Q8—31no—
254Q8—32no—
255Q8—62no—
256Q8—62no—
257Q8—61no—
258Q8—242no—
259Q8—242no—
260Q8#92411yescharacteristic
261D4—21no—
262D4—31no—
263D4—31no—
264D4—61no—
265D4—62no—
266D4—62no—
267D4A4⋊Q812yes—
268C2×C4—21no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—34no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—42no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—62no—
283C2×C4—61no—
284C2×C4—121no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—122no—
289C2×C4—124no—
290C2×C4—122no—
291C2×C4—241no—
292C2×C4GL(2,ℤ/4)11yescharacteristic
293C2×C4GL(2,ℤ/4)11yescharacteristic
294C2×C4C22×S411yescharacteristic
295C8—122no—
296C8—122no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—162no—
301C22—22no—
302C22—31no—
303C22—31no—
304C22—41no—
305C22—41no—
306C22—61no—
307C22—61no—
308C22C2×GL(2,ℤ/4)11yescharacteristic
309C4—21no—
310C4—31no—
311C4—31no—
312C4—41no—
313C4—61no—
314C4—61no—
315C4—62no—
316C4—121no—
317C4—242no—
318C4—242no—
319C4#543112yes—
320C4C2×A4⋊Q811yescharacteristic
321C4#591211yescharacteristic
322C3—41noSylow
323C2—42no—
324C2—61no—
325C2—61no—
326C2#8548011yescharacteristic
327C2C2×#543111yescharacteristic
328C2#8554211yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×2−1+4D3
prime radicalC2×2−1+4⋊C3C22
hypercenterC4×D4S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #85337C2semidirect
4. #73075C2semidirect
5. C4⋊GU(2,𝔽3)C2semidirect
29. 2−1+4⋊C3Q8semidirect
95. 2−1+4Dic6semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4⋊C32
#853372
#731552
#730752
C4⋊GU(2,𝔽3)2
#329703
#50074