Finite Group Databasebeta

← #11714026

Subgroups — #11714026

Order 768.

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

normal subgroups only

This group has 504 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×#994C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic, complemented
3#73153C212yesmaximal
4#73076C212yesmaximal, complemented
5C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
6#32969—31nomaximal, Sylow
7C2×#994C2211yescharacteristic
8#520⋊C3—21no—
9#520⋊C3C2211yescharacteristic
10C4×#41⋊C3—21no—
11C4×#41⋊C3C2211yescharacteristic
12#5426C2212yes—
13C2×GU(2,𝔽3)C2212yes—
14#5391—21no—
15#5386—21no—
16#5020—41nomaximal
17C4×H(5,𝔽2)D311yescharacteristic
18#3079—31no—
19#3079—31no—
20#2542—31no—
21#1859—32no—
22#1859—32no—
23#1856—32no—
24#1806—31no—
25#1806—31no—
26#1800—31no—
27#1792—32no—
28#994—21no—
29#994Q812yescomplemented
30C2×#41⋊C3—21no—
31C2×#41⋊C3C2311yescharacteristic
32C3×C4×Q8—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#835—42no—
40#808—42no—
41C4⋊(C3⋊C8)—41no—
42C2×H(5,𝔽2)C2×D311yescharacteristic
43C2×C4⋊Q8—31no—
44C2×C22⋊Q8—62no—
45#520—21no—
46#520—31no—
47#520C2×D311yescharacteristic
48#518—31no—
49#518—61no—
50C4×#41—21no—
51C4×#41—31no—
52C4×#41C2×D311yescharacteristic
53C2×C4×D4—31no—
54C2×C4×D4—61no—
55#477—61no—
56#477—61no—
57#476—61no—
58#476—61no—
59#475—61no—
60#475—61no—
61#474—61no—
62#474—61no—
63#428—32no—
64#428—32no—
65#423—31no—
66#423—31no—
67C2×#104—32no—
68C2×#104—32no—
69#417—32no—
70#417—32no—
71C2×#100—32no—
72#399—62no—
73#390—32no—
74#384—32no—
75C4×C4⋊C4—31no—
76#343—32no—
77#339—32no—
78#339—32no—
79C2×C3×Q8—41no—
80C2×Dic6—81no—
81#41⋊C3—22no—
82#41⋊C3C2×Q811yescharacteristic
83#41⋊C3#4111yescharacteristic
84C2×SL(2,𝔽3)C2×D411yescharacteristic
85C3×C4⋊C4—41no—
86C3×C4⋊C4—81no—
87C3×C42—41no—
88C3×C42—81no—
89C4⋊Dic3—42no—
90C4⋊Dic3—81no—
91C2×C3⋊C8—42no—
92H(5,𝔽2)—21no—
93H(5,𝔽2)—32no—
94H(5,𝔽2)—61no—
95H(5,𝔽2)Dic612yescomplemented
96C2×#41—21no—
97C2×#41—31no—
98C2×#41C22×D311yescharacteristic
99C22×Q8—61no—
100C22×D4—31no—
101C22×D4—61no—
102C23×C4—32no—
103C2×M16—32no—
104C2×M16—32no—
105C4⋊Q8—61no—
106C4⋊Q8—61no—
107C4⋊Q8—61no—
108C4⋊Q8—61no—
109C22⋊Q8—128no—
110C4×Q8—21no—
111C4×Q8—31no—
112C4×Q8#6711yescharacteristic
113C4×Q8#6711yescharacteristic
114C4×Q8S411yescharacteristic
115C4×D4—31no—
116C4×D4—31no—
117C4×D4—31no—
118C4×D4—31no—
119C4×D4—61no—
120C4×D4—61no—
121C4×D4—61no—
122C4×D4—61no—
123C4×D4—61no—
124C4×D4—61no—
125C4×D4—62no—
126C4×D4—61no—
127C4×D4—61no—
128#117—31no—
129#117—31no—
130#117—61no—
131#117—61no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—32no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—64no—
137C2×C4⋊C4—61no—
138C2×C22⋊C4—32no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—64no—
141C2×C42—31no—
142C2×C42—32no—
143C2×C42—61no—
144#107—62no—
145#107—62no—
146#106—62no—
147#106—62no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8—61no—
152#104—62no—
153#104—64no—
154#104—62no—
155#103—62no—
156#103—62no—
157#102—62no—
158#102—62no—
159#100—64no—
160#95—61no—
161C3×Q8—42no—
162C3×Q8—81no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×Dic3—82no—
169Dic6—162no—
170SL(2,𝔽3)C22⋊Q811yescharacteristic, complemented
171C3⋊C8—82no—
172C24—32no—
173#41—22no—
174#41—31no—
175#41—31no—
176#41—32no—
177#41—32no—
178#41—61no—
179#41—61no—
180#41—62no—
181#41C2×Dic611yescharacteristic
182#41#23511yescharacteristic
183C2×Q8—121no—
184C2×Q8—124no—
185C2×Q8—121no—
186C2×Q8C2×#6711yescharacteristic
187C2×Q8C2×S411yescharacteristic
188C2×D4—31no—
189C2×D4—32no—
190C2×D4—32no—
191C2×D4—31no—
192C2×D4—61no—
193C2×D4—61no—
194C2×D4—61no—
195C2×D4—61no—
196C2×D4—62no—
197C2×D4—62no—
198C2×D4—64no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—32no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—61no—
208C22×C4—62no—
209C22×C4—124no—
210M16—62no—
211M16—62no—
212M16—62no—
213M16—62no—
214C2×C8—62no—
215C2×C8—62no—
216C4⋊C4—21no—
217C4⋊C4—31no—
218C4⋊C4—31no—
219C4⋊C4—31no—
220C4⋊C4—61no—
221C4⋊C4—61no—
222C4⋊C4—62no—
223C4⋊C4—64no—
224C4⋊C4—62no—
225C4⋊C4—121no—
226C4⋊C4—128no—
227C4⋊C4—121no—
228C4⋊C4C2×S411yescharacteristic
229C22⋊C4—61no—
230C22⋊C4—61no—
231C22⋊C4—61no—
232C22⋊C4—62no—
233C22⋊C4—61no—
234C22⋊C4—128no—
235C42—21no—
236C42—31no—
237C42—31no—
238C42—31no—
239C42—61no—
240C42—61no—
241C42—62no—
242C42—62no—
243C42C2×S411yescharacteristic
244C22×C3—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—42no—
248C3×C4—81no—
249C3×C4—82no—
250C3×C4—161no—
251Dic3—162no—
252C23—31no—
253C23—34no—
254C23—62no—
255C23—61no—
256C23—122no—
257C23—122no—
258Q8—21no—
259Q8—31no—
260Q8—242no—
261Q8—242no—
262Q8#92411yescharacteristic
263Q8A4⋊Q812yes—
264D4—31no—
265D4—31no—
266D4—32no—
267D4—62no—
268D4—62no—
269D4—61no—
270D4—62no—
271D4—62no—
272D4—61no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—64no—
286C2×C4—61no—
287C2×C4—121no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—124no—
293C2×C4—122no—
294C2×C4—241no—
295C2×C4—242no—
296C2×C4GL(2,ℤ/4)11yescharacteristic
297C2×C4GL(2,ℤ/4)11yescharacteristic
298C2×C4C22×S411yescharacteristic
299C8—122no—
300C8—122no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C22—31no—
305C22—31no—
306C22—61no—
307C22—61no—
308C22—62no—
309C22—124no—
310C22—121no—
311C22—121no—
312C22—242no—
313C22C2×GL(2,ℤ/4)11yescharacteristic
314C4—21no—
315C4—22no—
316C4—31no—
317C4—31no—
318C4—41no—
319C4—61no—
320C4—61no—
321C4—124no—
322C4—121no—
323C4—242no—
324C4#542912yes—
325C4C2×A4⋊Q811yescharacteristic
326C4#591211yescharacteristic
327C3—41noSylow
328C2—61no—
329C2—61no—
330C2—122no—
331C2#8548011yescharacteristic
332C2C2×#542911yescharacteristic
333C2#8554211yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
FittingC4×H(5,𝔽2)D3
prime radicalC2×#994C22
hypercenterC4×Q8S4
hypocenterSL(2,𝔽3)C22⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. #85337C2semidirect
4. #73076C2semidirect
5. C4⋊GU(2,𝔽3)C2semidirect
29. #994Q8semidirect
95. H(5,𝔽2)Dic6semidirect
170. SL(2,𝔽3)C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#9942
#853372
#731532
#730762
C4⋊GU(2,𝔽3)2
#329693
#50204