Finite Group Databasebeta

← #11713987

Subgroups — #11713987

Order 768.

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

normal subgroups only

This group has 550 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
2C2×#5430C212yesmaximal
3#85336C211yesmaximal, characteristic, complemented
4#73077C212yesmaximal, complemented
5C4⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
6#32923—31nomaximal, Sylow
7C2×2−1+4⋊C3C2211yescharacteristic
8C2×#984C2212yes—
9#520⋊C3C2211yescharacteristic
10#519⋊C3—21no—
11C4×#41⋊C3C2211yescharacteristic
12C2×C4×SL(2,𝔽3)—21no—
13#5430—24no—
14C2×GU(2,𝔽3)C2212yes—
15C4⋊#226—21no—
16C4⋊#226—21no—
17#5019—41nomaximal
18C4×2−1+4D311yescharacteristic
19#3039—31no—
20#3039—31no—
21C2×#456—32no—
22#2546—31no—
23#1886—32no—
24#1882—32no—
25#1882—32no—
26#1806—31no—
27#1806—31no—
28#1801—31no—
292−1+4⋊C3—21no—
302−1+4⋊C3D412yes—
31C2×#41⋊C3C2311yescharacteristic
32C22×SL(2,𝔽3)—21no—
33#984—22no—
34#984—22no—
35C2×#226—22no—
36C3×C4×D4—41no—
37C2×#214—42no—
38#869—41no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)D411yescharacteristic
41C4×SL(2,𝔽3)D411yescharacteristic
42GU(2,𝔽3)—22no—
43#810—42no—
44C4⋊(C3⋊C8)—41no—
45C2×2−1+4C2×D311yescharacteristic
46C2×#137—32no—
47C2×#137—32no—
48C2×C4⋊Q8—32no—
49C2×C4⋊Q8—31no—
50#520—31no—
51#520—61no—
52#520C2×D311yescharacteristic
53#519—21no—
54#519—31no—
55C4×#41—31no—
56C4×#41—61no—
57C4×#41C2×D311yescharacteristic
58C2×C4×Q8—21no—
59C2×C4×Q8—31no—
60#462—61no—
61#462—61no—
62#462—61no—
63#462—61no—
64#461—61no—
65#461—61no—
66#461—61no—
67#461—61no—
68#456—64no—
69#456—64no—
70#423—31no—
71#423—31no—
72C2×#104—32no—
73C2×#104—32no—
74#419—32no—
75#419—32no—
76C2×#101—32no—
77#398—62no—
78#391—32no—
79#389—62no—
80C4×C4⋊C4—31no—
81#343—32no—
82#339—32no—
83#339—32no—
84C2×C3×D4—41no—
85C22×C3×C4—81no—
86C2×Dic6—42no—
87#41⋊C3C2×D411yescharacteristic
88#41⋊C3#4111yescharacteristic
89C2×SL(2,𝔽3)—22no—
90C2×SL(2,𝔽3)C2×D411yescharacteristic
91#226—42no—
92C3×C4⋊C4—41no—
93C3×C22⋊C4—81no—
94C3×C42—41no—
95#214—84no—
96C4⋊Dic3—81no—
97C4⋊Dic3—81no—
98C2×C3⋊C8—42no—
992−1+4—21no—
1002−1+4—32no—
1012−1+4—61no—
1022−1+4D1212yes—
103C2×#41—31no—
104C2×#41—61no—
105C2×#41C22×D311yescharacteristic
106C22×Q8—21no—
107C22×Q8—31no—
108C22×Q8—32no—
109#137—62no—
110#137—64no—
111#137—62no—
112#137—64no—
113#137—62no—
114#137—62no—
115C2×Q16—62no—
116C2×Q16—62no—
117C2×SD16—62no—
118C2×SD16—62no—
119C2×M16—32no—
120C2×M16—32no—
121C4⋊Q8—61no—
122C4⋊Q8—61no—
123C4⋊Q8—61no—
124C4⋊Q8—64no—
125C4⋊Q8—64no—
126C4⋊Q8—61no—
127C4×Q8—21no—
128C4×Q8—31no—
129C4×Q8—31no—
130C4×Q8—61no—
131C4×Q8—61no—
132C4×Q8—61no—
133C4×Q8—61no—
134C4×Q8—61no—
135C4×Q8#6711yescharacteristic
136C4×Q8#6711yescharacteristic
137C4×D4—31no—
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—61no—
141C4×D4—61no—
142C4×D4—62no—
143C4×D4—61no—
144C4×D4S411yescharacteristic
145#117—31no—
146#117—31no—
147#117—61no—
148#117—62no—
149#117—61no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—64no—
156C2×C4⋊C4—61no—
157C2×C42—31no—
158C2×C42—32no—
159C2×C42—32no—
160C2×C42—61no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#104—62no—
166#104—64no—
167#104—62no—
168#103—62no—
169#103—62no—
170#103—62no—
171#103—62no—
172#101—64no—
173#95—61no—
174#95—64no—
175C23×C3—81no—
176C3×D4—42no—
177C3×D4—81no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—81no—
182C2×C3×C4—162no—
183C2×Dic3—82no—
184Dic6—82no—
185Dic6—82no—
186SL(2,𝔽3)#12111yescharacteristic
187C3⋊C8—82no—
188#41—31no—
189#41—31no—
190#41—32no—
191#41—32no—
192#41—62no—
193#41—61no—
194#41—61no—
195#41—64no—
196#41—62no—
197#41C2×D1211yescharacteristic
198#41#23511yescharacteristic
199C2×Q8—22no—
200C2×Q8—32no—
201C2×Q8—32no—
202C2×Q8—31no—
203C2×Q8—61no—
204C2×Q8—61no—
205C2×Q8—61no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—62no—
209C2×Q8—62no—
210C2×Q8—62no—
211C2×Q8—62no—
212C2×Q8—62no—
213C2×Q8C2×#6711yescharacteristic
214C2×D4—31no—
215C2×D4—61no—
216C2×D4C2×S411yescharacteristic
217C22×C4—21no—
218C22×C4—31no—
219C22×C4—32no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—32no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—62no—
226C22×C4—62no—
227Q16—122no—
228Q16—124no—
229Q16—122no—
230SD16—122no—
231SD16—122no—
232SD16—124no—
233M16—62no—
234M16—62no—
235M16—62no—
236M16—62no—
237C2×C8—62no—
238C2×C8—62no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C4⋊C4—64no—
246C4⋊C4—61no—
247C4⋊C4—128no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4C2×S411yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C42—31no—
258C42—31no—
259C42—31no—
260C42—61no—
261C42—61no—
262C42—62no—
263C42—62no—
264C42—62no—
265C42—61no—
266C42C2×S411yescharacteristic
267C22×C3—41no—
268C22×C3—82no—
269C22×C3—161no—
270C22×C3—161no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—42no—
274C3×C4—81no—
275C3×C4—161no—
276Dic3—162no—
277C23—21no—
278C23—31no—
279Q8—31no—
280Q8—32no—
281Q8—62no—
282Q8—62no—
283Q8—61no—
284Q8—122no—
285Q8—122no—
286Q8—122no—
287Q8—122no—
288Q8#93011yescharacteristic
289D4—21no—
290D4—31no—
291D4—31no—
292D4—61no—
293D4—62no—
294D4—62no—
295D4C4⋊S412yes—
296C2×C4—21no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—34no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—42no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—62no—
311C2×C4—64no—
312C2×C4—61no—
313C2×C4—121no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—241no—
321C2×C4GL(2,ℤ/4)11yescharacteristic
322C2×C4GL(2,ℤ/4)11yescharacteristic
323C2×C4C22×S411yescharacteristic
324C8—122no—
325C8—122no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—162no—
330C22—22no—
331C22—31no—
332C22—31no—
333C22—41no—
334C22—41no—
335C22—61no—
336C22—61no—
337C22C2×GL(2,ℤ/4)11yescharacteristic
338C4—21no—
339C4—31no—
340C4—31no—
341C4—41no—
342C4—61no—
343C4—61no—
344C4—62no—
345C4—124no—
346C4—121no—
347C4—242no—
348C4#543012yescomplemented
349C4C2×C4⋊S411yescharacteristic
350C4#591211yescharacteristic
351C3—41noSylow
352C2—42no—
353C2—61no—
354C2—61no—
355C2#8548611yescharacteristic
356C2C2×#543011yescharacteristic
357C2#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)#121
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
3. #85336C2semidirect
4. #73077C2semidirect
5. C4⋊GU(2,𝔽3)C2semidirect
348. C4#5430semidirect

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
C2×#54302
#853362
#730772
C4⋊GU(2,𝔽3)2
#329233
#50194