Finite Group Databasebeta

← #11712610

Subgroups — #11712610

Order 768.

1111 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C22×2−1+4⋊C3C211yesmaximal, characteristic
2C2×#5426C211yesmaximal, characteristic, complemented
3C2×#5424C211yesmaximal, characteristic, complemented
4#73148C214yesmaximal, complemented
5#20603—31nomaximal, Sylow
6C2×2−1+4⋊C3—22no—
7C2×2−1+4⋊C3—22no—
8C2×2−1+4⋊C3C414yescomplemented
9C22×#41⋊C3C2211yescharacteristic
10C23×SL(2,𝔽3)C2212yes—
11#5426—22no—
12#5424—22no—
13C2×#859C2212yes—
14C2×#859C2212yes—
15#5221—41nomaximal
16C22×2−1+4D311yescharacteristic
17C2×#417—31no—
18C2×#419—31no—
19C2×#416—31no—
20C22×#103—31no—
21C2×#408—31no—
22#2311—31no—
23#1788—34no—
24#1787—34no—
25C22×C3×D4—41no—
262−1+4⋊C3—28no—
27C2×#41⋊C3C2×C412yes—
28C2×#41⋊C3D412yes—
29C2×#41⋊C3D412yes—
30C22×SL(2,𝔽3)—22no—
31C22×SL(2,𝔽3)—22no—
32C22×SL(2,𝔽3)C2×C414yescomplemented
33C22×SL(2,𝔽3)C2311yescharacteristic
34C2×#217—44no—
35C2×C4⋊Dic3—41no—
36C2×C4×Dic3—41no—
37#859—22no—
38#859—22no—
39C2×2−1+4—22no—
40C2×2−1+4—22no—
41C2×2−1+4—34no—
42C2×2−1+4—62no—
43C2×2−1+4—62no—
44C2×2−1+4Dic314yescomplemented
45C22×#41—31no—
46C22×#41—32no—
47C22×#41C2×D311yescharacteristic
48C23×Q8—31no—
49C23×Q8C2×D312yes—
50C22×M16—31no—
51C23×C8—31no—
52C2×#117—31no—
53C22×C4⋊C4—31no—
54#419—64no—
55#417—62no—
56#417—62no—
57#416—62no—
58#416—62no—
59C2×#103—34no—
60C2×#103—34no—
61C2×#103—32no—
62C2×#103—32no—
63C2×#103—64no—
64C2×#102—32no—
65C2×#102—32no—
66#408—64no—
67C2×C22⋊C8—34no—
68#391—61no—
69#390—61no—
70#386—62no—
71#380—34no—
72C24×C3—42no—
73C2×C3×D4—44no—
74C2×C3×D4—82no—
75C2×C3×D4—82no—
76C22×C3×C4—41no—
77C22×Dic3—42no—
78C22×Dic3—42no—
79#41⋊C3C22⋊C414yescomplemented
80C2×SL(2,𝔽3)—24no—
81C2×SL(2,𝔽3)C22×C411yescharacteristic
82C2×SL(2,𝔽3)C2×D411yescharacteristic
83C2×SL(2,𝔽3)C2×D411yescharacteristic
84#217—88no—
85C4⋊Dic3—82no—
86C4×Dic3—82no—
872−1+4—28no—
882−1+4—68no—
892−1+4—616no—
902−1+4—68no—
912−1+4—68no—
92C2×#41—34no—
93C2×#41—34no—
94C2×#41—32no—
95C2×#41—32no—
96C2×#41—32no—
97C2×#41—38no—
98C2×#41—62no—
99C2×#41—62no—
100C2×#41—62no—
101C2×#41—62no—
102C2×#41—64no—
103C2×#41—64no—
104C2×#41—64no—
105C2×#41—64no—
106C2×#41—64no—
107C2×#41C2×Dic312yes—
108C2×#41#6712yes—
109C2×#41#6712yes—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—31no—
113C22×Q8—32no—
114C22×Q8—34no—
115C22×Q8—34no—
116C22×Q8—34no—
117C22×Q8—34no—
118C22×Q8—62no—
119C22×Q8—62no—
120C22×Q8—62no—
121C22×Q8—62no—
122C22×Q8—62no—
123C22×Q8C2×Dic314yes—
124C22×Q8C22×D311yescharacteristic
125C22×D4—31no—
126C22×D4—32no—
127C22×D4S411yescharacteristic
128C23×C4—31no—
129C23×C4—32no—
130C23×C4—61no—
131C23×C4—61no—
132C2×M16—62no—
133C2×M16—62no—
134C2×M16—62no—
135C22×C8—32no—
136C22×C8—32no—
137C22×C8—62no—
138C22×C8—62no—
139C22×C8—62no—
140#117—62no—
141#117—62no—
142C2×C4⋊C4—32no—
143C2×C4⋊C4—32no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—64no—
149C2×C42—61no—
150#103—68no—
151#103—62no—
152#103—62no—
153#103—68no—
154#103—62no—
155#103—62no—
156#102—64no—
157#102—64no—
158C22⋊C8—68no—
159#95—64no—
160C23×C3—44no—
161C23×C3—41no—
162C23×C3—84no—
163C23×C3—82no—
164C23×C3—82no—
165C23×C3—84no—
166C3×D4—88no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×Dic3—44no—
171C2×Dic3—82no—
172C2×Dic3—82no—
173C2×Dic3—82no—
174C2×Dic3—84no—
175SL(2,𝔽3)C2×C22⋊C411yescharacteristic
176C24—31no—
177C24C2×S412yes—
178#41—34no—
179#41—34no—
180#41—34no—
181#41—64no—
182#41—68no—
183#41—68no—
184#41—68no—
185#41—68no—
186#41—616no—
187#41—616no—
188#41—616no—
189#41—616no—
190#41#21714yescomplemented
191C2×Q8—24no—
192C2×Q8—34no—
193C2×Q8—31no—
194C2×Q8—31no—
195C2×Q8—34no—
196C2×Q8—31no—
197C2×Q8—34no—
198C2×Q8—34no—
199C2×Q8—316no—
200C2×Q8—62no—
201C2×Q8—68no—
202C2×Q8—68no—
203C2×Q8—68no—
204C2×Q8—62no—
205C2×Q8—62no—
206C2×Q8—68no—
207C2×Q8—62no—
208C2×Q8—62no—
209C2×Q8—64no—
210C2×Q8—64no—
211C2×Q8—64no—
212C2×Q8C22×Dic311yescharacteristic
213C2×Q8C2×#6711yescharacteristic
214C2×Q8C2×#6711yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—32no—
218C2×D4—32no—
219C2×D4—32no—
220C2×D4—38no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—64no—
225C2×D4—64no—
226C2×D4SL(2,ℤ/4)14yes—
227C22×C4—31no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—32no—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4—32no—
234C22×C4—34no—
235C22×C4—34no—
236C22×C4—32no—
237C22×C4—64no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—64no—
243C22×C4—64no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—122no—
247C22×C4—242no—
248C22×C4C2×S411yescharacteristic
249M16—124no—
250C2×C8—62no—
251C2×C8—62no—
252C2×C8—62no—
253C2×C8—62no—
254C2×C8—62no—
255C2×C8—62no—
256C2×C8—124no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—122no—
262C22⋊C4—122no—
263C22⋊C4—244no—
264C42—122no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—84no—
271C22×C3—88no—
272C22×C3—162no—
273C22×C3—164no—
274C22×C3—162no—
275C3×C4—44no—
276Dic3—84no—
277Dic3—162no—
278C23—24no—
279C23—24no—
280C23—22no—
281C23—22no—
282C23—32no—
283C23—32no—
284C23—32no—
285C23—62no—
286C23—62no—
287C23C2×SL(2,ℤ/4)14yes—
288C23C22×S411yescharacteristic
289Q8—31no—
290Q8—31no—
291Q8—31no—
292Q8—38no—
293Q8—61no—
294Q8—64no—
295Q8—64no—
296Q8—64no—
297Q8—68no—
298Q8C2×#21711yescharacteristic
299D4—28no—
300D4—34no—
301D4—62no—
302D4—62no—
303D4—62no—
304D4—616no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—34no—
312C2×C4—34no—
313C2×C4—34no—
314C2×C4—34no—
315C2×C4—34no—
316C2×C4—64no—
317C2×C4—64no—
318C2×C4—64no—
319C2×C4—64no—
320C2×C4—616no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—122no—
324C2×C4—124no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—244no—
329C2×C4C2×SL(2,ℤ/4)12yes—
330C2×C4GL(2,ℤ/4)12yes—
331C2×C4GL(2,ℤ/4)12yes—
332C8—122no—
333C8—122no—
334C2×C3—42no—
335C2×C3—41no—
336C2×C3—42no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—164no—
340C22—28no—
341C22—24no—
342C22—34no—
343C22—42no—
344C22—44no—
345C22—42no—
346C22—64no—
347C22—62no—
348C22—64no—
349C22—62no—
350C22#592612yes—
351C22#592712yes—
352C22C22×SL(2,ℤ/4)11yescharacteristic
353C22C2×GL(2,ℤ/4)11yescharacteristic
354C22C2×GL(2,ℤ/4)11yescharacteristic
355C4—31no—
356C4—31no—
357C4—31no—
358C4—31no—
359C4—64no—
360C4—242no—
361C4—242no—
362C4#543214yes—
363C3—41noSylow
364C2—44no—
365C2—64no—
366C2#8553412yes—
367C2#8553611yescharacteristic
368C2#8553812yes—
369C2#8554011yescharacteristic
370C2C2×#543211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingC22×2−1+4D3
prime radicalC22×2−1+4⋊C3C2
cosocleC22×SL(2,𝔽3)C23
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)C2×C22⋊C4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5426C2semidirect
3. C2×#5424C2semidirect
4. #73148C2semidirect
8. C2×2−1+4⋊C3C4semidirect
32. C22×SL(2,𝔽3)C2×C4semidirect
44. C2×2−1+4Dic3semidirect
79. #41⋊C3C22⋊C4semidirect
190. #41#217semidirect

Maximal subgroups

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

subgroupindex
C22×2−1+4⋊C32
C2×#54262
C2×#54242
#731482
#206033
#52214