Finite Group Databasebeta

← #11712646

Subgroups — #11712646

Order 768.

649 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 101 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×A4×M16C211yesmaximal, characteristic
2#85482C211yesmaximal, characteristic
3C2×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4#73111C212yesmaximal, complemented
5#73110C212yesmaximal, complemented
6#20661—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8C22×SL(2,ℤ/4)—21no—
9A4×M16C414yescomplemented
10C2×C8×A4C2212yes—
11#5432—21no—
12C4⋊SL(2,ℤ/4)—21no—
13C4⋊SL(2,ℤ/4)C2211yescharacteristic
14C4⋊SL(2,ℤ/4)C2212yes—
15C4⋊SL(2,ℤ/4)C2211yescharacteristic
16C4×SL(2,ℤ/4)—21no—
17#5121—41nomaximal
18C23×M16D311yescharacteristic
19C2×#428—31no—
20C2×#428—31no—
21#2280—31no—
22C2×#380—31no—
23#1809—32no—
24#1809—32no—
25#1808—32no—
26#1807—32no—
27#1736—31no—
28#1736—31no—
29C23×A4D411yescharacteristic
30C2×C4×A4C2×C412yes—
31C2×C4×A4D411yescharacteristic
32C2×C4×A4Q812yes—
33C2×C4×A4C2311yescharacteristic
34C2×SL(2,ℤ/4)—21no—
35C2×SL(2,ℤ/4)—21no—
36C2×SL(2,ℤ/4)—22no—
37C2×SL(2,ℤ/4)—42no—
38C2×C3×M16—41no—
39#926—41no—
40C2×C4⋊Dic3—41no—
41C8×A4C2×C414yescomplemented
42#818—42no—
43#817—42no—
44C24×C4C2×D311yescharacteristic
45C22×M16—31no—
46C22×M16—32no—
47C22×M16—31no—
48C22×M16—34no—
49C22×M16Dic314yescomplemented
50C23×C8C2×D312yes—
51C2×#117—31no—
52C2×#117—31no—
53C22×C4⋊C4—31no—
54C22×C4⋊C4—31no—
55C22×C22⋊C4—61no—
56#428—64no—
57#428—61no—
58#428—62no—
59#428—61no—
60C2×#107—32no—
61C2×#107—32no—
62C2×#106—32no—
63C2×#106—32no—
64#381—61no—
65#380—32no—
66#380—31no—
67#380—31no—
68#380—31no—
69#380—32no—
70#380—31no—
71#380—61no—
72#380—61no—
73#379—61no—
74C4×C22⋊C4—61no—
75C4×C22⋊C4—61no—
76C2×#95—61no—
77#340—32no—
78#340—32no—
79#340—32no—
80#340—32no—
81C22×A4—21no—
82C22×A4—21no—
83C22×A4C4⋊C412yes—
84C22×A4C2×D411yescharacteristic
85C22×C3×C4—41no—
86C22×Dic3—81no—
87C4×A4C4⋊C412yes—
88C4×A4C22×C411yescharacteristic
89C4×A4C2×Q811yescharacteristic
90SL(2,ℤ/4)—41no—
91SL(2,ℤ/4)—42no—
92SL(2,ℤ/4)—41no—
93C3×M16—44no—
94C2×C3×C8—42no—
95#217—81no—
96C4⋊Dic3—41no—
97C4⋊Dic3—42no—
98C4⋊Dic3—41no—
99C4⋊Dic3—81no—
100C4×Dic3—81no—
101C25D1211yescharacteristic
102C23×C4—31no—
103C23×C4—32no—
104C23×C4—31no—
105C23×C4—32no—
106C23×C4—31no—
107C23×C4—31no—
108C23×C4—61no—
109C23×C4—61no—
110C23×C4Dic612yes—
111C23×C4D1211yescharacteristic
112C23×C4C2×Dic312yes—
113C23×C4C22×D311yescharacteristic
114C2×M16—31no—
115C2×M16—34no—
116C2×M16—34no—
117C2×M16—38no—
118C2×M16—34no—
119C2×M16—32no—
120C2×M16—61no—
121C2×M16—64no—
122C2×M16—62no—
123C2×M16S411yescharacteristic
124C22×C8—32no—
125C22×C8—34no—
126C22×C8—32no—
127C22×C8C2×Dic314yescomplemented
128#117—62no—
129#117—61no—
130#117—61no—
131#117—62no—
132#117—61no—
133#117—61no—
134C2×C4⋊C4—32no—
135C2×C4⋊C4—32no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—122no—
157C2×C22⋊C4—122no—
158C2×C42—61no—
159C2×C42—61no—
160#107—64no—
161#107—62no—
162#107—62no—
163#106—64no—
164#106—62no—
165#106—62no—
166#95—62no—
167#95—62no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172C23×C3—41no—
173C2×A4—22no—
174C2×A4C2×C4⋊C411yescharacteristic
175C2×A4AGL(1,ℤ/8)11yescharacteristic
176C2×A4#13711yescharacteristic
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C2×Dic3—82no—
184C2×Dic3—162no—
185C3×C8—44no—
186C24—21no—
187C24—21no—
188C24—31no—
189C24—31no—
190C24—32no—
191C24—61no—
192C24—61no—
193C24C4⋊Dic312yes—
194C24C2×D1211yescharacteristic
195C22×C4—31no—
196C22×C4—32no—
197C22×C4—32no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—32no—
208C22×C4—32no—
209C22×C4—32no—
210C22×C4—32no—
211C22×C4—32no—
212C22×C4—32no—
213C22×C4—32no—
214C22×C4—31no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—62no—
218C22×C4—61no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4—122no—
227C22×C4—121no—
228C22×C4—122no—
229C22×C4—122no—
230C22×C4—122no—
231C22×C4—122no—
232C22×C4—121no—
233C22×C4C4⋊Dic312yes—
234C22×C4C2×Dic611yescharacteristic
235C22×C4C22×Dic311yescharacteristic
236C22×C4C2×S411yescharacteristic
237M16—34no—
238M16—38no—
239M16—64no—
240M16SL(2,ℤ/4)14yescomplemented
241C2×C8—32no—
242C2×C8—34no—
243C2×C8—34no—
244C2×C8—34no—
245C2×C8—62no—
246C2×C8C2×S412yes—
247C4⋊C4—61no—
248C4⋊C4—62no—
249C4⋊C4—61no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—122no—
261C4⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—122no—
265C22⋊C4—122no—
266C22⋊C4—121no—
267C22⋊C4—122no—
268C22⋊C4—122no—
269C22⋊C4—124no—
270C22⋊C4—122no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C42—121no—
275C42—122no—
276C42—121no—
277C22×C3—41no—
278C22×C3—42no—
279C22×C3—81no—
280C22×C3—81no—
281A4#42811yescharacteristic, complemented
282C3×C4—41no—
283C3×C4—42no—
284C3×C4—41no—
285Dic3—162no—
286Dic3—161no—
287Dic3—161no—
288C23—22no—
289C23—31no—
290C23—31no—
291C23—31no—
292C23—31no—
293C23—32no—
294C23—31no—
295C23—32no—
296C23—32no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23—62no—
303C23—61no—
304C23—61no—
305C23—61no—
306C23—62no—
307C23—62no—
308C23—61no—
309C23—122no—
310C23#90811yescharacteristic
311C23#90911yescharacteristic
312C23C2×C4⋊Dic311yescharacteristic
313C23C4⋊S411yescharacteristic
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—242no—
354C2×C4—244no—
355C2×C4—242no—
356C2×C4A4⋊Q812yes—
357C2×C4C4⋊S411yescharacteristic
358C2×C4C2×SL(2,ℤ/4)12yes—
359C2×C4C22×S411yescharacteristic
360C8—34no—
361C8C2×SL(2,ℤ/4)14yescomplemented
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—41no—
365C2×C3—82no—
366C22—21no—
367C22—21no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—31no—
372C22—31no—
373C22—31no—
374C22—31no—
375C22—31no—
376C22—31no—
377C22—32no—
378C22—62no—
379C22—62no—
380C22—62no—
381C22—62no—
382C22—61no—
383C22—62no—
384C22—62no—
385C22—61no—
386C22—61no—
387C22—121no—
388C22—121no—
389C22#512111yescharacteristic, complemented
390C22C4⋊SL(2,ℤ/4)12yes—
391C22C2×C4⋊S411yescharacteristic
392C4—32no—
393C4—31no—
394C4—31no—
395C4—241no—
396C4—241no—
397C4—242no—
398C4—242no—
399C4—241no—
400C4—241no—
401C4C4⋊SL(2,ℤ/4)12yes—
402C4C2×A4⋊Q811yescharacteristic
403C4C22×SL(2,ℤ/4)11yescharacteristic
404C3—41noSylow
405C2—22no—
406C2—31no—
407C2—31no—
408C2—31no—
409C2—31no—
410C2—62no—
411C2#8544311yescharacteristic
412C2#8544411yescharacteristic
413C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC23×M16D3
socleC24C2×D12
prime radicalC23×A4D4
cosocleC2×C4×A4C23
hypercenterC2×M16S4
hypocenterA4#428
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #73111C2semidirect
5. #73110C2semidirect
9. A4×M16C4semidirect
41. C8×A4C2×C4semidirect
49. C22×M16Dic3semidirect
127. C22×C8C2×Dic3semidirect
240. M16SL(2,ℤ/4)semidirect
281. A4#428semidirect
361. C8C2×SL(2,ℤ/4)semidirect
389. C22#5121semidirect

Maximal subgroups

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

subgroupindex
C2×A4×M162
#854822
C2×C4⋊SL(2,ℤ/4)2
#731112
#731102
#206613
#51214