Finite Group Databasebeta

← #11713008

Subgroups — #11713008

Order 768.

551 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 69 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×#5400C211yesmaximal, characteristic
2#85307C211yesmaximal, characteristic
3C4⋊C4×S4C211yesmaximal, characteristic
4A4×#107C211yesmaximal, characteristic, complemented
5#73110C211yesmaximal, characteristic, complemented
6#73051C211yesmaximal, characteristic, complemented
7#73050C211yesmaximal, characteristic, complemented
8#22661—31nomaximal, Sylow
9C2×C4×S4—21no—
10C2×C4×S4C2211yescharacteristic
11C2×C8×A4C2211yescharacteristic
12A4×C4⋊C4C2211yescharacteristic
13A4×C4⋊C4C2211yescharacteristic
14(C2×S4)⋊C4—21no—
15C4⋊SL(2,ℤ/4)C2211yescharacteristic
16C4⋊SL(2,ℤ/4)C2211yescharacteristic
17SL(2,ℤ/4)⋊C4—21no—
18C4×SL(2,ℤ/4)—21no—
19C2×A4⋊C8C2211yescharacteristic
20#5400C414yescomplemented
21#4881—41nomaximal
22C2×#435—31no—
23C2×#428—31no—
24C22×#107D311yescharacteristic, complemented
25#2352—31no—
26D4×C4⋊C4—31no—
27#1949—31no—
28#1947—31no—
29#1854—31no—
30#1853—31no—
31#1852—31no—
32#1852—31no—
33#1847—31no—
34#1809—31no—
35#1807—31no—
36#1776—31no—
37C22×S4D411yescharacteristic
38C2×C4×A4—21no—
39C2×C4×A4—21no—
40C2×C4×A4C2311yescharacteristic
41C2×SL(2,ℤ/4)—21no—
42C2×SL(2,ℤ/4)—21no—
43C2×SL(2,ℤ/4)D411yescharacteristic
44C4×S4—42no—
45C4×S4C2×C412yes—
46C4×S4Q812yes—
47C2×ΓL(1,𝔽25)—41no—
48#892—41no—
49D3×C4⋊C4—41no—
50C8×A4C2×C412yescomplemented
51A4⋊C8C2×C412yescomplemented
52C3×#107—41no—
53#817—41no—
54#808—41no—
55#807—41no—
56C22×M16—31no—
57C23×C8C2×D311yescharacteristic
58C2×C4×D4—31no—
59C2×C4×D4—61no—
60C2×#117—31no—
61C22×C4⋊C4—31no—
62C22×C4⋊C4C2×D311yescharacteristic
63C22×C4⋊C4C2×D311yescharacteristic
64#435—34no—
65#435—34no—
66#428—61no—
67#428—61no—
68#428—61no—
69#428—61no—
70C2×#107—31no—
71C2×#107—31no—
72C2×#107—31no—
73C2×#107—31no—
74C2×#107—31no—
75C2×#107—31no—
76C2×#106—31no—
77C2×#106—31no—
78C2×#106—31no—
79C2×#106—31no—
80C2×C4⋊C8—31no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8—31no—
83C2×#97—31no—
84#389—61no—
85#388—61no—
86#387—61no—
87#386—61no—
88#385—61no—
89#385—61no—
90#384—31no—
91#382—31no—
92#380—31no—
93#380—31no—
94#380—31no—
95#380—31no—
96C4×C4⋊C4—31no—
97C4×C4⋊C4—31no—
98C4×C22⋊C4—61no—
99#340—31no—
100#340—31no—
101#340—31no—
102#340—31no—
103#340—31no—
104#340—31no—
105C22×A4C2×D411yescharacteristic
106C2×S4—21no—
107C2×S4—21no—
108C2×S4C4⋊C412yes—
109C2×C4×D3—41no—
110C2×C4×D3—81no—
111C4×A4—41no—
112C4×A4—41no—
113C4×A4C22×C411yescharacteristic
114C4×A4C2×Q811yescharacteristic
115SL(2,ℤ/4)—41no—
116SL(2,ℤ/4)—41no—
117SL(2,ℤ/4)C4⋊C412yes—
118C2×C3×C8—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121(C2×D3)⋊C4—81no—
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124Dic3⋊C4—81no—
125C4×Dic3—81no—
126C2×C3⋊C8—41no—
127ΓL(1,𝔽25)—44no—
128C22×D4—31no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—31no—
132C23×C4—61no—
133C23×C4C22×D311yescharacteristic
134C2×M16—34no—
135C2×M16—62no—
136C2×M16—61no—
137C2×M16—61no—
138C22×C8—31no—
139C22×C8—31no—
140C22×C8—32no—
141C22×C8—31no—
142C22×C8—31no—
143C22×C8C4×D312yescomplemented
144C4×D4—32no—
145C4×D4—32no—
146C4×D4—32no—
147C4×D4—32no—
148C4×D4—122no—
149C4×D4—122no—
150#117—61no—
151#117—61no—
152#117—121no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—31no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—31no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—121no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—61no—
173C2×C22⋊C4—61no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C42—31no—
178C2×C42—61no—
179C2×C42—61no—
180C2×C42—61no—
181#107—31no—
182#107—31no—
183#107—31no—
184#107—61no—
185#107—61no—
186#107—61no—
187#107—121no—
188#107S411yescharacteristic, complemented
189#106—61no—
190#106—62no—
191#106—61no—
192#106—61no—
193#106—61no—
194#106—121no—
195C4⋊C8—34no—
196C22⋊C8—34no—
197C22⋊C8—32no—
198C22⋊C8—32no—
199#97—34no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208C22×D3—41no—
209C2×A4C2×C4⋊C411yescharacteristic
210C2×A4AGL(1,ℤ/8)11yescharacteristic
211C2×A4#13711yescharacteristic
212S4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×Dic3—41no—
217C2×Dic3—81no—
218C2×Dic3—81no—
219C4×D3—42no—
220C4×D3—42no—
221C4×D3—162no—
222C3×C8—42no—
223C3⋊C8—42no—
224C24—31no—
225C24D3×D411yescharacteristic
226C2×D4—32no—
227C2×D4—32no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—32no—
240C22×C4—32no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—121no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4—121no—
259C22×C4—122no—
260C22×C4—241no—
261C22×C4C2×C4×D311yescharacteristic
262C22×C4D3×Q811yescharacteristic
263M16—64no—
264M16—64no—
265C2×C8—31no—
266C2×C8—32no—
267C2×C8—32no—
268C2×C8—32no—
269C2×C8—32no—
270C2×C8—32no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8C2×S411yescharacteristic
277C4⋊C4—32no—
278C4⋊C4—32no—
279C4⋊C4—31no—
280C4⋊C4—31no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—122no—
296C4⋊C4—241no—
297C4⋊C4C2×S411yescharacteristic
298C4⋊C4C2×S411yescharacteristic
299C22⋊C4—32no—
300C22⋊C4—32no—
301C22⋊C4—32no—
302C22⋊C4—32no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—122no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C22⋊C4—241no—
311C42—32no—
312C42—32no—
313C42—121no—
314C42—121no—
315C42—122no—
316C42—241no—
317C22×C3—41no—
318C2×D3—42no—
319C2×D3—81no—
320C2×D3—81no—
321A4#42811yescharacteristic, complemented
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—161no—
325C3×C4—161no—
326Dic3—42no—
327Dic3—161no—
328Dic3—161no—
329C23—31no—
330C23—31no—
331C23—31no—
332C23—31no—
333C23—32no—
334C23—61no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23D3×C4⋊C411yescharacteristic
341C23#91111yescharacteristic
342C23#91811yescharacteristic
343D4—62no—
344D4—62no—
345D4—62no—
346D4—62no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—32no—
357C2×C4—32no—
358C2×C4—32no—
359C2×C4—61no—
360C2×C4—62no—
361C2×C4—62no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—62no—
369C2×C4—62no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—242no—
393C2×C4—242no—
394C2×C4C22×S411yescharacteristic
395C8—32no—
396C8—62no—
397C8—62no—
398C8C4×S412yescomplemented
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402D3—82no—
403C22—31no—
404C22—31no—
405C22—31no—
406C22—31no—
407C22—31no—
408C22—31no—
409C22—31no—
410C22—31no—
411C22—31no—
412C22—61no—
413C22—62no—
414C22—62no—
415C22—62no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22—121no—
420C22#488111yescharacteristic, complemented
421C22D4×S411yescharacteristic
422C4—31no—
423C4—31no—
424C4—41no—
425C4—41no—
426C4—62no—
427C4—62no—
428C4—62no—
429C4—121no—
430C4—121no—
431C4—241no—
432C4—241no—
433C4—241no—
434C4—241no—
435C4C2×C4×S411yescharacteristic
436C4Q8×S411yescharacteristic
437C3—41noSylow
438C2—31no—
439C2—31no—
440C2—31no—
441C2—31no—
442C2—122no—
443C2C4⋊C4×S411yescharacteristic
444C2#8545711yescharacteristic
445C2#8546411yescharacteristic
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#107D3
socleC24D3×D4
prime radicalC22×S4D4
cosocleC2×C4×A4C23
hypercenter#107S4
hypocenterA4#428
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
4. A4×#107C2semidirect
5. #73110C2semidirect
6. #73051C2semidirect
7. #73050C2semidirect
20. #5400C4semidirect
24. C22×#107D3semidirect
50. C8×A4C2×C4semidirect
51. A4⋊C8C2×C4semidirect
143. C22×C8C4×D3semidirect
188. #107S4semidirect
321. A4#428semidirect
398. C8C4×S4semidirect
420. C22#4881semidirect

Maximal subgroups

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

subgroupindex
C2×#54002
#853072
C4⋊C4×S42
A4×#1072
#731102
#730512
#730502
#226613
#48814