Finite Group Databasebeta

← #11713967

Subgroups — #11713967

Order 768.

662 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 65 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×Q8×A4C211yesmaximal, characteristic, complemented
2C2×#5417C212yesmaximal, complemented
3#85289C211yesmaximal, characteristic
4#73053C212yesmaximal, complemented
5C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
6#32903—31nomaximal, Sylow
7C2×Q8×A4C2211yescharacteristic
8C2×C4⋊S4—21no—
9C2×C4⋊S4—21no—
10C2×C4⋊S4C2212yes—
11A4×C4⋊C4—21no—
12A4×C4⋊C4C2211yescharacteristic
13C42×A4—21no—
14C42×A4C2211yescharacteristic
15#5417—24no—
16C2×A4⋊C8C2212yes—
17#5027—41nomaximal
18C22×C4×Q8D311yescharacteristic, complemented
19C2×#460—31no—
20C2×#460—31no—
21C2×#448—32no—
22#2856—31no—
23#1880—32no—
24#1880—32no—
25#1874—32no—
26#1803—31no—
27#1803—31no—
28#1797—31no—
29C22×S4—22no—
30Q8×A4—21no—
31Q8×A4D412yescomplemented
32C2×C4×A4—21no—
33C2×C4×A4—21no—
34C2×C4×A4D411yescharacteristic
35C2×C4×A4D411yescharacteristic
36C2×C4×A4C2311yescharacteristic
37C4⋊S4—22no—
38C4⋊S4—22no—
39C4⋊S4—42no—
40C4⋊S4—42no—
41C3×C4×Q8—41no—
42C2×#215—42no—
43#874—41no—
44A4⋊C8—22no—
45#809—42no—
46C4⋊(C3⋊C8)—41no—
47C23×Q8C2×D311yescharacteristic
48C22×SD16—32no—
49C22×SD16—32no—
50C2×#127—31no—
51C2×#121—32no—
52C2×#121—61no—
53C2×#121—61no—
54C2×C4×Q8—31no—
55C2×C4×Q8—31no—
56C2×C4×Q8—31no—
57C2×C4×Q8—31no—
58C2×C4×Q8—61no—
59C2×C4×Q8—61no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×D311yescharacteristic
62C22×C42—21no—
63C22×C42C2×D311yescharacteristic
64#460—62no—
65#460—62no—
66#460—61no—
67#460—61no—
68#460—61no—
69#460—61no—
70#448—64no—
71#448—64no—
72C2×C4⋊C8—31no—
73C2×C4⋊C8—31no—
74C2×#103—32no—
75C2×#103—32no—
76C2×#102—32no—
77C2×#102—32no—
78C2×C22⋊C8—32no—
79#390—32no—
80#390—61no—
81#390—61no—
82#384—31no—
83#340—32no—
84#340—32no—
85#336—32no—
86C22×A4C2×D411yescharacteristic
87C2×S4—42no—
88C2×S4—42no—
89C2×S4—42no—
90C2×C3×Q8—41no—
91C2×D12—42no—
92C2×D12—81no—
93C2×D12—81no—
94C4×A4—22no—
95C4×A4—21no—
96C4×A4—41no—
97C4×A4SD1612yescomplemented
98C4×A4C2×D411yescharacteristic
99C4×A4#4111yescharacteristic
100C3×C4⋊C4—41no—
101C3×C4⋊C4—81no—
102C3×C42—41no—
103C3×C42—81no—
104#215—84no—
105C2×C3⋊C8—42no—
106C22×Q8—21no—
107C22×Q8—31no—
108C22×Q8—31no—
109C22×Q8—32no—
110C22×Q8—61no—
111C22×Q8—61no—
112C22×Q8D1212yescomplemented
113C22×D4—32no—
114C22×D4—61no—
115C22×D4—61no—
116C22×D4—62no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4#6711yescharacteristic
120C23×C4#6711yescharacteristic
121C23×C4C22×D311yescharacteristic
122C2×SD16—64no—
123C2×SD16—64no—
124C2×SD16—64no—
125C2×SD16—64no—
126C2×SD16—64no—
127C2×SD16—62no—
128C2×SD16—62no—
129C22×C8—32no—
130C22×C8—32no—
131#127—61no—
132#127—61no—
133#127—61no—
134#127—61no—
135#121—62no—
136#121—62no—
137#121—62no—
138#121—62no—
139#121—122no—
140#121—122no—
141#121—122no—
142#121—122no—
143C4×Q8—32no—
144C4×Q8—32no—
145C4×Q8—31no—
146C4×Q8—31no—
147C4×Q8—31no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×Q8—61no—
151C4×Q8—62no—
152C4×Q8—62no—
153C4×Q8S411yescharacteristic, complemented
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—62no—
167C2×C42—31no—
168C2×C42—31no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42—61no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179#103—64no—
180#103—64no—
181#102—64no—
182#102—62no—
183#102—62no—
184C22⋊C8—62no—
185C22⋊C8—62no—
186C22×D3—82no—
187C2×A4#12111yescharacteristic
188C2×A4C2×SD1611yescharacteristic
189C2×A4AGL(1,ℤ/8)11yescharacteristic
190S4—82no—
191C3×Q8—42no—
192C3×Q8—81no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198D12—82no—
199D12—82no—
200D12—162no—
201D12—162no—
202C3⋊C8—82no—
203C24—62no—
204C24C2×#6711yescharacteristic
205C2×Q8—32no—
206C2×Q8—32no—
207C2×Q8—32no—
208C2×Q8—32no—
209C2×Q8—32no—
210C2×Q8—31no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—61no—
216C2×Q8—61no—
217C2×Q8—62no—
218C2×Q8—64no—
219C2×Q8C2×S411yescharacteristic
220C2×D4—62no—
221C2×D4—62no—
222C2×D4—62no—
223C2×D4—62no—
224C2×D4—62no—
225C2×D4—62no—
226C2×D4—122no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—122no—
230C2×D4—122no—
231C2×D4—122no—
232C2×D4—122no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—122no—
236C2×D4—122no—
237C2×D4—122no—
238C2×D4—121no—
239C2×D4—121no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—32no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—41no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—62no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—62no—
259C22×C4—121no—
260C22×C4#21512yescomplemented
261C22×C4C2×D1211yescharacteristic
262C22×C4#23511yescharacteristic
263SD16—128no—
264SD16—124no—
265SD16—124no—
266C2×C8—62no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—62no—
271C2×C8—62no—
272C4⋊C4—21no—
273C4⋊C4—32no—
274C4⋊C4—31no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—62no—
279C4⋊C4—62no—
280C4⋊C4—62no—
281C4⋊C4—61no—
282C4⋊C4—122no—
283C4⋊C4—122no—
284C4⋊C4C2×S411yescharacteristic
285C22⋊C4—122no—
286C22⋊C4—122no—
287C22⋊C4—124no—
288C22⋊C4—124no—
289C42—21no—
290C42—31no—
291C42—31no—
292C42—31no—
293C42—61no—
294C42—61no—
295C42—61no—
296C42—62no—
297C42—61no—
298C42C2×S411yescharacteristic
299C22×C3—41no—
300C2×D3—162no—
301C2×D3—162no—
302C2×D3—162no—
303A4#46011yescharacteristic, complemented
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—42no—
307C3×C4—81no—
308C3×C4—82no—
309C3×C4—161no—
310C23—31no—
311C23—31no—
312C23—31no—
313C23—31no—
314C23—122no—
315C23—122no—
316C23—122no—
317C23—122no—
318C23—122no—
319C23—122no—
320C23—122no—
321C23#93011yescharacteristic
322C23C2×#21511yescharacteristic
323C23#94911yescharacteristic
324Q8—21no—
325Q8—34no—
326Q8—32no—
327Q8—62no—
328Q8—62no—
329Q8—62no—
330Q8—61no—
331Q8C4⋊S412yescomplemented
332D4—122no—
333D4—122no—
334D4—122no—
335D4—122no—
336D4—242no—
337D4—242no—
338D4—242no—
339D4—242no—
340D4—242no—
341D4—242no—
342D4—242no—
343D4—242no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—61no—
359C2×C4—62no—
360C2×C4—61no—
361C2×C4—62no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—62no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—62no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4GL(2,ℤ/4)11yescharacteristic
378C2×C4GL(2,ℤ/4)11yescharacteristic
379C2×C4C22×S411yescharacteristic
380C8—122no—
381C8—122no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—41no—
385D3—322no—
386C22—31no—
387C22—31no—
388C22—31no—
389C22—31no—
390C22—31no—
391C22—31no—
392C22—31no—
393C22—31no—
394C22—31no—
395C22—61no—
396C22—242no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22—242no—
401C22—242no—
402C22—242no—
403C22#502711yescharacteristic, complemented
404C22C2×GL(2,ℤ/4)11yescharacteristic
405C4—21no—
406C4—22no—
407C4—31no—
408C4—31no—
409C4—32no—
410C4—41no—
411C4—61no—
412C4—62no—
413C4—121no—
414C4—242no—
415C4#541712yescomplemented
416C4C2×C4⋊S411yescharacteristic
417C4#591211yescharacteristic
418C3—41noSylow
419C2—31no—
420C2—31no—
421C2—31no—
422C2—31no—
423C2—482no—
424C2#8548611yescharacteristic
425C2C2×#541711yescharacteristic
426C2#8551311yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×Q8D3
socleC24C2×#67
prime radical#85289C2
hypercenterC4×Q8S4
hypocenterA4#460
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×A4C2semidirect
2. C2×#5417C2semidirect
4. #73053C2semidirect
5. C4⋊(A4⋊C8)C2semidirect
18. C22×C4×Q8D3semidirect
31. Q8×A4D4semidirect
97. C4×A4SD16semidirect
112. C22×Q8D12semidirect
153. C4×Q8S4semidirect
260. C22×C4#215semidirect
303. A4#460semidirect
331. Q8C4⋊S4semidirect
403. C22#5027semidirect
415. C4#5417semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
C2×#54172
#852892
#730532
C4⋊(A4⋊C8)2
#329033
#50274