Finite Group Databasebeta

← #11712892

Subgroups — #11712892

Order 768.

686 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 85 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×D4×A4C211yesmaximal, characteristic
2C2×#5409C212yesmaximal, complemented
3#73141C212yesmaximal, complemented
4C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
5C4×A4⋊C8C211yesmaximal, characteristic, complemented
6#22401—31nomaximal, Sylow
7C2×D4×A4C411yescharacteristic
8C22×C4×A4C2212yes—
9A4×C4⋊C4C411yescharacteristic
10A4×C22⋊C4C412yes—
11C42×A4C2211yescharacteristic
12#5409—24no—
13C2×A4⋊C8C2211yescharacteristic
14C2×A4⋊C8C2212yes—
15C2×A4⋊C8C2211yescharacteristic
16#5012—41nomaximal
17C22×C4×D4D311yescharacteristic
18C2×#436—31no—
19C2×#436—31no—
20C2×#407—32no—
21#1954—31no—
22#1954—31no—
23#1870—32no—
24#1870—32no—
25#1797—31no—
26#1779—32no—
27C4×C22⋊C8—31no—
28C23×A4C2×C412yes—
29D4×A4—22no—
30C2×C4×A4—22no—
31C2×C4×A4—22no—
32C2×C4×A4C2×C412yes—
33C2×C4×A4C2×C411yescharacteristic
34C2×C4×A4C2×C411yescharacteristic
35C2×C4×A4C2311yescharacteristic
36C3×C4×D4—41no—
37C2×#208—42no—
38A4⋊C8—22no—
39A4⋊C8—21no—
40A4⋊C8D412yescomplemented
41#830—42no—
42C4⋊(C3⋊C8)—41no—
43C4×C3⋊C8—41no—
44C23×D4Dic311yescharacteristic
45C24×C4C2×D312yes—
46C22×M16—32no—
47C22×M16—32no—
48C2×C4×D4—31no—
49C2×C4×D4—32no—
50C2×C4×D4—32no—
51C2×C4×D4—31no—
52C2×C4×D4—31no—
53C2×C4×D4—31no—
54C22×C4⋊C4Dic311yescharacteristic
55C22×C22⋊C4Dic312yes—
56C22×C42C2×D311yescharacteristic
57#436—61no—
58#436—61no—
59#436—62no—
60#436—62no—
61#436—61no—
62#436—61no—
63C2×C4⋊C8—31no—
64C2×C4⋊C8—31no—
65#407—64no—
66#407—64no—
67C2×C22⋊C8—31no—
68C2×C22⋊C8—32no—
69C2×C22⋊C8—32no—
70C2×C22⋊C8—32no—
71C2×C22⋊C8—32no—
72C2×C22⋊C8—32no—
73C2×C22⋊C8—31no—
74C2×C22⋊C8—32no—
75C2×C22⋊C8—32no—
76C2×C4×C8—31no—
77C2×C4×C8—31no—
78#336—32no—
79#336—31no—
80#336—31no—
81C22×A4—22no—
82C22×A4—22no—
83C22×A4—22no—
84C22×A4C22×C411yescharacteristic
85C2×C3×D4—41no—
86C22×C3×C4—42no—
87C4×A4—22no—
88C4×A4—21no—
89C4×A4M1612yescomplemented
90C4×A4C2×D411yescharacteristic
91C4×A4#4111yescharacteristic
92C3×C4⋊C4—41no—
93C3×C22⋊C4—42no—
94C3×C42—41no—
95#208—84no—
96C2×C3⋊C8—41no—
97C2×C3⋊C8—41no—
98C2×C3⋊C8—42no—
99C25C2×Dic312yes—
100C22×D4—22no—
101C22×D4—31no—
102C22×D4—31no—
103C22×D4—32no—
104C22×D4—62no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—32no—
108C23×C4—32no—
109C23×C4—32no—
110C23×C4—32no—
111C23×C4—62no—
112C23×C4—62no—
113C23×C4C2×Dic311yescharacteristic
114C23×C4C2×Dic312yes—
115C23×C4C2×Dic311yescharacteristic
116C23×C4C22×D311yescharacteristic
117C2×M16—64no—
118C2×M16—64no—
119C2×M16—62no—
120C2×M16—62no—
121C2×M16—62no—
122C2×M16—62no—
123C2×M16—122no—
124C2×M16—122no—
125C22×C8—31no—
126C22×C8—31no—
127C22×C8—32no—
128C22×C8—32no—
129C22×C8—31no—
130C22×C8—31no—
131C4×D4—31no—
132C4×D4—32no—
133C4×D4—31no—
134C4×D4—32no—
135C4×D4—31no—
136C4×D4—61no—
137C4×D4—61no—
138C4×D4—61no—
139C4×D4—62no—
140C4×D4—62no—
141C4×D4S411yescharacteristic
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—32no—
145C2×C22⋊C4—32no—
146C2×C22⋊C4—32no—
147C2×C22⋊C4—32no—
148C2×C22⋊C4—32no—
149C2×C42—31no—
150C2×C42—31no—
151C2×C42—31no—
152C2×C42—31no—
153C4⋊C8—61no—
154C4⋊C8—61no—
155C4⋊C8—61no—
156C4⋊C8—61no—
157C22⋊C8—32no—
158C22⋊C8—32no—
159C22⋊C8—64no—
160C22⋊C8—64no—
161C22⋊C8—61no—
162C22⋊C8—62no—
163C22⋊C8—62no—
164C22⋊C8—62no—
165C22⋊C8—62no—
166C22⋊C8—62no—
167C22⋊C8—61no—
168C22⋊C8—62no—
169C22⋊C8—122no—
170C22⋊C8—122no—
171C4×C8—61no—
172C4×C8—61no—
173C4×C8—61no—
174C4×C8—61no—
175C23×C3—42no—
176C2×A4—42no—
177C2×A4C4×D411yescharacteristic
178C2×A4C2×M1611yescharacteristic
179C2×A4#13111yescharacteristic
180C3×D4—82no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—82no—
186C2×C3×C4—82no—
187C3⋊C8—42no—
188C3⋊C8—81no—
189C3⋊C8—82no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24—32no—
194C24—32no—
195C24—62no—
196C24—62no—
197C24—62no—
198C24C22×Dic311yescharacteristic
199C2×D4—34no—
200C2×D4—32no—
201C2×D4—31no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—61no—
205C2×D4—62no—
206C2×D4—64no—
207C2×D4—64no—
208C2×D4SL(2,ℤ/4)11yescharacteristic
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—32no—
218C22×C4—32no—
219C22×C4—32no—
220C22×C4—31no—
221C22×C4—32no—
222C22×C4—32no—
223C22×C4—31no—
224C22×C4—32no—
225C22×C4—62no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—62no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—61no—
240C22×C4—122no—
241C22×C4—122no—
242C22×C4#20812yescomplemented
243C22×C4D3×D411yescharacteristic
244C22×C4#23711yescharacteristic
245C22×C4C2×S412yes—
246M16—124no—
247M16—124no—
248M16—124no—
249M16—124no—
250C2×C8—32no—
251C2×C8—32no—
252C2×C8—61no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—61no—
256C2×C8—61no—
257C2×C8—62no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—61no—
267C2×C8—62no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—61no—
271C4⋊C4SL(2,ℤ/4)11yescharacteristic
272C22⋊C4—32no—
273C22⋊C4—32no—
274C22⋊C4—32no—
275C22⋊C4—62no—
276C22⋊C4SL(2,ℤ/4)12yes—
277C42—31no—
278C42—31no—
279C42—31no—
280C42—61no—
281C42C2×S411yescharacteristic
282C22×C3—41no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—82no—
286A4#43611yescharacteristic, complemented
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—42no—
290C3×C4—82no—
291C3×C4—81no—
292C23—31no—
293C23—31no—
294C23—31no—
295C23—31no—
296C23—34no—
297C23—32no—
298C23—42no—
299C23—62no—
300C23—62no—
301C23—62no—
302C23—62no—
303C23—62no—
304C23—62no—
305C23—62no—
306C23—62no—
307C23—62no—
308C23—62no—
309C23—62no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23—124no—
314C23C2×#20811yescharacteristic
315C23D4×Dic311yescharacteristic
316C23#94811yescharacteristic
317C23C2×SL(2,ℤ/4)12yes—
318D4—22no—
319D4—64no—
320D4—64no—
321D4—62no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—32no—
335C2×C4—31no—
336C2×C4—32no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—61no—
340C2×C4—62no—
341C2×C4—61no—
342C2×C4—62no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—62no—
349C2×C4—61no—
350C2×C4—62no—
351C2×C4—61no—
352C2×C4—62no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4C2×SL(2,ℤ/4)11yescharacteristic
356C2×C4C2×SL(2,ℤ/4)12yes—
357C2×C4C2×SL(2,ℤ/4)11yescharacteristic
358C2×C4C22×S411yescharacteristic
359C8—62no—
360C8—62no—
361C8—121no—
362C8—122no—
363C8—122no—
364C8—121no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—162no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—31no—
373C22—31no—
374C22—31no—
375C22—31no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—61no—
382C22—64no—
383C22—64no—
384C22—62no—
385C22—62no—
386C22—62no—
387C22—122no—
388C22—122no—
389C22—122no—
390C22—122no—
391C22—122no—
392C22—122no—
393C22#501211yescharacteristic, complemented
394C22C22×SL(2,ℤ/4)11yescharacteristic
395C4—21no—
396C4—22no—
397C4—32no—
398C4—31no—
399C4—31no—
400C4—61no—
401C4—62no—
402C4#540912yescomplemented
403C4D4×S411yescharacteristic
404C4#591411yescharacteristic
405C3—41noSylow
406C2—31no—
407C2—31no—
408C2—31no—
409C2—31no—
410C2—42no—
411C2—122no—
412C2C2×#540911yescharacteristic
413C2D4×SL(2,ℤ/4)11yescharacteristic
414C2#8551111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×C4×D4D3
socleC24C22×Dic3
prime radicalC2×D4×A4C4
cosocleC2×C4×A4C23
hypercenterC4×D4S4
hypocenterA4#436
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5409C2semidirect
3. #73141C2semidirect
4. C4⋊(A4⋊C8)C2semidirect
5. C4×A4⋊C8C2semidirect
40. A4⋊C8D4semidirect
89. C4×A4M16semidirect
242. C22×C4#208semidirect
286. A4#436semidirect
393. C22#5012semidirect
402. C4#5409semidirect

Maximal subgroups

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

subgroupindex
C4×D4×A42
C2×#54092
#731412
C4⋊(A4⋊C8)2
C4×A4⋊C82
#224013
#50124