Finite Group Databasebeta

← #11713036

Subgroups — #11713036

Order 768.

449 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#85287C211yesmaximal, characteristic
2A4×C4⋊C8C211yesmaximal, characteristic, complemented
3(C2×S4)⋊C8C211yesmaximal, characteristic
4(C2×S4)⋊C8C211yesmaximal, characteristic
5#73108C211yesmaximal, characteristic, complemented
6C8×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
7C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
8#22754—31nomaximal, Sylow
9C2×C4×S4C2211yescharacteristic
10C2×C8×A4C2211yescharacteristic
11C2×C8×A4C2211yescharacteristic
12C42×A4C2211yescharacteristic
13(C2×S4)⋊C4C411yescharacteristic
14(C2×S4)⋊C4C411yescharacteristic
15SL(2,ℤ/4)⋊C4C411yescharacteristic
16SL(2,ℤ/4)⋊C4C411yescharacteristic
17C4×SL(2,ℤ/4)C2211yescharacteristic
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#4840—41nomaximal
21C2×#433—31no—
22C22×C4⋊C8D311yescharacteristic, complemented
23#2290—31no—
24#1924—31no—
25#1923—31no—
26#1917—31no—
27#1916—31no—
28#1869—31no—
29#1869—31no—
30#1832—31no—
31#1798—31no—
32#1797—31no—
33#1768—31no—
34#1752—31no—
35C8×C22⋊C4—31no—
36C22×S4C2×C411yescharacteristic
37C2×C4×A4C2×C411yescharacteristic
38C2×C4×A4C2×C411yescharacteristic
39C2×C4×A4C2311yescharacteristic
40C2×SL(2,ℤ/4)C2×C411yescharacteristic
41C2×SL(2,ℤ/4)C2×C411yescharacteristic
42C2×SL(2,ℤ/4)C2×C411yescharacteristic
43C4×S4—21no—
44C4×S4—21no—
45#872—41no—
46C8×A4—21no—
47C8×A4—21no—
48A4⋊C8—21no—
49A4⋊C8—21no—
50C3×C4⋊C8—41no—
51(C2×D3)⋊C8—41no—
52(C2×D3)⋊C8—41no—
53#815—41no—
54C8×Dic3—41no—
55C4⋊(C3⋊C8)—41no—
56C23×C8C2×D311yescharacteristic
57C23×C8C2×D311yescharacteristic
58C2×C4×D4—31no—
59C2×#117—31no—
60C22×C42C2×D311yescharacteristic
61#433—61no—
62#433—61no—
63#433—61no—
64#433—61no—
65C2×C4⋊C8—31no—
66C2×C4⋊C8—31no—
67C2×C4⋊C8—31no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8—31no—
70C2×C4⋊C8—31no—
71C2×C22⋊C8—31no—
72C2×C22⋊C8—31no—
73C2×C22⋊C8—31no—
74C2×C22⋊C8—31no—
75C2×C22⋊C8—31no—
76C2×C22⋊C8—31no—
77C2×#97—31no—
78C2×#97—31no—
79C2×C4×C8—31no—
80C2×C4×C8—31no—
81#388—31no—
82#388—31no—
83#387—31no—
84#387—31no—
85#386—31no—
86#386—31no—
87#385—31no—
88#385—31no—
89C4×C4⋊C4—31no—
90C4×C22⋊C4—31no—
91C4×C22⋊C4—31no—
92#376—31no—
93#336—31no—
94#336—31no—
95#336—31no—
96#336—31no—
97#336—31no—
98#336—31no—
99#336—31no—
100#336—31no—
101#336—31no—
102#336—31no—
103C22×A4C22×C411yescharacteristic
104C2×S4—21no—
105C2×S4—21no—
106C2×S4—21no—
107C2×C4×D3—41no—
108C4×A4—21no—
109C4×A4—21no—
110C4×A4#4111yescharacteristic
111C4×A4#4111yescharacteristic
112SL(2,ℤ/4)—21no—
113SL(2,ℤ/4)—21no—
114SL(2,ℤ/4)—21no—
115C2×C3×C8—41no—
116C2×C3×C8—41no—
117C3×C42—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122C4×Dic3—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C22×D4—31no—
126C23×C4—31no—
127C23×C4C4×D311yescharacteristic
128C23×C4C4×D311yescharacteristic
129C23×C4C22×D311yescharacteristic
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—31no—
133C22×C8—31no—
134C22×C8—31no—
135C22×C8—31no—
136C22×C8—31no—
137C22×C8—31no—
138C22×C8—31no—
139C22×C8—31no—
140C22×C8—61no—
141C22×C8—61no—
142C4×D4—61no—
143C4×D4—61no—
144C4×D4—61no—
145C4×D4—61no—
146#117—61no—
147#117—61no—
148#117—61no—
149#117—61no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C42—31no—
164C2×C42—31no—
165C2×C42—31no—
166C2×C42—31no—
167C2×C42—31no—
168C2×C42—31no—
169C2×C42—31no—
170C4⋊C8—31no—
171C4⋊C8—31no—
172C4⋊C8—31no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178C4⋊C8S411yescharacteristic, complemented
179C22⋊C8—61no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C22⋊C8—61no—
187C22⋊C8—121no—
188C22⋊C8—121no—
189#97—61no—
190#97—61no—
191#97—121no—
192C4×C8—61no—
193C4×C8—61no—
194C4×C8—121no—
195#95—31no—
196#95—31no—
197#95—31no—
198#95—31no—
199#95—31no—
200#95—31no—
201#95—31no—
202#95—31no—
203#95—31no—
204#95—31no—
205C22×D3—41no—
206C2×A4#11711yescharacteristic
207C2×A4#13111yescharacteristic
208C2×A4#13111yescharacteristic
209S4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C4×D3—81no—
217C4×D3—81no—
218C3×C8—81no—
219C3×C8—81no—
220C3⋊C8—81no—
221C3⋊C8—81no—
222C24—31no—
223C24C2×C4×D311yescharacteristic
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—31no—
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#23711yescharacteristic
256C22×C4#23911yescharacteristic
257C2×C8—31no—
258C2×C8—31no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8C2×S411yescharacteristic
280C2×C8C2×S411yescharacteristic
281C4⋊C4—62no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—121no—
289C22⋊C4—62no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—121no—
303C42—31no—
304C42—31no—
305C42—31no—
306C42—61no—
307C42—61no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42—61no—
312C42—61no—
313C42C2×S411yescharacteristic
314C22×C3—41no—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318A4#43311yescharacteristic, complemented
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—81no—
322C3×C4—81no—
323Dic3—81no—
324Dic3—81no—
325Dic3—81no—
326C23—31no—
327C23—31no—
328C23—31no—
329C23—31no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—61no—
335C23—61no—
336C23—61no—
337C23#89211yescharacteristic
338C23#90111yescharacteristic
339C23#90711yescharacteristic
340D4—121no—
341D4—121no—
342D4—121no—
343D4—121no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—32no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4C4×S411yescharacteristic
388C2×C4C4×S411yescharacteristic
389C2×C4C22×S411yescharacteristic
390C8—21no—
391C8—21no—
392C8—61no—
393C8—61no—
394C8—121no—
395C8—121no—
396C8—121no—
397C8—121no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401D3—161no—
402C22—31no—
403C22—31no—
404C22—31no—
405C22—31no—
406C22—31no—
407C22—31no—
408C22—31no—
409C22—31no—
410C22—31no—
411C22—61no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22#484011yescharacteristic, complemented
420C22C2×C4×S411yescharacteristic
421C4—21no—
422C4—21no—
423C4—31no—
424C4—31no—
425C4—61no—
426C4—61no—
427C4—121no—
428C4—121no—
429C4—121no—
430C4—121no—
431C4—121no—
432C4—121no—
433C4—121no—
434C4#591411yescharacteristic
435C4#591911yescharacteristic
436C3—41noSylow
437C2—31no—
438C2—31no—
439C2—31no—
440C2—31no—
441C2—241no—
442C2#8530711yescharacteristic
443C2#8543711yescharacteristic
444C2#8544211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×C4⋊C8D3
socleC24C2×C4×D3
prime radicalC22×S4C2×C4
cosocleC2×C4×A4C23
hypercenterC4⋊C8S4
hypocenterA4#433
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. A4×C4⋊C8C2semidirect
5. #73108C2semidirect
6. C8×SL(2,ℤ/4)C2semidirect
7. C4⋊(A4⋊C8)C2semidirect
22. C22×C4⋊C8D3semidirect
178. C4⋊C8S4semidirect
318. A4#433semidirect
419. C22#4840semidirect

Maximal subgroups

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

subgroupindex
#852872
A4×C4⋊C82
(C2×S4)⋊C82
#731082
C8×SL(2,ℤ/4)2
C4⋊(A4⋊C8)2
#227543
#48404