Finite Group Databasebeta

← #11713971

Subgroups — #11713971

Order 768.

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

normal subgroups only

This group has 572 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
2C2×#5416C212yesmaximal
3#85283C211yesmaximal, characteristic
4#73052C212yesmaximal
5C4⋊(A4⋊C8)C211yesmaximal, characteristic
6#32906—31nomaximal, Sylow
7C2×Q8×A4C2211yescharacteristic
8C2×A4⋊Q8C2212yes—
9A4×C4⋊C4—21no—
10A4×C4⋊C4C2211yescharacteristic
11C42×A4—21no—
12C42×A4C2211yescharacteristic
13#5416—24no—
14C4⋊SL(2,ℤ/4)—21no—
15C4⋊SL(2,ℤ/4)—21no—
16C2×A4⋊C8C2212yes—
17#5031—41nomaximal
18C22×C4×Q8D311yescharacteristic
19C2×#462—31no—
20C2×#462—31no—
21C2×#451—32no—
22#2857—31no—
23#1878—32no—
24#1878—32no—
25#1872—32no—
26#1803—31no—
27#1803—31no—
28#1797—31no—
29Q8×A4—21no—
30Q8×A4D412yes—
31C2×C4×A4—21no—
32C2×C4×A4—21no—
33C2×C4×A4D411yescharacteristic
34C2×C4×A4D411yescharacteristic
35C2×C4×A4C2311yescharacteristic
36C2×SL(2,ℤ/4)—22no—
37A4⋊Q8—22no—
38A4⋊Q8—22no—
39C3×C4×Q8—41no—
40C2×#216—42no—
41#869—41no—
42A4⋊C8—22no—
43#810—42no—
44C4⋊(C3⋊C8)—41no—
45C23×Q8C2×D311yescharacteristic
46C22×Q16—32no—
47C22×Q16—32no—
48C2×C4⋊Q8—31no—
49C2×C22⋊Q8—32no—
50C2×C4×Q8—31no—
51C2×C4×Q8—31no—
52C2×C4×Q8—31no—
53C2×C4×Q8—31no—
54C2×C4×Q8—61no—
55C2×C4×Q8—61no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×D311yescharacteristic
58C22×C42—21no—
59C22×C42C2×D311yescharacteristic
60#462—62no—
61#462—62no—
62#462—61no—
63#462—61no—
64#462—61no—
65#462—61no—
66#451—64no—
67#451—64no—
68C2×C4⋊C8—31no—
69C2×C4⋊C8—31no—
70C2×#103—32no—
71C2×#103—32no—
72C2×#103—32no—
73C2×#103—32no—
74C2×C22⋊C8—32no—
75#391—32no—
76#389—61no—
77#389—61no—
78#384—31no—
79#380—61no—
80#380—61no—
81#340—32no—
82#340—32no—
83#336—32no—
84C22×A4C2×D411yescharacteristic
85C2×C3×Q8—41no—
86C2×Dic6—42no—
87C4×A4—22no—
88C4×A4—21no—
89C4×A4—41no—
90C4×A4Q1612yescomplemented
91C4×A4C2×D411yescharacteristic
92C4×A4#4111yescharacteristic
93SL(2,ℤ/4)—42no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—81no—
96C3×C42—41no—
97C3×C42—81no—
98#216—84no—
99C4⋊Dic3—81no—
100C4⋊Dic3—81no—
101C2×C3⋊C8—42no—
102C22×Q8—21no—
103C22×Q8—31no—
104C22×Q8—31no—
105C22×Q8—32no—
106C22×Q8—32no—
107C22×Q8—61no—
108C22×Q8—61no—
109C22×Q8D1212yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4#6711yescharacteristic
113C23×C4#6711yescharacteristic
114C23×C4C22×D311yescharacteristic
115C2×Q16—64no—
116C2×Q16—64no—
117C2×Q16—64no—
118C2×Q16—64no—
119C2×Q16—64no—
120C2×Q16—62no—
121C2×Q16—62no—
122C22×C8—32no—
123C22×C8—32no—
124C4⋊Q8—61no—
125C4⋊Q8—61no—
126C4⋊Q8—61no—
127C4⋊Q8—61no—
128C22⋊Q8—62no—
129C22⋊Q8—62no—
130C22⋊Q8—62no—
131C22⋊Q8—62no—
132C4×Q8—32no—
133C4×Q8—32no—
134C4×Q8—31no—
135C4×Q8—31no—
136C4×Q8—31no—
137C4×Q8—61no—
138C4×Q8—61no—
139C4×Q8—61no—
140C4×Q8—62no—
141C4×Q8—62no—
142C4×Q8S411yescharacteristic
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—32no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—62no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C22⋊C4—62no—
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—31no—
160C2×C42—31no—
161C2×C42—61no—
162C2×C42—61no—
163C2×C42—61no—
164C2×C42—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C4⋊C8—61no—
169#103—64no—
170#103—62no—
171#103—64no—
172#103—64no—
173#103—62no—
174C22⋊C8—62no—
175C22⋊C8—62no—
176#95—62no—
177#95—62no—
178C2×A4#12111yescharacteristic
179C2×A4C2×Q1611yescharacteristic
180C2×A4#13711yescharacteristic
181C3×Q8—42no—
182C3×Q8—81no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×Dic3—82no—
189Dic6—82no—
190Dic6—82no—
191C3⋊C8—82no—
192C24C2×#6711yescharacteristic
193C2×Q8—32no—
194C2×Q8—32no—
195C2×Q8—32no—
196C2×Q8—32no—
197C2×Q8—32no—
198C2×Q8—31no—
199C2×Q8—61no—
200C2×Q8—61no—
201C2×Q8—61no—
202C2×Q8—62no—
203C2×Q8—61no—
204C2×Q8—61no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—62no—
209C2×Q8—62no—
210C2×Q8—62no—
211C2×Q8—64no—
212C2×Q8—62no—
213C2×Q8C2×S411yescharacteristic
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—32no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—41no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—62no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—121no—
235C22×C4#21612yescomplemented
236C22×C4C2×D1211yescharacteristic
237C22×C4#23511yescharacteristic
238Q16—128no—
239Q16—124no—
240Q16—124no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—62no—
246C2×C8—62no—
247C4⋊C4—21no—
248C4⋊C4—32no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—61no—
257C4⋊C4—122no—
258C4⋊C4—122no—
259C4⋊C4—122no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C4⋊C4—122no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4C2×S411yescharacteristic
266C22⋊C4—122no—
267C22⋊C4—122no—
268C42—21no—
269C42—31no—
270C42—31no—
271C42—31no—
272C42—61no—
273C42—61no—
274C42—61no—
275C42—62no—
276C42—61no—
277C42C2×S411yescharacteristic
278C22×C3—41no—
279A4#46211yescharacteristic, complemented
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—42no—
283C3×C4—81no—
284C3×C4—82no—
285C3×C4—161no—
286Dic3—162no—
287C23—31no—
288C23—31no—
289C23—31no—
290C23—31no—
291C23#93011yescharacteristic
292C23C2×#21611yescharacteristic
293C23#95111yescharacteristic
294Q8—21no—
295Q8—34no—
296Q8—32no—
297Q8—62no—
298Q8—62no—
299Q8—62no—
300Q8—61no—
301Q8—122no—
302Q8—122no—
303Q8—122no—
304Q8—122no—
305Q8C4⋊S412yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—32no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—31no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—62no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—62no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4GL(2,ℤ/4)11yescharacteristic
343C2×C4GL(2,ℤ/4)11yescharacteristic
344C2×C4C22×S411yescharacteristic
345C8—122no—
346C8—122no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C22—31no—
351C22—31no—
352C22—31no—
353C22—31no—
354C22—31no—
355C22—31no—
356C22—31no—
357C22—31no—
358C22—31no—
359C22—61no—
360C22#503111yescharacteristic, complemented
361C22C2×GL(2,ℤ/4)11yescharacteristic
362C4—21no—
363C4—22no—
364C4—31no—
365C4—31no—
366C4—32no—
367C4—41no—
368C4—61no—
369C4—62no—
370C4—121no—
371C4—242no—
372C4—242no—
373C4#541612yescomplemented
374C4C2×C4⋊S411yescharacteristic
375C4#591211yescharacteristic
376C3—41noSylow
377C2—31no—
378C2—31no—
379C2—31no—
380C2—31no—
381C2#8548611yescharacteristic
382C2C2×#541611yescharacteristic
383C2#8551211yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×Q8D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenterC4×Q8S4
hypocenterA4#462
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
90. C4×A4Q16semidirect
235. C22×C4#216semidirect
279. A4#462semidirect
360. C22#5031semidirect
373. C4#5416semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
C2×#54162
#852832
#730522
C4⋊(A4⋊C8)2
#329063
#50314