Finite Group Databasebeta

← #11713995

Subgroups — #11713995

Order 768.

597 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 59 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×A4×M16C211yesmaximal, characteristic
2#85480C212yesmaximal
3C2×#5398C211yesmaximal, characteristic, complemented
4#73110C211yesmaximal, characteristic, complemented
5#73109C212yesmaximal, complemented
6#32937—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8C2×A4⋊Q8C2212yes—
9A4×M16—22no—
10C2×C8×A4C2211yescharacteristic
11C2×C8×A4C2211yescharacteristic
12#5432—22no—
13C4⋊SL(2,ℤ/4)C2212yes—
14SL(2,ℤ/4)⋊C4—22no—
15#5398—22no—
16#5137—41nomaximal
17C23×M16D311yescharacteristic
18C2×#470—31no—
19C2×#470—31no—
20C2×#467—31no—
21#2848—32no—
22#1894—32no—
23#1894—32no—
24#1892—32no—
25#1809—31no—
26#1809—31no—
27#1807—31no—
28C23×A4D411yescharacteristic
29C2×C4×A4—21no—
30C2×C4×A4—21no—
31C2×C4×A4D411yescharacteristic
32C2×C4×A4C2311yescharacteristic
33C2×SL(2,ℤ/4)—22no—
34C2×SL(2,ℤ/4)—22no—
35A4⋊Q8—22no—
36A4⋊Q8—22no—
37C2×C3×M16—41no—
38#924—42no—
39C2×Dic12—41no—
40C8×A4—21no—
41C8×A4D412yes—
42#817—41no—
43#816—42no—
44C24×C4C2×D311yescharacteristic
45C22×Q16—31no—
46C22×M16—22no—
47C22×M16—31no—
48C22×M16—31no—
49C22×M16—31no—
50C22×M16—31no—
51C22×M16—62no—
52C23×C8C2×D311yescharacteristic
53C23×C8C2×D311yescharacteristic
54C2×C22⋊Q8—32no—
55C2×C22⋊Q8—32no—
56C2×C22⋊Q8—32no—
57#470—64no—
58#470—61no—
59#470—61no—
60#470—61no—
61#470—61no—
62#467—62no—
63#467—62no—
64C2×#107—31no—
65C2×#106—31no—
66C2×#106—31no—
67C2×#103—32no—
68C2×#103—32no—
69C2×#103—32no—
70#385—62no—
71#385—62no—
72#385—62no—
73#380—32no—
74#380—32no—
75#380—32no—
76#379—62no—
77#340—32no—
78#340—32no—
79#340—32no—
80C22×A4—21no—
81C22×A4—22no—
82C22×A4C2×D411yescharacteristic
83C22×C3×C4—41no—
84C2×Dic6—42no—
85C4×A4—21no—
86C4×A4C2×D411yescharacteristic
87C4×A4#4111yescharacteristic
88SL(2,ℤ/4)—42no—
89SL(2,ℤ/4)—42no—
90C3×M16—82no—
91C2×C3×C8—41no—
92C2×C3×C8—41no—
93#217—82no—
94C4⋊Dic3—42no—
95Dic3⋊C4—82no—
96Dic12—82no—
97C25D1211yescharacteristic
98C22×Q8—32no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—31no—
102C23×C4—31no—
103C23×C4—31no—
104C23×C4—31no—
105C23×C4—61no—
106C23×C4—61no—
107C23×C4D1211yescharacteristic
108C23×C4C22×D311yescharacteristic
109C2×Q16—62no—
110C2×Q16—61no—
111C2×Q16—62no—
112C2×Q16—61no—
113C2×M16—34no—
114C2×M16—31no—
115C2×M16—31no—
116C2×M16—31no—
117C2×M16—61no—
118C2×M16—62no—
119C2×M16—62no—
120C2×M16—62no—
121C2×M16—62no—
122C2×M16—62no—
123C2×M16—62no—
124C2×M16—62no—
125C2×M16S411yescharacteristic
126C22×C8—21no—
127C22×C8—31no—
128C22×C8—31no—
129C22×C8—31no—
130C22×C8—32no—
131C22×C8—31no—
132C22×C8—61no—
133C22×C8#6712yes—
134C22⋊Q8—62no—
135C22⋊Q8—64no—
136C22⋊Q8—62no—
137C22⋊Q8—62no—
138C22⋊Q8—62no—
139C22⋊Q8—64no—
140C22⋊Q8—62no—
141C22⋊Q8—62no—
142C22⋊Q8—62no—
143C22⋊Q8—62no—
144C2×C4⋊C4—32no—
145C2×C4⋊C4—32no—
146C2×C4⋊C4—32no—
147C2×C4⋊C4—62no—
148C2×C4⋊C4—62no—
149C2×C4⋊C4—62no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4—62no—
156#107—62no—
157#106—62no—
158#106—61no—
159#106—61no—
160#103—64no—
161#103—64no—
162#103—62no—
163#103—62no—
164#95—62no—
165#95—62no—
166#95—62no—
167C23×C3—41no—
168C2×A4—41no—
169C2×A4#12111yescharacteristic
170C2×A4#13712yes—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×Dic3—82no—
176C2×Dic3—82no—
177Dic6—82no—
178Dic6—82no—
179C3×C8—42no—
180C3×C8—81no—
181C24—22no—
182C24—21no—
183C24—31no—
184C24—31no—
185C24—61no—
186C24—62no—
187C24C2×D1211yescharacteristic
188C2×Q8—62no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×Q8—62no—
192C2×Q8—62no—
193C2×Q8—62no—
194C22×C4—21no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4—62no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—62no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4#23511yescharacteristic
225C22×C4C2×#6711yescharacteristic
226C22×C4C2×S411yescharacteristic
227Q16—122no—
228Q16—122no—
229M16—22no—
230M16—62no—
231M16—62no—
232M16—64no—
233M16—62no—
234C2×C8—31no—
235C2×C8—32no—
236C2×C8—32no—
237C2×C8—32no—
238C2×C8—32no—
239C2×C8—31no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—61no—
243C2×C8—61no—
244C2×C8C2×S411yescharacteristic
245C2×C8C2×S411yescharacteristic
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—62no—
252C4⋊C4—122no—
253C4⋊C4—122no—
254C4⋊C4—124no—
255C4⋊C4—122no—
256C4⋊C4—122no—
257C22⋊C4—122no—
258C22⋊C4—122no—
259C22⋊C4—124no—
260C22⋊C4—122no—
261C22⋊C4—124no—
262C22⋊C4—124no—
263C22⋊C4—122no—
264C22⋊C4—122no—
265C22⋊C4—122no—
266C22×C3—41no—
267C22×C3—81no—
268C22×C3—82no—
269A4#47011yescharacteristic, complemented
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—81no—
273Dic3—162no—
274Dic3—162no—
275C23—31no—
276C23—31no—
277C23—32no—
278C23—32no—
279C23—31no—
280C23—41no—
281C23—62no—
282C23—61no—
283C23—61no—
284C23—62no—
285C23—62no—
286C23—62no—
287C23—61no—
288C23—122no—
289C23—121no—
290C23—122no—
291C23#90912yes—
292C23#93011yescharacteristic
293C23C4⋊S411yescharacteristic
294Q8—122no—
295Q8—122no—
296Q8—122no—
297Q8—122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4C4⋊S411yescharacteristic
334C2×C4C22×S411yescharacteristic
335C8—21no—
336C8—32no—
337C8—61no—
338C8GL(2,ℤ/4)12yes—
339C2×C3—42no—
340C2×C3—41no—
341C2×C3—161no—
342C22—22no—
343C22—21no—
344C22—32no—
345C22—31no—
346C22—32no—
347C22—32no—
348C22—31no—
349C22—31no—
350C22—61no—
351C22—64no—
352C22—62no—
353C22—61no—
354C22—121no—
355C22—121no—
356C22—122no—
357C22—121no—
358C22—121no—
359C22#513711yescharacteristic, complemented
360C22C2×C4⋊S411yescharacteristic
361C4—21no—
362C4—31no—
363C4—31no—
364C4—61no—
365C4—242no—
366C4—242no—
367C4—242no—
368C4—242no—
369C4#591211yescharacteristic
370C4C2×GL(2,ℤ/4)11yescharacteristic
371C3—41noSylow
372C2—31no—
373C2—32no—
374C2—31no—
375C2—41no—
376C2—121no—
377C2#8544412yes—
378C2#8548611yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC23×M16D3
socleC24C2×D12
prime radicalC23×A4D4
hypercenterC2×M16S4
hypocenterA4#470
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C2×#5398C2semidirect
4. #73110C2semidirect
5. #73109C2semidirect
269. A4#470semidirect
359. C22#5137semidirect

Maximal subgroups

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

subgroupindex
C2×A4×M162
#854802
C2×#53982
#731102
#731092
#329373
#51374