Finite Group Databasebeta

← C2×#85290 #11714463

Subgroups — C2×#85290 #11714463

Order 768.

1933 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1933 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C4⋊S4C211yesmaximal, characteristic
2C22×A4⋊Q8C211yesmaximal, characteristic, complemented
3C2×C42×A4C211yesmaximal, characteristic, complemented
4C2×(C2×S4)⋊C4C214yesmaximal, complemented
5#85290C218yesmaximal, direct product factor
6C2×#2850—31nomaximal, Sylow
7C23×S4C2212yes—
8C22×C4×A4C2212yes—
9C22×C4×A4C2211yescharacteristic
10C22×SL(2,ℤ/4)C2212yes—
11C2×C4⋊S4—24no—
12C2×C4⋊S4C2214yes—
13C2×A4⋊Q8—24no—
14C2×A4⋊Q8C2214yescomplemented
15C2×#875—41nomaximal
16C42×A4C2214yescomplemented
17(C2×S4)⋊C4C22116yescomplemented
18C22×#124—31no—
19C22×C22⋊Q8—31no—
20C22×#121—31no—
21C23×C42D311yescharacteristic, complemented
22#2850—38no—
23#2850—38no—
24C2×#391—31no—
25C2×#390—31no—
26C2×#388—34no—
27C2×#386—34no—
28C2×#382—31no—
29C23×A4C2311yescharacteristic
30C22×S4—24no—
31C22×S4—28no—
32C22×S4C2314yes—
33C22×D12—41no—
34C22×Dic6—41no—
35C2×C4×A4—24no—
36C2×C4×A4D414yes—
37C2×C4×A4C2314yes—
38C2×C4×A4C2312yes—
39C2×SL(2,ℤ/4)—24no—
40C2×SL(2,ℤ/4)C2314yes—
41C4⋊S4—28no—
42A4⋊Q8—28no—
43C2×C3×C42—41no—
44C2×(C2×D3)⋊C4—44no—
45#875—48no—
46C23×Q8—31no—
47C23×D4—31no—
48C23×D4—32no—
49C24×C4C2×D312yes—
50C24×C4C2×D311yescharacteristic
51C2×#124—38no—
52C2×#124—61no—
53C2×#124—62no—
54C2×#124—64no—
55C2×#124—61no—
56C2×C22⋊Q8—34no—
57C2×C22⋊Q8—34no—
58C2×C22⋊Q8—64no—
59C2×C22⋊Q8—64no—
60C2×#121—34no—
61C2×#121—34no—
62C2×#121—64no—
63C2×#121—64no—
64C22×C4⋊C4—31no—
65C22×C4⋊C4—31no—
66C22×C4⋊C4—32no—
67C22×C22⋊C4—32no—
68C22×C22⋊C4—34no—
69C22×C22⋊C4—34no—
70C22×C22⋊C4—32no—
71C22×C42—31no—
72C22×C42—34no—
73C22×C42—32no—
74C22×C42—31no—
75C22×C42C2×D314yescomplemented
76#391—38no—
77#390—38no—
78#388—332no—
79#386—316no—
80#386—316no—
81#382—38no—
82C2×#95—34no—
83C2×#95—34no—
84C23×D3—42no—
85C22×A4C2×D412yes—
86C22×A4#4114yes—
87C22×A4C2411yescharacteristic
88C2×S4—24no—
89C2×S4—28no—
90C2×S4—48no—
91C22×C3×C4—41no—
92C22×C3×C4—42no—
93C22×Dic3—42no—
94C2×D12—44no—
95C2×D12—84no—
96C2×Dic6—44no—
97C2×Dic6—84no—
98C4×A4—24no—
99C4×A4C2×D414yes—
100SL(2,ℤ/4)—24no—
101C3×C42—44no—
102(C2×D3)⋊C4—416no—
103C25—32no—
104C25C22×D311yescharacteristic
105C22×Q8—34no—
106C22×Q8—64no—
107C22×Q8—61no—
108C22×Q8—62no—
109C22×Q8—64no—
110C22×Q8—61no—
111C22×D4—34no—
112C22×D4—34no—
113C22×D4—34no—
114C22×D4—64no—
115C22×D4—61no—
116C22×D4—62no—
117C22×D4—64no—
118C22×D4—64no—
119C22×D4—68no—
120C22×D4—64no—
121C22×D4—68no—
122C22×D4—61no—
123C23×C4—24no—
124C23×C4—31no—
125C23×C4—31no—
126C23×C4—32no—
127C23×C4—32no—
128C23×C4—34no—
129C23×C4—32no—
130C23×C4—34no—
131C23×C4—32no—
132C23×C4—32no—
133C23×C4—32no—
134C23×C4—64no—
135C23×C4D1214yes—
136C23×C4C22×D314yes—
137C23×C4C22×D312yes—
138#124—68no—
139#124—616no—
140#124—68no—
141C22⋊Q8—616no—
142C22⋊Q8—68no—
143C22⋊Q8—68no—
144#121—616no—
145#121—68no—
146#121—68no—
147C2×C4⋊C4—34no—
148C2×C4⋊C4—34no—
149C2×C4⋊C4—38no—
150C2×C4⋊C4—64no—
151C2×C4⋊C4—64no—
152C2×C4⋊C4—68no—
153C2×C22⋊C4—38no—
154C2×C22⋊C4—34no—
155C2×C22⋊C4—316no—
156C2×C22⋊C4—316no—
157C2×C22⋊C4—34no—
158C2×C22⋊C4—616no—
159C2×C22⋊C4—68no—
160C2×C22⋊C4—64no—
161C2×C22⋊C4—68no—
162C2×C22⋊C4—64no—
163C2×C22⋊C4—64no—
164C2×C22⋊C4—64no—
165C2×C42—32no—
166C2×C42—31no—
167C2×C42—34no—
168C2×C42—34no—
169C2×C42—38no—
170C2×C42—34no—
171C2×C42—62no—
172C2×C42—64no—
173C2×C42—61no—
174C2×C42S411yescharacteristic, complemented
175#95—316no—
176#95—316no—
177C23×C3—41no—
178C22×D3—44no—
179C22×D3—84no—
180C22×D3—88no—
181C2×A4#12414yescomplemented
182C2×A4C22×D411yescharacteristic
183C2×A4C2×#4112yes—
184S4—44no—
185C2×C3×C4—44no—
186C2×C3×C4—42no—
187C2×C3×C4—44no—
188C2×C3×C4—84no—
189C2×Dic3—44no—
190C2×Dic3—84no—
191D12—88no—
192Dic6—88no—
193C24—31no—
194C24—31no—
195C24—32no—
196C24—34no—
197C24—34no—
198C24—62no—
199C24—64no—
200C24—68no—
201C24—64no—
202C24—68no—
203C24—62no—
204C24C2×D1212yes—
205C24#23514yes—
206C24C23×D311yescharacteristic
207C2×Q8—68no—
208C2×Q8—64no—
209C2×Q8—68no—
210C2×Q8—64no—
211C2×Q8—124no—
212C2×Q8—128no—
213C2×Q8—124no—
214C2×D4—68no—
215C2×D4—64no—
216C2×D4—68no—
217C2×D4—68no—
218C2×D4—64no—
219C2×D4—616no—
220C2×D4—68no—
221C2×D4—64no—
222C2×D4—68no—
223C2×D4—64no—
224C2×D4—124no—
225C2×D4—128no—
226C2×D4—128no—
227C2×D4—1216no—
228C2×D4—128no—
229C2×D4—124no—
230C22×C4—24no—
231C22×C4—31no—
232C22×C4—34no—
233C22×C4—34no—
234C22×C4—32no—
235C22×C4—32no—
236C22×C4—34no—
237C22×C4—32no—
238C22×C4—34no—
239C22×C4—38no—
240C22×C4—34no—
241C22×C4—38no—
242C22×C4—32no—
243C22×C4—34no—
244C22×C4—34no—
245C22×C4—34no—
246C22×C4—34no—
247C22×C4—34no—
248C22×C4—64no—
249C22×C4—64no—
250C22×C4—64no—
251C22×C4—61no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—64no—
255C22×C4—68no—
256C22×C4—68no—
257C22×C4—64no—
258C22×C4—62no—
259C22×C4—64no—
260C22×C4—64no—
261C22×C4—64no—
262C22×C4—68no—
263C22×C4—68no—
264C22×C4—64no—
265C22×C4—62no—
266C22×C4C2×D1214yes—
267C22×C4C2×S412yes—
268C22×C4C2×S411yescharacteristic
269C4⋊C4—68no—
270C4⋊C4—68no—
271C4⋊C4—616no—
272C22⋊C4—632no—
273C22⋊C4—616no—
274C22⋊C4—616no—
275C22⋊C4—68no—
276C22⋊C4—64no—
277C22⋊C4—64no—
278C22⋊C4—616no—
279C42—38no—
280C42—34no—
281C42—64no—
282C42C2×S414yescomplemented
283C22×C3—42no—
284C22×C3—44no—
285C22×C3—41no—
286C2×D3—84no—
287C2×D3—88no—
288C2×D3—168no—
289A4C2×#12411yescharacteristic, complemented
290C3×C4—44no—
291C3×C4—84no—
292Dic3—84no—
293C23—32no—
294C23—32no—
295C23—34no—
296C23—34no—
297C23—31no—
298C23—31no—
299C23—32no—
300C23—31no—
301C23—34no—
302C23—32no—
303C23—34no—
304C23—38no—
305C23—61no—
306C23—62no—
307C23—64no—
308C23—64no—
309C23—68no—
310C23—64no—
311C23—68no—
312C23—64no—
313C23—128no—
314C23—124no—
315C23—128no—
316C23—1216no—
317C23—128no—
318C23—124no—
319C23—128no—
320C23#87514yescomplemented
321C23C22×D1211yescharacteristic
322C23C2×#23512yes—
323C23C22×S411yescharacteristic
324Q8—128no—
325Q8—128no—
326D4—128no—
327D4—1216no—
328D4—124no—
329D4—128no—
330D4—124no—
331D4—128no—
332C2×C4—24no—
333C2×C4—34no—
334C2×C4—32no—
335C2×C4—34no—
336C2×C4—34no—
337C2×C4—38no—
338C2×C4—34no—
339C2×C4—38no—
340C2×C4—34no—
341C2×C4—64no—
342C2×C4—64no—
343C2×C4—68no—
344C2×C4—62no—
345C2×C4—64no—
346C2×C4—68no—
347C2×C4—64no—
348C2×C4—64no—
349C2×C4—64no—
350C2×C4—64no—
351C2×C4—64no—
352C2×C4—68no—
353C2×C4—68no—
354C2×C4—64no—
355C2×C4—64no—
356C2×C4—128no—
357C2×C4—128no—
358C2×C4—124no—
359C2×C4—124no—
360C2×C4C4⋊S414yes—
361C2×C4C22×S414yes—
362C2×C4C22×S412yes—
363C2×C3—44no—
364C2×C3—41no—
365C2×C3—42no—
366D3—164no—
367C22—34no—
368C22—31no—
369C22—32no—
370C22—34no—
371C22—34no—
372C22—38no—
373C22—31no—
374C22—31no—
375C22—32no—
376C22—32no—
377C22—32no—
378C22—34no—
379C22—61no—
380C22—62no—
381C22—64no—
382C22—124no—
383C22—124no—
384C22—128no—
385C22—124no—
386C22—128no—
387C22—248no—
388C22—248no—
389C22C2×#87511yescharacteristic, complemented
390C22C2×C4⋊S412yes—
391C22#591214yes—
392C22C23×S411yescharacteristic
393C4—24no—
394C4—34no—
395C4—64no—
396C4—124no—
397C4—124no—
398C4—124no—
399C4C2×C4⋊S414yes—
400C3—41noSylow
401C2—31no—
402C2—31no—
403C2—32no—
404C2—34no—
405C2—244no—
406C2#8529014yesdirect product factor
407C2C22×C4⋊S411yescharacteristic
408C2C2×#591212yes—
subgroupG/H
centerC23C22×S4
commutator, cosocleC22×A4C24
FrattiniC22C23×S4
FittingC23×C42D3
socleC25C22×D3
prime radicalC22×C4⋊S4C2
hypercenterC2×C42S4
hypocenterA4C2×#124
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C22×A4⋊Q8C2semidirect
3. C2×C42×A4C2semidirect
4. C2×(C2×S4)⋊C4C2semidirect
5. #85290C2direct (N × H)
14. C2×A4⋊Q8C22semidirect
16. C42×A4C22semidirect
17. (C2×S4)⋊C4C22semidirect
21. C23×C42D3semidirect
75. C22×C42C2×D3semidirect
174. C2×C42S4semidirect
181. C2×A4#124semidirect
282. C42C2×S4semidirect
289. A4C2×#124semidirect
320. C23#875semidirect
389. C22C2×#875semidirect
406. C2#85290direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊S42
C22×A4⋊Q82
C2×C42×A42
C2×(C2×S4)⋊C42
#852902
C2×#28503
C2×#8754