Finite Group Databasebeta

← #10943891

Subgroups — #10943891

Order 768.

3606 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 3606 conjugacy classes of subgroups — too many to draw. Only its 255 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×#1007C211yesmaximal, characteristic, complemented
2C3×#2436C211yesmaximal, characteristic, complemented
3C2×#5062C214yesmaximal, complemented
4#82624C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#69692C214yesmaximal, complemented
7#69454C212yesmaximal, complemented
8#32889—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#1007—22no—
11C2×#1007—216no—
12C2×#1007—24no—
13C2×#1007C2214yes—
14C22×#239—22no—
15C22×D3×Q8—22no—
16C22×#235—22no—
17C22×#235C2212yes—
18C2×C3×#124C2212yescomplemented
19C2×C3×#121C2211yescharacteristic
20C2×#932C2214yescomplemented
21C3×#393—22no—
22C3×#391C2212yescomplemented
23C3×#386—22no—
24C3×C4×C22⋊C4C2211yescharacteristic
25C2×#836C2214yescomplemented
26#5207C2212yes—
27#5062—216no—
28#5040—22no—
29#5037—24no—
30#5036—22no—
31#4966C2214yescomplemented
32C2×#805C2218yescomplemented
33C22×2−1+4—31no—
34C2×#454—34no—
35#3006—32no—
36#2436D311yescharacteristic, complemented
37#1793—34no—
38#1789—32no—
39#1784—31no—
40C22×C3×Q8—28no—
41C22×C3×Q8—22no—
42C22×C3×Q8—22no—
43C22×C3×Q8D412yes—
44C22×C3×Q8C2312yes—
45C22×C3×D4—21no—
46C22×C3×D4C2311yescharacteristic
47C23×C3×C4D412yes—
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50#1007—232no—
51#1007—232no—
52#1007—232no—
53#1007—28no—
54#1007—216no—
55#1007D4116yescomplemented
56C2×#239—232no—
57C2×#239—28no—
58C2×#239—28no—
59C2×D3×Q8—28no—
60C2×D3×Q8—232no—
61C2×D3×Q8—28no—
62C2×#235—28no—
63C2×#235—232no—
64C2×#235—28no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—216no—
68C2×#235D414yes—
69C2×#235C2314yes—
70C22×D12—22no—
71C22×D12D412yes—
72C22×C4×D3—24no—
73C22×C4×D3—22no—
74C22×Dic6—22no—
75C22×Dic6D412yes—
76C3×#124—28no—
77C3×#121—22no—
78C3×#121—22no—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42C2312yes—
84C2×#214—24no—
85#932—216no—
86C2×#213—24no—
87C2×#208C2314yes—
88C3×#95—21no—
89C3×#95—24no—
90#836—28no—
91#830—22no—
92#830—22no—
93#810—24no—
94#809—24no—
95#805—216no—
96C2×2−1+4—34no—
97C2×2−1+4—64no—
98C2×2−1+4—616no—
99C2×2−1+4—62no—
100C22×#41—32no—
101C22×#41—62no—
102C22×#41—62no—
103C23×Q8—62no—
104C23×Q8C2×D311yescharacteristic
105C2×AGL(1,ℤ/8)—34no—
106C2×#124C2×D312yescomplemented
107C2×#121C2×D311yescharacteristic
108#454—616no—
109#449—64no—
110#448—62no—
111#447—62no—
112C2×#104—38no—
113#417—34no—
114C2×#101—34no—
115#407—32no—
116#393—22no—
117#391C2×D312yescomplemented
118#386—22no—
119C4×C22⋊C4C2×D311yescharacteristic
120C24×C3—21no—
121C24×C3C2×D411yescharacteristic
122C23×D3—22no—
123C2×C3×Q8—216no—
124C2×C3×Q8—216no—
125C2×C3×Q8—24no—
126C2×C3×Q8—28no—
127C2×C3×Q8—24no—
128C2×C3×Q8—24no—
129C2×C3×Q8C2×D418yes—
130C2×C3×D4—22no—
131C2×C3×D4—24no—
132C2×C3×D4—44no—
133C2×C3×D4—41no—
134C2×C3×D4—41no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—28no—
138C22×C3×C4—24no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D414yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C2×#67—216no—
147C2×#67—24no—
148C2×#67—24no—
149C22×Dic3—22no—
150#239—264no—
151#239—232no—
152#239—232no—
153D3×Q8—264no—
154D3×Q8—232no—
155D3×Q8—232no—
156#235—264no—
157#235—232no—
158#235—216no—
159#235—216no—
160#235—232no—
161#235—28no—
162#235—28no—
163#235—28no—
164#235C2×D4116yes—
165C2×D12—216no—
166C2×D12—28no—
167C2×D12—22no—
168C2×D12—28no—
169C2×D12C2×D414yes—
170C2×C4×D3—232no—
171C2×C4×D3—216no—
172C2×C4×D3—216no—
173C2×C4×D3—24no—
174C2×C4×D3—24no—
175C2×Dic6—216no—
176C2×Dic6—28no—
177C2×Dic6—22no—
178C2×Dic6—28no—
179C2×Dic6C2×D414yes—
180C3×C4⋊C4—22no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—48no—
184C3×C42—24no—
185#214—48no—
186#213—48no—
187#208—28no—
188C2×C3⋊C8—24no—
1892−1+4—316no—
1902−1+4—616no—
1912−1+4—632no—
1922−1+4—632no—
1932−1+4—632no—
1942−1+4—68no—
195C2×#41—34no—
196C2×#41—34no—
197C2×#41—616no—
198C2×#41—68no—
199C2×#41—632no—
200C2×#41—68no—
201C2×#41—632no—
202C2×#41—68no—
203C2×#41—62no—
204C2×#41—62no—
205C2×#41—68no—
206C22×Q8—28no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—32no—
210C22×Q8—62no—
211C22×Q8—68no—
212C22×Q8—632no—
213C22×Q8—68no—
214C22×Q8#6712yes—
215C22×Q8C22×D312yes—
216C22×D4—21no—
217C22×D4—32no—
218C22×D4—62no—
219C22×D4—62no—
220C22×D4C22×D311yescharacteristic
221C23×C4—64no—
222C23×C4—62no—
223C23×C4#6712yes—
224C23×C4C22×D311yescharacteristic
225AGL(1,ℤ/8)—616no—
226C2×SD16—64no—
227C2×D8—64no—
228C2×M16—34no—
229#124—28no—
230#121—22no—
231#121—22no—
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—24no—
236C2×C42C22×D312yes—
237#104—616no—
238#103—64no—
239#102—64no—
240#101—68no—
241C22⋊C8—62no—
242C22⋊C8—62no—
243#95—21no—
244#95—24no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—41no—
252C23×C3#12011yescharacteristic
253C23×C3#12011yescharacteristic
254C23×C3C22×D411yescharacteristic
255C22×D3—28no—
256C22×D3—24no—
257C22×D3—44no—
258C22×D3—44no—
259C3×Q8—216no—
260C3×Q8—216no—
261C3×D4—44no—
262C3×D4—82no—
263C3×D4—82no—
264C2×C3×C4—28no—
265C2×C3×C4—28no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4#12012yes—
277C2×C3×C4#12018yes—
278C2×C3×C4C22×D412yes—
279C2×C3×C4C22×D414yes—
280#67—216no—
281#67—216no—
282C2×Dic3—28no—
283C2×Dic3—24no—
284D12—232no—
285D12—28no—
286D12—24no—
287D12#12018yes—
288C4×D3—264no—
289C4×D3—216no—
290C4×D3—216no—
291Dic6—232no—
292Dic6—28no—
293Dic6—24no—
294Dic6#12018yes—
295C3⋊C8—44no—
296C24—21no—
297C24—62no—
298C24C2×#6711yescharacteristic
299#41—316no—
300#41—68no—
301#41—68no—
302#41—664no—
303#41—616no—
304#41—632no—
305#41—616no—
306#41—632no—
307#41—664no—
308#41—632no—
309#41—632no—
310#41—68no—
311C2×Q8—216no—
312C2×Q8—216no—
313C2×Q8—24no—
314C2×Q8—28no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—34no—
318C2×Q8—68no—
319C2×Q8—616no—
320C2×Q8—68no—
321C2×Q8—632no—
322C2×Q8—664no—
323C2×Q8—632no—
324C2×Q8—62no—
325C2×Q8D3×D418yes—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—34no—
329C2×D4—44no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—68no—
333C2×D4—616no—
334C2×D4—68no—
335C2×D4—616no—
336C2×D4—64no—
337C2×D4—64no—
338C2×D4—62no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—632no—
346C22×C4—616no—
347C22×C4—616no—
348C22×C4—62no—
349C22×C4—64no—
350C22×C4—64no—
351C22×C4D3×D412yes—
352C22×C4D3×D412yes—
353C22×C4C2×#6714yes—
354C22×C4C2×#6711yescharacteristic
355C22×C4C23×D311yescharacteristic
356SD16—128no—
357D8—128no—
358M16—68no—
359C2×C8—64no—
360C4⋊C4—22no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—48no—
364C42—24no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—44no—
373C22×C3—84no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3#45412yescomplemented
378C22×C3C2×#12012yes—
379C22×C3C2×#12011yescharacteristic
380C2×D3—28no—
381C2×D3—416no—
382C2×D3—44no—
383C2×D3—44no—
384C3×C4—28no—
385C3×C4—22no—
386C3×C4—82no—
387C3×C4—81no—
388C3×C4C2×#12014yes—
389Dic3—28no—
390C23—22no—
391C23—22no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—68no—
398C23—64no—
399C23—124no—
400C23—124no—
401C23#93711yescharacteristic
402C23#93711yescharacteristic
403C23C22×#6711yescharacteristic
404Q8—216no—
405Q8—216no—
406Q8—38no—
407Q8—68no—
408Q8—632no—
409Q8—64no—
410D4—38no—
411D4—44no—
412D4—68no—
413D4—632no—
414D4—616no—
415D4—616no—
416D4—64no—
417D4—82no—
418D4—82no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—664no—
432C2×C4—616no—
433C2×C4—68no—
434C2×C4—64no—
435C2×C4—616no—
436C2×C4#93718yes—
437C2×C4#93712yes—
438C2×C4C2×D3×D414yes—
439C2×C4C2×D3×D412yes—
440C8—124no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—41no—
444C2×C3—161no—
445C2×C3#284611yescharacteristic
446C2×C3C2×#45411yescharacteristic
447C2×C3#302011yescharacteristic
448D3—48no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—44no—
457C22—68no—
458C22—84no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—1216no—
463C22—124no—
464C22—124no—
465C22#506212yescomplemented
466C22C2×#93712yes—
467C22C2×#93711yescharacteristic
468C4—28no—
469C4—22no—
470C4—68no—
471C4—82no—
472C4—81no—
473C4D3×#12014yes—
474C3#3288911yescharacteristic, complemented, Sylow
475C2—22no—
476C2—22no—
477C2—41no—
478C2—128no—
479C2—161no—
480C2#8261211yescharacteristic
481C2C2×#506211yescharacteristic
482C2#8341511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2436C2
socleC22×C3C2×#120
hypercenter#2436D3
hypocenterC3#32889
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
1. C22×#1007C2semidirect
2. C3×#2436C2semidirect
3. C2×#5062C2semidirect
4. #82624C2semidirect
5. #72133C2semidirect
6. #69692C2semidirect
7. #69454C2semidirect
18. C2×C3×#124C22semidirect
20. C2×#932C22semidirect
22. C3×#391C22semidirect
25. C2×#836C22semidirect
31. #4966C22semidirect
32. C2×#805C22semidirect
36. #2436D3semidirect
55. #1007D4semidirect
106. C2×#124C2×D3semidirect
117. #391C2×D3semidirect
377. C22×C3#454semidirect
465. C22#5062semidirect
474. C3#32889semidirect

Maximal subgroups

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

subgroupindex
C22×#10072
C3×#24362
C2×#50622
#826242
#721332
#696922
#694542
#328893