Finite Group Databasebeta

← #10828462

Subgroups — #10828462

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C2×#4968C212yesmaximal, complemented
4#71560C212yesmaximal, complemented
5#71531C211yesmaximal, characteristic, complemented
6#69627C212yesmaximal, complemented
7#69626C214yesmaximal, complemented
8#69605C212yesmaximal, complemented
9#24061—31nomaximal, Sylow
10C22×#235—21no—
11C22×C3×C4⋊C4—21no—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2212yes—
17#5477—28no—
18C2×#875—21no—
19C2×#874C2211yescharacteristic
20C2×C4×D12—21no—
21C2×C4×D12C2212yes—
22C2×#869C2211yescharacteristic
23C3×#389—21no—
24C3×#389—21no—
25C3×#389C2212yes—
26C3×#389C2212yes—
27C2×C3×#95—21no—
28C2×#830C2212yes—
29#4968—28no—
30C2×#809C2214yes—
31C2×#809C2214yes—
32C2×#808C2214yes—
33#4529C2214yes—
34#4524C2212yes—
35#4524C2212yes—
36C2×#532—31no—
37C2×#481—32no—
38C2×#389D311yescharacteristic, complemented
39#2058—32no—
40#2041—32no—
41#1874—32no—
42#1864—34no—
43#1817—31no—
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50C2×#235—28no—
51C2×#235—44no—
52C2×#235—44no—
53C22×D12—21no—
54C22×D12—21no—
55C22×D12C2312yes—
56C22×C4×D3—22no—
57C22×Dic6—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—44no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42—24no—
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72#930—28no—
73#930—24no—
74#930D418yes—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2312yes—
79C22×C3⋊C8C2314yes—
80#875—28no—
81#874—24no—
82C4×D12—28no—
83C4×D12—24no—
84C4×D12—24no—
85#869—24no—
86C3×#95—22no—
87C3×#95—22no—
88#830—24no—
89#809—28no—
90#809—28no—
91#808—28no—
92C22×#41—61no—
93#532—68no—
94C2×C4⋊Q8—31no—
95C2×#127—31no—
96C2×#124—61no—
97C2×#121—32no—
98C2×#121—61no—
99C2×C4×D4—32no—
100C2×C4×D4—61no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2×D311yescharacteristic
103C22×C4⋊C4C2×D311yescharacteristic
104C22×C42C2×D311yescharacteristic
105#481—68no—
106C2×#106—34no—
107C2×#102—34no—
108C2×#102—34no—
109C2×C22⋊C8—32no—
110#389—21no—
111#389—21no—
112#389C2×D312yes—
113#389C2×D312yes—
114C2×#95—21no—
115#340—34no—
116#336—32no—
117#336—32no—
118C24×C3C2×D411yescharacteristic
119C23×D3—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—41no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4SD1614yes—
136C22×C3×C4C2×D414yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C2×#67—28no—
144C2×#67—44no—
145C2×#67—44no—
146C22×Dic3—22no—
147#235—432no—
148C2×D12—24no—
149C2×D12—24no—
150C2×D12—24no—
151C2×D12—24no—
152C2×D12—22no—
153C2×D12—42no—
154C2×D12—42no—
155C2×D12—42no—
156C2×D12—42no—
157C2×D12C2×D414yes—
158C2×C4×D3—28no—
159C2×C4×D3—44no—
160C2×C4×D3—44no—
161C2×Dic6—24no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—44no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4#4114yescomplemented
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—22no—
174C3×C42—22no—
175(C2×D3)⋊C4—28no—
176(C2×D3)⋊C4—28no—
177C4⋊Dic3—24no—
178C4⋊Dic3—22no—
179C4⋊Dic3C2×D414yes—
180C2×C3⋊C8—24no—
181C2×C3⋊C8—24no—
182C2×C3⋊C8—24no—
183C2×#41—68no—
184C2×#41—124no—
185C2×#41—124no—
186C22×Q8—61no—
187C22×D4—32no—
188C22×D4—62no—
189C22×D4—61no—
190C22×D4—61no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—62no—
194C23×C4#6711yescharacteristic
195C23×C4#6711yescharacteristic
196C23×C4C22×D311yescharacteristic
197C22×C8—34no—
198C4⋊Q8—64no—
199#127—64no—
200#124—68no—
201#121—38no—
202#121—68no—
203#121—64no—
204C4×D4—64no—
205C4×D4—64no—
206C4×D4—68no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—32no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4C22×D312yes—
217C2×C4⋊C4C22×D312yes—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C42—24no—
221C2×C42#6711yescharacteristic
222C2×C42#6711yescharacteristic
223C2×C42C22×D311yescharacteristic
224C2×C42C22×D311yescharacteristic
225#106—68no—
226#102—68no—
227#102—68no—
228C22⋊C8—64no—
229#95—22no—
230#95—22no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3#12012yes—
236C23×C3C2×SD1612yes—
237C23×C3AGL(1,ℤ/8)12yes—
238C23×C3C22×D411yescharacteristic
239C22×D3—24no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—42no—
245C22×D3—42no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—44no—
261C2×C3×C4—48no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—84no—
268C2×C3×C4#12012yes—
269C2×C3×C4C2×SD1612yes—
270C2×C3×C4C2×SD1612yes—
271C2×C3×C4AGL(1,ℤ/8)12yes—
272C2×C3×C4C22×D412yes—
273C2×C3×C4C2×#4112yes—
274C2×C3×C42−1+412yes—
275#67—416no—
276C2×Dic3—24no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279D12—28no—
280D12—48no—
281D12—48no—
282D12—44no—
283C4×D3—416no—
284Dic6—48no—
285C3⋊C8—44no—
286C24—62no—
287C24C2×#6711yescharacteristic
288#41—1232no—
289C2×Q8—64no—
290C2×Q8—122no—
291C2×Q8—122no—
292C2×D4—34no—
293C2×D4—64no—
294C2×D4—64no—
295C2×D4—68no—
296C2×D4—64no—
297C2×D4—62no—
298C2×D4—64no—
299C2×D4—122no—
300C2×D4—122no—
301C2×D4—124no—
302C2×D4—124no—
303C2×D4—122no—
304C2×D4—122no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—68no—
321C22×C4—62no—
322C22×C4—124no—
323C22×C4—124no—
324C22×C4#21514yes—
325C22×C4D3×D414yes—
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C23×D311yescharacteristic
332C2×C8—64no—
333C2×C8—64no—
334C2×C8—64no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—34no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—64no—
343C4⋊C4—62no—
344C4⋊C4#23514yescomplemented
345C22⋊C4—68no—
346C22⋊C4—68no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3#45014yes—
358C22×C3C2×#12011yescharacteristic
359C22×C3#52511yescharacteristic
360C22×C3#53611yescharacteristic
361C22×C3C22×SD1611yescharacteristic
362C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
363C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
364C22×C3#57511yescharacteristic
365C2×D3—44no—
366C2×D3—44no—
367C2×D3—44no—
368C2×D3—82no—
369C2×D3—82no—
370C2×D3—82no—
371C2×D3—82no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—41no—
378C3×C4—82no—
379C3×C4#54714yes—
380Dic3—44no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—64no—
386C23—122no—
387C23—122no—
388C23—122no—
389C23—122no—
390C23—122no—
391C23—122no—
392C23#93712yes—
393C23C2×#21512yes—
394C23#94912yes—
395C23C22×#6711yescharacteristic
396Q8—128no—
397D4—68no—
398D4—128no—
399D4—128no—
400D4—1216no—
401D4—124no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—28no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—64no—
424C2×C4—84no—
425C2×C4—1216no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4#93212yes—
429C2×C4#93712yes—
430C2×C4C2×#21512yes—
431C2×C4C2×#21512yes—
432C2×C4C2×#23512yes—
433C2×C4C2×D3×D412yes—
434C2×C4#100712yes—
435C8—124no—
436C2×C3—24no—
437C2×C3#240611yescharacteristic
438C2×C3C2×#45011yescharacteristic
439C2×C3#300611yescharacteristic
440C2×C3#310311yescharacteristic
441C2×C3#310911yescharacteristic
442C2×C3#322711yescharacteristic
443C2×C3#323411yescharacteristic
444D3—84no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—124no—
452C22—124no—
453C22—124no—
454C22—242no—
455C22—242no—
456C22—242no—
457C22—242no—
458C22#504614yes—
459C22#579111yescharacteristic
460C22C2×#93211yescharacteristic
461C22C2×#93711yescharacteristic
462C22C22×#21511yescharacteristic
463C22#581611yescharacteristic
464C22C2×#94911yescharacteristic
465C22#582211yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—44no—
471C4—41no—
472C4—82no—
473C4—124no—
474C4#550714yes—
475C3#2406111yescharacteristic, complemented, Sylow
476C2—24no—
477C2—244no—
478C2#8219211yescharacteristic
479C2#8231011yescharacteristic
480C2#8244211yescharacteristic
481C2#8262411yescharacteristic
482C2C2×#504611yescharacteristic
483C2#8301911yescharacteristic
484C2#8321611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#389C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenterC2×#389D3
hypocenterC3#24061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
3. C2×#4968C2semidirect
4. #71560C2semidirect
5. #71531C2semidirect
6. #69627C2semidirect
7. #69626C2semidirect
8. #69605C2semidirect
38. C2×#389D3semidirect
170. C3×C4⋊C4#41semidirect
344. C4⋊C4#235semidirect
475. C3#24061semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C2×C3×#3892
C2×#49682
#715602
#715312
#696272
#696262
#696052
#240613