Finite Group Databasebeta

← #10849219

Subgroups — #10849219

Order 768.

1346 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5718C211yesmaximal, characteristic
2C2×#5070C211yesmaximal, characteristic, complemented
3C2×#4842C211yesmaximal, characteristic, complemented
4#73721C211yesmaximal, characteristic, complemented
5C3×#1982C211yesmaximal, characteristic, complemented
6#70472C211yesmaximal, characteristic, complemented
7#70177C214yesmaximal, complemented
8#70170C214yesmaximal, complemented
9C4×#809C211yesmaximal, characteristic, complemented
10#25649—31nomaximal, Sylow
11C22×#237—21no—
12C2×C3×#127C2211yescharacteristic
13C2×#940C2212yes—
14C2×#939—22no—
15C2×#938—21no—
16C2×D4×Dic3C2212yes—
17#5718—24no—
18#5718—24no—
19C2×C4×D12C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2214yes—
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#832C2214yes—
26C2×#821C2212yes—
27C2×#817C2212yes—
28#5070—24no—
29#4984C2214yes—
30C2×#809C2212yes—
31#4935—21no—
32#4929—21no—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4842—24no—
35#4656—22no—
36#4550C2214yes—
37#4529C2212yes—
38C2×#550—31no—
39C2×#486—31no—
40C2×#476—31no—
41#2567—31no—
42#2041—34no—
43#1984—31no—
44#1982D311yescharacteristic, complemented
45#1864—34no—
46C4×#102—31no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4C2312yes—
50C22×#67—22no—
51C22×#67—22no—
52C23×Dic3—22no—
53C2×#237—28no—
54C2×#237—44no—
55C2×#237—44no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6—21no—
59C22×Dic6D411yescharacteristic
60C22×C3×C8C2312yes—
61C3×#127—22no—
62C3×#127D414yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#217—22no—
67#940—24no—
68#940—24no—
69#939—216no—
70#938—28no—
71D4×Dic3—24no—
72D4×Dic3—24no—
73C2×(C2×D3)⋊C4—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4D412yes—
80C2×C4×Dic3C2312yes—
81C22×C3⋊C8C2312yes—
82C4×D12—22no—
83C4×D12D414yes—
84#869—22no—
85#869—22no—
86C4×Dic6—22no—
87C4×Dic6—22no—
88C3×C4⋊C8—22no—
89C3×#102—28no—
90C3×#95—21no—
91#832—28no—
92C3⋊#95—24no—
93#821—24no—
94#817—24no—
95#809—24no—
96C4×C3⋊C8—22no—
97C22×#41—61no—
98#550—64no—
99#550—64no—
100C2×C4⋊Q8—31no—
101C2×#127—32no—
102C2×#127C2×D311yescharacteristic
103C2×#121—61no—
104C2×#121—62no—
105C2×C4×Q8—31no—
106C2×C4×D4—32no—
107C2×C4×D4—31no—
108#486—64no—
109#476—64no—
110C2×#106—32no—
111C2×C4⋊C8C2×D311yescharacteristic
112C2×#102—34no—
113C2×#102—32no—
114C2×#102—32no—
115C2×#102C2×D314yes—
116C2×C4×C8—31no—
117#395—62no—
118#391—61no—
119#389—34no—
120#384—61no—
121C4×C4⋊C4C2×D311yescharacteristic
122#340—32no—
123#336—34no—
124C24×C3—22no—
125C23×D3—21no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—22no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C2×C3×D4—82no—
134C2×C3×D4C2×D414yes—
135C22×C3×C4—22no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2411yescharacteristic
139C2×#67—28no—
140C2×#67—28no—
141C2×#67—44no—
142C2×#67—44no—
143C2×#67—84no—
144C22×Dic3—28no—
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—44no—
148C22×Dic3—44no—
149C22×Dic3C2×D412yes—
150C22×Dic3C2×D412yes—
151#237—432no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—21no—
155C2×D12C2×D412yes—
156C2×C4×D3—24no—
157C2×C4×D3—42no—
158C2×C4×D3—42no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—42no—
162C2×Dic6—82no—
163C2×Dic6SD1614yes—
164C2×C3×C8—22no—
165C2×C3×C8—24no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4#4114yescomplemented
168C3×C42—42no—
169C3×C42C2×D412yes—
170C3×C42C2×D412yes—
171#217—28no—
172(C2×D3)⋊C4—24no—
173C4⋊Dic3—22no—
174C4⋊Dic3—21no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—42no—
178C4⋊Dic3—84no—
179C4⋊Dic3C2×D412yes—
180Dic3⋊C4—24no—
181C4×Dic3—22no—
182C4×Dic3—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×#41—68no—
187C2×#41—124no—
188C2×#41—124no—
189C22×Q8—31no—
190C22×Q8—61no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—31no—
194C22×D4—62no—
195C22×D4—62no—
196C22×D4C22×D312yes—
197C23×C4—62no—
198C23×C4—61no—
199C22×C8—32no—
200C22×C8C22×D312yes—
201C4⋊Q8—62no—
202C4⋊Q8—62no—
203#127—22no—
204#127—64no—
205#127—64no—
206#127D1214yes—
207#121—68no—
208#121—616no—
209C4×Q8—62no—
210C4×Q8—62no—
211C4×D4—34no—
212C4×D4—64no—
213C4×D4—64no—
214C4×D4—62no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4C22×D312yes—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—61no—
224C2×C42—21no—
225C2×C42—32no—
226C2×C42C22×D311yescharacteristic
227#106—64no—
228C4⋊C8—22no—
229#102—28no—
230#102—64no—
231#102—68no—
232#102—64no—
233C4×C8—62no—
234#95—21no—
235#95—64no—
236C23×C3—24no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3C22×D411yescharacteristic
242C22×D3—22no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C3×D4—28no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—48no—
251C3×D4—84no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4#12012yes—
257C2×C3×C4#12012yes—
258C2×C3×C4C2×SD1612yes—
259C2×C3×C4AGL(1,ℤ/8)12yes—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C2×#4112yes—
263C2×C3×C42−1+412yes—
264#67—416no—
265#67—88no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—24no—
271C2×Dic3—416no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—42no—
275C2×Dic3—44no—
276C2×Dic3C2×SD1614yes—
277D12—22no—
278D12—22no—
279D12—42no—
280C4×D3—48no—
281Dic6—28no—
282Dic6—84no—
283C3×C8—42no—
284C3⋊C8—42no—
285C24—22no—
286C24—61no—
287#41—1232no—
288C2×Q8—34no—
289C2×Q8—62no—
290C2×Q8—62no—
291C2×Q8—122no—
292C2×Q8—242no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—32no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—68no—
302C2×D4—68no—
303C2×D4—61no—
304C2×D4—62no—
305C2×D4—62no—
306C2×D4—82no—
307C2×D4—124no—
308C2×D4—124no—
309C2×D4—244no—
310C2×D4C2×D1214yes—
311C22×C4—22no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—68no—
315C22×C4—61no—
316C22×C4—64no—
317C22×C4—62no—
318C22×C4—124no—
319C22×C4—124no—
320C22×C4—122no—
321C22×C4—122no—
322C22×C4D3×D411yescharacteristic
323C22×C4D3×D411yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—62no—
328C2×C8—62no—
329C2×C8—62no—
330C4⋊C4—22no—
331C4⋊C4—32no—
332C4⋊C4—64no—
333C4⋊C4—62no—
334C4⋊C4—61no—
335C4⋊C4—62no—
336C4⋊C4—62no—
337C4⋊C4—122no—
338C4⋊C4—244no—
339C4⋊C4#23714yescomplemented
340C22⋊C4—68no—
341C22⋊C4—64no—
342C42—42no—
343C42—62no—
344C42—62no—
345C42C2×D1212yes—
346C42D3×D412yes—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—84no—
351C22×C3—84no—
352C22×C3C2×#12011yescharacteristic
353C22×C3#54812yes—
354C22×C3C22×SD1611yescharacteristic
355C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
356C22×C3#57612yes—
357C2×D3—42no—
358C2×D3—42no—
359C2×D3—42no—
360C2×D3—82no—
361C2×D3—82no—
362C3×C4—22no—
363C3×C4—44no—
364C3×C4#45014yes—
365C3×C4#53211yescharacteristic
366C3×C4#53611yescharacteristic
367C3×C4#54712yes—
368Dic3—24no—
369Dic3—42no—
370Dic3—42no—
371Dic3—82no—
372C23—24no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—62no—
378C23—121no—
379C23—121no—
380C23—122no—
381C23—122no—
382C23C2×D3×D411yescharacteristic
383Q8—68no—
384Q8—244no—
385D4—28no—
386D4—42no—
387D4—42no—
388D4—48no—
389D4—62no—
390D4—62no—
391D4—84no—
392D4—1216no—
393D4—122no—
394D4—248no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—34no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—64no—
405C2×C4—1216no—
406C2×C4—121no—
407C2×C4—121no—
408C2×C4—128no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4#88212yes—
412C2×C4#88212yes—
413C2×C4#91112yes—
414C2×C4D3×SD1612yes—
415C2×C4C22×D1211yescharacteristic
416C2×C4C2×D3×D411yescharacteristic
417C2×C4C2×#23712yes—
418C2×C4#101012yes—
419C8—42no—
420C8—122no—
421C2×C3—84no—
422C2×C3#243211yescharacteristic
423C2×C3C2×#45011yescharacteristic
424C2×C3#300811yescharacteristic
425C2×C3#336212yes—
426C2×C3#336612yes—
427D3—82no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—84no—
432C22—84no—
433C22—122no—
434C22—122no—
435C22—122no—
436C22—242no—
437C22—242no—
438C22C2×#88211yescharacteristic
439C22#559812yes—
440C22C2×#91111yescharacteristic
441C22C2×D3×SD1611yescharacteristic
442C22#577412yes—
443C4—22no—
444C4—44no—
445C4—64no—
446C4—122no—
447C4—122no—
448C4—242no—
449C4#477614yes—
450C4#555512yes—
451C4#567511yescharacteristic
452C4#571611yescharacteristic
453C3#2564911yescharacteristic, complemented, Sylow
454C2—84no—
455C2—242no—
456C2#7644511yescharacteristic
457C2C2×#477611yescharacteristic
458C2#7892811yescharacteristic
459C2#8032012yes—
460C2#8036912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1982C2
socleC23×C3C22×D4
prime radicalC2×#5718C2
hypercenter#1982D3
hypocenterC3#25649
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. C2×#5070C2semidirect
3. C2×#4842C2semidirect
4. #73721C2semidirect
5. C3×#1982C2semidirect
6. #70472C2semidirect
7. #70177C2semidirect
8. #70170C2semidirect
9. C4×#809C2semidirect
44. #1982D3semidirect
167. C3×C4⋊C4#41semidirect
339. C4⋊C4#237semidirect
453. C3#25649semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×#50702
C2×#48422
#737212
C3×#19822
#704722
#701772
#701702
C4×#8092
#256493