Finite Group Databasebeta

← #10801883

Subgroups — #10801883

Order 768.

1170 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×#5135C211yesmaximal, characteristic
2C2×Dic3×M16C211yesmaximal, characteristic, complemented
3C2×#4869C211yesmaximal, characteristic, complemented
4#73865C212yesmaximal, complemented
5C3×#1809C211yesmaximal, characteristic, complemented
6#70628C211yesmaximal, characteristic, complemented
7#70605C211yesmaximal, characteristic, complemented
8#70559C211yesmaximal, characteristic, complemented
9#69517C212yesmaximal, complemented
10#68852C212yesmaximal, complemented
11#68782C212yesmaximal, complemented
12#21538—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14C2×#930C2212yes—
15C22×C4×Dic3C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C22×D24C2211yescharacteristic
18C2×#893C2212yes—
19C2×#67⋊C4—22no—
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C3×#380C2212yes—
23#5135C418yescomplemented
24Dic3×M16—24no—
25C2×#821C2212yes—
26C2×#817C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×#815C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4988C2212yes—
32Dic3×C4⋊C4C2212yes—
33#4972C2212yes—
34C2×#809C2212yes—
35C2×#809C2212yes—
36#4958—22no—
37#4869—24no—
38#4669—22no—
39#4666—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#469—31no—
44C2×#442—31no—
45C2×C4×M16—31no—
46C4×#121—32no—
47#1970—31no—
48#1941—31no—
49#1934—32no—
50#1895—32no—
51#1809D311yescharacteristic, complemented
52#1776—31no—
53#1762—32no—
54C23×C3×C4C2311yescharacteristic
55C22×#67—22no—
56C23×Dic3D412yes—
57C22×D12C2312yes—
58C22×C4×D3—22no—
59C2×C3×M16—22no—
60C2×C3×M16—22no—
61C2×C3×M16C2×C414yescomplemented
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67#930—24no—
68#930C2×C418yescomplemented
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×C4⋊Dic3C2312yes—
74C2×Dic3⋊C4—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3D412yes—
79C2×C4×Dic3C2311yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C2×#208—22no—
82C2×#208—22no—
83C2×#208—22no—
84C22×C3⋊C8C2311yescharacteristic
85C22×C3⋊C8C2311yescharacteristic
86C2×D24—22no—
87C2×D24—22no—
88C2×D24C2×C414yescomplemented
89#893—24no—
90#893—24no—
91#67⋊C4—48no—
92C3×#107—22no—
93C3×#106—22no—
94C3×#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#821C2×C418yescomplemented
98#817C2×C414yescomplemented
99#815—22no—
100#815D414yescomplemented
101C8×Dic3—22no—
102C8×Dic3—22no—
103#809—24no—
104#809—24no—
105C22×D8—31no—
106C22×M16—31no—
107C22×M16C2×D311yescharacteristic
108C2×#121—32no—
109C2×C4×D4—32no—
110C2×C4×D4—62no—
111C22×C42—31no—
112#469—38no—
113#442—64no—
114C2×#107C2×D311yescharacteristic
115C2×#106—31no—
116C2×#106C2×D311yescharacteristic
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C4×M16—64no—
121C2×#97—31no—
122C2×#97—31no—
123C2×C4×C8—31no—
124C2×C4×C8—31no—
125#390—32no—
126#389—32no—
127#388—62no—
128#386—62no—
129#380C2×D312yes—
130C4×C4⋊C4—32no—
131C4×C22⋊C4—62no—
132#340—32no—
133#340—32no—
134#340C2×D312yes—
135C24×C3C2×D411yescharacteristic
136C23×D3—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4C22×C412yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4112yes—
146C22×C3×C4C2411yescharacteristic
147C2×#67—28no—
148C2×#67—44no—
149C2×#67—44no—
150C22×Dic3—22no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—42no—
154C22×Dic3C2×D412yes—
155C22×Dic3C2×D412yes—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12C22×C414yes—
162C2×C4×D3—44no—
163C2×C4×D3—44no—
164C2×C4×D3—44no—
165C3×M16—28no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8C22×C412yes—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C2×D412yes—
171C2×C3×C8#4112yes—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—44no—
177(C2×D3)⋊C4—28no—
178(C2×D3)⋊C4—44no—
179(C2×D3)⋊C4—44no—
180(C2×D3)⋊C4—44no—
181C4⋊Dic3—22no—
182C4⋊Dic3C22×C414yes—
183Dic3⋊C4—44no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—24no—
187C4×Dic3—44no—
188C4×Dic3C2×D412yes—
189C4×Dic3#4112yes—
190#208—28no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8C2×D412yes—
196C2×C3⋊C8#4112yes—
197D24—28no—
198C22×D4—32no—
199C22×D4—62no—
200C23×C4—32no—
201C23×C4—62no—
202C23×C4C22×D311yescharacteristic
203C2×D8—34no—
204C2×D8—62no—
205C2×D8—62no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—62no—
209C2×M16—62no—
210C2×M16—62no—
211C2×M16C4×D314yescomplemented
212C22×C8—31no—
213C22×C8—31no—
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D311yescharacteristic
216#121—38no—
217#121—64no—
218C4×D4—64no—
219C4×D4—64no—
220C4×D4—128no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4C22×D312yes—
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—62no—
230C2×C42—31no—
231C2×C42—32no—
232C2×C42—31no—
233C2×C42—62no—
234C2×C42—62no—
235C2×C42—62no—
236#107—22no—
237#106—22no—
238#106—34no—
239#102—38no—
240#102—64no—
241#102—64no—
242#97—34no—
243#97—62no—
244C4×C8—62no—
245C4×C8—62no—
246#95—22no—
247#95—62no—
248#95—62no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3C4×D412yes—
254C23×C3C22×D411yescharacteristic
255C22×D3—24no—
256C22×D3—42no—
257C22×D3—42no—
258C22×D3—42no—
259C22×D3—42no—
260C22×D3—42no—
261C22×D3—42no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4C4×D412yes—
275C2×C3×C4C23×C411yescharacteristic
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281#67—416no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—24no—
287C2×Dic3—48no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3#12114yes—
291C2×Dic3AGL(1,ℤ/8)14yes—
292D12—24no—
293D12—24no—
294D12—42no—
295D12—42no—
296C4×D3—88no—
297C3×C8—22no—
298C3×C8C4×D414yescomplemented
299C3⋊C8—22no—
300C3⋊C8—22no—
301C24—62no—
302C24D3×D411yescharacteristic
303C2×D4—34no—
304C2×D4—68no—
305C2×D4—62no—
306C2×D4—62no—
307C2×D4—62no—
308C2×D4—62no—
309C2×D4—62no—
310C2×D4—124no—
311C2×D4—124no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—62no—
323C22×C4—124no—
324C22×C4—124no—
325C22×C4—124no—
326C22×C4—122no—
327C22×C4C2×C4×D312yes—
328C22×C4D3×D411yescharacteristic
329C22×C4#23912yes—
330C22×C4C23×D311yescharacteristic
331D8—68no—
332M16—28no—
333M16—68no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—32no—
337C2×C8—32no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8C2×C4×D312yes—
343C2×C8C2×C4×D312yes—
344C2×C8D3×D412yes—
345C2×C8#23712yes—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—34no—
351C4⋊C4—62no—
352C4⋊C4—124no—
353C22⋊C4—44no—
354C22⋊C4—68no—
355C22⋊C4—124no—
356C22⋊C4—124no—
357C22⋊C4—124no—
358C42—32no—
359C42—32no—
360C42—64no—
361C42—61no—
362C42—61no—
363C42—124no—
364C22×C3—24no—
365C22×C3—22no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—42no—
369C22×C3C2×C4×D411yescharacteristic
370C22×C3C2×#12111yescharacteristic
371C22×C3#53211yescharacteristic
372C22×C3C2×AGL(1,ℤ/8)12yes—
373C22×C3#57512yes—
374C2×D3—44no—
375C2×D3—44no—
376C2×D3—44no—
377C2×D3—82no—
378C2×D3—82no—
379C2×D3—82no—
380C2×D3—82no—
381C3×C4—22no—
382C3×C4—82no—
383C3×C4—82no—
384C3×C4C2×C4×D411yescharacteristic
385C3×C4C4×#4111yescharacteristic
386C3×C4#52511yescharacteristic
387C3×C4#52911yescharacteristic
388Dic3—24no—
389Dic3—24no—
390Dic3—44no—
391C23—22no—
392C23—22no—
393C23—21no—
394C23—21no—
395C23—64no—
396C23—122no—
397C23—122no—
398C23—122no—
399C23—122no—
400C23—122no—
401C23—122no—
402C23#89312yes—
403C23C2×D3×D411yescharacteristic
404D4—64no—
405D4—64no—
406D4—1216no—
407D4—122no—
408D4—122no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—34no—
416C2×C4—34no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—62no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—64no—
428C2×C4—128no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—248no—
432C2×C4#89312yes—
433C2×C4C22×C4×D311yescharacteristic
434C2×C4C2×D3×D411yescharacteristic
435C2×C4C2×#23711yescharacteristic
436C2×C4C2×#23911yescharacteristic
437C2×C4D3×#4111yescharacteristic
438C2×C4D3×#4111yescharacteristic
439C8—22no—
440C8—62no—
441C8—62no—
442C8#67⋊C414yescomplemented
443C2×C3—42no—
444C2×C3C4×#12111yescharacteristic
445C2×C3C4×AGL(1,ℤ/8)12yes—
446C2×C3#305512yes—
447C2×C3#318311yescharacteristic
448C2×C3#318411yescharacteristic
449D3—84no—
450C22—24no—
451C22—22no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—124no—
456C22—124no—
457C22—124no—
458C22—242no—
459C22—242no—
460C22—242no—
461C22—242no—
462C22C2×#89311yescharacteristic
463C22#559611yescharacteristic
464C22D3×#12111yescharacteristic
465C22D3×AGL(1,ℤ/8)12yes—
466C22#577712yes—
467C4—22no—
468C4—64no—
469C4—64no—
470C4—82no—
471C4—82no—
472C4—124no—
473C4C2×#67⋊C411yescharacteristic
474C4#552211yescharacteristic
475C4#562411yescharacteristic
476C4#562711yescharacteristic
477C3#2153811yescharacteristic, complemented, Sylow
478C2—42no—
479C2—244no—
480C2#7386511yescharacteristic
481C2#7919912yes—
482C2#8080512yes—
483C2#8085311yescharacteristic
484C2#8089211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1809C2
socleC23×C3C22×D4
prime radicalC2×#5135C2
cosocleC22×C3×C4C24
hypercenter#1809D3
hypocenterC3#21538
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×M16C2semidirect
3. C2×#4869C2semidirect
4. #73865C2semidirect
5. C3×#1809C2semidirect
6. #70628C2semidirect
7. #70605C2semidirect
8. #70559C2semidirect
9. #69517C2semidirect
10. #68852C2semidirect
11. #68782C2semidirect
23. #5135C4semidirect
51. #1809D3semidirect
61. C2×C3×M16C2×C4semidirect
68. #930C2×C4semidirect
88. C2×D24C2×C4semidirect
97. #821C2×C4semidirect
98. #817C2×C4semidirect
100. #815D4semidirect
211. C2×M16C4×D3semidirect
298. C3×C8C4×D4semidirect
442. C8#67⋊C4semidirect
477. C3#21538semidirect

Maximal subgroups

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

subgroupindex
C2×#51352
C2×Dic3×M162
C2×#48692
#738652
C3×#18092
#706282
#706052
#705592
#695172
#688522
#687822
#215383