Finite Group Databasebeta

← #10801890

Subgroups — #10801890

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×C3×C4×M16C211yesmaximal, characteristic, complemented
2C2×#5135C211yesmaximal, characteristic
3C4×#930C212yesmaximal, complemented
4C2×#4711C211yesmaximal, characteristic, complemented
5#71990C212yesmaximal, complemented
6#71988C211yesmaximal, characteristic, complemented
7#69814C211yesmaximal, characteristic, complemented
8#69812C212yesmaximal, complemented
9#69776C211yesmaximal, characteristic, complemented
10#69769C211yesmaximal, characteristic, complemented
11#69768C212yesmaximal, complemented
12#21538—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#930C2212yes—
17C2×C4×#67—22no—
18C22×D24C2211yescharacteristic
19C2×C4×D12C2212yes—
20C3×C4×M16—24no—
21C2×C3×#97C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25#5213—22no—
26#5212C2212yes—
27#5135C418yescomplemented
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#818C2211yescharacteristic
32C2×#817C2211yescharacteristic
33C2×#817C2211yescharacteristic
34#4940—22no—
35#4939C2212yes—
36C4×(C2×D3)⋊C4—22no—
37C4×C4⋊Dic3C2212yes—
38#4928C2212yes—
39#4711—24no—
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43C2×#469—31no—
44C2×#442—31no—
45C2×C4×M16D311yescharacteristic, complemented
46C4×#121—32no—
47#1970—31no—
48#1941—31no—
49#1934—32no—
50#1895—32no—
51#1809—31no—
52#1776—31no—
53#1762—32no—
54C23×C3×C4D412yes—
55C23×C3×C4C2311yescharacteristic
56C22×#67—22no—
57C22×D12C2312yes—
58C22×C4×D3—22no—
59C2×C3×M16—22no—
60C2×C3×M16—22no—
61C2×C3×M16—22no—
62C2×C3×M16—22no—
63C2×C3×M16—22no—
64C2×C3×M16C2×C414yescomplemented
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42D412yes—
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75#930—24no—
76#930C2×C418yescomplemented
77C4×#67—48no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×C4×Dic3—22no—
87C2×D24—22no—
88C2×D24—22no—
89C2×D24C2×C414yescomplemented
90C4×D12—24no—
91C4×D12—24no—
92C3×#97—22no—
93C3×#97D414yescomplemented
94C3×C4×C8—22no—
95C3×C4×C8—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#821—24no—
100#821—24no—
101#821C2×C418yescomplemented
102#818—22no—
103#817—22no—
104#817C2×C414yescomplemented
105C22×D8—31no—
106C22×M16C2×D311yescharacteristic
107C22×M16C2×D311yescharacteristic
108C2×#121—32no—
109C2×C4×D4—32no—
110C2×C4×D4—62no—
111C22×C42C2×D311yescharacteristic
112#469—38no—
113#442—64no—
114C2×#107—31no—
115C2×#106—31no—
116C2×#106—31no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—32no—
120C4×M16—24no—
121C2×#97C2×D311yescharacteristic
122C2×#97C2×D311yescharacteristic
123C2×C4×C8C2×D311yescharacteristic
124C2×C4×C8C2×D311yescharacteristic
125#390—32no—
126#389—32no—
127#388—62no—
128#386—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131C4×C22⋊C4—62no—
132#340—32no—
133#340—32no—
134#340—32no—
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—24no—
142C22×C3×C4—24no—
143C22×C3×C4—42no—
144C22×C3×C4C22×C412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4#4112yes—
149C22×C3×C4C2411yescharacteristic
150C2×#67—28no—
151C2×#67—44no—
152C2×#67—44no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
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—
166C3×M16—28no—
167C2×C3×C8—21no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8C22×C412yes—
174C2×C3×C8C22×C412yes—
175C2×C3×C8C2×D412yes—
176C2×C3×C8C2×D412yes—
177C2×C3×C8#4112yes—
178C2×C3×C8#4112yes—
179C3×C42—24no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42C2×D412yes—
183C3×C42#4112yes—
184#217—44no—
185(C2×D3)⋊C4—28no—
186(C2×D3)⋊C4—44no—
187(C2×D3)⋊C4—44no—
188(C2×D3)⋊C4—44no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3C22×C414yes—
195Dic3⋊C4—44no—
196C4×Dic3—44no—
197D24—28no—
198C22×D4—32no—
199C22×D4—62no—
200C23×C4—62no—
201C23×C4D1212yes—
202C23×C4C22×D311yescharacteristic
203C2×D8—34no—
204C2×D8—62no—
205C2×D8—62no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16C4×D314yescomplemented
212C22×C8C22×D311yescharacteristic
213C22×C8C22×D311yescharacteristic
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—32no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—62no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—62no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—62no—
233C2×C42D1212yes—
234C2×C42C22×D311yescharacteristic
235C2×C42C22×D311yescharacteristic
236#107—62no—
237#106—34no—
238#106—62no—
239#102—38no—
240#102—64no—
241#102—64no—
242#97—22no—
243#97#6714yescomplemented
244C4×C8—22no—
245C4×C8—22no—
246#95—62no—
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—22no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—48no—
273C2×C3×C4C4×D412yes—
274C2×C3×C4#12114yes—
275C2×C3×C4AGL(1,ℤ/8)14yes—
276C2×C3×C4C23×C411yescharacteristic
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282#67—416no—
283C2×Dic3—24no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292D12—24no—
293D12—24no—
294D12—42no—
295D12—42no—
296C4×D3—88no—
297C3×C8—22no—
298C3×C8—22no—
299C3×C8—22no—
300C3×C8C4×D414yescomplemented
301C24—62no—
302C24C2×D1211yescharacteristic
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—24no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—124no—
323C22×C4—124no—
324C22×C4—124no—
325C22×C4C2×C4×D312yes—
326C22×C4C2×D1211yescharacteristic
327C22×C4C2×D1212yes—
328C22×C4C2×D1212yes—
329C22×C4#23512yes—
330C22×C4C23×D311yescharacteristic
331D8—68no—
332M16—28no—
333M16—28no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8C2×C4×D312yes—
341C2×C8C2×C4×D312yes—
342C2×C8#23512yes—
343C2×C8#23512yes—
344C2×C8C2×#6712yes—
345C2×C8C2×#6712yes—
346C4⋊C4—34no—
347C4⋊C4—62no—
348C4⋊C4—62no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—124no—
353C22⋊C4—68no—
354C22⋊C4—124no—
355C22⋊C4—124no—
356C22⋊C4—124no—
357C22⋊C4—124no—
358C42—24no—
359C42—21no—
360C42—21no—
361C42—124no—
362C42#23512yes—
363C42C2×#6712yes—
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—24no—
382C3×C4—24no—
383C3×C4—22no—
384C3×C4C2×C4×D411yescharacteristic
385C3×C4C4×#4111yescharacteristic
386C3×C4#52511yescharacteristic
387C3×C4#52911yescharacteristic
388Dic3—44no—
389Dic3—82no—
390Dic3—82no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—21no—
395C23—64no—
396C23—122no—
397C23—122no—
398C23—122no—
399C23—122no—
400C23—122no—
401C23—122no—
402C23C4×D1212yes—
403C23C22×D1211yescharacteristic
404D4—64no—
405D4—64no—
406D4—1216no—
407D4—122no—
408D4—122no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—48no—
420C2×C4—64no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—248no—
430C2×C4C4×D1212yes—
431C2×C4#90814yes—
432C2×C4#93014yes—
433C2×C4C22×C4×D311yescharacteristic
434C2×C4C2×#23511yescharacteristic
435C2×C4C2×#23511yescharacteristic
436C2×C4C2×#23511yescharacteristic
437C2×C4C2×#23511yescharacteristic
438C2×C4C22×#6711yescharacteristic
439C8—22no—
440C8—22no—
441C8—22no—
442C8C4×#6714yescomplemented
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—22no—
451C22—24no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—124no—
456C22—124no—
457C22—124no—
458C22—242no—
459C22—242no—
460C22—242no—
461C22—242no—
462C22C2×C4×D1211yescharacteristic
463C22#547711yescharacteristic
464C22C2×#90812yes—
465C22#574812yes—
466C22C2×#93011yescharacteristic
467C4—24no—
468C4—24no—
469C4—22no—
470C4—124no—
471C4—242no—
472C4—242no—
473C4C4×#23511yescharacteristic
474C4#547911yescharacteristic
475C4C2×C4×#6711yescharacteristic
476C4#579111yescharacteristic
477C3#2153811yescharacteristic, complemented, Sylow
478C2—42no—
479C2—244no—
480C2C4×#90812yes—
481C2#7853411yescharacteristic
482C2#7853511yescharacteristic
483C2C4×#93011yescharacteristic
484C2#8377212yes—
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×M16C2
socleC23×C3C22×D4
prime radicalC2×#5135C2
cosocleC22×C3×C4C24
hypercenterC2×C4×M16D3
hypocenterC3#21538
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×M16C2semidirect
3. C4×#930C2semidirect
4. C2×#4711C2semidirect
5. #71990C2semidirect
6. #71988C2semidirect
7. #69814C2semidirect
8. #69812C2semidirect
9. #69776C2semidirect
10. #69769C2semidirect
11. #69768C2semidirect
27. #5135C4semidirect
45. C2×C4×M16D3semidirect
64. C2×C3×M16C2×C4semidirect
76. #930C2×C4semidirect
89. C2×D24C2×C4semidirect
93. C3×#97D4semidirect
101. #821C2×C4semidirect
104. #817C2×C4semidirect
211. C2×M16C4×D3semidirect
243. #97#67semidirect
300. C3×C8C4×D4semidirect
442. C8C4×#67semidirect
477. C3#21538semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×M162
C2×#51352
C4×#9302
C2×#47112
#719902
#719882
#698142
#698122
#697762
#697692
#697682
#215383