Finite Group Databasebeta

← #10830132

Subgroups — #10830132

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic
3#82480C211yesmaximal, characteristic, complemented
4C2×#5030C211yesmaximal, characteristic, complemented
5#71765C211yesmaximal, characteristic
6#71729C212yesmaximal, complemented
7#71722C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71627C211yesmaximal, characteristic, complemented
10#71612C212yesmaximal, complemented
11#71607C212yesmaximal, complemented
12#24202—31nomaximal, Sylow
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124—21no—
17C2×C3×#124C2212yes—
18C2×C3×#123—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4C2212yes—
23C22×#216C2211yescharacteristic
24C22×#214C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26#5230—21no—
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5075—24no—
33#5030—24no—
34#4985—21no—
35#4978—22no—
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4936C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559D311yescharacteristic
47C2×#460—31no—
48C2×#441—31no—
49#2581—31no—
50#2042—32no—
51#2015—32no—
52#1985—31no—
53#1961—31no—
54#1877—32no—
55#1864—32no—
56#1805—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C23×C3×C4—22no—
63C22×Dic6C2311yescharacteristic
64C3×#127—22no—
65C3×#127D414yes—
66C3×#124—28no—
67C3×#124—24no—
68C3×#124—24no—
69C3×#123—216no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8D414yes—
74C3×C4×D4—28no—
75C3×C4×D4—24no—
76C3×C4×D4—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D412yes—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C2×#216—22no—
88C2×#216—22no—
89C2×#216—22no—
90C2×#214—24no—
91C2×#214—24no—
92C2×#214—42no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2312yes—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—21no—
97C2×C4×Dic3—21no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101C4×Dic6—22no—
102C4×Dic6—22no—
103#835—22no—
104#835—42no—
105#832—24no—
106#832—24no—
107#832—42no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#810—22no—
114#808—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×Q16—31no—
118C22×SD16—32no—
119#559—24no—
120#559—24no—
121C2×#127C2×D311yescharacteristic
122C2×#124—21no—
123C2×#124C2×D312yes—
124C2×#123—22no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—21no—
129C2×C4×D4C2×D312yes—
130#460—64no—
131#441—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#97—31no—
141#400—61no—
142#395—61no—
143#391—32no—
144#389—32no—
145#387—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—42no—
158C2×C3×Q8C2×D412yes—
159C2×C3×D4—24no—
160C2×C3×D4—24no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—44no—
167C2×C3×D4—82no—
168C2×C3×D4C2×D414yes—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—28no—
172C22×C3×C4—84no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6#4112yes—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—28no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#216—28no—
201#214—48no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206Dic3⋊C4—44no—
207Dic3⋊C4—42no—
208C4×Dic3—42no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8#4112yes—
218C22×Q8—31no—
219C22×Q8#6711yescharacteristic
220C22×Q8C22×D311yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C22×D312yes—
224C23×C4—22no—
225C2×Q16—62no—
226C2×Q16—62no—
227C2×Q16—62no—
228C2×SD16—64no—
229C2×SD16—64no—
230C2×SD16—122no—
231C22×C8—31no—
232C22×C8—32no—
233C22×C8—31no—
234#127—22no—
235#127#6714yes—
236#124—28no—
237#124—24no—
238#124—24no—
239#123—216no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—62no—
244C4×Q8—62no—
245C4×Q8#6714yes—
246C4×D4—28no—
247C4×D4—24no—
248C4×D4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4#6711yescharacteristic
255C2×C4⋊C4#6712yes—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C42—21no—
261C2×C42—61no—
262C2×C42C22×D312yes—
263C2×C42C22×D311yescharacteristic
264#106—62no—
265C4⋊C8—62no—
266#103—62no—
267#103—62no—
268#103—122no—
269#102—64no—
270#102—64no—
271#102—122no—
272#97—62no—
273#95—61no—
274#95—61no—
275#95—62no—
276#95—62no—
277#95—62no—
278C23×C3—24no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—44no—
282C23×C3—44no—
283C23×C3C22×D411yescharacteristic
284C3×Q8—22no—
285C3×Q8—22no—
286C3×Q8—44no—
287C3×Q8—41no—
288C3×Q8—41no—
289C3×D4—24no—
290C3×D4—24no—
291C3×D4—48no—
292C3×D4—42no—
293C3×D4—42no—
294C3×D4—84no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4—88no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4H(5,𝔽2)11yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316C2×Dic3—22no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324Dic6—22no—
325Dic6—22no—
326Dic6—41no—
327Dic6—41no—
328C3⋊C8—22no—
329C3⋊C8—41no—
330C3⋊C8—42no—
331C24—22no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—32no—
339C2×Q8—42no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8C2×#6712yes—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—44no—
354C2×D4—82no—
355C2×D4C2×#6714yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—28no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—84no—
363C22×C4C2×#6712yes—
364C22×C4C2×#6712yes—
365C22×C4C2×#6711yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C23×D311yescharacteristic
368Q16—68no—
369SD16—128no—
370C2×C8—32no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—62no—
374C2×C8—61no—
375C2×C8—62no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—62no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—124no—
388C4⋊C4—122no—
389C4⋊C4C2×#6712yes—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—122no—
397C42C2×#6712yes—
398C42C2×#6712yes—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—84no—
403C22×C3—84no—
404C22×C3C2×#12011yescharacteristic
405C22×C3#54811yescharacteristic
406C22×C3#55011yescharacteristic
407C22×C3#57512yes—
408C22×C3#57712yes—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4#52511yescharacteristic
416C3×C4#53711yescharacteristic
417C3×C4#54611yescharacteristic
418C3×C4#54711yescharacteristic
419Dic3—42no—
420Dic3—82no—
421Dic3—81no—
422C23—24no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23C22×#6711yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—44no—
431Q8—41no—
432Q8—41no—
433Q8—62no—
434Q8—62no—
435Q8—121no—
436Q8—121no—
437D4—24no—
438D4—24no—
439D4—48no—
440D4—42no—
441D4—42no—
442D4—84no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—62no—
456C2×C4—88no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4#95312yes—
465C2×C4#95312yes—
466C2×C4D3×#4111yescharacteristic
467C2×C4D3×#4111yescharacteristic
468C2×C4#100911yescharacteristic
469C2×C4#101011yescharacteristic
470C2×C4C22×#6711yescharacteristic
471C2×C4C22×#6711yescharacteristic
472C8—62no—
473C8—122no—
474C8—121no—
475C2×C3—84no—
476C2×C3#240711yescharacteristic
477C2×C3#300511yescharacteristic
478C2×C3#300911yescharacteristic
479C2×C3#337112yes—
480C2×C3#340112yes—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—84no—
485C22—84no—
486C22#582212yes—
487C22#583111yescharacteristic
488C22#583411yescharacteristic
489C22#583612yes—
490C22C2×#95311yescharacteristic
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—122no—
498C4—242no—
499C4—241no—
500C4#582411yescharacteristic
501C4#582811yescharacteristic
502C4#582911yescharacteristic
503C4#583311yescharacteristic
504C3#2420211yescharacteristic, complemented, Sylow
505C2—84no—
506C2#8371212yes—
507C2#8371812yes—
508C2#8408111yescharacteristic
509C2#8409711yescharacteristic
510C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#24202
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
3. #82480C2semidirect
4. C2×#5030C2semidirect
6. #71729C2semidirect
7. #71722C2semidirect
8. #71635C2semidirect
9. #71627C2semidirect
10. #71612C2semidirect
11. #71607C2semidirect
504. C3#24202semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50752
#824802
C2×#50302
#717652
#717292
#717222
#716352
#716272
#716122
#716072
#242023