Finite Group Databasebeta

← #10827214

Subgroups — #10827214

Order 768.

1258 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic
2#82933C212yesmaximal, complemented
3C2×#4973C211yesmaximal, characteristic, complemented
4#71905C212yesmaximal, complemented
5#71881C212yesmaximal, complemented
6#71711C212yesmaximal, complemented
7#71666C214yesmaximal, complemented
8#71562C211yesmaximal, characteristic, complemented
9#23949—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C3×#554—22no—
12C3×#553—22no—
13C3×#552—22no—
14C3×#547—22no—
15C3×#544C2211yescharacteristic
16C3×#539—22no—
17C3×#537C2211yescharacteristic
18C3×#525—22no—
19C3×#525C2212yes—
20C3×#520C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C2×#951C2214yes—
23C22×C4×Dic3C2211yescharacteristic
24C2×#837C2214yes—
25C2×C3⋊#95—22no—
26#5207C2212yes—
27#5050—22no—
28#5050—22no—
29#5040—22no—
30#5040—22no—
31#5004C2214yes—
32#4973—22no—
33#4973C2212yes—
34#4973C2212yes—
35#4532C2214yes—
36#4530C2212yes—
37#4530C2212yes—
38#3492D311yescharacteristic
39#3004—32no—
40C2×#391—31no—
41#2028—32no—
42#2007—32no—
43#1896—32no—
44#1882—34no—
45#1821—31no—
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3—21no—
52C22×Dic6—21no—
53C22×Dic6—24no—
54C22×Dic6—22no—
55C22×Dic6—22no—
56C22×Dic6—42no—
57C22×Dic6C2312yes—
58C3×C4⋊Q8—22no—
59C3×#126—24no—
60C3×#125—24no—
61C3×#125—22no—
62C3×#123—24no—
63C3×#123—24no—
64C3×#123—22no—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8—24no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8D412yes—
70C3×C22⋊Q8D412yes—
71C3×#121—22no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#121D412yes—
75C3×#121D412yes—
76C3×#120—22no—
77C3×C4×Q8—22no—
78C3×C4×D4—24no—
79C3×C4×D4—22no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117C2312yes—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92#951—28no—
93#951—44no—
94C2×#216—24no—
95C2×#214—24no—
96C2×C4×Dic3—21no—
97C2×C4×Dic3—42no—
98C2×C4×Dic3C2312yes—
99C2×#208C2314yes—
100#837—44no—
101C3⋊#95—24no—
102C3⋊#95—24no—
103#830—22no—
104#830—22no—
105#810—24no—
106#810—24no—
107C23×Q8—31no—
108C2×#137—34no—
109#554—22no—
110#553—22no—
111#552—22no—
112#547—22no—
113#544C2×D311yescharacteristic
114#539—22no—
115#537C2×D311yescharacteristic
116#525—22no—
117#525C2×D312yes—
118#520C2×D312yes—
119C22×C4⋊C4C2×D311yescharacteristic
120C22×C42—31no—
121#451—62no—
122#451—62no—
123#448—62no—
124#448—62no—
125C2×#104—34no—
126#419—34no—
127#407—32no—
128#391—32no—
129#391—32no—
130#391—62no—
131C2×#95—62no—
132#343—32no—
133#343—32no—
134#339—34no—
135C24×C3C2×D411yescharacteristic
136C3×#41—24no—
137C3×#41—24no—
138C3×#41—84no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×D412yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4C2×D412yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—44no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—24no—
160C22×Dic3—24no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—42no—
164C22×Dic3—42no—
165C22×Dic3—42no—
166C22×Dic3—41no—
167C22×Dic3—41no—
168C2×Dic6—24no—
169C2×Dic6—24no—
170C2×Dic6—24no—
171C2×Dic6—22no—
172C2×Dic6—48no—
173C2×Dic6—48no—
174C2×Dic6—42no—
175C2×Dic6—42no—
176C2×Dic6#4114yescomplemented
177C3×C4⋊C4—24no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4C2×D412yes—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C42—24no—
196C3×C42—22no—
197#216—48no—
198#214—48no—
199C4×Dic3—24no—
200C4×Dic3—42no—
201#208—24no—
202#208—24no—
203C2×C3⋊C8—24no—
204C2×#41—22no—
205C2×#41C22×D312yes—
206C22×Q8—32no—
207C22×Q8—64no—
208C22×Q8—61no—
209C22×Q8—62no—
210C22×Q8—62no—
211C22×Q8—122no—
212C23×C4—61no—
213C23×C4—61no—
214C23×C4#6712yes—
215C23×C4C22×D311yescharacteristic
216#137—68no—
217#137—124no—
218C2×Q16—64no—
219C2×SD16—64no—
220C2×M16—34no—
221C4⋊Q8—22no—
222#126—24no—
223#125—24no—
224#125—22no—
225#123—24no—
226#123—24no—
227#123—22no—
228C22⋊Q8—22no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8#6712yes—
233C22⋊Q8#6712yes—
234#121—22no—
235#121—22no—
236#121—22no—
237#121#6712yes—
238#121#6712yes—
239#120—22no—
240C4×Q8—22no—
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246#117—22no—
247#117—22no—
248#117C22×D312yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4#6712yes—
253C2×C4⋊C4C22×D311yescharacteristic
254C2×C4⋊C4C22×D311yescharacteristic
255C2×C42—32no—
256C2×C42—61no—
257C2×C42—122no—
258#104—124no—
259#103—64no—
260#103—64no—
261C22⋊C8—62no—
262C22⋊C8—62no—
263#95—64no—
264#95—64no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3#12011yescharacteristic
271C23×C3#12011yescharacteristic
272C23×C3C22×D411yescharacteristic
273C3×Q8—24no—
274C3×Q8—82no—
275C3×D4—24no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—44no—
279C3×D4—44no—
280C3×D4—82no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—44no—
291C2×C3×C4—44no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—82no—
295C2×C3×C4—84no—
296C2×C3×C4#12012yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C4C2×#4112yes—
299C2×C3×C4H(5,𝔽2)12yes—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—44no—
303C2×Dic3—44no—
304C2×Dic3—48no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309Dic6—28no—
310Dic6—48no—
311Dic6—44no—
312C3⋊C8—44no—
313C24C2×#6711yescharacteristic
314#41—24no—
315#41—24no—
316#41—84no—
317C2×Q8—22no—
318C2×Q8—34no—
319C2×Q8—64no—
320C2×Q8—62no—
321C2×Q8—64no—
322C2×Q8—64no—
323C2×Q8—128no—
324C2×Q8—128no—
325C2×Q8—122no—
326C2×Q8—122no—
327C2×Q8C2×#6712yes—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4C2×#6712yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—44no—
340C22×C4—64no—
341C22×C4—64no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4—122no—
347C22×C4—121no—
348C22×C4—121no—
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6712yes—
351C22×C4C2×#6712yes—
352C22×C4C2×#6712yes—
353C22×C4C2×#6712yes—
354C22×C4C23×D311yescharacteristic
355Q16—128no—
356SD16—128no—
357M16—64no—
358M16—64no—
359C2×C8—64no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4C2×#6712yes—
370C4⋊C4C2×#6712yes—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C42—24no—
379C42—22no—
380C42—64no—
381C42—122no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3#45512yes—
394C22×C3#45612yes—
395C22×C3C2×#12011yescharacteristic
396C22×C3#52511yescharacteristic
397C22×C3#53511yescharacteristic
398C3×C4—22no—
399C3×C4—42no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—42no—
404C3×C4#54614yes—
405Dic3—44no—
406Dic3—44no—
407C23—22no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23#95311yescharacteristic
413C23#95311yescharacteristic
414C23C22×#6711yescharacteristic
415Q8—24no—
416Q8—68no—
417Q8—82no—
418Q8—128no—
419Q8—124no—
420D4—24no—
421D4—44no—
422D4—44no—
423D4—42no—
424D4—42no—
425D4—82no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—124no—
444C2×C4—124no—
445C2×C4—128no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4#95312yes—
451C2×C4D3×#4112yes—
452C2×C4#100912yes—
453C2×C4C22×#6712yes—
454C8—124no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—82no—
460C2×C3#240411yescharacteristic
461C2×C3C2×#45511yescharacteristic
462C2×C3C2×#45611yescharacteristic
463C22—24no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22#524212yes—
475C22#524612yes—
476C22#582811yescharacteristic
477C22#583011yescharacteristic
478C22C2×#95311yescharacteristic
479C4—22no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—124no—
486C4—124no—
487C4#582914yes—
488C3#2394911yescharacteristic, complemented, Sylow
489C2—21no—
490C2—22no—
491C2—22no—
492C2—21no—
493C2—82no—
494C2#8409611yescharacteristic
495C2C2×#524211yescharacteristic
496C2C2×#524611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3492C2
socleC22×C3C2×#120
hypercenter#3492D3
hypocenterC3#23949
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
2. #82933C2semidirect
3. C2×#4973C2semidirect
4. #71905C2semidirect
5. #71881C2semidirect
6. #71711C2semidirect
7. #71666C2semidirect
8. #71562C2semidirect
176. C2×Dic6#41semidirect
488. C3#23949semidirect

Maximal subgroups

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

subgroupindex
C3×#34922
#829332
C2×#49732
#719052
#718812
#717112
#716662
#715622
#239493