Finite Group Databasebeta

← #10831744

Subgroups — #10831744

Order 768.

1248 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
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×#5091C211yesmaximal, characteristic, complemented
3#82480C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic
5#71794C212yesmaximal, complemented
6#71788C211yesmaximal, characteristic, complemented
7#71782C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71612C212yesmaximal
10#71606C212yesmaximal
11#71605C211yesmaximal, characteristic
12#24342—31nomaximal, Sylow
13C23×C3×Q8—21no—
14C3×D4×Q8—24no—
15C3×D4×Q8—24no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—22no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25#5230—21no—
26C2×#832C2211yescharacteristic
27C2×#832C2212yes—
28#5091—24no—
29#5018—24no—
30#4985—21no—
31#4978—22no—
32C2×#810C2212yes—
33C2×#810C2211yescharacteristic
34C2×#807C2212yes—
35C2×#807C2211yescharacteristic
36#4936C2211yescharacteristic
37#4929C2212yes—
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×D4×Q8D311yescharacteristic
47C2×#477—31no—
48C2×#440—31no—
49#2581—31no—
50#2057—32no—
51#2024—32no—
52#1993—31no—
53#1985—31no—
54#1877—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8—28no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—48no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C22×Dic6C2311yescharacteristic
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8—24no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yes—
70C3×C22⋊Q8—28no—
71C3×C22⋊Q8—216no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C3×C4×D4—22no—
77C3×C4×D4D414yes—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#214—22no—
89C2×#214—22no—
90C2×#214Q814yescomplemented
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3—21no—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C22×C3⋊C8C2311yescharacteristic
99C4×Dic6—22no—
100C4×Dic6—22no—
101#832—22no—
102#832—24no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108#810—22no—
109#810—24no—
110#807—22no—
111#807—24no—
112C4⋊(C3⋊C8)—22no—
113#803—22no—
114C23×Q8—21no—
115C22×SD16—31no—
116D4×Q8—24no—
117D4×Q8—24no—
118C2×C4⋊Q8C2×D312yes—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—21no—
122C2×C4×Q8—31no—
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×D4C2×D312yes—
125C2×C4×D4C2×D311yescharacteristic
126#477—64no—
127#440—64no—
128C2×#107—32no—
129C2×#107—31no—
130C2×C4⋊C8—31no—
131C2×#103—32no—
132C2×#103—31no—
133C2×#102—32no—
134C2×#102—31no—
135C2×#97—31no—
136#400—61no—
137#395—61no—
138#391—32no—
139#389—32no—
140#387—62no—
141#382—31no—
142#340—32no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—32no—
147C24×C3—21no—
148C2×C3×Q8—28no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C2×C3×Q8—24no—
152C2×C3×Q8—44no—
153C2×C3×Q8—44no—
154C2×C3×Q8—432no—
155C2×C3×Q8—44no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4C2×D412yes—
160C2×C3×D4C2×Q812yes—
161C2×C3×D4#4112yes—
162C22×C3×C4—28no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—24no—
166C22×C3×C4—44no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6C2×Q812yescomplemented
183C3×C4⋊C4—28no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4C2×D414yes—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197#214—28no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202Dic3⋊C4—42no—
203Dic3⋊C4—44no—
204C4×Dic3—42no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8C2×Q812yescomplemented
214C22×Q8—28no—
215C22×Q8—22no—
216C22×Q8—21no—
217C22×Q8—31no—
218C22×Q8—48no—
219C22×Q8#6711yescharacteristic
220C22×D4C22×D311yescharacteristic
221C23×C4—22no—
222C23×C4—21no—
223C2×SD16—34no—
224C2×SD16—62no—
225C2×SD16—62no—
226C22×C8—31no—
227C22×C8—31no—
228C22×C8—32no—
229C4⋊Q8—24no—
230C4⋊Q8—24no—
231C4⋊Q8—22no—
232C4⋊Q8#6714yes—
233C22⋊Q8—28no—
234C22⋊Q8—216no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—62no—
238C4×Q8—62no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4#6714yes—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4#6712yes—
250C2×C4⋊C4#6711yescharacteristic
251C2×C4⋊C4C22×D312yes—
252C2×C4⋊C4C22×D311yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C42—61no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#107—64no—
260C4⋊C8—62no—
261#103—62no—
262#103—64no—
263#102—64no—
264#102—62no—
265#97—62no—
266#95—61no—
267#95—61no—
268#95—62no—
269#95—62no—
270#95—62no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3C22×D411yescharacteristic
277C3×Q8—28no—
278C3×Q8—48no—
279C3×Q8—416no—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4C22⋊Q814yes—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—48no—
293C2×C3×C4—416no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4AGL(1,ℤ/8)14yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×Q811yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×C3×C42−1+411yescharacteristic
306C2×Dic3—22no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314Dic6—22no—
315Dic6—22no—
316Dic6—41no—
317Dic6—41no—
318C3⋊C8—22no—
319C3⋊C8—41no—
320C3⋊C8—42no—
321C24—21no—
322C2×Q8—28no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—32no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—432no—
330C2×Q8—44no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—61no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4D3×Q812yes—
340C2×D4#23912yes—
341C2×D4C2×#6712yes—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—61no—
350C22×C4—62no—
351C22×C4—61no—
352C22×C4C2×#6712yes—
353C22×C4C2×#6712yes—
354C22×C4C2×#6711yescharacteristic
355C22×C4C2×#6711yescharacteristic
356C22×C4C23×D311yescharacteristic
357SD16—68no—
358C2×C8—32no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—62no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—62no—
365C2×C8—61no—
366C2×C8—62no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—61no—
376C4⋊C4—61no—
377C4⋊C4—124no—
378C4⋊C4—122no—
379C4⋊C4C2×#6714yes—
380C4⋊C4C2×#6712yes—
381C22⋊C4—28no—
382C22⋊C4—24no—
383C42—22no—
384C42—22no—
385C42—122no—
386C42C2×#6712yes—
387C42C2×#6712yes—
388C22×C3—24no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3C2×#12011yescharacteristic
394C22×C3#54811yescharacteristic
395C22×C3D4×Q811yescharacteristic
396C22×C3C2×AGL(1,ℤ/8)12yes—
397C22×C3#57812yes—
398C3×C4—22no—
399C3×C4—24no—
400C3×C4—22no—
401C3×C4—44no—
402C3×C4—42no—
403C3×C4—42no—
404C3×C4C2×C22⋊Q811yescharacteristic
405C3×C4#53611yescharacteristic
406C3×C4#54811yescharacteristic
407C3×C4D4×Q811yescharacteristic
408Dic3—42no—
409Dic3—82no—
410Dic3—81no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—42no—
416C23C22×#6711yescharacteristic
417Q8—28no—
418Q8—416no—
419Q8—48no—
420Q8—62no—
421Q8—62no—
422Q8—121no—
423Q8—121no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4#94614yes—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—416no—
437C2×C4—48no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—62no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4#93214yes—
450C2×C4#95312yes—
451C2×C4#95312yes—
452C2×C4C2×D3×Q811yescharacteristic
453C2×C4C2×#23911yescharacteristic
454C2×C4#100711yescharacteristic
455C2×C4#100711yescharacteristic
456C2×C4C22×#6711yescharacteristic
457C2×C4C22×#6711yescharacteristic
458C8—62no—
459C8—121no—
460C8—122no—
461C2×C3—44no—
462C2×C3#240911yescharacteristic
463C2×C3#300311yescharacteristic
464C2×C3#301111yescharacteristic
465C2×C3#336712yes—
466C2×C3#338612yes—
467C22—24no—
468C22—42no—
469C22—44no—
470C22—44no—
471C22—42no—
472C22C2×#93212yes—
473C22Q8×#6711yescharacteristic
474C22#581711yescharacteristic
475C22#583812yes—
476C22C2×#95311yescharacteristic
477C4—24no—
478C4—22no—
479C4—22no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—122no—
484C4—241no—
485C4—242no—
486C4C2×#94611yescharacteristic
487C4Q8×#6711yescharacteristic
488C4#581611yescharacteristic
489C4#581711yescharacteristic
490C3#2434211yescharacteristic, complemented, Sylow
491C2—44no—
492C2#8355812yes—
493C2#8356312yes—
494C2#8403811yescharacteristic
495C2#8406811yescharacteristic
496C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×D4×Q8D3
hypocenterC3#24342
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#5091C2semidirect
3. #82480C2semidirect
5. #71794C2semidirect
6. #71788C2semidirect
7. #71782C2semidirect
8. #71635C2semidirect
90. C2×#214Q8semidirect
182. C2×Dic6C2×Q8semidirect
213. C2×C3⋊C8C2×Q8semidirect
490. C3#24342semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×#50912
#824802
C2×#50182
#717942
#717882
#717822
#716352
#716122
#716062
#716052
#243423