Finite Group Databasebeta

← #10729045

Subgroups — #10729045

Order 768.

2150 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×C3⋊#95C211yesmaximal, characteristic
3C2×#4946C212yesmaximal
4#72134C212yesmaximal, complemented
5#72127C211yesmaximal, characteristic
6#71568C212yesmaximal, complemented
7#71566C214yesmaximal, complemented
8#71544C212yesmaximal
9#14914—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×Dic3C2211yescharacteristic
13C24×Dic3C2211yescharacteristic
14C2×C3×C22⋊Q8—24no—
15C2×C3×C22⋊Q8—28no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×#217C2212yes—
20C22×Dic3⋊C4C2214yes—
21C2×C3⋊#95—24no—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2212yes—
29#4946—28no—
30C22×C22⋊Q8D311yescharacteristic
31C2×#385—32no—
32C22×#95—31no—
33#1446—32no—
34#1444—32no—
35#1441—34no—
36C22⋊#95—31no—
37C22⋊#95—32no—
38C25×C3C2311yescharacteristic
39C22×C3×Q8—24no—
40C22×C3×Q8—22no—
41C22×C3×Q8—24no—
42C22×C3×Q8—42no—
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—42no—
46C23×C3×C4D414yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×Dic3—22no—
52C23×Dic3—22no—
53C23×Dic3—22no—
54C23×Dic3—24no—
55C23×Dic3—24no—
56C23×Dic3—42no—
57C23×Dic3—42no—
58C23×Dic3—42no—
59C23×Dic3Q814yescomplemented
60C23×Dic3C2312yes—
61C23×Dic3C2314yes—
62C23×Dic3C2312yes—
63C3×C22⋊Q8—232no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4D418yes—
72C2×#217—24no—
73C2×#217—24no—
74C2×#217Q818yescomplemented
75C2×Dic3⋊C4—28no—
76C2×Dic3⋊C4—28no—
77C2×Dic3⋊C4—44no—
78C3⋊#95—28no—
79C3⋊#95—24no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—42no—
83C3⋊#95—44no—
84C3⋊#95—44no—
85C23×Q8C2×D311yescharacteristic
86C24×C4—31no—
87C24×C4—31no—
88C24×C4C2×D311yescharacteristic
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C22×C4⋊C4—34no—
92C22×C4⋊C4C2×D312yes—
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C22⋊C4—32no—
95C22×C22⋊C4C2×D312yes—
96#385—68no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—34no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—34no—
103C2×#95—34no—
104C2×#95—64no—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3C2×D414yes—
110C24×C3C2×D412yes—
111C24×C3C2×Q814yes—
112C24×C3C2×Q812yes—
113C24×C3#4112yes—
114C24×C3C2411yescharacteristic
115C2×C3×Q8—28no—
116C2×C3×Q8—416no—
117C2×C3×Q8—44no—
118C2×C3×Q8—44no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—28no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—24no—
133C22×C3×C4—28no—
134C22×C3×C4—44no—
135C22×C3×C4—416no—
136C22×C3×C4C2×D414yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D414yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4#4114yes—
141C22×Dic3—28no—
142C22×Dic3—28no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—48no—
158C22×Dic3—416no—
159C22×Dic3—44no—
160C22×Dic3—44no—
161C22×Dic3—44no—
162C22×Dic3—48no—
163C22×Dic3—44no—
164C22×Dic3C2×Q818yes—
165C22×Dic3C2×Q814yes—
166C22×Dic3#4114yes—
167C3×C4⋊C4—28no—
168C3×C4⋊C4—216no—
169C3×C22⋊C4—216no—
170#217—216no—
171Dic3⋊C4—416no—
172C25C22×D311yescharacteristic
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—42no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—34no—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4—34no—
183C23×C4—42no—
184C23×C4—64no—
185C23×C4—62no—
186C23×C4—62no—
187C23×C4—62no—
188C23×C4—64no—
189C23×C4—122no—
190C23×C4—122no—
191C23×C4—122no—
192C23×C4#6714yes—
193C23×C4C22×D312yes—
194C23×C4C22×D312yes—
195C23×C4C22×D311yescharacteristic
196C23×C4C22×D311yescharacteristic
197C22⋊Q8—232no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—68no—
204C2×C4⋊C4—68no—
205C2×C4⋊C4—124no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—38no—
209C2×C22⋊C4—64no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4#6718yes—
212#95—64no—
213#95—64no—
214#95—64no—
215#95—68no—
216#95—122no—
217#95—124no—
218#95—124no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—24no—
222C23×C3—24no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—28no—
228C23×C3—28no—
229C23×C3—28no—
230C23×C3—28no—
231C23×C3#12014yes—
232C23×C3C22⋊Q818yes—
233C23×C3C22⋊Q814yes—
234C23×C3C4⋊Q814yes—
235C23×C3C4⋊Q818yes—
236C23×C3C22×D411yescharacteristic
237C23×C3C22×D412yes—
238C23×C3C22×Q811yescharacteristic
239C23×C3C22×Q812yes—
240C23×C3C2×#4112yes—
241C23×C3C2×#4111yescharacteristic
242C23×C3C2×#4112yes—
243C23×C3H(5,𝔽2)12yes—
244C23×C3H(5,𝔽2)12yes—
245C3×Q8—416no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—28no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—416no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4#12314yes—
265C2×Dic3—28no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—28no—
269C2×Dic3—28no—
270C2×Dic3—28no—
271C2×Dic3—28no—
272C2×Dic3—416no—
273C2×Dic3—416no—
274C2×Dic3—42no—
275C2×Dic3—44no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—44no—
280C2×Dic3—44no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283C2×Dic3—44no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24D3×Q814yes—
289C24D3×Q812yes—
290C24#23912yes—
291C24C2×#6712yes—
292C24C2×#6714yes—
293C24C23×D311yescharacteristic
294C2×Q8—28no—
295C2×Q8—416no—
296C2×Q8—44no—
297C2×Q8—44no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—28no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—28no—
313C22×C4—38no—
314C22×C4—34no—
315C22×C4—34no—
316C22×C4—44no—
317C22×C4—416no—
318C22×C4—68no—
319C22×C4—68no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—64no—
324C22×C4—62no—
325C22×C4—64no—
326C22×C4—62no—
327C22×C4—64no—
328C22×C4—64no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—64no—
333C22×C4—64no—
334C22×C4—128no—
335C22×C4—1216no—
336C22×C4—128no—
337C22×C4—124no—
338C22×C4—124no—
339C22×C4—124no—
340C22×C4—124no—
341C22×C4#23714yes—
342C22×C4C2×#6714yes—
343C22×C4C2×#6712yes—
344C22×C4C2×#6714yes—
345C22×C4C2×#6712yes—
346C4⋊C4—28no—
347C4⋊C4—216no—
348C4⋊C4—1216no—
349C22⋊C4—216no—
350C22⋊C4—616no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—216no—
357C22×C3—24no—
358C22×C3—216no—
359C22×C3#39518yes—
360C22×C3C2×#12011yescharacteristic
361C22×C3C2×C22⋊Q811yescharacteristic
362C22×C3C2×C22⋊Q812yes—
363C22×C3C2×#12311yescharacteristic
364C22×C3#52511yescharacteristic
365C22×C3#52912yes—
366C22×C3C2×C4⋊Q811yescharacteristic
367C22×C3C2×C4⋊Q812yes—
368C22×C3#53312yes—
369C22×C3#53411yescharacteristic
370C22×C3#53511yescharacteristic
371C22×C3#53812yes—
372C22×C3#54312yes—
373C22×C3#54614yes—
374C22×C3#54614yes—
375C22×C3#55414yes—
376C22×C3#55414yes—
377C3×C4—24no—
378C3×C4—44no—
379C3×C4—42no—
380C3×C4—44no—
381Dic3—44no—
382Dic3—44no—
383Dic3—48no—
384C23—24no—
385C23—28no—
386C23—28no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—28no—
391C23—22no—
392C23—28no—
393C23—22no—
394C23—24no—
395C23—24no—
396C23#94414yes—
397C23#94418yes—
398C23#94618yes—
399C23#94614yes—
400C23#95314yes—
401C23C2×#23712yes—
402C23#100412yes—
403C23C2×D3×Q811yescharacteristic
404C23C2×D3×Q812yes—
405C23C2×#23911yescharacteristic
406C23D3×#4112yes—
407C23#100912yes—
408C23C22×#6711yescharacteristic
409C23C22×#6712yes—
410Q8—416no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—416no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—68no—
430C2×C4—68no—
431C2×C4—68no—
432C2×C4—68no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—68no—
436C2×C4—1216no—
437C2×C4—1216no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—122no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4—124no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—124no—
448C2×C4#93514yes—
449C2×C3—28no—
450C2×C3C2×#39511yescharacteristic
451C2×C3#240411yescharacteristic
452C2×C3#240511yescharacteristic
453C2×C3#245014yes—
454C2×C3#250211yescharacteristic
455C2×C3#254914yes—
456C2×C3#258311yescharacteristic
457C2×C3#262611yescharacteristic
458C2×C3#285511yescharacteristic
459C22—216no—
460C22—24no—
461C22—24no—
462C22—216no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22#523018yes—
468C22#562212yes—
469C22#562412yes—
470C22#563412yes—
471C22#563514yes—
472C22#563614yes—
473C22#564412yes—
474C22C2×#93511yescharacteristic
475C22#580411yescharacteristic
476C22#580614yes—
477C22C2×#94411yescharacteristic
478C22C2×#94412yes—
479C22C2×#94612yes—
480C22C2×#94611yescharacteristic
481C22#582811yescharacteristic
482C22#582914yes—
483C22#583011yescharacteristic
484C22C2×#95311yescharacteristic
485C4—24no—
486C4—42no—
487C4—44no—
488C4—44no—
489C4—128no—
490C4—124no—
491C4—124no—
492C3#1491411yescharacteristic, complemented, Sylow
493C2—28no—
494C2#8276514yes—
495C2#8277511yescharacteristic
496C2#8278114yes—
497C2#8278511yescharacteristic
498C2#8281011yescharacteristic
499C2#8282211yescharacteristic
500C2#8404711yescharacteristic
501C2C2×#523011yescharacteristic
502C2#8409611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C22⋊Q8C2
prime radicalC25×C3C23
hypercenterC22×C22⋊Q8D3
hypocenterC3#14914
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. #72134C2semidirect
6. #71568C2semidirect
7. #71566C2semidirect
59. C23×Dic3Q8semidirect
74. C2×#217Q8semidirect
492. C3#14914semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×C3⋊#952
C2×#49462
#721342
#721272
#715682
#715662
#715442
#149143