Finite Group Databasebeta

← #10729289

Subgroups — #10729289

Order 768.

2062 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
1C22×#924C211yesmaximal, characteristic
2C22×C3×#95C211yesmaximal, characteristic
3C2×#5211C212yesmaximal
4#72127C212yesmaximal
5#72127C211yesmaximal, characteristic
6#69424C212yesmaximal, complemented
7#69415C212yesmaximal, complemented
8#69414C214yesmaximal, complemented
9#14934—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C2×#924—24no—
18C2×#924—28no—
19C22×Dic3⋊C4C2212yes—
20C2×C3×#95—24no—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23#5211—28no—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C22×C22⋊Q8—31no—
31C2×#381—32no—
32C22×#95D311yescharacteristic
33#1446—32no—
34#1442—34no—
35#1442—32no—
36C22⋊#95—32no—
37C22⋊#95—31no—
38C25×C3C2311yescharacteristic
39C23×C3×C4—24no—
40C23×C3×C4—22no—
41C23×C3×C4—22no—
42C23×C3×C4—24no—
43C23×C3×C4—22no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—42no—
49C23×C3×C4—42no—
50C23×C3×C4D414yes—
51C23×C3×C4Q814yescomplemented
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3C2314yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C22×Dic6—22no—
60C22×Dic6—24no—
61C22×Dic6—24no—
62C22×Dic6—42no—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—24no—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217D418yes—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—24no—
71#924—232no—
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—24no—
75C3×#95—24no—
76C3×#95—24no—
77C3⋊#95—28no—
78C3⋊#95—24no—
79C3⋊#95—28no—
80C3⋊#95—28no—
81C3⋊#95—44no—
82C3⋊#95—42no—
83C23×Q8—31no—
84C24×C4C2×D311yescharacteristic
85C24×C4C2×D311yescharacteristic
86C24×C4C2×D311yescharacteristic
87C2×C22⋊Q8—68no—
88C2×C22⋊Q8—64no—
89C22×C4⋊C4—32no—
90C22×C4⋊C4—31no—
91C22×C22⋊C4—32no—
92C22×C22⋊C4—32no—
93#381—68no—
94C2×#95—24no—
95C2×#95—34no—
96C2×#95—32no—
97C2×#95—34no—
98C2×#95—34no—
99C2×#95—32no—
100C2×#95—34no—
101C2×#95C2×D312yes—
102C2×#95C2×D312yes—
103C24×C3—22no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3C2×D414yes—
108C24×C3C2×D412yes—
109C24×C3C2×Q812yes—
110C24×C3#4112yes—
111C24×C3#4114yes—
112C24×C3C2411yescharacteristic
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—28no—
116C22×C3×C4—28no—
117C22×C3×C4—216no—
118C22×C3×C4—28no—
119C22×C3×C4—24no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—28no—
131C22×C3×C4—44no—
132C22×C3×C4—416no—
133C22×C3×C4—416no—
134C22×C3×C4—44no—
135C22×C3×C4—44no—
136C22×C3×C4—48no—
137C22×C3×C4—44no—
138C22×C3×C4—44no—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×Q814yes—
142C22×C3×C4#4114yes—
143C22×C3×C4#4114yes—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3C2×D414yes—
160C22×Dic3C2×D414yes—
161C2×Dic6—28no—
162C2×Dic6—416no—
163C2×Dic6—44no—
164C2×Dic6—44no—
165#217—216no—
166#217—48no—
167C4⋊Dic3—28no—
168Dic3⋊C4—216no—
169C25C22×D311yescharacteristic
170C22×Q8—64no—
171C22×Q8—64no—
172C22×Q8—62no—
173C22×Q8—122no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—32no—
184C23×C4—32no—
185C23×C4—34no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4Dic614yescomplemented
189C23×C4#6714yes—
190C23×C4C22×D312yes—
191C23×C4C22×D312yes—
192C23×C4C22×D311yescharacteristic
193C23×C4C22×D311yescharacteristic
194C22⋊Q8—632no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—64no—
199C2×C4⋊C4—62no—
200C2×C22⋊C4—38no—
201C2×C22⋊C4—64no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4—64no—
204C2×C22⋊C4—64no—
205C2×C22⋊C4—64no—
206#95—24no—
207#95—24no—
208#95—68no—
209#95—68no—
210#95—68no—
211#95—64no—
212#95—124no—
213#95—122no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—28no—
223C23×C3—28no—
224C23×C3—28no—
225C23×C3—28no—
226C23×C3#12014yes—
227C23×C3C22⋊Q818yes—
228C23×C3C22⋊Q814yes—
229C23×C3#12418yes—
230C23×C3#12414yes—
231C23×C3C22×D412yes—
232C23×C3C22×D411yescharacteristic
233C23×C3C22×Q811yescharacteristic
234C23×C3C2×#4111yescharacteristic
235C23×C3C2×#4112yes—
236C23×C3C2×#4112yes—
237C23×C3C2×#4112yes—
238C23×C3H(5,𝔽2)12yes—
239C23×C32−1+412yes—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—28no—
243C2×C3×C4—22no—
244C2×C3×C4—28no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—28no—
250C2×C3×C4—416no—
251C2×C3×C4—416no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—416no—
260C2×C3×C4#12314yes—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—28no—
264C2×Dic3—28no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—44no—
270C2×Dic3—44no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273C2×Dic3—44no—
274C2×Dic3—44no—
275C2×Dic3—44no—
276C2×Dic3—44no—
277C2×Dic3—44no—
278Dic6—416no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24C2×Dic612yes—
284C24#23512yes—
285C24D3×D414yes—
286C24#23714yes—
287C24C2×#6712yes—
288C24C23×D311yescharacteristic
289C2×Q8—68no—
290C2×Q8—1216no—
291C2×Q8—124no—
292C2×Q8—124no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—28no—
296C22×C4—216no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—22no—
300C22×C4—22no—
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—28no—
311C22×C4—34no—
312C22×C4—34no—
313C22×C4—44no—
314C22×C4—416no—
315C22×C4—416no—
316C22×C4—44no—
317C22×C4—48no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—64no—
326C22×C4—64no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—64no—
330C22×C4—64no—
331C22×C4—64no—
332C22×C4—64no—
333C22×C4—64no—
334C22×C4—64no—
335C22×C4—64no—
336C22×C4C2×Dic614yes—
337C22×C4#23514yes—
338C22×C4#23514yes—
339C22×C4C2×#6712yes—
340C22×C4C2×#6712yes—
341C4⋊C4—616no—
342C4⋊C4—68no—
343C22⋊C4—616no—
344C22⋊C4—128no—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—24no—
350C22×C3—216no—
351C22×C3—216no—
352C22×C3—24no—
353C22×C3#39318yes—
354C22×C3C2×#12011yescharacteristic
355C22×C3C2×C22⋊Q812yes—
356C22×C3C2×C22⋊Q811yescharacteristic
357C22×C3C2×#12311yescharacteristic
358C22×C3#52511yescharacteristic
359C22×C3C2×#12412yes—
360C22×C3C2×#12411yescharacteristic
361C22×C3#52912yes—
362C22×C3#53312yes—
363C22×C3#53411yescharacteristic
364C22×C3#53611yescharacteristic
365C22×C3#53912yes—
366C22×C3#54412yes—
367C22×C3#54614yes—
368C22×C3#54714yes—
369C22×C3#55114yes—
370C22×C3#55214yes—
371C3×C4—24no—
372C3×C4—44no—
373C3×C4—44no—
374C3×C4—42no—
375Dic3—44no—
376Dic3—44no—
377Dic3—84no—
378C23—24no—
379C23—24no—
380C23—28no—
381C23—28no—
382C23—24no—
383C23—28no—
384C23—28no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23#87818yes—
391C23#88518yes—
392C23#92414yes—
393C23#93614yes—
394C23#93714yes—
395C23C22×Dic611yescharacteristic
396C23C2×#23512yes—
397C23C2×#23512yes—
398C23C2×#23511yescharacteristic
399C23C2×D3×D412yes—
400C23C2×#23712yes—
401C23#100412yes—
402C23#100712yes—
403C23C22×#6711yescharacteristic
404Q8—1216no—
405C2×C4—28no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—416no—
416C2×C4—416no—
417C2×C4—416no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—68no—
426C2×C4—68no—
427C2×C4—64no—
428C2×C4—64no—
429C2×C4—124no—
430C2×C4—124no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4—124no—
442C2×C4#92914yes—
443C2×C3—28no—
444C2×C3C2×#39311yescharacteristic
445C2×C3#240511yescharacteristic
446C2×C3#240611yescharacteristic
447C2×C3#243814yes—
448C2×C3#248814yes—
449C2×C3#250311yescharacteristic
450C2×C3#256111yescharacteristic
451C2×C3#262711yescharacteristic
452C2×C3#285511yescharacteristic
453C22—24no—
454C22—24no—
455C22—24no—
456C22—216no—
457C22—216no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22#494718yes—
462C22#547012yes—
463C22#547912yes—
464C22C2×#87812yes—
465C22#549014yes—
466C22C2×#88512yes—
467C22#549514yes—
468C22#550012yes—
469C22#550714yes—
470C22#551314yes—
471C22#551512yes—
472C22C2×#92411yescharacteristic
473C22C2×#92911yescharacteristic
474C22#579111yescharacteristic
475C22C2×#93611yescharacteristic
476C22C2×#93711yescharacteristic
477C22#580411yescharacteristic
478C22#581611yescharacteristic
479C4—24no—
480C4—44no—
481C4—44no—
482C4—42no—
483C4—124no—
484C4—124no—
485C4—244no—
486C3#1493411yescharacteristic, complemented, Sylow
487C2—28no—
488C2#7335014yes—
489C2#7335214yes—
490C2#8207511yescharacteristic
491C2C2×#494711yescharacteristic
492C2#8211611yescharacteristic
493C2#8212611yescharacteristic
494C2#8217911yescharacteristic
495C2#8218311yescharacteristic
496C2#8219211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#95C2
prime radicalC25×C3C23
hypercenterC22×#95D3
hypocenterC3#14934
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
6. #69424C2semidirect
7. #69415C2semidirect
8. #69414C2semidirect
51. C23×C3×C4Q8semidirect
188. C23×C4Dic6semidirect
486. C3#14934semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C22×C3×#952
C2×#52112
#721272
#694242
#694152
#694142
#149343