Finite Group Databasebeta

← #10732584

Subgroups — #10732584

Order 768.

1686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C2×#5212C212yesmaximal
3C2×#4928C212yesmaximal, complemented
4C2×#4936C211yesmaximal, characteristic, complemented
5#71547C214yesmaximal
6#71535C211yesmaximal, characteristic, complemented
7#69421C214yesmaximal, complemented
8#15223—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C22×C3×C22⋊C4C2212yes—
11C22×C3×C42C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C22×#217C2214yes—
14C22×C4⋊Dic3C2212yes—
15C22×C4⋊Dic3C2212yes—
16C22×Dic3⋊C4C2214yes—
17C3×C4×C22⋊C4—24no—
18C2×C3×#95C2212yes—
19#5212—28no—
20C2×C3⋊#95C2214yes—
21C2×C3⋊#95C2214yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2214yes—
24#4936—24no—
25#4928—28no—
26C2×#389—32no—
27C2×#382—31no—
28C2×#380—32no—
29C2×C4×C22⋊C4D311yescharacteristic
30#1445—34no—
31#1439—34no—
32C4⋊#95—31no—
33C25×C3C2311yescharacteristic
34C23×C3×C4—24no—
35C23×C3×C4—22no—
36C23×C3×C4—22no—
37C23×C3×C4—24no—
38C23×C3×C4Q814yes—
39C23×C3×C4C2312yes—
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2314yes—
44C23×Dic3C2314yes—
45C2×C3×C22⋊C4—24no—
46C2×C3×C22⋊C4—24no—
47C2×C3×C22⋊C4—24no—
48C2×C3×C42—24no—
49C2×C3×C42—24no—
50C2×C3×C42—22no—
51C2×C3×C42D414yes—
52C2×#217—28no—
53C2×#217—28no—
54C2×#217—44no—
55C2×C4⋊Dic3—24no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—24no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3D418yes—
61C2×Dic3⋊C4—28no—
62C2×Dic3⋊C4—28no—
63C2×Dic3⋊C4—44no—
64C3×#95—24no—
65C3⋊#95—28no—
66C3⋊#95—28no—
67C3⋊#95—28no—
68C3⋊#95—44no—
69C24×C4C2×D311yescharacteristic
70C22×C4⋊C4—32no—
71C22×C4⋊C4—32no—
72C22×C4⋊C4—34no—
73C22×C22⋊C4—34no—
74C22×C22⋊C4C2×D312yes—
75C22×C42C2×D311yescharacteristic
76C22×C42C2×D311yescharacteristic
77#389—68no—
78#382—64no—
79#380—68no—
80C4×C22⋊C4—24no—
81C2×#95—34no—
82C2×#95—34no—
83C2×#95—34no—
84C2×#95—34no—
85C2×#95C2×D312yes—
86C24×C3—22no—
87C24×C3—21no—
88C24×C3—22no—
89C24×C3—21no—
90C24×C3—22no—
91C24×C3—22no—
92C24×C3—22no—
93C24×C3C2×Q812yes—
94C24×C3#4114yes—
95C24×C3C2411yescharacteristic
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—21no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—216no—
113C22×C3×C4—28no—
114C22×C3×C4—48no—
115C22×C3×C4—44no—
116C22×C3×C4—44no—
117C22×C3×C4C2×D414yes—
118C22×C3×C4C2×D414yes—
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×Q812yes—
121C22×C3×C4C2×Q812yes—
122C22×C3×C4#4114yes—
123C22×C3×C4#4112yes—
124C22×C3×C4#4114yes—
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3—24no—
128C22×Dic3—24no—
129C22×Dic3—24no—
130C22×Dic3—24no—
131C22×Dic3—24no—
132C22×Dic3—24no—
133C22×Dic3—24no—
134C22×Dic3—24no—
135C22×Dic3—24no—
136C22×Dic3—24no—
137C22×Dic3—24no—
138C22×Dic3C2×D418yes—
139C3×C22⋊C4—216no—
140C3×C42—28no—
141C3×C42—44no—
142#217—416no—
143C4⋊Dic3—216no—
144C4⋊Dic3—48no—
145Dic3⋊C4—416no—
146C25C22×D311yescharacteristic
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—34no—
152C23×C4—34no—
153C23×C4Dic614yes—
154C23×C4C22×D311yescharacteristic
155C23×C4C22×D312yes—
156C23×C4C22×D312yes—
157C23×C4C22×D311yescharacteristic
158C2×C4⋊C4—38no—
159C2×C4⋊C4—64no—
160C2×C4⋊C4—64no—
161C2×C4⋊C4—68no—
162C2×C4⋊C4—68no—
163C2×C4⋊C4—64no—
164C2×C4⋊C4—64no—
165C2×C4⋊C4—64no—
166C2×C4⋊C4—124no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—68no—
171C2×C22⋊C4—68no—
172C2×C22⋊C4—124no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42#6714yes—
177#95—24no—
178#95—68no—
179#95—68no—
180#95—68no—
181#95—124no—
182C23×C3—24no—
183C23×C3—22no—
184C23×C3—24no—
185C23×C3—24no—
186C23×C3—22no—
187C23×C3—24no—
188C23×C3—24no—
189C23×C3—24no—
190C23×C3—24no—
191C23×C3—24no—
192C23×C3—42no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3#12514yes—
203C23×C3C22×D412yes—
204C23×C3C22×D411yescharacteristic
205C23×C3C22×Q811yescharacteristic
206C23×C3C2×#4111yescharacteristic
207C23×C3C2×#4112yes—
208C23×C3C2×#4112yes—
209C23×C3C2×#4112yes—
210C23×C3H(5,𝔽2)12yes—
211C23×C32−1+412yes—
212C2×C3×C4—24no—
213C2×C3×C4—22no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—416no—
232C2×C3×C4C22⋊Q818yes—
233C2×C3×C4C22⋊Q814yes—
234C2×C3×C4#12318yes—
235C2×C3×C4#12314yes—
236C2×C3×C4#12714yes—
237C2×Dic3—28no—
238C2×Dic3—28no—
239C2×Dic3—28no—
240C2×Dic3—44no—
241C2×Dic3—44no—
242C2×Dic3—44no—
243C2×Dic3—44no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—44no—
247C2×Dic3—44no—
248C2×Dic3—44no—
249C2×Dic3—44no—
250C2×Dic3—44no—
251C24—22no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24C2×Dic612yes—
259C24#23514yes—
260C24C23×D311yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—216no—
278C22×C4—28no—
279C22×C4—38no—
280C22×C4—48no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—64no—
284C22×C4—64no—
285C22×C4—64no—
286C22×C4—64no—
287C22×C4—64no—
288C22×C4—64no—
289C22×C4—64no—
290C22×C4—64no—
291C22×C4—64no—
292C22×C4—64no—
293C22×C4—64no—
294C22×C4—64no—
295C22×C4—64no—
296C22×C4C2×Dic612yes—
297C22×C4C2×Dic612yes—
298C22×C4#23514yes—
299C22×C4#23512yes—
300C22×C4D3×D414yes—
301C22×C4#23714yes—
302C22×C4C2×#6714yes—
303C22×C4C2×#6712yes—
304C4⋊C4—616no—
305C4⋊C4—1216no—
306C4⋊C4—128no—
307C22⋊C4—216no—
308C22⋊C4—1216no—
309C42—28no—
310C42—44no—
311C22×C3—24no—
312C22×C3—28no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3—44no—
319C22×C3—44no—
320C22×C3C2×C22⋊Q812yes—
321C22×C3C2×C22⋊Q811yescharacteristic
322C22×C3C2×#12312yes—
323C22×C3C2×#12311yescharacteristic
324C22×C3#52512yes—
325C22×C3C2×#12511yescharacteristic
326C22×C3#52912yes—
327C22×C3C2×#12711yescharacteristic
328C22×C3#53311yescharacteristic
329C22×C3#53712yes—
330C22×C3#54011yescharacteristic
331C22×C3#54111yescharacteristic
332C22×C3#54412yes—
333C22×C3#54612yes—
334C22×C3#54612yes—
335C22×C3#54812yes—
336C22×C3#54812yes—
337C22×C3#55212yes—
338C22×C3#55312yes—
339C22×C3#55512yes—
340C22×C3#55612yes—
341C3×C4—42no—
342C3×C4—44no—
343C3×C4—44no—
344C3×C4#39918yes—
345Dic3—48no—
346Dic3—84no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23#87014yes—
368C23C22×Dic611yescharacteristic
369C23C2×#23512yes—
370C23C2×#23512yes—
371C23C2×#23511yescharacteristic
372C23C2×D3×D412yes—
373C23C2×#23712yes—
374C23#100912yes—
375C23#101012yes—
376C23C22×#6711yescharacteristic
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—416no—
397C2×C4—68no—
398C2×C4—68no—
399C2×C4—68no—
400C2×C4—124no—
401C2×C4—124no—
402C2×C4—124no—
403C2×C4—124no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4—124no—
408C2×C4—124no—
409C2×C4—124no—
410C2×C4—124no—
411C2×C4#87818yes—
412C2×C4#88318yes—
413C2×C4#92414yes—
414C2×C4#93514yes—
415C2×C4#94014yes—
416C2×C3—44no—
417C2×C3C2×#39911yescharacteristic
418C2×C3#242012yes—
419C2×C3#242112yes—
420C2×C3#243912yes—
421C2×C3#244212yes—
422C2×C3#248512yes—
423C2×C3#254012yes—
424C2×C3#262911yescharacteristic
425C2×C3#285111yescharacteristic
426C22—28no—
427C22—24no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—44no—
432C22—42no—
433C22—44no—
434C22—44no—
435C22C2×#87011yescharacteristic
436C22#547011yescharacteristic
437C22#547912yes—
438C22C2×#87812yes—
439C22C2×#88312yes—
440C22#551811yescharacteristic
441C22#551912yes—
442C22#553511yescharacteristic
443C22#554112yes—
444C22#554212yes—
445C22#555012yes—
446C22#555012yes—
447C22#555212yes—
448C22#555212yes—
449C22#556312yes—
450C22#556512yes—
451C22C2×#92411yescharacteristic
452C22#579112yes—
453C22C2×#93511yescharacteristic
454C22C2×#94011yescharacteristic
455C22#583312yes—
456C4—42no—
457C4—44no—
458C4—44no—
459C4—128no—
460C4—244no—
461C4#494918yes—
462C3#1522311yescharacteristic, complemented, Sylow
463C2—44no—
464C2#7348312yes—
465C2#7349212yes—
466C2#7349512yes—
467C2#7358412yes—
468C2#8207411yescharacteristic
469C2C2×#494911yescharacteristic
470C2#8226511yescharacteristic
471C2#8237212yes—
472C2#8237712yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C22⋊C4C2
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#15223
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#4928C2semidirect
4. C2×#4936C2semidirect
6. #71535C2semidirect
7. #69421C2semidirect
462. C3#15223semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C2×#52122
C2×#49282
C2×#49362
#715472
#715352
#694212
#152233