Finite Group Databasebeta

← #10728721

Subgroups — #10728721

Order 768.

2086 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×#390C211yesmaximal, characteristic
3C2×#5222C212yesmaximal
4C2×#4947C212yesmaximal, complemented
5#71566C212yesmaximal, complemented
6#71546C212yesmaximal, complemented
7#71545C214yesmaximal, complemented
8#71534C211yesmaximal, characteristic, complemented
9#14883—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C23×Dic6C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C22×#217C2214yes—
17C22×#217C2214yes—
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2212yes—
21C2×C4×Dic6—24no—
22C2×C4×Dic6—28no—
23C3×#390—24no—
24#5222—28no—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2214yes—
30#4947—28no—
31C22×C4×Q8—31no—
32C2×#394—32no—
33C2×#393—32no—
34C2×#390D311yescharacteristic
35#1441—32no—
36#1439—32no—
37#1437—34no—
38#1434—31no—
39C25×C3C2312yes—
40C22×C3×D4—24no—
41C22×C3×D4—22no—
42C22×C3×D4—24no—
43C22×C3×D4—42no—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2314yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C22×Dic6—22no—
52C22×Dic6—22no—
53C22×Dic6—24no—
54C22×Dic6—24no—
55C22×Dic6D414yescomplemented
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—42no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4—28no—
63C2×C3×C42—24no—
64C2×C3×C42D414yes—
65C2×#217—28no—
66C2×#217—28no—
67C2×#217—28no—
68C2×#217—28no—
69C2×#217—28no—
70C2×#217—44no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—24no—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4D418yes—
76C2×C4×Dic3—24no—
77C2×C4×Dic3—24no—
78C2×C4×Dic3—24no—
79C4×Dic6—232no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82C3⋊#95—24no—
83C3⋊#95—28no—
84C3⋊#95—44no—
85C23×Q8—31no—
86C23×D4C2×D311yescharacteristic
87C2×C4×Q8—68no—
88C2×C4×Q8—64no—
89C22×C4⋊C4—32no—
90C22×C4⋊C4—31no—
91C22×C4⋊C4C2×D311yescharacteristic
92C22×C22⋊C4—34no—
93C22×C22⋊C4—34no—
94C22×C22⋊C4C2×D312yes—
95C22×C22⋊C4C2×D312yes—
96C22×C42—32no—
97C22×C42C2×D311yescharacteristic
98#394—68no—
99#393—68no—
100#390—24no—
101C2×#95—32no—
102C2×#95—34no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—34no—
106C24×C3—22no—
107C24×C3—24no—
108C24×C3—22no—
109C24×C3—24no—
110C24×C3—24no—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—24no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3#4114yes—
117C24×C3C2411yescharacteristic
118C2×C3×D4—28no—
119C2×C3×D4—416no—
120C2×C3×D4—44no—
121C2×C3×D4—44no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4C2×D414yes—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4#4112yes—
139C22×C3×C4#4114yes—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3C2×D414yes—
156C22×Dic3C2×D414yes—
157C22×Dic3C2×D414yes—
158C22×Dic3#4114yes—
159C2×Dic6—28no—
160C2×Dic6—216no—
161C2×Dic6—28no—
162C2×Dic6—28no—
163C2×Dic6—216no—
164C3×C4⋊C4—44no—
165C3×C22⋊C4—48no—
166C3×C22⋊C4—48no—
167C3×C42—28no—
168#217—416no—
169#217—416no—
170C4⋊Dic3—28no—
171Dic3⋊C4—216no—
172C4×Dic3—216no—
173C25C22×D312yes—
174C22×Q8—34no—
175C22×Q8—64no—
176C22×Q8—64no—
177C22×Q8—62no—
178C22×Q8—62no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—42no—
183C23×C4—32no—
184C23×C4—32no—
185C23×C4—34no—
186C23×C4C22×D312yes—
187C23×C4C22×D311yescharacteristic
188C23×C4C22×D311yescharacteristic
189C23×C4C22×D311yescharacteristic
190C4×Q8—632no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—38no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—62no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—68no—
204C2×C22⋊C4—68no—
205C2×C22⋊C4—68no—
206C2×C22⋊C4—68no—
207C2×C22⋊C4—68no—
208C2×C22⋊C4—124no—
209C2×C42—24no—
210C2×C42—64no—
211C2×C42—64no—
212C2×C42—64no—
213C2×C42#6714yes—
214#95—68no—
215#95—64no—
216#95—64no—
217#95—64no—
218#95—124no—
219C23×C3—24no—
220C23×C3—28no—
221C23×C3—28no—
222C23×C3—24no—
223C23×C3—24no—
224C23×C3—42no—
225C23×C3—44no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—44no—
229C23×C3—42no—
230C23×C3—44no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3—44no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—42no—
241C23×C3—44no—
242C23×C3C22×D412yes—
243C23×C3C22×D411yescharacteristic
244C23×C3C22×D411yescharacteristic
245C23×C3C2×#4112yes—
246C23×C3C2×#4111yescharacteristic
247C23×C3C2×#4112yes—
248C23×C3C2×#4112yes—
249C23×C3H(5,𝔽2)12yes—
250C23×C32−1+412yes—
251C3×D4—416no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4#12014yes—
268C2×C3×C4#12114yes—
269C2×C3×C4#12414yes—
270C2×Dic3—24no—
271C2×Dic3—24no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274C2×Dic3—24no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—28no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—44no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283C2×Dic3—44no—
284C2×Dic3—44no—
285C2×Dic3—44no—
286C2×Dic3—44no—
287C2×Dic3—44no—
288C2×Dic3#12118yes—
289Dic6—232no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24#23514yes—
301C24C23×D311yescharacteristic
302C2×Q8—616no—
303C2×Q8—68no—
304C2×Q8—68no—
305C2×Q8—616no—
306C2×Q8—68no—
307C2×D4—28no—
308C2×D4—416no—
309C2×D4—44no—
310C2×D4—44no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—34no—
323C22×C4—34no—
324C22×C4—34no—
325C22×C4—34no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—64no—
331C22×C4—62no—
332C22×C4—64no—
333C22×C4—62no—
334C22×C4—64no—
335C22×C4—64no—
336C22×C4—64no—
337C22×C4—64no—
338C22×C4—64no—
339C22×C4—64no—
340C22×C4—64no—
341C22×C4D3×D414yes—
342C22×C4D3×D412yes—
343C22×C4#23714yes—
344C22×C4#23712yes—
345C22×C4C2×#6712yes—
346C22×C4C2×#6712yes—
347C22×C4C2×#6712yes—
348C4⋊C4—44no—
349C4⋊C4—68no—
350C4⋊C4—616no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—1216no—
354C22⋊C4—1216no—
355C42—28no—
356C42—616no—
357C22×C3—44no—
358C22×C3—44no—
359C22×C3—48no—
360C22×C3—48no—
361C22×C3—44no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—84no—
365C22×C3—82no—
366C22×C3—84no—
367C22×C3—82no—
368C22×C3C2×#12011yescharacteristic
369C22×C3C2×#12111yescharacteristic
370C22×C3C2×#12112yes—
371C22×C3#52511yescharacteristic
372C22×C3#52511yescharacteristic
373C22×C3#52512yes—
374C22×C3C2×#12411yescharacteristic
375C22×C3#52914yes—
376C22×C3#53511yescharacteristic
377C22×C3#53611yescharacteristic
378C22×C3#54214yes—
379C22×C3#54614yes—
380C22×C3#54812yes—
381C22×C3#54814yes—
382C22×C3#55012yes—
383C22×C3#55212yes—
384C22×C3#55912yes—
385C3×C4—24no—
386C3×C4—24no—
387C3×C4—24no—
388C3×C4—82no—
389Dic3—28no—
390Dic3—44no—
391Dic3—84no—
392C23—28no—
393C23—24no—
394C23—28no—
395C23—24no—
396C23—24no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23C2×#23512yes—
416C23C2×D3×D411yescharacteristic
417C23C2×D3×D412yes—
418C23C2×#23711yescharacteristic
419C23C2×#23712yes—
420C23#100412yes—
421C23D3×#4112yes—
422C23#101012yes—
423C23C22×#6711yescharacteristic
424Q8—632no—
425D4—416no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—38no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—64no—
445C2×C4—64no—
446C2×C4—64no—
447C2×C4—64no—
448C2×C4—68no—
449C2×C4—64no—
450C2×C4—124no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—124no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4—124no—
457C2×C4—124no—
458C2×C4—124no—
459C2×C4—124no—
460C2×C4#93614yes—
461C2×C4#93714yes—
462C2×C4#93814yes—
463C2×C3—84no—
464C2×C3#239811yescharacteristic
465C2×C3#240411yescharacteristic
466C2×C3#242614yes—
467C2×C3#243611yescharacteristic
468C2×C3#245814yes—
469C2×C3#253611yescharacteristic
470C2×C3#254811yescharacteristic
471C2×C3#263211yescharacteristic
472C2×C3#284711yescharacteristic
473C22—48no—
474C22—48no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—84no—
479C22—82no—
480C22—82no—
481C22—84no—
482C22—82no—
483C22—82no—
484C22#549014yes—
485C22#553814yes—
486C22#554112yes—
487C22#555212yes—
488C22#559814yes—
489C22D3×#12112yes—
490C22#560612yes—
491C22#566914yes—
492C22#571312yes—
493C22#571812yes—
494C22#579111yescharacteristic
495C22C2×#93611yescharacteristic
496C22C2×#93711yescharacteristic
497C22C2×#93811yescharacteristic
498C22#580511yescharacteristic
499C22#582511yescharacteristic
500C22#582811yescharacteristic
501C4—24no—
502C4—24no—
503C4—24no—
504C4—68no—
505C4—82no—
506C4—124no—
507C4—244no—
508C3#1488311yescharacteristic, complemented, Sylow
509C2—84no—
510C2#7631214yes—
511C2#7632614yes—
512C2#8212511yescharacteristic
513C2#8234811yescharacteristic
514C2#8237511yescharacteristic
515C2#8265411yescharacteristic
516C2#8272311yescharacteristic
517C2#8297711yescharacteristic
518C2#8317911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#14883
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic6C2semidirect
4. C2×#4947C2semidirect
5. #71566C2semidirect
6. #71546C2semidirect
7. #71545C2semidirect
8. #71534C2semidirect
55. C22×Dic6D4semidirect
508. C3#14883semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic62
C2×C3×#3902
C2×#52222
C2×#49472
#715662
#715462
#715452
#715342
#148833