Finite Group Databasebeta

← #10727947

Subgroups — #10727947

Order 768.

2190 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic3⋊C4C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3C2×#4950C212yesmaximal
4C2×#4661C212yesmaximal, complemented
5C3×C22⋊#95C211yesmaximal, characteristic
6#71544C212yesmaximal
7#69411C214yesmaximal, complemented
8#69410C212yesmaximal, complemented
9#14821—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×Dic3C2211yescharacteristic
12C24×Dic3C2211yescharacteristic
13C22×C3×C22⋊C4C2212yes—
14C22×#217C2212yes—
15C22×C4⋊Dic3C2214yes—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4—24no—
19C22×Dic3⋊C4—28no—
20C22×Dic3⋊C4C2212yes—
21C22×Dic3⋊C4C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24#5212—24no—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29#4950—28no—
30#4661—28no—
31C23×C4⋊C4—31no—
32C2×#398—32no—
33C2×#385—32no—
34C2×#380—31no—
35#1441—32no—
36#1441—34no—
37C22⋊#95—32no—
38C22⋊#95D311yescharacteristic
39C25×C3C2311yescharacteristic
40C23×C3×C4—24no—
41C23×C3×C4—24no—
42C23×C3×C4—42no—
43C23×C3×C4D414yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2314yes—
46C23×C3×C4C2311yescharacteristic
47C23×Dic3—22no—
48C23×Dic3—22no—
49C23×Dic3—24no—
50C23×Dic3—24no—
51C23×Dic3—22no—
52C23×Dic3—24no—
53C23×Dic3—24no—
54C23×Dic3—42no—
55C23×Dic3D414yes—
56C23×Dic3C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4Q818yes—
63C2×#217—24no—
64C2×#217—24no—
65C2×#217—24no—
66C2×C4⋊Dic3—24no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—28no—
70C2×C4⋊Dic3—28no—
71C2×C4⋊Dic3—28no—
72C2×C4⋊Dic3—42no—
73C2×Dic3⋊C4—232no—
74C2×Dic3⋊C4—24no—
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—24no—
77C2×Dic3⋊C4—24no—
78C2×Dic3⋊C4—24no—
79C2×Dic3⋊C4D418yes—
80C3×#95—24no—
81C3×#95—42no—
82C3⋊#95—24no—
83C3⋊#95—28no—
84C3⋊#95—28no—
85C3⋊#95—44no—
86C24×C4—31no—
87C24×C4—31no—
88C24×C4C2×D311yescharacteristic
89C22×C4⋊C4—32no—
90C22×C4⋊C4—31no—
91C22×C4⋊C4—34no—
92C22×C4⋊C4—32no—
93C22×C4⋊C4—31no—
94C22×C4⋊C4—68no—
95C22×C4⋊C4—64no—
96C22×C22⋊C4—32no—
97C22×C22⋊C4C2×D312yes—
98#398—68no—
99#385—68no—
100#380—64no—
101C2×#95—34no—
102C2×#95—32no—
103C2×#95—34no—
104C2×#95—34no—
105C2×#95C2×D312yes—
106C2×#95C2×D312yes—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3C2×D412yes—
112C24×C3C2×D412yes—
113C24×C3C2×Q814yes—
114C24×C3#4114yes—
115C24×C3#4112yes—
116C24×C3C2411yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—28no—
120C22×C3×C4—24no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—24no—
127C22×C3×C4—22no—
128C22×C3×C4—24no—
129C22×C3×C4—28no—
130C22×C3×C4—44no—
131C22×C3×C4—416no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D414yes—
135C22×C3×C4C2×Q818yes—
136C22×Dic3—28no—
137C22×Dic3—216no—
138C22×Dic3—28no—
139C22×Dic3—28no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—216no—
145C22×Dic3—28no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—22no—
151C22×Dic3—24no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—24no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C22×Dic3—416no—
161C22×Dic3—44no—
162C22×Dic3—44no—
163C22×Dic3C2×D414yes—
164C22×Dic3C2×D414yes—
165C22×Dic3C2×D414yes—
166C22×Dic3#4114yes—
167C3×C22⋊C4—216no—
168#217—48no—
169C4⋊Dic3—44no—
170C4⋊Dic3—44no—
171C4⋊Dic3—416no—
172Dic3⋊C4—216no—
173Dic3⋊C4—216no—
174C25C22×D311yescharacteristic
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—32no—
178C23×C4—34no—
179C23×C4—32no—
180C23×C4—32no—
181C23×C4—32no—
182C23×C4—42no—
183C23×C4—64no—
184C23×C4—64no—
185C23×C4—64no—
186C23×C4—62no—
187C23×C4—62no—
188C23×C4—62no—
189C23×C4—64no—
190C23×C4—122no—
191C23×C4D1214yes—
192C23×C4C22×D314yes—
193C23×C4C22×D311yescharacteristic
194C23×C4C22×D311yescharacteristic
195C2×C4⋊C4—38no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—632no—
199C2×C4⋊C4—64no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—68no—
203C2×C4⋊C4—68no—
204C2×C4⋊C4—68no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4—64no—
208C2×C4⋊C4—122no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—64no—
214C2×C22⋊C4Dic618yes—
215#95—24no—
216#95—42no—
217#95—68no—
218#95—68no—
219#95—64no—
220#95—124no—
221C23×C3—22no—
222C23×C3—24no—
223C23×C3—22no—
224C23×C3—24no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—28no—
230C23×C3—28no—
231C23×C3—28no—
232C23×C3—28no—
233C23×C3#12114yes—
234C23×C3C22⋊Q818yes—
235C23×C3#12314yes—
236C23×C3#12518yes—
237C23×C3C4⋊Q814yes—
238C23×C3C22×D411yescharacteristic
239C23×C3C22×D412yes—
240C23×C3C22×D411yescharacteristic
241C23×C3C22×Q812yes—
242C23×C3C2×#4112yes—
243C23×C3C2×#4111yescharacteristic
244C23×C3C2×#4112yes—
245C23×C3H(5,𝔽2)12yes—
246C23×C32−1+412yes—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—28no—
250C2×C3×C4—22no—
251C2×C3×C4—28no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—28no—
255C2×C3×C4—28no—
256C2×C3×C4—416no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4#12014yes—
263C2×Dic3—232no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—24no—
271C2×Dic3—28no—
272C2×Dic3—416no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—44no—
276C2×Dic3—42no—
277C2×Dic3—44no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—44no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—44no—
285C2×Dic3—44no—
286C2×Dic3#12118yes—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24C2×Dic614yes—
292C24C2×D1212yes—
293C24D3×D412yes—
294C24#23714yes—
295C24#23912yes—
296C24C23×D311yescharacteristic
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—28no—
310C22×C4—34no—
311C22×C4—34no—
312C22×C4—34no—
313C22×C4—34no—
314C22×C4—44no—
315C22×C4—416no—
316C22×C4—616no—
317C22×C4—68no—
318C22×C4—68no—
319C22×C4—616no—
320C22×C4—68no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—68no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—64no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—64no—
334C22×C4—62no—
335C22×C4—64no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—64no—
339C22×C4—64no—
340C22×C4—1216no—
341C22×C4—124no—
342C22×C4—124no—
343C22×C4C2×Dic618yes—
344C22×C4C2×D1212yes—
345C22×C4C2×D1212yes—
346C22×C4D3×D414yes—
347C4⋊C4—616no—
348C4⋊C4—616no—
349C4⋊C4—124no—
350C4⋊C4—1216no—
351C4⋊C4—124no—
352C22⋊C4—216no—
353C22⋊C4—128no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—216no—
359C22×C3—24no—
360C22×C3—216no—
361C22×C3—24no—
362C22×C3#39818yescomplemented
363C22×C3C2×#12011yescharacteristic
364C22×C3C2×#12111yescharacteristic
365C22×C3C2×#12112yes—
366C22×C3C2×C22⋊Q812yes—
367C22×C3C2×#12311yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3#52512yes—
370C22×C3C2×#12512yes—
371C22×C3C2×C4⋊Q811yescharacteristic
372C22×C3#53411yescharacteristic
373C22×C3#53611yescharacteristic
374C22×C3#53712yes—
375C22×C3#53912yes—
376C22×C3#54614yes—
377C22×C3D4×Q814yes—
378C22×C3#55214yes—
379C22×C3#55414yes—
380C3×C4—24no—
381C3×C4—42no—
382C3×C4—48no—
383Dic3—28no—
384Dic3—44no—
385Dic3—82no—
386Dic3—82no—
387C23—24no—
388C23—28no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23#86914yes—
400C23#87818yes—
401C23#88614yes—
402C23#89018yes—
403C23#89514yes—
404C23C22×Dic612yes—
405C23C22×D1211yescharacteristic
406C23C2×D3×D411yescharacteristic
407C23C2×D3×D412yes—
408C23C2×#23712yes—
409C23#100412yes—
410C23C2×#23911yescharacteristic
411C23D3×#4112yes—
412C23#101012yes—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—38no—
423C2×C4—416no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—632no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—68no—
438C2×C4—1216no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—124no—
450C2×C4—124no—
451C2×C4—124no—
452C2×C4#88214yes—
453C2×C3—28no—
454C2×C3C2×#39811yescharacteristic
455C2×C3#240511yescharacteristic
456C2×C3#241714yes—
457C2×C3#243411yescharacteristic
458C2×C3#247314yes—
459C2×C3#248411yescharacteristic
460C2×C3#258311yescharacteristic
461C2×C3#261811yescharacteristic
462C2×C3#284711yescharacteristic
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—216no—
470C22—216no—
471C22#466118yescomplemented
472C22C2×#86911yescharacteristic
473C22C2×#87812yes—
474C22C2×#88211yescharacteristic
475C22C2×#88611yescharacteristic
476C22#549311yescharacteristic
477C22C2×#89012yes—
478C22C2×#89511yescharacteristic
479C22#552411yescharacteristic
480C22#552611yescharacteristic
481C22D4×Dic614yes—
482C22#553914yes—
483C22#559114yes—
484C22#559712yes—
485C22D3×#12112yes—
486C22#562712yes—
487C22#564512yes—
488C22#565014yes—
489C4—24no—
490C4—42no—
491C4—48no—
492C4—68no—
493C4—124no—
494C4—242no—
495C4—242no—
496C3#1482111yescharacteristic, complemented, Sylow
497C2—28no—
498C2C2×#466111yescharacteristic
499C2#7349011yescharacteristic
500C2#7354711yescharacteristic
501C2#7382311yescharacteristic
502C2#7403211yescharacteristic
503C2#7404811yescharacteristic
504C2#7407411yescharacteristic
505C2#7444314yes—
506C2#7469314yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×C22⋊#95C2
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#14821
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. C2×#4661C2semidirect
7. #69411C2semidirect
8. #69410C2semidirect
362. C22×C3#398semidirect
471. C22#4661semidirect
496. C3#14821semidirect

Maximal subgroups

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

subgroupindex
C23×Dic3⋊C42
C2×#52122
C2×#49502
C2×#46612
C3×C22⋊#952
#715442
#694112
#694102
#148213