Finite Group Databasebeta

← #10732145

Subgroups — #10732145

Order 768.

1746 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2C2×#5222C212yesmaximal
3C2×#4985C212yesmaximal, complemented
4C2×#4928C211yesmaximal, characteristic, complemented
5#71546C212yesmaximal, complemented
6#71546C212yesmaximal, complemented
7#71545C214yesmaximal, complemented
8C4×C3⋊#95C211yesmaximal, characteristic, complemented
9#15170—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×C3×C22⋊C4C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×#217C2214yes—
16C22×#217C2214yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2212yes—
21C3×#390—24no—
22#5222—28no—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2212yes—
28#4985—28no—
29#4928—24no—
30C2×#400—32no—
31C2×#394—32no—
32C2×#390D311yescharacteristic
33C2×#389—31no—
34#1439—32no—
35#1439—32no—
36#1437—34no—
37C4×#95—31no—
38C25×C3C2312yes—
39C22×C3×D4—24no—
40C22×C3×D4—22no—
41C22×C3×D4—24no—
42C22×C3×D4—42no—
43C23×C3×C4C2312yes—
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2314yes—
49C23×Dic3C2312yes—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C22⋊C4—24no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4—28no—
57C2×C3×C42—42no—
58C2×C3×C42D414yes—
59C2×#217—28no—
60C2×#217—28no—
61C2×#217—28no—
62C2×#217—28no—
63C2×#217—44no—
64C2×#217—44no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—24no—
67C2×C4⋊Dic3—22no—
68C2×C4⋊Dic3—42no—
69C2×Dic3⋊C4—24no—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—24no—
72C2×C4×Dic3—24no—
73C2×C4×Dic3—42no—
74C2×C4×Dic3—42no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—28no—
78C3⋊#95—28no—
79C3⋊#95D418yes—
80C23×D4C2×D311yescharacteristic
81C22×C4⋊C4—32no—
82C22×C4⋊C4—31no—
83C22×C4⋊C4—31no—
84C22×C4⋊C4C2×D311yescharacteristic
85C22×C22⋊C4—34no—
86C22×C22⋊C4—34no—
87C22×C22⋊C4C2×D312yes—
88C22×C22⋊C4C2×D312yes—
89C22×C42—32no—
90C22×C42C2×D311yescharacteristic
91#400—68no—
92#394—68no—
93#390—24no—
94#389—64no—
95C2×#95—32no—
96C2×#95—34no—
97C2×#95—34no—
98C2×#95—32no—
99C2×#95—32no—
100C24×C3—24no—
101C24×C3—22no—
102C24×C3—24no—
103C24×C3—24no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—22no—
108C24×C3—24no—
109C24×C3—22no—
110C24×C3#4114yes—
111C24×C3C2411yescharacteristic
112C2×C3×D4—28no—
113C2×C3×D4—416no—
114C2×C3×D4—44no—
115C2×C3×D4—44no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—24no—
124C22×C3×C4—22no—
125C22×C3×C4—24no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4C2×D414yes—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D412yes—
132C22×C3×C4#4114yes—
133C22×C3×C4#4114yes—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—22no—
140C22×Dic3—24no—
141C22×Dic3—24no—
142C22×Dic3—24no—
143C22×Dic3—24no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3C2×D418yes—
152C22×Dic3#4114yes—
153C3×C4⋊C4—44no—
154C3×C22⋊C4—48no—
155C3×C22⋊C4—48no—
156C3×C42—44no—
157#217—416no—
158#217—416no—
159C4⋊Dic3—28no—
160C4⋊Dic3—44no—
161Dic3⋊C4—48no—
162C4×Dic3—48no—
163C25C22×D312yes—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—42no—
168C23×C4—32no—
169C23×C4—34no—
170C23×C4—32no—
171C23×C4C22×D312yes—
172C23×C4C22×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C23×C4C22×D311yescharacteristic
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—64no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4—64no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—122no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—68no—
190C2×C22⋊C4—68no—
191C2×C22⋊C4—68no—
192C2×C22⋊C4—68no—
193C2×C22⋊C4—124no—
194C2×C22⋊C4—124no—
195C2×C42—42no—
196C2×C42—64no—
197C2×C42—122no—
198C2×C42—122no—
199C2×C42#6714yes—
200#95—38no—
201#95—68no—
202#95—68no—
203#95—64no—
204#95—64no—
205C23×C3—24no—
206C23×C3—28no—
207C23×C3—28no—
208C23×C3—24no—
209C23×C3—24no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—44no—
214C23×C3—42no—
215C23×C3—44no—
216C23×C3—44no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—44no—
220C23×C3—44no—
221C23×C3—44no—
222C23×C3—44no—
223C23×C3—44no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3C22×D412yes—
229C23×C3C22×D411yescharacteristic
230C23×C3C2×#4112yes—
231C23×C3C2×#4112yes—
232C23×C3C2×#4112yes—
233C23×C3C2×#4112yes—
234C23×C3H(5,𝔽2)12yes—
235C23×C3H(5,𝔽2)11yescharacteristic
236C23×C32−1+411yescharacteristic
237C3×D4—416no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—28no—
247C2×C3×C4—28no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—44no—
252C2×C3×C4—44no—
253C2×C3×C4#12414yes—
254C2×C3×C4#12414yes—
255C2×C3×C4#12714yes—
256C2×Dic3—24no—
257C2×Dic3—28no—
258C2×Dic3—24no—
259C2×Dic3—28no—
260C2×Dic3—28no—
261C2×Dic3—24no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—44no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—44no—
274C2×Dic3—44no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24#23514yes—
286C24C23×D311yescharacteristic
287C2×D4—28no—
288C2×D4—416no—
289C2×D4—44no—
290C2×D4—44no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—38no—
304C22×C4—34no—
305C22×C4—64no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—64no—
309C22×C4—64no—
310C22×C4—64no—
311C22×C4—64no—
312C22×C4—62no—
313C22×C4—64no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—64no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—64no—
321C22×C4—62no—
322C22×C4D3×D414yes—
323C22×C4#23714yes—
324C22×C4#23714yes—
325C22×C4C2×#6712yes—
326C22×C4C2×#6712yes—
327C22×C4C2×#6712yes—
328C4⋊C4—44no—
329C4⋊C4—68no—
330C4⋊C4—124no—
331C4⋊C4—128no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—1216no—
335C22⋊C4—1216no—
336C42—44no—
337C42—128no—
338C22×C3—44no—
339C22×C3—44no—
340C22×C3—44no—
341C22×C3—48no—
342C22×C3—48no—
343C22×C3—82no—
344C22×C3—82no—
345C22×C3—82no—
346C22×C3—82no—
347C22×C3—84no—
348C22×C3—84no—
349C22×C3#52511yescharacteristic
350C22×C3#52511yescharacteristic
351C22×C3C2×#12411yescharacteristic
352C22×C3C2×#12411yescharacteristic
353C22×C3#52914yes—
354C22×C3C2×#12711yescharacteristic
355C22×C3#53214yes—
356C22×C3#53411yescharacteristic
357C22×C3#53711yescharacteristic
358C22×C3#53812yes—
359C22×C3#53912yes—
360C22×C3#54612yes—
361C22×C3#54614yes—
362C22×C3#54812yes—
363C22×C3#55212yes—
364C22×C3#55312yes—
365C22×C3#55314yes—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—82no—
370Dic3—44no—
371Dic3—48no—
372Dic3—82no—
373C23—28no—
374C23—24no—
375C23—28no—
376C23—24no—
377C23—24no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23C2×#23512yes—
397C23C2×D3×D412yes—
398C23C2×#23712yes—
399C23C2×#23712yes—
400C23#100412yes—
401C23D3×#4112yes—
402C23#100911yescharacteristic
403C23#101011yescharacteristic
404C23C22×#6711yescharacteristic
405D4—416no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—64no—
422C2×C4—68no—
423C2×C4—64no—
424C2×C4—68no—
425C2×C4—68no—
426C2×C4—64no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—124no—
430C2×C4—122no—
431C2×C4—124no—
432C2×C4—122no—
433C2×C4—124no—
434C2×C4—124no—
435C2×C4—124no—
436C2×C4—124no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4#93614yes—
441C2×C4#93614yes—
442C2×C4#94014yes—
443C2×C3—84no—
444C2×C3#239811yescharacteristic
445C2×C3#253911yescharacteristic
446C2×C3#254011yescharacteristic
447C2×C3#255814yes—
448C2×C3#255914yes—
449C2×C3#256211yescharacteristic
450C2×C3#259111yescharacteristic
451C2×C3#263911yescharacteristic
452C2×C3#285811yescharacteristic
453C22—48no—
454C22—48no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—82no—
459C22—84no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22#549014yes—
465C22#553814yes—
466C22#554112yes—
467C22#556512yes—
468C22#560114yes—
469C22#560212yes—
470C22#561712yes—
471C22#567612yes—
472C22#567712yes—
473C22#571614yes—
474C22#579111yescharacteristic
475C22C2×#93611yescharacteristic
476C22C2×#93611yescharacteristic
477C22C2×#94011yescharacteristic
478C22#580411yescharacteristic
479C22#582811yescharacteristic
480C22#583311yescharacteristic
481C4—24no—
482C4—24no—
483C4—24no—
484C4—82no—
485C4—124no—
486C4—128no—
487C4—242no—
488C3#1517011yescharacteristic, complemented, Sylow
489C2—84no—
490C2#7635814yes—
491C2#7635914yes—
492C2#8211511yescharacteristic
493C2#8234811yescharacteristic
494C2#8238311yescharacteristic
495C2#8267411yescharacteristic
496C2#8267911yescharacteristic
497C2#8299811yescharacteristic
498C2#8318411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#15170
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
3. C2×#4985C2semidirect
4. C2×#4928C2semidirect
5. #71546C2semidirect
6. #71546C2semidirect
7. #71545C2semidirect
8. C4×C3⋊#95C2semidirect
488. C3#15170semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C2×#52222
C2×#49852
C2×#49282
#715462
#715452
C4×C3⋊#952
#151703