Finite Group Databasebeta

← #10725633

Subgroups — #10725633

Order 768.

1138 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic
2#71565C211yesmaximal, characteristic
3#71565C212yesmaximal
4#71563C211yesmaximal, characteristic
5#69412C212yesmaximal
6#69407C212yesmaximal
7#69407C212yesmaximal
8#69407C212yesmaximal
9#69401C212yesmaximal
10#14603—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic3C2212yes—
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4×Dic3C2212yes—
17C22×C4×Dic3C2211yescharacteristic
18C3×#400C418yescomplemented
19C2×C3×#95C2211yescharacteristic
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×#400D311yescharacteristic
34#1442—32no—
35#1441—32no—
36#1441—31no—
37#1436—32no—
38#1436—32no—
39#1436—32no—
40C4⋊#95—31no—
41#1434—32no—
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C2×C3×C4⋊C4—41no—
52C2×C3×C4⋊C4—41no—
53C2×C3×C4⋊C4—41no—
54C2×C3×C4⋊C4—41no—
55C2×C3×C4⋊C4C2×C414yes—
56C2×C3×C4⋊C4C2×C414yes—
57C2×C4⋊Dic3—24no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—41no—
60C2×C4⋊Dic3—41no—
61C2×C4⋊Dic3—42no—
62C2×C4⋊Dic3—42no—
63C2×C4⋊Dic3—41no—
64C2×C4⋊Dic3Q814yes—
65C2×C4×Dic3—22no—
66C2×C4×Dic3—24no—
67C2×C4×Dic3—42no—
68C2×C4×Dic3—42no—
69C2×C4×Dic3—41no—
70C2×C4×Dic3—41no—
71C3×#95C2×C414yes—
72C3×#95C2×C418yes—
73C3×#95C2×C418yes—
74C3⋊#95—24no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—24no—
78C3⋊#95—42no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95—42no—
82C3⋊#95—42no—
83C3⋊#95—41no—
84C3⋊#95—42no—
85C22×C4⋊C4—31no—
86C22×C4⋊C4—32no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4C2×D311yescharacteristic
89C22×C4⋊C4C2×D311yescharacteristic
90C22×C42—32no—
91C22×C42—31no—
92#400Dic318yescomplemented
93C2×#95—32no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—32no—
97C2×#95—32no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—31no—
103C2×#95—32no—
104C2×#95C2×D311yescharacteristic
105C2×#95C2×D312yes—
106C2×#95C2×D312yes—
107C24×C3C2411yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4C22×C414yes—
133C22×C3×C4C22×C414yes—
134C22×C3×C4C22×C412yes—
135C22×C3×C4C22×C412yes—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C2×Q812yes—
138C22×C3×C4#4112yes—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3C2×Q814yes—
165C22×Dic3#4114yes—
166C22×Dic3#4114yes—
167C3×C4⋊C4—44no—
168C3×C4⋊C4—44no—
169C4⋊Dic3—44no—
170C4⋊Dic3—48no—
171C4⋊Dic3—44no—
172C4×Dic3—48no—
173C4×Dic3—44no—
174C23×C4—32no—
175C23×C4—32no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4C22×D312yes—
179C23×C4C22×D311yescharacteristic
180C23×C4C22×D311yescharacteristic
181C23×C4C22×D312yes—
182C23×C4C22×D311yescharacteristic
183C2×C4⋊C4—34no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—64no—
189C2×C4⋊C4—64no—
190C2×C4⋊C4—121no—
191C2×C4⋊C4—121no—
192C2×C4⋊C4—122no—
193C2×C4⋊C4—121no—
194C2×C4⋊C4—122no—
195C2×C4⋊C4C2×Dic314yes—
196C2×C4⋊C4C2×Dic314yes—
197C2×C42—64no—
198C2×C42—62no—
199C2×C42—121no—
200C2×C42—121no—
201C2×C42—122no—
202C2×C42—122no—
203#95—64no—
204#95—64no—
205#95—64no—
206#95—64no—
207#95—121no—
208#95—122no—
209#95—122no—
210#95—122no—
211#95—122no—
212#95—122no—
213#95—122no—
214#95C2×Dic314yes—
215#95C2×Dic318yes—
216#95C2×Dic318yes—
217C23×C3C23×C411yescharacteristic
218C23×C3C22×Q811yescharacteristic
219C23×C3C2×#4111yescharacteristic
220C23×C3C2×#4112yes—
221C23×C3C2×#4112yes—
222C23×C3H(5,𝔽2)11yescharacteristic
223C23×C3H(5,𝔽2)11yescharacteristic
224C23×C3H(5,𝔽2)11yescharacteristic
225C23×C3H(5,𝔽2)12yes—
226C23×C32−1+411yescharacteristic
227C23×C32−1+412yes—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4C4×Q814yes—
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—24no—
267C2×Dic3—24no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C24C23×D311yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—34no—
313C22×C4—34no—
314C22×C4—34no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4D3×Q812yes—
341C22×C4#23912yes—
342C22×C4C22×Dic314yes—
343C22×C4C22×Dic314yes—
344C22×C4C22×Dic312yes—
345C22×C4C22×Dic312yes—
346C22×C4C22×Dic312yes—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—124no—
350C4⋊C4—128no—
351C4⋊C4—124no—
352C42—124no—
353C42—128no—
354C22×C3C2×C4×Q811yescharacteristic
355C22×C3C4×#4112yes—
356C22×C3#51811yescharacteristic
357C22×C3#52011yescharacteristic
358C22×C3#52012yes—
359C22×C3#53811yescharacteristic
360C22×C3#53812yes—
361C22×C3#53911yescharacteristic
362C22×C3#53912yes—
363C22×C3#54112yes—
364C22×C3#54212yes—
365C22×C3#54311yescharacteristic
366C22×C3#54411yescharacteristic
367C22×C3#55112yes—
368C22×C3#55212yes—
369C22×C3#55312yes—
370C22×C3#55412yes—
371C22×C3#55511yescharacteristic
372C22×C3#55511yescharacteristic
373C22×C3#55712yes—
374C22×C3#56011yescharacteristic
375C22×C3#56212yes—
376C22×C3#56311yescharacteristic
377C3×C4—24no—
378C3×C4—44no—
379C3×C4—44no—
380C3×C4—42no—
381C3×C4—42no—
382Dic3—44no—
383Dic3—44no—
384Dic3—44no—
385Dic3—82no—
386C23#100411yescharacteristic
387C23#100411yescharacteristic
388C23C2×D3×Q811yescharacteristic
389C23C2×#23911yescharacteristic
390C23D3×#4112yes—
391C23D3×#4112yes—
392C23#100911yescharacteristic
393C23#100912yes—
394C23#101011yescharacteristic
395C23#101012yes—
396C23C23×Dic311yescharacteristic
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—64no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4Q8×Dic314yes—
456C2×C3#232911yescharacteristic
457C2×C3#233412yes—
458C2×C3#237012yes—
459C2×C3#237211yescharacteristic
460C2×C3#237611yescharacteristic
461C2×C3#275012yes—
462C2×C3#275512yes—
463C2×C3#276911yescharacteristic
464C2×C3#278411yescharacteristic
465C2×C3#279511yescharacteristic
466C2×C3#280911yescharacteristic
467C22#562311yescharacteristic
468C22#563411yescharacteristic
469C22#564411yescharacteristic
470C22#564511yescharacteristic
471C22#565111yescharacteristic
472C22#566611yescharacteristic
473C22#566712yes—
474C22#567011yescharacteristic
475C22#567011yescharacteristic
476C22#567912yes—
477C22#568512yes—
478C22#569012yes—
479C22#569412yes—
480C22#569512yes—
481C22#570112yes—
482C22#570412yes—
483C22#570712yes—
484C22#571012yes—
485C22#579811yescharacteristic
486C22C2×Q8×Dic311yescharacteristic
487C22Dic3×#4112yes—
488C22#582711yescharacteristic
489C22#582712yes—
490C4—24no—
491C4—44no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—124no—
496C4—124no—
497C4—124no—
498C4—242no—
499C3#1460311yescharacteristic, complemented, Sylow
500C2#7825412yes—
501C2#7827311yescharacteristic
502C2#7829211yescharacteristic
503C2#7830311yescharacteristic
504C2#7831411yescharacteristic
505C2#7839212yes—
506C2#8274511yescharacteristic
507C2#8284011yescharacteristic
508C2#8295611yescharacteristic
509C2#8302512yes—
510C2#8307412yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#400C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#400D3
hypocenterC3#14603
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
18. C3×#400C4semidirect
92. #400Dic3semidirect
499. C3#14603semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4002
#715652
#715632
#694122
#694072
#694012
#146033