Finite Group Databasebeta

← #10725604

Subgroups — #10725604

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#398C211yesmaximal, characteristic
2#71569C211yesmaximal, characteristic
3#71565C212yesmaximal
4#71565C212yesmaximal
5#71564C211yesmaximal, characteristic
6#71563C212yesmaximal
7#69407C212yesmaximal
8#69407C212yesmaximal
9#69407C212yesmaximal
10#14600—31nomaximal, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C4⋊Dic3C2212yes—
15C22×C4⋊Dic3C2212yes—
16C22×C4×Dic3C2212yes—
17C22×C4×Dic3C2211yescharacteristic
18C3×#398C418yescomplemented
19C2×C3×#95C2211yescharacteristic
20C2×C3×#95C2212yes—
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2211yescharacteristic
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×#398D311yescharacteristic
34#1445—31no—
35#1441—32no—
36#1441—32no—
37#1436—32no—
38#1436—32no—
39#1436—32no—
40#1436—31no—
41C4⋊#95—32no—
42C23×C3×C4C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C2×C3×C4⋊C4—42no—
52C2×C3×C4⋊C4—42no—
53C2×C3×C4⋊C4—41no—
54C2×C3×C4⋊C4—41no—
55C2×C3×C4⋊C4—41no—
56C2×C3×C4⋊C4—41no—
57C2×C3×C4⋊C4C2×C414yes—
58C2×C3×C4⋊C4C2×C414yes—
59C2×C3×C4⋊C4C2×C418yes—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3—42no—
62C2×C4⋊Dic3—42no—
63C2×C4⋊Dic3—42no—
64C2×C4⋊Dic3—42no—
65C2×C4⋊Dic3Q818yes—
66C2×C4×Dic3—24no—
67C2×C4×Dic3—42no—
68C2×C4×Dic3—42no—
69C2×C4×Dic3—41no—
70C2×C4×Dic3—42no—
71C3×#95C2×C418yes—
72C3×#95C2×C414yes—
73C3⋊#95—24no—
74C3⋊#95—24no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—41no—
78C3⋊#95—42no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95—41no—
82C3⋊#95—42no—
83C3⋊#95—42no—
84C3⋊#95—41no—
85C22×C4⋊C4—32no—
86C22×C4⋊C4—32no—
87C22×C4⋊C4C2×D312yes—
88C22×C4⋊C4C2×D311yescharacteristic
89C22×C4⋊C4C2×D311yescharacteristic
90C22×C42—32no—
91C22×C42—31no—
92#398Dic318yescomplemented
93C2×#95—32no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—32no—
97C2×#95—31no—
98C2×#95—32no—
99C2×#95—31no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—31no—
105C2×#95C2×D312yes—
106C2×#95C2×D311yescharacteristic
107C24×C3C2411yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4C22×C412yes—
131C22×C3×C4C22×C414yes—
132C22×C3×C4C22×C412yes—
133C22×C3×C4C22×C412yes—
134C22×C3×C4C22×C414yes—
135C22×C3×C4C2×Q814yes—
136C22×C3×C4#4114yes—
137C22×Dic3—22no—
138C22×Dic3—22no—
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×Dic3C2×Q814yes—
163C22×Dic3C2×Q814yes—
164C22×Dic3#4114yes—
165C3×C4⋊C4—48no—
166C3×C4⋊C4—44no—
167C3×C4⋊C4—44no—
168C4⋊Dic3—48no—
169C4⋊Dic3—48no—
170C4×Dic3—48no—
171C4×Dic3—44no—
172C23×C4—32no—
173C23×C4—32no—
174C23×C4—32no—
175C23×C4—32no—
176C23×C4C22×D312yes—
177C23×C4C22×D311yescharacteristic
178C23×C4C22×D311yescharacteristic
179C23×C4C22×D311yescharacteristic
180C23×C4C22×D312yes—
181C2×C4⋊C4—38no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—64no—
189C2×C4⋊C4—122no—
190C2×C4⋊C4—122no—
191C2×C4⋊C4—122no—
192C2×C4⋊C4—122no—
193C2×C4⋊C4C2×Dic314yes—
194C2×C4⋊C4C2×Dic314yes—
195C2×C4⋊C4C2×Dic318yes—
196C2×C42—64no—
197C2×C42—122no—
198C2×C42—122no—
199C2×C42—121no—
200C2×C42—122no—
201#95—64no—
202#95—64no—
203#95—64no—
204#95—64no—
205#95—121no—
206#95—122no—
207#95—121no—
208#95—122no—
209#95—121no—
210#95—122no—
211#95—122no—
212#95—122no—
213#95C2×Dic318yes—
214#95C2×Dic314yes—
215C23×C3C23×C411yescharacteristic
216C23×C3C22×Q812yes—
217C23×C3C2×#4112yes—
218C23×C3C2×#4112yes—
219C23×C3H(5,𝔽2)12yes—
220C23×C3H(5,𝔽2)11yescharacteristic
221C23×C3H(5,𝔽2)11yescharacteristic
222C23×C3H(5,𝔽2)12yes—
223C23×C32−1+412yes—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4C4×Q818yes—
253C2×Dic3—24no—
254C2×Dic3—24no—
255C2×Dic3—24no—
256C2×Dic3—24no—
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
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—
281C24C23×D311yescharacteristic
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—34no—
305C22×C4—34no—
306C22×C4—34no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—62no—
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×C4D3×Q814yes—
333C22×C4#23914yes—
334C22×C4C22×Dic312yes—
335C22×C4C22×Dic312yes—
336C22×C4C22×Dic314yes—
337C22×C4C22×Dic312yes—
338C22×C4C22×Dic314yes—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—128no—
343C4⋊C4—128no—
344C42—128no—
345C42—124no—
346C22×C3C2×C4×Q812yes—
347C22×C3C4×#4111yescharacteristic
348C22×C3#51811yescharacteristic
349C22×C3#51811yescharacteristic
350C22×C3#52012yes—
351C22×C3#53812yes—
352C22×C3#53812yes—
353C22×C3#53912yes—
354C22×C3#54012yes—
355C22×C3#54312yes—
356C22×C3#54412yes—
357C22×C3#55212yes—
358C22×C3#55312yes—
359C22×C3#55412yes—
360C22×C3#55412yes—
361C22×C3#55511yescharacteristic
362C22×C3#55712yes—
363C22×C3Q8211yescharacteristic
364C22×C3#56011yescharacteristic
365C22×C3#56212yes—
366C22×C3#56311yescharacteristic
367C3×C4—28no—
368C3×C4—42no—
369C3×C4—44no—
370C3×C4—42no—
371C3×C4—42no—
372Dic3—44no—
373Dic3—44no—
374Dic3—44no—
375Dic3—82no—
376C23#100412yes—
377C23#100411yescharacteristic
378C23#100411yescharacteristic
379C23C2×D3×Q812yes—
380C23C2×#23912yes—
381C23D3×#4112yes—
382C23#100912yes—
383C23#101012yes—
384C23C23×Dic311yescharacteristic
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—64no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—64no—
417C2×C4—64no—
418C2×C4—64no—
419C2×C4—64no—
420C2×C4—64no—
421C2×C4—64no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4Q8×Dic318yes—
442C2×C3#232912yes—
443C2×C3#233311yescharacteristic
444C2×C3#236911yescharacteristic
445C2×C3#237012yes—
446C2×C3#237611yescharacteristic
447C2×C3#278012yes—
448C2×C3#278112yes—
449C2×C3#279112yes—
450C2×C3#280612yes—
451C22#561312yes—
452C22#561512yes—
453C22#562312yes—
454C22#563412yes—
455C22#564412yes—
456C22#564512yes—
457C22#565111yescharacteristic
458C22#566611yescharacteristic
459C22#566712yes—
460C22#568512yes—
461C22#569012yes—
462C22#569412yes—
463C22#569512yes—
464C22#572211yescharacteristic
465C22#573012yes—
466C22#573411yescharacteristic
467C22#579811yescharacteristic
468C22#579811yescharacteristic
469C22C2×Q8×Dic312yes—
470C22Dic3×#4111yescharacteristic
471C22#582712yes—
472C4—28no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—44no—
477C4—124no—
478C4—124no—
479C4—124no—
480C4—242no—
481C3#1460011yescharacteristic, complemented, Sylow
482C2#7790112yes—
483C2#7790812yes—
484C2#7792212yes—
485C2#7800012yes—
486C2#8263511yescharacteristic
487C2#8274512yes—
488C2#8284011yescharacteristic
489C2#8302512yes—
490C2#8323011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#398C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#398D3
hypocenterC3#14600
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
18. C3×#398C4semidirect
92. #398Dic3semidirect
481. C3#14600semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3982
#715692
#715652
#715642
#715632
#694072
#146003