Finite Group Databasebeta

← #10723935

Subgroups — #10723935

Order 768.

1210 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2#72134C212yesmaximal
3#71564C212yesmaximal
4#71535C211yesmaximal, characteristic
5#71534C211yesmaximal, characteristic
6#69416C212yesmaximal
7#69412C212yesmaximal
8#69407C212yesmaximal
9#69401C212yesmaximal
10#14469—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2212yes—
15C22×C4×Dic3C2212yes—
16C3×#391C418yescomplemented
17C2×C3×#95C2212yes—
18C2×C3×#95C2212yes—
19C2×C3⋊#95C2212yes—
20C2×C3⋊#95C2211yescharacteristic
21C2×C3⋊#95C2212yes—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2211yescharacteristic
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×#391D311yescharacteristic
34#1446—32no—
35#1443—32no—
36#1442—32no—
37#1436—32no—
38#1436—32no—
39C4⋊#95—31no—
40#1434—31no—
41#1434—32no—
42C22×C3×Q8—22no—
43C22×C3×Q8—22no—
44C22×C3×Q8—42no—
45C22×C3×Q8—42no—
46C22×C3×Q8C2×C414yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C2×C3×C4⋊C4—42no—
57C2×C3×C4⋊C4C2×C414yes—
58C2×C3×C42—42no—
59C2×C3×C42C2×C414yes—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3—24no—
62C2×C4⋊Dic3—42no—
63C2×C4×Dic3—24no—
64C2×C4×Dic3—42no—
65C2×C4×Dic3—42no—
66C3×#95C2×C418yes—
67C3×#95C2×C418yes—
68C3⋊#95—24no—
69C3⋊#95—24no—
70C3⋊#95—22no—
71C3⋊#95—24no—
72C3⋊#95—42no—
73C3⋊#95—42no—
74C3⋊#95—41no—
75C3⋊#95—42no—
76C3⋊#95—42no—
77C3⋊#95—42no—
78C3⋊#95—41no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95Q814yescomplemented
82C23×Q8C2×D311yescharacteristic
83C22×C4⋊C4—32no—
84C22×C4⋊C4C2×D311yescharacteristic
85C22×C42—32no—
86C22×C42C2×D311yescharacteristic
87#391Dic318yescomplemented
88C2×#95—31no—
89C2×#95—32no—
90C2×#95—32no—
91C2×#95—31no—
92C2×#95—32no—
93C2×#95—31no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—32no—
97C2×#95—31no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95C2×D312yes—
103C2×#95C2×D312yes—
104C24×C3C2411yescharacteristic
105C2×C3×Q8—28no—
106C2×C3×Q8—44no—
107C2×C3×Q8—44no—
108C2×C3×Q8—48no—
109C2×C3×Q8—48no—
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—22no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
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—21no—
130C22×C3×C4C22×C412yes—
131C22×C3×C4C22×C414yes—
132C22×C3×C4C22×C414yes—
133C22×C3×C4C22×C412yes—
134C22×C3×C4C22×C412yes—
135C22×C3×C4C2×Q812yes—
136C22×C3×C4#4112yes—
137C22×C3×C4#4112yes—
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×Dic3—22no—
165C22×Dic3C2×Q814yes—
166C22×Dic3#4114yes—
167C3×C4⋊C4—44no—
168C3×C42—44no—
169C4⋊Dic3—48no—
170C4×Dic3—48no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8C2×Dic314yes—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4C22×D312yes—
181C23×C4C22×D312yes—
182C23×C4C22×D311yescharacteristic
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D311yescharacteristic
185C2×C4⋊C4—42no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4—64no—
188C2×C4⋊C4—122no—
189C2×C4⋊C4C2×Dic314yes—
190C2×C42—42no—
191C2×C42—64no—
192C2×C42—122no—
193C2×C42—122no—
194C2×C42C2×Dic314yes—
195#95—34no—
196#95—64no—
197#95—64no—
198#95—64no—
199#95—62no—
200#95—122no—
201#95—122no—
202#95—122no—
203#95—122no—
204#95—122no—
205#95—122no—
206#95—121no—
207#95—121no—
208#95—122no—
209#95C2×Dic318yes—
210#95C2×Dic318yes—
211C23×C3C23×C411yescharacteristic
212C23×C3C22×Q811yescharacteristic
213C23×C3C2×#4111yescharacteristic
214C23×C3C2×#4111yescharacteristic
215C23×C3C2×#4111yescharacteristic
216C23×C3C2×#4112yes—
217C23×C3H(5,𝔽2)12yes—
218C23×C3H(5,𝔽2)12yes—
219C23×C3H(5,𝔽2)11yescharacteristic
220C23×C32−1+412yes—
221C23×C32−1+411yescharacteristic
222C3×Q8—416no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
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—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
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#11714yes—
251C2×C3×C4C4×Q814yes—
252C2×Dic3—24no—
253C2×Dic3—24no—
254C2×Dic3—24no—
255C2×Dic3—24no—
256C2×Dic3—24no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
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—
280C24C23×D311yescharacteristic
281C2×Q8—28no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—48no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
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—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—34no—
307C22×C4—34no—
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×C4—62no—
333C22×C4—62no—
334C22×C4#23512yes—
335C22×C4#23512yes—
336C22×C4D3×Q812yes—
337C22×C4#23912yes—
338C22×C4C22×Dic312yes—
339C22×C4C22×Dic314yes—
340C22×C4C22×Dic314yes—
341C22×C4C22×Dic312yes—
342C22×C4C22×Dic312yes—
343C4⋊C4—44no—
344C4⋊C4—128no—
345C42—44no—
346C42—128no—
347C22×C3C2×#11711yescharacteristic
348C22×C3C2×C4×Q811yescharacteristic
349C22×C3C4×#4111yescharacteristic
350C22×C3#51811yescharacteristic
351C22×C3#51811yescharacteristic
352C22×C3#51911yescharacteristic
353C22×C3#52011yescharacteristic
354C22×C3#53812yes—
355C22×C3#53812yes—
356C22×C3#53812yes—
357C22×C3#54212yes—
358C22×C3#54212yes—
359C22×C3#54412yes—
360C22×C3#55112yes—
361C22×C3#55212yes—
362C22×C3#55412yes—
363C22×C3#55411yescharacteristic
364C22×C3#55612yes—
365C22×C3#55611yescharacteristic
366C22×C3#55711yescharacteristic
367C22×C3#55911yescharacteristic
368C22×C3#56212yes—
369C22×C3#56312yes—
370C3×C4—24no—
371C3×C4—24no—
372C3×C4—44no—
373C3×C4—44no—
374C3×C4—42no—
375Dic3—44no—
376Dic3—44no—
377Dic3—82no—
378Dic3—82no—
379C23C2×#23511yescharacteristic
380C23C2×#23511yescharacteristic
381C23#100412yes—
382C23#100412yes—
383C23C2×D3×Q811yescharacteristic
384C23C2×#23911yescharacteristic
385C23#100712yes—
386C23D3×#4112yes—
387C23#100911yescharacteristic
388C23#101011yescharacteristic
389C23C23×Dic311yescharacteristic
390Q8—416no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—64no—
419C2×C4—64no—
420C2×C4—64no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—64no—
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×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—122no—
446C2×C4#92614yes—
447C2×C4Q8×Dic314yes—
448C2×C3#231211yescharacteristic
449C2×C3#233111yescharacteristic
450C2×C3#233711yescharacteristic
451C2×C3#235511yescharacteristic
452C2×C3#235611yescharacteristic
453C2×C3#236811yescharacteristic
454C2×C3#237811yescharacteristic
455C2×C3#275112yes—
456C2×C3#275912yes—
457C2×C3#277412yes—
458C2×C3#278812yes—
459C22#549412yes—
460C22#549412yes—
461C22#556711yescharacteristic
462C22#556812yes—
463C22#557911yescharacteristic
464C22#558112yes—
465C22#558411yescharacteristic
466C22#558511yescharacteristic
467C22#563612yes—
468C22#563712yes—
469C22#564612yes—
470C22#564712yes—
471C22#566912yes—
472C22#567812yes—
473C22#572912yes—
474C22#573312yes—
475C22C2×#92611yescharacteristic
476C22#579811yescharacteristic
477C22#579811yescharacteristic
478C22C2×Q8×Dic311yescharacteristic
479C22#581211yescharacteristic
480C22Dic3×#4111yescharacteristic
481C22#582711yescharacteristic
482C4—24no—
483C4—24no—
484C4—44no—
485C4—42no—
486C4—44no—
487C4—124no—
488C4—124no—
489C4—242no—
490C4—242no—
491C3#1446911yescharacteristic, complemented, Sylow
492C2#7658312yes—
493C2#7658812yes—
494C2#7659012yes—
495C2#7665812yes—
496C2#8209311yescharacteristic
497C2#8246211yescharacteristic
498C2#8246311yescharacteristic
499C2#8274911yescharacteristic
500C2#8275011yescharacteristic
501C2#8295411yescharacteristic
502C2#8322811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#391C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#391D3
hypocenterC3#14469
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
16. C3×#391C4semidirect
81. C3⋊#95Q8semidirect
87. #391Dic3semidirect
491. C3#14469semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
#721342
#715642
#715352
#715342
#694162
#694122
#694072
#694012
#144693