Finite Group Databasebeta

← #10723944

Subgroups — #10723944

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2#71569C212yesmaximal
3#71569C212yesmaximal
4#71564C212yesmaximal
5#71564C212yesmaximal
6#71535C211yesmaximal, characteristic
7#71534C211yesmaximal, characteristic
8#69412C212yesmaximal
9#69407C212yesmaximal
10#14470—31nomaximal, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3C2212yes—
16C22×C4×Dic3C2212yes—
17C3×#389C418yescomplemented
18C2×C3×#95C2212yes—
19C2×C3⋊#95C2212yes—
20C2×C3⋊#95C2211yescharacteristic
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⋊#95C2211yescharacteristic
32C2×C3⋊#95C2211yescharacteristic
33C2×#389D311yescharacteristic
34#1445—32no—
35#1445—32no—
36#1442—32no—
37#1436—32no—
38#1436—32no—
39#1436—32no—
40C4⋊#95—31no—
41#1434—31no—
42C23×C3×C4C2312yes—
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3C2312yes—
48C23×Dic3C2312yes—
49C23×Dic3C2312yes—
50C23×Dic3C2312yes—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—42no—
54C2×C3×C4⋊C4—42no—
55C2×C3×C4⋊C4—42no—
56C2×C3×C4⋊C4C2×C414yes—
57C2×C3×C4⋊C4C2×C418yes—
58C2×C3×C4⋊C4C2×C414yes—
59C2×C3×C42—42no—
60C2×C3×C42C2×C414yes—
61C2×C4⋊Dic3—24no—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3—42no—
64C2×C4×Dic3—42no—
65C2×C4×Dic3—42no—
66C2×C4×Dic3—42no—
67C3×#95C2×C418yes—
68C3⋊#95—24no—
69C3⋊#95—24no—
70C3⋊#95—22no—
71C3⋊#95—24no—
72C3⋊#95—24no—
73C3⋊#95—41no—
74C3⋊#95—42no—
75C3⋊#95—42no—
76C3⋊#95—42no—
77C3⋊#95—42no—
78C3⋊#95—41no—
79C3⋊#95—42no—
80C3⋊#95—42no—
81C3⋊#95Q814yes—
82C22×C4⋊C4—32no—
83C22×C4⋊C4C2×D311yescharacteristic
84C22×C4⋊C4C2×D311yescharacteristic
85C22×C4⋊C4C2×D312yes—
86C22×C42—32no—
87C22×C42C2×D311yescharacteristic
88#389Dic318yescomplemented
89C2×#95—31no—
90C2×#95—32no—
91C2×#95—31no—
92C2×#95—32no—
93C2×#95—32no—
94C2×#95—32no—
95C2×#95—32no—
96C2×#95—31no—
97C2×#95—31no—
98C2×#95—32no—
99C2×#95—32no—
100C2×#95—32no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95C2×D312yes—
104C24×C3C2411yescharacteristic
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
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×C4C22×C414yes—
126C22×C3×C4C22×C414yes—
127C22×C3×C4C22×C412yes—
128C22×C3×C4C22×C412yes—
129C22×C3×C4C22×C412yes—
130C22×C3×C4C2×Q812yes—
131C22×C3×C4#4112yes—
132C22×C3×C4#4112yes—
133C22×C3×C4#4112yes—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
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×Dic3C2×Q814yes—
161C22×Dic3#4114yes—
162C3×C4⋊C4—28no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4—48no—
165C3×C42—44no—
166C4⋊Dic3—48no—
167C4×Dic3—48no—
168C23×C4—32no—
169C23×C4—32no—
170C23×C4—32no—
171C23×C4—32no—
172C23×C4C22×D312yes—
173C23×C4C22×D311yescharacteristic
174C23×C4C22×D312yes—
175C23×C4C22×D311yescharacteristic
176C23×C4C22×D311yescharacteristic
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4—64no—
184C2×C4⋊C4—122no—
185C2×C4⋊C4C2×Dic314yes—
186C2×C4⋊C4C2×Dic318yes—
187C2×C4⋊C4C2×Dic314yes—
188C2×C42—42no—
189C2×C42—122no—
190C2×C42—122no—
191C2×C42—122no—
192C2×C42C2×Dic314yes—
193#95—34no—
194#95—64no—
195#95—64no—
196#95—64no—
197#95—64no—
198#95—62no—
199#95—122no—
200#95—122no—
201#95—122no—
202#95—122no—
203#95—122no—
204#95—121no—
205#95—121no—
206#95—122no—
207#95C2×Dic318yes—
208C23×C3C23×C411yescharacteristic
209C23×C3C22×Q811yescharacteristic
210C23×C3C2×#4111yescharacteristic
211C23×C3C2×#4111yescharacteristic
212C23×C3C2×#4111yescharacteristic
213C23×C3C2×#4112yes—
214C23×C3H(5,𝔽2)12yes—
215C23×C3H(5,𝔽2)12yes—
216C23×C3H(5,𝔽2)11yescharacteristic
217C23×C32−1+412yes—
218C23×C32−1+411yescharacteristic
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
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#11714yes—
247C2×C3×C4C4×Q814yes—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—24no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
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—
276C24C23×D311yescharacteristic
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—34no—
298C22×C4—34no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
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#23512yes—
326C22×C4#23512yes—
327C22×C4D3×Q812yes—
328C22×C4#23912yes—
329C22×C4C22×Dic314yes—
330C22×C4C22×Dic314yes—
331C22×C4C22×Dic312yes—
332C22×C4C22×Dic312yes—
333C22×C4C22×Dic312yes—
334C4⋊C4—28no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—128no—
338C42—44no—
339C42—128no—
340C22×C3C2×#11711yescharacteristic
341C22×C3C2×C4×Q811yescharacteristic
342C22×C3C4×#4111yescharacteristic
343C22×C3#51811yescharacteristic
344C22×C3#51811yescharacteristic
345C22×C3#51911yescharacteristic
346C22×C3#52011yescharacteristic
347C22×C3#53912yes—
348C22×C3#53912yes—
349C22×C3#53912yes—
350C22×C3#54112yes—
351C22×C3#54212yes—
352C22×C3#54412yes—
353C22×C3#55112yes—
354C22×C3#55312yes—
355C22×C3#55412yes—
356C22×C3#55411yescharacteristic
357C22×C3#55611yescharacteristic
358C22×C3#55711yescharacteristic
359C22×C3#55912yes—
360C22×C3#55911yescharacteristic
361C22×C3#56012yes—
362C22×C3#56212yes—
363C3×C4—24no—
364C3×C4—24no—
365C3×C4—44no—
366C3×C4—42no—
367C3×C4—44no—
368Dic3—44no—
369Dic3—44no—
370Dic3—82no—
371Dic3—82no—
372C23C2×#23511yescharacteristic
373C23C2×#23511yescharacteristic
374C23#100412yes—
375C23#100412yes—
376C23C2×D3×Q811yescharacteristic
377C23C2×#23911yescharacteristic
378C23#100712yes—
379C23D3×#4112yes—
380C23#100911yescharacteristic
381C23#101011yescharacteristic
382C23C23×Dic311yescharacteristic
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—64no—
411C2×C4—64no—
412C2×C4—64no—
413C2×C4—64no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
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#92614yes—
439C2×C4Q8×Dic314yes—
440C2×C3#231311yescharacteristic
441C2×C3#233111yescharacteristic
442C2×C3#233611yescharacteristic
443C2×C3#235511yescharacteristic
444C2×C3#235711yescharacteristic
445C2×C3#236811yescharacteristic
446C2×C3#237711yescharacteristic
447C2×C3#279612yes—
448C2×C3#279812yes—
449C2×C3#280212yes—
450C2×C3#280712yes—
451C22#551512yes—
452C22#551512yes—
453C22#555812yes—
454C22#555912yes—
455C22#556711yescharacteristic
456C22#557911yescharacteristic
457C22#558411yescharacteristic
458C22#558511yescharacteristic
459C22#562012yes—
460C22#562112yes—
461C22#563612yes—
462C22#563712yes—
463C22#569212yes—
464C22#569612yes—
465C22#572912yes—
466C22#573312yes—
467C22C2×#92611yescharacteristic
468C22#579811yescharacteristic
469C22#579811yescharacteristic
470C22C2×Q8×Dic311yescharacteristic
471C22#581211yescharacteristic
472C22Dic3×#4111yescharacteristic
473C22#582711yescharacteristic
474C4—24no—
475C4—24no—
476C4—44no—
477C4—42no—
478C4—44no—
479C4—124no—
480C4—124no—
481C4—242no—
482C4—242no—
483C3#1447011yescharacteristic, complemented, Sylow
484C2#7617012yes—
485C2#7617512yes—
486C2#7628012yes—
487C2#7628112yes—
488C2#8215511yescharacteristic
489C2#8233811yescharacteristic
490C2#8246211yescharacteristic
491C2#8263911yescharacteristic
492C2#8274911yescharacteristic
493C2#8302411yescharacteristic
494C2#8322811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#389C2
socle, prime radical, cosocleC24×C3C24
hypercenterC2×#389D3
hypocenterC3#14470
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
17. C3×#389C4semidirect
88. #389Dic3semidirect
483. C3#14470semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
#715692
#715642
#715352
#715342
#694122
#694072
#144703