Finite Group Databasebeta

← #10731950

Subgroups — #10731950

Order 768.

1410 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4984C212yesmaximal
2C2×#4928C211yesmaximal, characteristic
3C2×C4×C4⋊Dic3C211yesmaximal, characteristic
4C2×#4661C212yesmaximal
5C3×C4⋊#95C211yesmaximal, characteristic
6#71563C212yesmaximal
7#69415C214yesmaximal
8#69410C212yesmaximal
9#15149—31nomaximal, Sylow
10C22×C3×C4⋊C4C2212yes—
11C22×C3×C42C2211yescharacteristic
12C22×C4⋊Dic3C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2214yes—
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×Dic3⋊C4C2212yes—
18C22×C4×Dic3C2212yes—
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3⋊#95C2214yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2212yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2214yes—
26#4984—28no—
27C4×C4⋊Dic3—24no—
28#4928—24no—
29#4661—28no—
30C2×#398—32no—
31C2×#389—31no—
32C2×#389—32no—
33C2×C4×C4⋊C4—31no—
34#1442—34no—
35#1441—32no—
36C4⋊#95—32no—
37C4⋊#95D311yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C23×C3×C4C2314yes—
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3C2312yes—
43C23×Dic3C2312yes—
44C23×Dic3C2312yes—
45C23×Dic3C2312yes—
46C2×C3×C4⋊C4—24no—
47C2×C3×C4⋊C4—24no—
48C2×C3×C4⋊C4Q818yes—
49C2×C3×C42—24no—
50C2×C3×C42D414yes—
51C2×C4⋊Dic3—22no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3—22no—
54C2×C4⋊Dic3—22no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—22no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—28no—
60C2×C4⋊Dic3—28no—
61C2×C4⋊Dic3—44no—
62C2×C4⋊Dic3D414yes—
63C2×C4⋊Dic3D414yes—
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4—24no—
66C2×Dic3⋊C4—24no—
67C2×C4×Dic3—24no—
68C2×C4×Dic3—24no—
69C2×C4×Dic3—24no—
70C3×#95—24no—
71C3×#95—42no—
72C3⋊#95—24no—
73C3⋊#95—24no—
74C3⋊#95—28no—
75C3⋊#95—28no—
76C3⋊#95—28no—
77C22×C4⋊C4—31no—
78C22×C4⋊C4—31no—
79C22×C4⋊C4—32no—
80C22×C4⋊C4—31no—
81C22×C4⋊C4—31no—
82C22×C4⋊C4—34no—
83C22×C4⋊C4C2×D312yes—
84C22×C42—32no—
85C22×C42C2×D311yescharacteristic
86#398—68no—
87#389—68no—
88#389—64no—
89C4×C4⋊C4—64no—
90C2×#95—34no—
91C2×#95—34no—
92C2×#95—34no—
93C2×#95—32no—
94C2×#95—32no—
95C2×#95C2×D312yes—
96C2×#95C2×D312yes—
97C24×C3C2411yescharacteristic
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—22no—
104C22×C3×C4—24no—
105C22×C3×C4—21no—
106C22×C3×C4—24no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—24no—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×Q818yes—
116C22×C3×C4C2×Q814yes—
117C22×C3×C4#4114yes—
118C22×C3×C4#4112yes—
119C22×C3×C4#4112yes—
120C22×Dic3—22no—
121C22×Dic3—22no—
122C22×Dic3—22no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—22no—
127C22×Dic3—22no—
128C22×Dic3—22no—
129C22×Dic3—22no—
130C22×Dic3—22no—
131C22×Dic3—22no—
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—24no—
141C22×Dic3—22no—
142C22×Dic3—24no—
143C22×Dic3C2×D414yes—
144C22×Dic3C2×D414yes—
145C22×Dic3#4114yes—
146C3×C4⋊C4—216no—
147C3×C42—28no—
148C4⋊Dic3—28no—
149C4⋊Dic3—28no—
150C4⋊Dic3—28no—
151C4⋊Dic3—416no—
152C4⋊Dic3—44no—
153Dic3⋊C4—48no—
154C4×Dic3—48no—
155C23×C4—32no—
156C23×C4—32no—
157C23×C4—32no—
158C23×C4—32no—
159C23×C4C22×D314yes—
160C23×C4C22×D311yescharacteristic
161C23×C4C22×D311yescharacteristic
162C23×C4C22×D311yescharacteristic
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—34no—
166C2×C4⋊C4—34no—
167C2×C4⋊C4—64no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—62no—
172C2×C4⋊C4—62no—
173C2×C4⋊C4—68no—
174C2×C4⋊C4—68no—
175C2×C4⋊C4—64no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4—64no—
178C2×C4⋊C4—64no—
179C2×C4⋊C4—64no—
180C2×C4⋊C4—124no—
181C2×C4⋊C4Dic618yes—
182C2×C42—24no—
183C2×C42—64no—
184C2×C42—64no—
185C2×C42—64no—
186C2×C42D1214yes—
187#95—24no—
188#95—42no—
189#95—68no—
190#95—68no—
191#95—68no—
192#95—64no—
193#95—64no—
194C23×C3C22×D411yescharacteristic
195C23×C3C22×D411yescharacteristic
196C23×C3C22×D411yescharacteristic
197C23×C3C22×Q812yes—
198C23×C3C2×#4112yes—
199C23×C3C2×#4111yescharacteristic
200C23×C3C2×#4111yescharacteristic
201C23×C3C2×#4112yes—
202C23×C3H(5,𝔽2)11yescharacteristic
203C23×C32−1+411yescharacteristic
204C23×C32−1+412yes—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—28no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—28no—
214C2×C3×C4—28no—
215C2×C3×C4—44no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—44no—
219C2×C3×C4—44no—
220C2×C3×C4—44no—
221C2×C3×C4#12114yes—
222C2×C3×C4#12114yes—
223C2×C3×C4C22⋊Q818yes—
224C2×C3×C4#12314yes—
225C2×C3×C4#12518yes—
226C2×C3×C4C4⋊Q814yes—
227C2×Dic3—24no—
228C2×Dic3—24no—
229C2×Dic3—24no—
230C2×Dic3—24no—
231C2×Dic3—24no—
232C2×Dic3—24no—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—44no—
252C2×Dic3—44no—
253C24C23×D311yescharacteristic
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—34no—
267C22×C4—34no—
268C22×C4—34no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—64no—
290C22×C4—62no—
291C22×C4—64no—
292C22×C4C2×Dic618yes—
293C22×C4C2×Dic614yes—
294C22×C4C2×D1212yes—
295C22×C4C2×D1212yes—
296C22×C4C2×D1212yes—
297C22×C4D3×D412yes—
298C22×C4D3×D412yes—
299C22×C4#23714yes—
300C22×C4#23912yes—
301C22×C4#23912yes—
302C4⋊C4—216no—
303C4⋊C4—68no—
304C4⋊C4—68no—
305C4⋊C4—68no—
306C4⋊C4—1216no—
307C4⋊C4—124no—
308C4⋊C4—128no—
309C42—28no—
310C42—128no—
311C22×C3C2×#12111yescharacteristic
312C22×C3C2×#12111yescharacteristic
313C22×C3C2×C22⋊Q812yes—
314C22×C3C2×#12311yescharacteristic
315C22×C3#52511yescharacteristic
316C22×C3C2×#12512yes—
317C22×C3#52912yes—
318C22×C3C2×C4⋊Q811yescharacteristic
319C22×C3#53212yes—
320C22×C3#53611yescharacteristic
321C22×C3#53711yescharacteristic
322C22×C3#54112yes—
323C22×C3#54212yes—
324C22×C3D4211yescharacteristic
325C22×C3#54711yescharacteristic
326C22×C3#54711yescharacteristic
327C22×C3#54812yes—
328C22×C3D4×Q812yes—
329C22×C3#55212yes—
330C22×C3#55412yes—
331C22×C3#55511yescharacteristic
332C22×C3#55712yes—
333C22×C3#55712yes—
334C3×C4—24no—
335C3×C4—42no—
336C3×C4—48no—
337C3×C4#39818yescomplemented
338Dic3—44no—
339Dic3—44no—
340Dic3—44no—
341Dic3—82no—
342C23C22×Dic612yes—
343C23C22×D1211yescharacteristic
344C23C2×D3×D411yescharacteristic
345C23C2×D3×D411yescharacteristic
346C23C2×#23712yes—
347C23#100411yescharacteristic
348C23C2×#23911yescharacteristic
349C23C2×#23911yescharacteristic
350C23#100711yescharacteristic
351C23D3×#4112yes—
352C23#101012yes—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—64no—
372C2×C4—64no—
373C2×C4—64no—
374C2×C4—64no—
375C2×C4—64no—
376C2×C4—64no—
377C2×C4—64no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4#86914yes—
396C2×C4#87818yes—
397C2×C4#88614yes—
398C2×C4#89018yes—
399C2×C4#89514yes—
400C2×C4#89514yes—
401C2×C3C2×#39811yescharacteristic
402C2×C3#245212yes—
403C2×C3#245312yes—
404C2×C3#250612yes—
405C2×C3#250812yes—
406C2×C3#253611yescharacteristic
407C2×C3#253711yescharacteristic
408C2×C3#254611yescharacteristic
409C2×C3#258411yescharacteristic
410C2×C3#262511yescharacteristic
411C2×C3#285411yescharacteristic
412C22C2×#86911yescharacteristic
413C22C2×#87812yes—
414C22C2×#88611yescharacteristic
415C22C2×#89012yes—
416C22C2×#89511yescharacteristic
417C22C2×#89511yescharacteristic
418C22#551011yescharacteristic
419C22#552411yescharacteristic
420C22#552611yescharacteristic
421C22D4×Dic612yes—
422C22#553912yes—
423C22#556912yes—
424C22#557012yes—
425C22#559612yes—
426C22#560312yes—
427C22#562412yes—
428C22#562512yes—
429C22#567712yes—
430C22#569412yes—
431C22#571711yescharacteristic
432C22#572611yescharacteristic
433C22#572611yescharacteristic
434C22#573411yescharacteristic
435C4—24no—
436C4—48no—
437C4—42no—
438C4—124no—
439C4—124no—
440C4—124no—
441C4—242no—
442C4#466118yescomplemented
443C3#1514911yescharacteristic, complemented, Sylow
444C2C2×#466111yescharacteristic
445C2#7371111yescharacteristic
446C2#7372711yescharacteristic
447C2#7395111yescharacteristic
448C2#7399911yescharacteristic
449C2#7424011yescharacteristic
450C2#7436811yescharacteristic
451C2#7603112yes—
452C2#7604012yes—
453C2#7632312yes—
454C2#7651212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC3×C4⋊#95C2
hypercenterC4⋊#95D3
hypocenterC3#15149
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
337. C3×C4#398semidirect
442. C4#4661semidirect
443. C3#15149semidirect

Maximal subgroups

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

subgroupindex
C2×#49842
C2×#49282
C2×C4×C4⋊Dic32
C2×#46612
C3×C4⋊#952
#715632
#694152
#694102
#151493