Finite Group Databasebeta

← #10701413

Subgroups — #10701413

Order 768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2273C211yesmaximal, characteristic
2C2×#4946C211yesmaximal, characteristic
3#71544C212yesmaximal
4C2×#4526C211yesmaximal, characteristic
5C2×#4526C212yesmaximal
6#10918—31nomaximal, Sylow
7C24×Dic3C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×C3×C22⋊C4C2212yes—
10C22×#217C2212yes—
11C22×#217C2211yescharacteristic
12C22×Dic3⋊C4—21no—
13C3×C4×C22⋊C4—21no—
14C3×C4×C22⋊C4—21no—
15C3×C4×C22⋊C4—24no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95—24no—
18#4946—22no—
19#4946C414yescomplemented
20#4526—22no—
21#4526C418yescomplemented
22C2×#385—31no—
23#2273D311yescharacteristic
24C2×#342—32no—
25C2×#342—31no—
26C22⋊#95—32no—
27C25×C3C2311yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D414yes—
31C23×Dic3—22no—
32C23×Dic3—21no—
33C23×Dic3—21no—
34C23×Dic3—21no—
35C23×Dic3—22no—
36C23×Dic3—22no—
37C23×Dic3—44no—
38C23×Dic3C2×C412yes—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4C2×C412yes—
46C2×C3×C22⋊C4C2×C414yes—
47C2×C3×C22⋊C4D412yes—
48C2×C3×C22⋊C4Q814yes—
49C2×C3×C42—22no—
50C2×C3×C42—22no—
51C2×#217—21no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—22no—
58C2×#217C2×C414yes—
59C2×#217C2×C412yes—
60C2×#217C2×C414yes—
61C2×Dic3⋊C4—24no—
62C2×Dic3⋊C4—44no—
63C3×#95—22no—
64C3×#95—22no—
65C3⋊#95—24no—
66C3⋊#95—48no—
67C24×C4—31no—
68C22×C4⋊C4—61no—
69C22×C22⋊C4—31no—
70C22×C22⋊C4—32no—
71C22×C22⋊C4C2×D312yes—
72C22×C22⋊C4C2×D311yescharacteristic
73#385—34no—
74#385—62no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4—21no—
77C4×C22⋊C4—21no—
78C2×#95—61no—
79C2×#95—64no—
80#342—38no—
81#342—62no—
82C24×C3—42no—
83C24×C3—42no—
84C24×C3C22×C411yescharacteristic
85C24×C3C22×C412yes—
86C24×C3C2×D411yescharacteristic
87C24×C3C2×D411yescharacteristic
88C24×C3C2×D412yes—
89C24×C3C2×Q812yes—
90C24×C3#4112yes—
91C24×C3#4112yes—
92C24×C3#4112yes—
93C22×C3×C4—24no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—24no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—44no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4—44no—
103C22×C3×C4C22⋊C412yes—
104C22×C3×C4C22⋊C412yes—
105C22×C3×C4C4⋊C418yescomplemented
106C22×Dic3—28no—
107C22×Dic3—24no—
108C22×Dic3—22no—
109C22×Dic3—22no—
110C22×Dic3—24no—
111C22×Dic3—24no—
112C22×Dic3—24no—
113C22×Dic3—44no—
114C22×Dic3—44no—
115C22×Dic3—48no—
116C22×Dic3—42no—
117C22×Dic3—42no—
118C22×Dic3—44no—
119C22×Dic3—42no—
120C22×Dic3—41no—
121C22×Dic3—42no—
122C22×Dic3—41no—
123C22×Dic3—44no—
124C22×Dic3—44no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C3×C22⋊C4—24no—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—24no—
133C3×C42—82no—
134C3×C42—82no—
135#217—22no—
136#217—24no—
137#217—24no—
138#217—24no—
139#217—42no—
140#217—42no—
141#217C4214yescomplemented
142Dic3⋊C4—48no—
143C25C22×D311yescharacteristic
144C23×C4—32no—
145C23×C4—61no—
146C23×C4—61no—
147C23×C4—62no—
148C23×C4—61no—
149C23×C4—62no—
150C23×C4—62no—
151C23×C4—124no—
152C23×C4#6714yes—
153C23×C4#6711yescharacteristic
154C23×C4#6711yescharacteristic
155C2×C4⋊C4—64no—
156C2×C4⋊C4—124no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—34no—
164C2×C22⋊C4—32no—
165C2×C22⋊C4—34no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4Dic614yes—
174C2×C22⋊C4C4×D314yes—
175C2×C22⋊C4C4×D312yes—
176C2×C22⋊C4D1212yes—
177C2×C42—22no—
178C2×C42—22no—
179#95—22no—
180#95—22no—
181#95—64no—
182#95—128no—
183C23×C3—22no—
184C23×C3—24no—
185C23×C3—22no—
186C23×C3—24no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—42no—
190C23×C3—44no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3—44no—
196C23×C3—42no—
197C23×C3C2×C4211yescharacteristic
198C23×C3C2×C22⋊C411yescharacteristic
199C23×C3C2×C4⋊C412yes—
200C23×C3#11711yescharacteristic
201C23×C3#11712yes—
202C23×C3C4×D412yes—
203C23×C3C4×D412yes—
204C23×C3C4×D412yes—
205C23×C3C4×D412yes—
206C23×C3C4×Q812yes—
207C23×C3C4×Q812yes—
208C23×C3#12011yescharacteristic
209C23×C3#12112yes—
210C23×C3C22⋊Q812yes—
211C23×C3C22⋊Q812yes—
212C23×C3#12312yes—
213C23×C3#12311yescharacteristic
214C23×C3#12411yescharacteristic
215C23×C3#12512yes—
216C23×C3#12612yes—
217C23×C3C4⋊Q811yescharacteristic
218C2×C3×C4—28no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—44no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—48no—
228C2×C3×C4—84no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×Dic3—24no—
232C2×Dic3—28no—
233C2×Dic3—24no—
234C2×Dic3—42no—
235C2×Dic3—48no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—44no—
239C2×Dic3—48no—
240C2×Dic3—44no—
241C2×Dic3—42no—
242C2×Dic3—44no—
243C2×Dic3—48no—
244C2×Dic3—44no—
245C2×Dic3—84no—
246C2×Dic3—82no—
247C2×Dic3—82no—
248C24—42no—
249C24—42no—
250C24C2×Dic612yes—
251C24C2×C4×D311yescharacteristic
252C24C2×C4×D312yes—
253C24C2×D1211yescharacteristic
254C24#23512yes—
255C24#23512yes—
256C24#23512yes—
257C24C2×#6712yes—
258C24C2×#6711yescharacteristic
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—68no—
270C22×C4—64no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—64no—
274C22×C4—64no—
275C22×C4—64no—
276C22×C4—124no—
277C22×C4—124no—
278C22×C4—128no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4—124no—
282C22×C4—122no—
283C22×C4—121no—
284C22×C4—122no—
285C22×C4—121no—
286C22×C4—124no—
287C22×C4—124no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4Dic3⋊C418yescomplemented
291C22×C4(C2×D3)⋊C412yes—
292C22×C4(C2×D3)⋊C412yes—
293C4⋊C4—128no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—34no—
301C22⋊C4—62no—
302C22⋊C4—64no—
303C22⋊C4—64no—
304C22⋊C4—64no—
305C22⋊C4—122no—
306C22⋊C4—122no—
307C42—82no—
308C42—82no—
309C22×C3—24no—
310C22×C3—24no—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—24no—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—22no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3C4×C22⋊C411yescharacteristic
334C22×C3C4×C4⋊C412yes—
335C22×C3#38512yes—
336C22×C3#38611yescharacteristic
337C22×C3#38712yes—
338C22×C3#38712yes—
339C22×C3#38812yes—
340C22×C3#38912yes—
341C22×C3#39111yescharacteristic
342C22×C3B(4,𝔽2)12yes—
343C22×C3#45812yes—
344C3×C4—48no—
345C3×C4—42no—
346C3×C4—42no—
347Dic3—44no—
348Dic3—44no—
349Dic3—84no—
350C23—22no—
351C23—24no—
352C23—22no—
353C23—24no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—44no—
358C23—41no—
359C23—44no—
360C23—41no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23C4×Dic612yes—
365C23C4×Dic612yes—
366C23#86911yescharacteristic
367C23#87012yes—
368C23C42×D311yescharacteristic
369C23#87212yes—
370C23#87211yescharacteristic
371C23C4×D1212yes—
372C23#87511yescharacteristic
373C23#87612yes—
374C23C2×Dic3⋊C412yes—
375C23#92412yes—
376C23#92412yes—
377C23C2×(C2×D3)⋊C411yescharacteristic
378C23C4×#6712yes—
379C23C4×#6712yes—
380C23C4×#6712yes—
381C23#92911yescharacteristic
382C23#92912yes—
383C23#93012yes—
384C23#95311yescharacteristic
385C2×C4—28no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—64no—
396C2×C4—68no—
397C2×C4—64no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—84no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—124no—
404C2×C4—124no—
405C2×C4—128no—
406C2×C4—122no—
407C2×C4—128no—
408C2×C4—122no—
409C2×C4—124no—
410C2×C4—128no—
411C2×C4—124no—
412C2×C4—242no—
413C2×C4—242no—
414C2×C4—244no—
415C2×C3—21no—
416C2×C3—22no—
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3#143611yescharacteristic
423C2×C3#186714yes—
424C2×C3#188911yescharacteristic
425C2×C3#189011yescharacteristic
426C22—21no—
427C22—22no—
428C22—21no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—42no—
441C22—42no—
442C22—41no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—42no—
447C22—42no—
448C22—41no—
449C22—42no—
450C22#492812yes—
451C22#492911yescharacteristic
452C22C4×Dic3⋊C412yes—
453C22#493312yes—
454C22#493312yes—
455C22C4×(C2×D3)⋊C411yescharacteristic
456C22#494012yes—
457C22#521012yes—
458C22#521311yescharacteristic
459C22#524312yes—
460C22#524412yes—
461C4—42no—
462C4—42no—
463C4—48no—
464C4—124no—
465C4—124no—
466C4—244no—
467C3#1091811yescharacteristic, complemented, Sylow
468C2—21no—
469C2—22no—
470C2—22no—
471C2—21no—
472C2—22no—
473C2—22no—
474C2—22no—
475C2#7153611yescharacteristic
476C2#7165411yescharacteristic
477C2#7165511yescharacteristic
478C2#7165714yes—
subgroupG/H
centerC23#953
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC3×#2273C2
socleC23×C3#120
prime radical, cosocleC25×C3C23
hypercenter#2273D3
hypocenterC3#10918
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ #2273

Complements

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

NH ≅ G/Ntype
19. #4946C4semidirect
21. #4526C4semidirect
105. C22×C3×C4C4⋊C4semidirect
141. #217C42semidirect
290. C22×C4Dic3⋊C4semidirect
467. C3#10918semidirect

Maximal subgroups

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

subgroupindex
C3×#22732
C2×#49462
#715442
C2×#45262
#109183