Finite Group Databasebeta

← #10944104

Subgroups — #10944104

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5812C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic
3C2×#5228C211yesmaximal, characteristic
4C2×#5182C212yesmaximal, complemented
5#82786C211yesmaximal, characteristic
6#72065C212yesmaximal, complemented
7#71693C211yesmaximal, characteristic
8#70239C212yesmaximal, complemented
9#70222C212yesmaximal, complemented
10#69931C211yesmaximal, characteristic
11#69909C211yesmaximal, characteristic
12#32915—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16#5812—28no—
17C2×Q8×Dic3—22no—
18C2×Q8×Dic3C2212yes—
19C2×Q8×Dic3C2212yes—
20C2×#944C2211yescharacteristic
21C22×#216C2212yes—
22C2×#926—22no—
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C3×#451—24no—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29#5229C2211yescharacteristic
30#5228—24no—
31C2×#835C2212yes—
32C2×#835C2212yes—
33C2×#830C2211yescharacteristic
34#5182—28no—
35C2×#816C2212yes—
36C2×Dic3⋊C8C2212yes—
37#4982—22no—
38#4977C2211yescharacteristic
39#4973C2211yescharacteristic
40#4952—22no—
41#4551C2212yes—
42#4550C2212yes—
43#4529C2212yes—
44C2×#519—31no—
45C2×#462—32no—
46C2×#451—31no—
47C2×#451D311yescharacteristic
48#2621—31no—
49#1878—32no—
50#1878—32no—
51#1873—31no—
52#1873—31no—
53#1805—32no—
54#1798—31no—
55C22×C3×Q8—24no—
56C22×C3×Q8—24no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8D414yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2312yes—
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C22×Dic6C2311yescharacteristic
65C2×C3×Q16—24no—
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C22×C3×C8C2312yes—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C22⋊C4—21no—
74C2×#217—21no—
75C2×#217—21no—
76C2×#217D412yes—
77Q8×Dic3—216no—
78Q8×Dic3—24no—
79Q8×Dic3—24no—
80Q8×Dic3—24no—
81Q8×Dic3D418yes—
82#944—24no—
83C2×#216—24no—
84C2×#216—24no—
85C2×#216—24no—
86#926—216no—
87#926—24no—
88#926—24no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3D412yes—
92C2×C4⋊Dic3C2311yescharacteristic
93C2×C4⋊Dic3C2312yes—
94#924—22no—
95#924—22no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—22no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3C2311yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C3×#103—24no—
105C3×C22⋊C8—22no—
106#835—24no—
107#835—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111#830—22no—
112#816—24no—
113Dic3⋊C8—24no—
114C23×Q8C2×D311yescharacteristic
115C22×Q16—32no—
116C22×Q16C2×D312yes—
117C2×C4⋊Q8—31no—
118C2×C22⋊Q8—31no—
119C2×C22⋊Q8C2×D311yescharacteristic
120#519—68no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—62no—
124C2×#117—31no—
125C2×#117—31no—
126C2×#117—62no—
127#462—68no—
128#451—24no—
129#451—64no—
130C2×C4⋊C8—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134C2×#103C2×D312yes—
135C2×C22⋊C8—31no—
136C2×C22⋊C8C2×D311yescharacteristic
137#398—62no—
138#397—62no—
139#391—31no—
140#391—31no—
141#382—31no—
142#340—32no—
143#340—32no—
144#336—32no—
145C24×C3C2×D411yescharacteristic
146C2×C3×Q8—24no—
147C2×C3×Q8—216no—
148C2×C3×Q8—24no—
149C2×C3×Q8—216no—
150C2×C3×Q8—24no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8C2×D414yes—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8#4114yes—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D412yes—
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3C2×D412yes—
177C22×Dic3C2×D412yes—
178C2×Dic6—22no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—22no—
182C3×Q16—48no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—42no—
189C3×C22⋊C4—42no—
190#217—24no—
191#217—24no—
192#217—42no—
193#216—48no—
194C4⋊Dic3—28no—
195C4⋊Dic3—22no—
196C4⋊Dic3—28no—
197C4⋊Dic3—24no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3C2×D414yes—
201Dic3⋊C4—42no—
202Dic3⋊C4—44no—
203C4×Dic3—28no—
204C4×Dic3—24no—
205C4×Dic3—22no—
206C4×Dic3—28no—
207C4×Dic3C2×D414yes—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8—22no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—31no—
216C22×Q8#6714yes—
217C22×Q8C22×D311yescharacteristic
218C22×Q8C22×D312yes—
219C23×C4—21no—
220C23×C4C22×D311yescharacteristic
221C2×Q16—24no—
222C2×Q16—24no—
223C2×Q16—24no—
224C2×Q16—64no—
225C2×Q16—64no—
226C2×Q16—64no—
227C22×C8—32no—
228C22×C8C22×D312yes—
229C4⋊Q8—64no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—62no—
233C22⋊Q8—62no—
234C4×Q8—38no—
235C4×Q8—64no—
236C4×Q8—64no—
237C4×Q8—616no—
238C4×Q8—64no—
239#117—64no—
240#117—64no—
241#117—616no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—61no—
255C2×C42—31no—
256C2×C42—31no—
257C2×C42—62no—
258C2×C42—62no—
259C2×C42—61no—
260C4⋊C8—64no—
261#103—24no—
262#103—64no—
263#103—64no—
264#103—64no—
265C22⋊C8—22no—
266C22⋊C8—62no—
267#95—62no—
268#95—62no—
269#95—62no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3#12112yes—
275C23×C3C22×D411yescharacteristic
276C3×Q8—216no—
277C3×Q8—24no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8#12118yes—
283C2×C3×C4—216no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—21no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4#12112yes—
297C2×C3×C4C22×D412yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4112yes—
300C2×C3×C4C2×#4111yescharacteristic
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—24no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3#12014yes—
315Dic6—42no—
316Dic6—42no—
317C3×C8—42no—
318C3⋊C8—42no—
319C24D3×D411yescharacteristic
320C2×Q8—24no—
321C2×Q8—216no—
322C2×Q8—24no—
323C2×Q8—216no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8—62no—
333C2×Q8—61no—
334C2×Q8—62no—
335C2×Q8—61no—
336C2×Q8D3×D414yes—
337C2×Q8#23714yes—
338C2×Q8C2×#6714yes—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4D3×D411yescharacteristic
353C22×C4#23712yes—
354C22×C4C2×#6712yes—
355C22×C4C23×D311yescharacteristic
356Q16—48no—
357Q16—128no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—62no—
362C2×C8—62no—
363C2×C8—62no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—34no—
367C4⋊C4—42no—
368C4⋊C4—64no—
369C4⋊C4—68no—
370C4⋊C4—68no—
371C4⋊C4—62no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—124no—
375C4⋊C4—122no—
376C22⋊C4—42no—
377C22⋊C4—64no—
378C22⋊C4—64no—
379C22⋊C4—122no—
380C42—34no—
381C42—68no—
382C42—68no—
383C42—64no—
384C42—62no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3C2×#12011yescharacteristic
391C22×C3C2×#12111yescharacteristic
392C22×C3#52511yescharacteristic
393C22×C3#57812yes—
394C22×C3#57812yes—
395C3×C4—28no—
396C3×C4—22no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4C2×#12112yes—
400C3×C4#52512yes—
401Dic3—24no—
402Dic3—42no—
403Dic3—44no—
404Dic3—81no—
405Dic3—81no—
406C23—22no—
407C23—22no—
408C23—21no—
409C23—21no—
410C23#93912yes—
411C23C2×D3×D411yescharacteristic
412Q8—216no—
413Q8—24no—
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—122no—
419Q8—122no—
420Q8#93918yes—
421C2×C4—216no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—34no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—64no—
440C2×C4—62no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4#93912yes—
449C2×C4C2×D3×D412yes—
450C2×C4C2×#23712yes—
451C2×C4C2×#23711yescharacteristic
452C2×C4C22×#6711yescharacteristic
453C8—42no—
454C8—122no—
455C2×C3—42no—
456C2×C3#284711yescharacteristic
457C2×C3#301112yes—
458C2×C3#304411yescharacteristic
459C2×C3#304411yescharacteristic
460C2×C3#312712yes—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—42no—
465C22—42no—
466C22#558611yescharacteristic
467C22D3×#12011yescharacteristic
468C22#576612yes—
469C22#577912yes—
470C22C2×#93911yescharacteristic
471C4—28no—
472C4—22no—
473C4—64no—
474C4—81no—
475C4—81no—
476C4—122no—
477C4—124no—
478C4—241no—
479C4—241no—
480C4#558612yes—
481C4C2×#93912yes—
482C3#3291511yescharacteristic, complemented, Sylow
483C2—42no—
484C2#8017712yes—
485C2#8018512yes—
486C2#8260111yescharacteristic
487C2#8387711yescharacteristic
488C2#8397011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#451D3
hypocenterC3#32915
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
4. C2×#5182C2semidirect
6. #72065C2semidirect
8. #70239C2semidirect
9. #70222C2semidirect
482. C3#32915semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
C2×C3×#4512
C2×#52282
C2×#51822
#827862
#720652
#716932
#702392
#702222
#699312
#699092
#329153