Finite Group Databasebeta

← #10849296

Subgroups — #10849296

Order 768.

1012 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic
2C2×#5009C211yesmaximal, characteristic
3C2×#4703C211yesmaximal, characteristic, complemented
4#73727C211yesmaximal, characteristic, complemented
5C3×#1762C211yesmaximal, characteristic
6#70178C212yesmaximal
7#70178C212yesmaximal
8#69838C211yesmaximal, characteristic, complemented
9#69486C211yesmaximal, characteristic, complemented
10#68907C212yesmaximal, complemented
11#68906C212yesmaximal, complemented
12#25656—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#936C2212yes—
15C2×#926—21no—
16C2×#924—21no—
17#5542—24no—
18#5542—24no—
19C2×#887C2212yes—
20C2×#879—22no—
21C2×#870C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#97C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#832C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#816C2212yes—
30C2×#816C2212yes—
31#5009—24no—
32#4984C2212yes—
33#4984C2212yes—
34C2×#810C2212yes—
35C2×#810C2212yes—
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4928—22no—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4703—24no—
41#4661—21no—
42#4656—21no—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
47C2×#491—31no—
48C2×#478—31no—
49#2584—31no—
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1981—31no—
54#1865—32no—
55#1863—32no—
56#1762D311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C22×Dic6D411yescharacteristic
60C22×Dic6C2312yes—
61C22×C3×C8C2312yes—
62C3×C4×D4—22no—
63C3×C4×D4D414yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—21no—
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69C2×#217—22no—
70C2×#217—22no—
71#936—24no—
72#936—24no—
73#926—28no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81#924—28no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4D412yes—
86C2×C4×Dic3—22no—
87C2×C4×Dic3C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90#887—24no—
91#887—24no—
92#879—216no—
93#870—22no—
94#870—22no—
95#869—22no—
96#869D414yes—
97C4×Dic6—22no—
98C4×Dic6—22no—
99C3×#102—24no—
100C3×#97—22no—
101C3×#95—21no—
102#832—22no—
103#832—22no—
104C3⋊#95—22no—
105C3⋊#95—22no—
106#816—24no—
107#816—24no—
108#810—24no—
109#810—24no—
110#808—22no—
111#807—22no—
112C4⋊(C3⋊C8)—22no—
113#556—64no—
114#556—64no—
115C2×C4⋊Q8—31no—
116C2×#126—62no—
117C2×#125—31no—
118C2×#124—32no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×D4C2×D311yescharacteristic
123C2×#117—61no—
124#491—64no—
125#478—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8—31no—
129C2×#103—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#102—31no—
134C2×#102—31no—
135C2×#102C2×D312yes—
136C2×#97C2×D311yescharacteristic
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340C2×D312yes—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C2×C3×D4—21no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D412yes—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—24no—
154C22×C3×C4—44no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D412yes—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—42no—
170C2×Dic6C2×D414yes—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4C2×D412yes—
177C3×C4⋊C4#4114yes—
178C3×C22⋊C4—24no—
179C3×C42—81no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182#217—28no—
183#217—28no—
184C4⋊Dic3—22no—
185C4⋊Dic3—24no—
186C4⋊Dic3—24no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—44no—
192C4⋊Dic3—82no—
193Dic3⋊C4—28no—
194Dic3⋊C4—24no—
195Dic3⋊C4—42no—
196Dic3⋊C4—42no—
197C4×Dic3—28no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—31no—
205C22×Q8—32no—
206C22×D4C22×D311yescharacteristic
207C23×C4—21no—
208C22×C8—31no—
209C22×C8—31no—
210C22×C8C22×D312yes—
211C4⋊Q8—34no—
212C4⋊Q8—62no—
213#126—616no—
214#125—62no—
215#125—62no—
216#124—64no—
217#124—64no—
218C22⋊Q8—68no—
219C4×Q8—64no—
220C4×Q8—64no—
221C4×Q8—62no—
222C4×Q8—62no—
223C4×D4—22no—
224C4×D4D1214yes—
225#117—68no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—61no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C42—21no—
243C2×C42—32no—
244C2×C42—62no—
245C2×C42C22×D311yescharacteristic
246#107—62no—
247#106—62no—
248C4⋊C8—62no—
249#103—64no—
250#103—64no—
251#103—64no—
252#103—64no—
253#102—24no—
254#102—62no—
255#102—62no—
256#97—22no—
257#95—21no—
258#95—62no—
259#95—62no—
260C23×C3—22no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3C22×D411yescharacteristic
266C3×D4—22no—
267C3×D4—22no—
268C3×D4—42no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—48no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4#12012yes—
279C2×C3×C4#12012yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C42−1+412yes—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—24no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298Dic6—24no—
299Dic6—24no—
300Dic6—44no—
301Dic6—42no—
302Dic6—42no—
303C3×C8—42no—
304C3⋊C8—41no—
305C3⋊C8—41no—
306C24—21no—
307C2×Q8—34no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—122no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—22no—
318C2×D4C2×D1212yes—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—32no—
323C22×C4—32no—
324C22×C4—44no—
325C22×C4—62no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—61no—
335C2×C8—61no—
336C2×C8—62no—
337C2×C8—62no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—68no—
341C4⋊C4—62no—
342C4⋊C4—64no—
343C4⋊C4—64no—
344C4⋊C4—64no—
345C4⋊C4—61no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—61no—
349C4⋊C4—124no—
350C4⋊C4—122no—
351C4⋊C4—122no—
352C4⋊C4—242no—
353C4⋊C4C2×D1212yes—
354C4⋊C4#23514yes—
355C22⋊C4—24no—
356C22⋊C4—68no—
357C22⋊C4—68no—
358C42—68no—
359C42—62no—
360C42—62no—
361C42—81no—
362C42C2×D1212yes—
363C42D3×D412yes—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3C2×#12011yescharacteristic
370C22×C3#54812yes—
371C22×C3#57512yes—
372C22×C3#57711yescharacteristic
373C22×C3#57811yescharacteristic
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—42no—
378C3×C4#53211yescharacteristic
379C3×C4#53611yescharacteristic
380C3×C4#54712yes—
381Dic3—44no—
382Dic3—42no—
383Dic3—42no—
384Dic3—81no—
385Dic3—81no—
386C23—22no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23C2×D3×D411yescharacteristic
392Q8—64no—
393Q8—64no—
394Q8—124no—
395Q8—122no—
396Q8—122no—
397D4—22no—
398D4—22no—
399D4—42no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—64no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4#88212yes—
424C2×C4#88212yes—
425C2×C4C22×D1211yescharacteristic
426C2×C4C2×#23512yes—
427C2×C4C2×D3×D411yescharacteristic
428C2×C4#101012yes—
429C8—42no—
430C8—121no—
431C8—121no—
432C2×C3—82no—
433C2×C3#243211yescharacteristic
434C2×C3#300511yescharacteristic
435C2×C3#301311yescharacteristic
436C2×C3#337612yes—
437C2×C3#337712yes—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22C2×#88211yescharacteristic
444C22#555212yes—
445C22#577312yes—
446C22#577511yescharacteristic
447C22#577911yescharacteristic
448C4—22no—
449C4—22no—
450C4—42no—
451C4—44no—
452C4—124no—
453C4—122no—
454C4—122no—
455C4—241no—
456C4—241no—
457C4#547711yescharacteristic
458C4#552611yescharacteristic
459C4#555112yes—
460C3#2565611yescharacteristic, complemented, Sylow
461C2—82no—
462C2#7348911yescharacteristic
463C2#7888711yescharacteristic
464C2#7892711yescharacteristic
465C2#7975812yes—
466C2#7985412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1762C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1762D3
hypocenterC3#25656
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
3. C2×#4703C2semidirect
4. #73727C2semidirect
8. #69838C2semidirect
9. #69486C2semidirect
10. #68907C2semidirect
11. #68906C2semidirect
460. C3#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C2×#50092
C2×#47032
#737272
C3×#17622
#701782
#698382
#694862
#689072
#689062
#256563