Finite Group Databasebeta

← #10849292

Subgroups — #10849292

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×#5584C211yesmaximal, characteristic
2C3×#2584C211yesmaximal, characteristic, complemented
3C2×#5066C211yesmaximal, characteristic
4C2×#5069C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69632C212yesmaximal, complemented
7#69631C212yesmaximal, complemented
8#69628C212yesmaximal
9#69628C212yesmaximal
10#69486C211yesmaximal, characteristic, complemented
11#69474C211yesmaximal, characteristic
12#25656—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—21no—
16#5584—24no—
17#5584—24no—
18C2×#898—22no—
19C2×#892—21no—
20C2×#875C2212yes—
21C2×C4×D12C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6C2212yes—
24C3×#398—21no—
25C3×#395—21no—
26C3×#389—22no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30#5069—24no—
31#5066—24no—
32C2×#810C2212yes—
33C2×#810C2212yes—
34C2×#810C2212yes—
35C2×#810C2212yes—
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2212yes—
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556—31no—
47C2×#491—31no—
48C2×#478—31no—
49#2584D311yescharacteristic, complemented
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1981—31no—
54#1865—32no—
55#1863—32no—
56#1762—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6C2312yes—
61C3×#125—22no—
62C3×#125—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#946—28no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C22×C3⋊C8C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89#898—216no—
90#892—28no—
91#875—24no—
92#875—24no—
93C4×D12—22no—
94C4×D12D414yes—
95#869—22no—
96#869D414yes—
97C4×Dic6—24no—
98C4×Dic6—24no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102#810—24no—
103#810—24no—
104#810—24no—
105#810—24no—
106#809—22no—
107#809—22no—
108#809—24no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113#556—64no—
114#556—64no—
115C2×C4⋊Q8—31no—
116C2×#126—62no—
117C2×#125C2×D311yescharacteristic
118C2×#124—32no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—32no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—31no—
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×#102—32no—
136C2×#97—31no—
137#398—21no—
138#395—21no—
139#389—22no—
140#389C2×D312yes—
141#389C2×D312yes—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—32no—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—42no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C22×Dic3—21no—
158C22×Dic3—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—21no—
162C2×D12C2×D412yes—
163C2×C4×D3—24no—
164C2×C4×D3—44no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—22no—
170C2×Dic6C2×D414yes—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—82no—
182C3×C4⋊C4#4114yes—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—81no—
186C3×C42C2×D412yes—
187C3×C42C2×D412yes—
188(C2×D3)⋊C4—28no—
189(C2×D3)⋊C4—28no—
190(C2×D3)⋊C4—24no—
191C4⋊Dic3—24no—
192C4⋊Dic3—24no—
193C4⋊Dic3—21no—
194C4⋊Dic3C2×D412yes—
195Dic3⋊C4—28no—
196C4×Dic3—28no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—32no—
205C22×Q8#6711yescharacteristic
206C22×D4—31no—
207C23×C4—61no—
208C22×C8—32no—
209C22×C8—31no—
210C22×C8—31no—
211C4⋊Q8—34no—
212C4⋊Q8—62no—
213#126—616no—
214#125—22no—
215#125—22no—
216#124—64no—
217#124—64no—
218C22⋊Q8—68no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—64no—
222C4×Q8—64no—
223C4×D4—34no—
224C4×D4—62no—
225#117—68no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4—62no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4#6711yescharacteristic
235C2×C4⋊C4#6712yes—
236C2×C4⋊C4C22×D311yescharacteristic
237C2×C4⋊C4C22×D311yescharacteristic
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—62no—
242C2×C42—21no—
243C2×C42—62no—
244C2×C42C22×D312yes—
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—62no—
254#102—62no—
255#102—64no—
256#97—62no—
257#95—21no—
258#95—22no—
259#95—22no—
260C23×C3C22×D411yescharacteristic
261C22×D3—22no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—41no—
265C22×D3—41no—
266C3×Q8—44no—
267C2×C3×C4—22no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C42−1+412yes—
290C2×Dic3—22no—
291C2×Dic3—24no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295D12—22no—
296D12—22no—
297D12—42no—
298C4×D3—48no—
299Dic6—24no—
300Dic6—24no—
301Dic6—42no—
302Dic6—42no—
303C3⋊C8—41no—
304C3⋊C8—41no—
305C3⋊C8—42no—
306C24—61no—
307C2×Q8—22no—
308C2×Q8—34no—
309C2×Q8—42no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×Q8—62no—
313C2×Q8—62no—
314C2×Q8—62no—
315C2×D4—32no—
316C2×D4—61no—
317C2×D4—62no—
318C2×D4—62no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—64no—
325C22×C4—124no—
326C22×C4C2×#6712yes—
327C22×C4C2×#6712yes—
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—62no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8—62no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—32no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—68no—
350C4⋊C4—64no—
351C4⋊C4—64no—
352C4⋊C4—61no—
353C4⋊C4—82no—
354C4⋊C4#23914yes—
355C22⋊C4—68no—
356C22⋊C4—68no—
357C22⋊C4—64no—
358C42—22no—
359C42—22no—
360C42—68no—
361C42—81no—
362C42D3×D412yes—
363C42D3×D412yes—
364C22×C3C2×#12011yescharacteristic
365C22×C3#54812yes—
366C22×C3#57512yes—
367C22×C3#57711yescharacteristic
368C22×C3#57811yescharacteristic
369C2×D3—42no—
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—82no—
373C2×D3—82no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—44no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—81no—
380C3×C4—81no—
381C3×C4#53211yescharacteristic
382C3×C4#53611yescharacteristic
383C3×C4#54712yes—
384Dic3—44no—
385Dic3—42no—
386C23—62no—
387C23—122no—
388C23—122no—
389C23—121no—
390C23—121no—
391C23C22×#6711yescharacteristic
392Q8—44no—
393Q8—64no—
394Q8—64no—
395Q8—122no—
396Q8—122no—
397D4—62no—
398D4—62no—
399D4—122no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—62no—
418C2×C4—64no—
419C2×C4—128no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4#93712yes—
424C2×C4#93712yes—
425C2×C4C2×D3×D411yescharacteristic
426C2×C4C2×D3×D411yescharacteristic
427C2×C4C2×#23912yes—
428C2×C4#100712yes—
429C8—121no—
430C8—121no—
431C8—122no—
432C2×C3#243211yescharacteristic
433C2×C3#300511yescharacteristic
434C2×C3#301311yescharacteristic
435C2×C3#337612yes—
436C2×C3#337712yes—
437D3—82no—
438C22—122no—
439C22—122no—
440C22—122no—
441C22—242no—
442C22—242no—
443C22C2×#93711yescharacteristic
444C22#581712yes—
445C22#582212yes—
446C22#583611yescharacteristic
447C22#583811yescharacteristic
448C4—22no—
449C4—22no—
450C4—44no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—81no—
455C4—124no—
456C4—122no—
457C4#572411yescharacteristic
458C4#572511yescharacteristic
459C4#572612yes—
460C3#2565611yescharacteristic, complemented, Sylow
461C2—242no—
462C2#8292811yescharacteristic
463C2#8325511yescharacteristic
464C2#8342711yescharacteristic
465C2#8364512yes—
466C2#8370012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2584C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2584D3
hypocenterC3#25656
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C3×#2584C2semidirect
4. C2×#5069C2semidirect
5. #71635C2semidirect
6. #69632C2semidirect
7. #69631C2semidirect
10. #69486C2semidirect
49. #2584D3semidirect
460. C3#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25842
C2×#50662
C2×#50692
#716352
#696322
#696312
#696282
#694862
#694742
#256563