Finite Group Databasebeta

← #10849314

Subgroups — #10849314

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C3×#2584C211yesmaximal, characteristic, complemented
3C2×#5063C211yesmaximal, characteristic, complemented
4C2×#5067C211yesmaximal, characteristic
5#71635C211yesmaximal, characteristic, complemented
6#69628C212yesmaximal, complemented
7#69628C212yesmaximal, complemented
8#69626C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69487C211yesmaximal, characteristic
11#69475C211yesmaximal, characteristic, complemented
12#25658—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947—21no—
16#5585—24no—
17#5585—24no—
18C2×#894—22no—
19C2×#893—21no—
20C2×#875C2212yes—
21C2×#874C2211yescharacteristic
22C2×C4×D12C2212yes—
23C2×C4×Dic6C2211yescharacteristic
24C3×#398—21no—
25C3×#395—21no—
26C3×#389—22no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×C4×C4⋊C4C2211yescharacteristic
30#5067—24no—
31#5063—24no—
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559—31no—
47C2×#490—31no—
48C2×#479—31no—
49#2584D311yescharacteristic, complemented
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1763—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×Dic6C2311yescharacteristic
63C3×#125—22no—
64C3×#125—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C42—21no—
77C2×C3×C42C2312yes—
78C2×C3×C42C2311yescharacteristic
79#947—28no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—21no—
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2312yes—
87C22×C3⋊C8C2311yescharacteristic
88C22×C3⋊C8C2311yescharacteristic
89#894—216no—
90#893—28no—
91#875—24no—
92#875—24no—
93#874—22no—
94#874D414yes—
95C4×D12—24no—
96C4×D12—24no—
97C4×Dic6—22no—
98C4×Dic6D414yes—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102#810—22no—
103#810—22no—
104#810—24no—
105#809—24no—
106#809—24no—
107#809—24no—
108#809—24no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113#559—64no—
114#559—64no—
115C2×#127—31no—
116C2×#125C2×D311yescharacteristic
117C2×#124—32no—
118C2×#124—61no—
119C2×#123—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124#490—64no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8—31no—
129C2×#103—31no—
130C2×#103—31no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
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—22no—
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—21no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—24no—
164C2×D12—22no—
165C2×D12—24no—
166C2×D12—44no—
167C2×D12—81no—
168C2×D12—81no—
169C2×D12C2×D414yes—
170C2×C4×D3—28no—
171C2×C4×D3—82no—
172C2×C4×D3—82no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6C2×D412yes—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—44no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—82no—
188C3×C4⋊C4#4114yes—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—81no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194(C2×D3)⋊C4—28no—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4—28no—
197C4⋊Dic3—21no—
198C4⋊Dic3C2×D412yes—
199Dic3⋊C4—24no—
200C4×Dic3—24no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8—31no—
209C22×Q8#6711yescharacteristic
210C22×D4—32no—
211C22×D4—61no—
212C22×D4—61no—
213C23×C4—62no—
214C22×C8—32no—
215C22×C8—31no—
216C22×C8—31no—
217#127—34no—
218#127—62no—
219#125—22no—
220#125—22no—
221#124—64no—
222#124—64no—
223#124—68no—
224#123—616no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—34no—
228C4×Q8—62no—
229C4×D4—64no—
230C4×D4—64no—
231C4×D4—68no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4#6711yescharacteristic
239C2×C4⋊C4#6712yes—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C42—21no—
247C2×C42—61no—
248C2×C42C22×D312yes—
249C2×C42C22×D311yescharacteristic
250#107—62no—
251#106—62no—
252C4⋊C8—62no—
253#103—62no—
254#103—62no—
255#103—64no—
256#102—64no—
257#102—64no—
258#102—64no—
259#102—64no—
260#97—62no—
261#95—21no—
262#95—22no—
263#95—22no—
264C23×C3C22×D411yescharacteristic
265C22×D3—24no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C3×Q8—44no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4#12012yes—
291C2×C3×C4#12012yes—
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4112yes—
295C2×C3×C42−1+412yes—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301D12—24no—
302D12—24no—
303D12—48no—
304D12—42no—
305D12—42no—
306D12—84no—
307C4×D3—88no—
308Dic6—22no—
309Dic6—22no—
310Dic6—42no—
311C3⋊C8—41no—
312C3⋊C8—41no—
313C3⋊C8—42no—
314C24—62no—
315C2×Q8—22no—
316C2×Q8—32no—
317C2×Q8—42no—
318C2×Q8—61no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×D4—34no—
322C2×D4—64no—
323C2×D4—64no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—124no—
330C2×D4—241no—
331C2×D4—241no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—68no—
336C22×C4—61no—
337C22×C4—61no—
338C22×C4—242no—
339C22×C4—242no—
340C22×C4C2×#6712yes—
341C22×C4C2×#6712yes—
342C22×C4C2×#6711yescharacteristic
343C22×C4C2×#6711yescharacteristic
344C22×C4C23×D311yescharacteristic
345C2×C8—62no—
346C2×C8—61no—
347C2×C8—61no—
348C2×C8—62no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—32no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—64no—
364C4⋊C4—61no—
365C4⋊C4—82no—
366C4⋊C4#23914yes—
367C22⋊C4—68no—
368C22⋊C4—68no—
369C22⋊C4—68no—
370C42—22no—
371C42—22no—
372C42—64no—
373C42—81no—
374C42D3×D412yes—
375C42D3×D412yes—
376C22×C3C2×#12011yescharacteristic
377C22×C3#54812yes—
378C22×C3#57512yes—
379C22×C3#57611yescharacteristic
380C22×C3#57711yescharacteristic
381C2×D3—44no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—82no—
385C2×D3—82no—
386C2×D3—82no—
387C2×D3—82no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—44no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4#53211yescharacteristic
396C3×C4#53611yescharacteristic
397C3×C4#54712yes—
398Dic3—42no—
399Dic3—42no—
400C23—64no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23C22×#6711yescharacteristic
408Q8—44no—
409Q8—62no—
410Q8—62no—
411Q8—122no—
412D4—64no—
413D4—64no—
414D4—128no—
415D4—122no—
416D4—122no—
417D4—244no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—122no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—248no—
441C2×C4#93712yes—
442C2×C4#93712yes—
443C2×C4C2×D3×D411yescharacteristic
444C2×C4C2×D3×D411yescharacteristic
445C2×C4C2×#23912yes—
446C2×C4#100712yes—
447C8—121no—
448C8—121no—
449C8—122no—
450C2×C3#243211yescharacteristic
451C2×C3#300511yescharacteristic
452C2×C3#301211yescharacteristic
453C2×C3#337012yes—
454C2×C3#337112yes—
455D3—84no—
456C22—124no—
457C22—124no—
458C22—124no—
459C22—242no—
460C22—242no—
461C22—242no—
462C22—242no—
463C22C2×#93711yescharacteristic
464C22#581712yes—
465C22#582212yes—
466C22#583511yescharacteristic
467C22#583711yescharacteristic
468C4—22no—
469C4—22no—
470C4—44no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—122no—
476C4—122no—
477C4#572411yescharacteristic
478C4#572511yescharacteristic
479C4#572612yes—
480C3#2565811yescharacteristic, complemented, Sylow
481C2—244no—
482C2#8292811yescharacteristic
483C2#8325511yescharacteristic
484C2#8342411yescharacteristic
485C2#8363912yes—
486C2#8369112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2584C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2584D3
hypocenterC3#25658
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
3. C2×#5063C2semidirect
5. #71635C2semidirect
6. #69628C2semidirect
7. #69628C2semidirect
8. #69626C2semidirect
9. #69625C2semidirect
11. #69475C2semidirect
49. #2584D3semidirect
480. C3#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#25842
C2×#50632
C2×#50672
#716352
#696282
#696262
#696252
#694872
#694752
#256583