Finite Group Databasebeta

← #10849290

Subgroups — #10849290

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×C3×#556C211yesmaximal, characteristic
2C2×#5082C211yesmaximal, characteristic, complemented
3#82483C211yesmaximal, characteristic, complemented
4C2×#5009C211yesmaximal, characteristic
5#71778C211yesmaximal, characteristic, complemented
6#71722C212yesmaximal
7#71722C212yesmaximal
8#71635C211yesmaximal, characteristic, complemented
9#71629C212yesmaximal, complemented
10#71628C212yesmaximal, complemented
11#71609C211yesmaximal, characteristic
12#25656—31nomaximal, Sylow
13C3×#556—24no—
14C3×#556—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#124C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C2×#870C2211yescharacteristic
24#5230—21no—
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×#835C2212yes—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2212yes—
32#5082—24no—
33#5009—24no—
34#4983—21no—
35#4976—22no—
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C4×C4⋊Dic3C2211yescharacteristic
39#4928C2212yes—
40#4928C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556D311yescharacteristic
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#1762—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8D414yes—
63C3×#126—216no—
64C3×#124—24no—
65C3×#124—24no—
66C3×C22⋊Q8—28no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C3×#117—28no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4D412yes—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—22no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—21no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97#870—22no—
98#870—22no—
99#835—24no—
100#835—24no—
101#835—24no—
102#835—24no—
103#832—22no—
104#832—22no—
105#832—24no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109#808—22no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113#556—24no—
114#556—24no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#126—22no—
117C2×#125—31no—
118C2×#124C2×D312yes—
119C2×C22⋊Q8—21no—
120C2×C4×Q8C2×D312yes—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4C2×D311yescharacteristic
123C2×#117—21no—
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—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4—31no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×Q8C2×D414yes—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—44no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4C2×D412yes—
178C3×C22⋊C4—28no—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C42—28no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186C4⋊Dic3—22no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3#4114yes—
192Dic3⋊C4—44no—
193Dic3⋊C4—42no—
194Dic3⋊C4—42no—
195Dic3⋊C4—82no—
196C4×Dic3—81no—
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#6711yescharacteristic
205C22×Q8C22×D312yes—
206C22×D4C22×D311yescharacteristic
207C23×C4—21no—
208C22×C8—32no—
209C22×C8—31no—
210C22×C8—31no—
211C4⋊Q8—22no—
212C4⋊Q8#6714yes—
213#126—216no—
214#125—62no—
215#125—62no—
216#124—24no—
217#124—24no—
218C22⋊Q8—28no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×D4—22no—
224C4×D4#6714yes—
225#117—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—61no—
233C2×C4⋊C4—61no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4#6711yescharacteristic
237C2×C4⋊C4#6712yes—
238C2×C4⋊C4C22×D311yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42—61no—
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—64no—
254#102—62no—
255#102—62no—
256#97—62no—
257#95—61no—
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×Q8—24no—
267C3×Q8—24no—
268C3×Q8—44no—
269C3×Q8—42no—
270C3×Q8—42no—
271C3×D4—22no—
272C3×D4—22no—
273C3×D4—42no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—48no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4#12012yes—
290C2×C3×C4#12012yes—
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C42−1+412yes—
295C2×Dic3—24no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C3⋊C8—41no—
304C3⋊C8—41no—
305C3⋊C8—42no—
306C24—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—42no—
314C2×Q8C2×#6714yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4C2×#6712yes—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—44no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4C2×#6712yes—
327C22×C4C2×#6712yes—
328C22×C4C2×#6711yescharacteristic
329C22×C4C2×#6711yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—61no—
337C2×C8—61no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—21no—
344C4⋊C4—34no—
345C4⋊C4—62no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—124no—
351C4⋊C4—122no—
352C4⋊C4—122no—
353C4⋊C4—242no—
354C4⋊C4C2×#6712yes—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C42—28no—
359C42—22no—
360C42—22no—
361C42—241no—
362C42C2×#6712yes—
363C42C2×#6712yes—
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—42no—
379C3×C4—42no—
380C3×C4#53211yescharacteristic
381C3×C4#53611yescharacteristic
382C3×C4#54712yes—
383Dic3—44no—
384Dic3—81no—
385Dic3—81no—
386C23—22no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23C22×#6711yescharacteristic
392Q8—24no—
393Q8—24no—
394Q8—44no—
395Q8—42no—
396Q8—42no—
397D4—22no—
398D4—22no—
399D4—42no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—64no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4#95312yes—
424C2×C4#95312yes—
425C2×C4D3×#4112yes—
426C2×C4#101012yes—
427C2×C4C22×#6711yescharacteristic
428C2×C4C22×#6711yescharacteristic
429C8—122no—
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—
443C22#582212yes—
444C22#583112yes—
445C22#583711yescharacteristic
446C22#583811yescharacteristic
447C22C2×#95311yescharacteristic
448C4—22no—
449C4—22no—
450C4—42no—
451C4—42no—
452C4—44no—
453C4—42no—
454C4—124no—
455C4—241no—
456C4—241no—
457C4#582412yes—
458C4#582511yescharacteristic
459C4#583211yescharacteristic
460C3#2565611yescharacteristic, complemented, Sylow
461C2—82no—
462C2#8366912yes—
463C2#8367812yes—
464C2#8408111yescharacteristic
465C2#8409311yescharacteristic
466C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#556C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#556D3
hypocenterC3#25656
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#5082C2semidirect
3. #82483C2semidirect
5. #71778C2semidirect
8. #71635C2semidirect
9. #71629C2semidirect
10. #71628C2semidirect
460. C3#25656semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×#50822
#824832
C2×#50092
#717782
#717222
#716352
#716292
#716282
#716092
#256563