Finite Group Databasebeta

← #10849298

Subgroups — #10849298

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×#5732C211yesmaximal, characteristic
2C2×#5082C211yesmaximal, characteristic, complemented
3C2×#4847C211yesmaximal, characteristic
4#73727C211yesmaximal, characteristic, complemented
5C3×#1981C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70245C212yesmaximal
8#70245C212yesmaximal
9#70238C212yesmaximal, complemented
10#70236C212yesmaximal, complemented
11#69474C211yesmaximal, characteristic
12#25656—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947C2212yes—
15C2×Q8×Dic3C2212yes—
16#5732—24no—
17#5732—24no—
18C2×#898—22no—
19C2×#897—21no—
20C2×#892—21no—
21C2×C4×D12C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34#5082—24no—
35#4984C2212yes—
36#4984C2212yes—
37C2×#809C2212yes—
38#4928—22no—
39C2×#803C2211yescharacteristic
40#4847—24no—
41#4661—21no—
42#4656—21no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#491—31no—
48C2×#478—31no—
49#2584—31no—
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1981D311yescharacteristic, complemented
54#1865—32no—
55#1863—32no—
56#1762—31no—
57C22×C3×Q8C2312yes—
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×Dic6D411yescharacteristic
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70#947—24no—
71#947—24no—
72Q8×Dic3—24no—
73Q8×Dic3—24no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4D412yes—
87C2×C4×Dic3—22no—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2312yes—
90#898—216no—
91#897—28no—
92#892—28no—
93C4×D12—22no—
94C4×D12D414yes—
95#870—22no—
96#870—22no—
97C4×Dic6—22no—
98C4×Dic6—22no—
99C3×C4⋊C8—22no—
100C3×#103—24no—
101C3×#103—24no—
102C3×#95—21no—
103#835—24no—
104#835—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#821—22no—
108#821—22no—
109#818—22no—
110#817—22no—
111#809—24no—
112#803—22no—
113#556—64no—
114#556—64no—
115C2×C4⋊Q8C2×D311yescharacteristic
116C2×#126—62no—
117C2×#125—31no—
118C2×#124—32no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—31no—
121C2×C4×Q8—32no—
122C2×C4×D4—31no—
123C2×#117—61no—
124#491—64no—
125#478—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8C2×D311yescharacteristic
129C2×#103—32no—
130C2×#103—32no—
131C2×#103C2×D312yes—
132C2×#103C2×D312yes—
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⋊C4C2×D311yescharacteristic
143#340—32no—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8C2×D414yes—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3C2×D412yes—
162C22×Dic3C2×D412yes—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—21no—
166C2×D12C2×D412yes—
167C2×C4×D3—24no—
168C2×C4×D3—44no—
169C2×Dic6—22no—
170C2×Dic6—42no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4#4114yes—
179C3×C42—81no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182(C2×D3)⋊C4—28no—
183(C2×D3)⋊C4—28no—
184(C2×D3)⋊C4—24no—
185C4⋊Dic3—28no—
186C4⋊Dic3—22no—
187C4⋊Dic3—21no—
188C4⋊Dic3—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—44no—
193C4⋊Dic3—82no—
194C4⋊Dic3C2×D412yes—
195Dic3⋊C4—24no—
196Dic3⋊C4—42no—
197Dic3⋊C4—42no—
198C4×Dic3—28no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×Q8—31no—
205C22×Q8C22×D312yes—
206C22×D4—31no—
207C23×C4—61no—
208C22×C8—32no—
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211C4⋊Q8—22no—
212C4⋊Q8D1214yes—
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—34no—
224C4×D4—62no—
225#117—68no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4C22×D312yes—
239C2×C22⋊C4—62no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—61no—
242C2×C42—21no—
243C2×C42—32no—
244C2×C42—62no—
245C2×C42C22×D311yescharacteristic
246#107—62no—
247#106—62no—
248C4⋊C8—22no—
249#103—24no—
250#103—24no—
251#103—64no—
252#103—64no—
253#102—62no—
254#102—62no—
255#102—64no—
256#97—62no—
257#95—21no—
258#95—62no—
259#95—62no—
260C23×C3C22×D411yescharacteristic
261C22×D3—22no—
262C22×D3—42no—
263C22×D3—42no—
264C22×D3—41no—
265C22×D3—41no—
266C3×Q8—24no—
267C3×Q8—24no—
268C3×Q8—42no—
269C3×Q8—42no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4#12012yes—
280C2×C3×C4#12012yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4112yes—
284C2×C3×C42−1+412yes—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—42no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298D12—22no—
299D12—22no—
300D12—42no—
301C4×D3—48no—
302Dic6—44no—
303C3×C8—41no—
304C3×C8—41no—
305C3⋊C8—42no—
306C24—61no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—62no—
313C2×Q8—122no—
314C2×Q8C2×D1214yes—
315C2×D4—32no—
316C2×D4—61no—
317C2×D4—62no—
318C2×D4—62no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—32no—
322C22×C4—32no—
323C22×C4—61no—
324C22×C4—64no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—124no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8—62no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—32no—
342C4⋊C4—68no—
343C4⋊C4—62no—
344C4⋊C4—64no—
345C4⋊C4—61no—
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⋊C4#23714yes—
355C22⋊C4—68no—
356C22⋊C4—68no—
357C22⋊C4—64no—
358C42—68no—
359C42—62no—
360C42—62no—
361C42—81no—
362C42C2×D1212yes—
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—44no—
378C3×C4#53211yescharacteristic
379C3×C4#53611yescharacteristic
380C3×C4#54712yes—
381Dic3—42no—
382Dic3—42no—
383Dic3—42no—
384Dic3—81no—
385Dic3—81no—
386C23—62no—
387C23—121no—
388C23—121no—
389C23—122no—
390C23—122no—
391C23C2×D3×D411yescharacteristic
392Q8—24no—
393Q8—24no—
394Q8—42no—
395Q8—42no—
396Q8—124no—
397D4—62no—
398D4—62no—
399D4—122no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—128no—
417C2×C4—122no—
418C2×C4—121no—
419C2×C4—121no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—122no—
423C2×C4#88212yes—
424C2×C4#88212yes—
425C2×C4C22×D1211yescharacteristic
426C2×C4C2×D3×D411yescharacteristic
427C2×C4C2×#23712yes—
428C2×C4#101012yes—
429C8—41no—
430C8—41no—
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×#88211yescharacteristic
444C22#559812yes—
445C22#576211yescharacteristic
446C22#576611yescharacteristic
447C22#577712yes—
448C4—22no—
449C4—22no—
450C4—44no—
451C4—44no—
452C4—122no—
453C4—122no—
454C4—122no—
455C4—241no—
456C4—241no—
457C4#555512yes—
458C4#567511yescharacteristic
459C4#571611yescharacteristic
460C3#2565611yescharacteristic, complemented, Sylow
461C2—242no—
462C2#7644511yescharacteristic
463C2#7879511yescharacteristic
464C2#7895911yescharacteristic
465C2#8041912yes—
466C2#8046712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1981C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1981D3
hypocenterC3#25656
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
2. C2×#5082C2semidirect
4. #73727C2semidirect
5. C3×#1981C2semidirect
6. #70473C2semidirect
9. #70238C2semidirect
10. #70236C2semidirect
53. #1981D3semidirect
460. C3#25656semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×#50822
C2×#48472
#737272
C3×#19812
#704732
#702452
#702382
#702362
#694742
#256563