Finite Group Databasebeta

← #10848846

Subgroups — #10848846

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5698C211yesmaximal, characteristic, complemented
2C2×#5068C211yesmaximal, characteristic, complemented
3C2×#4715C211yesmaximal, characteristic, complemented
4#74370C211yesmaximal, characteristic
5C3×#1992C211yesmaximal, characteristic, complemented
6#69830C211yesmaximal, characteristic, complemented
7#69579C211yesmaximal, characteristic, complemented
8#68909C212yesmaximal, complemented
9#68908C212yesmaximal, complemented
10#68888C212yesmaximal, complemented
11#68887C212yesmaximal
12#25612—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C22×Dic12C2212yes—
15C22×#204C2212yes—
16#5698—24no—
17#5698—24no—
18C2×#898—22no—
19C2×#897C2212yes—
20C2×#897C2211yescharacteristic
21C2×#895C2211yescharacteristic
22C2×#890—22no—
23C2×#887C2212yes—
24C2×#875C2211yescharacteristic
25C2×#872C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×#384C2211yescharacteristic
29C2×(C2×D3)⋊C8C2212yes—
30#5068—24no—
31#4994—22no—
32#4991—22no—
33#4988C2211yescharacteristic
34#4973C2211yescharacteristic
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×C4×C3⋊C8C2211yescharacteristic
40#4715—24no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43#4550C2212yes—
44C2×#541—31no—
45C2×#497—31no—
46C2×#488—31no—
47#2620—31no—
48#2001—32no—
49#2000—32no—
50#1992D311yescharacteristic, complemented
51#1956—31no—
52#1880—32no—
53#1878—32no—
54#1750—31no—
55C22×D12—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3C2311yescharacteristic
58C22×Dic6C2311yescharacteristic
59C22×Dic6C2312yes—
60C22×C3×C8C2312yes—
61C3×#125—22no—
62C3×#125D414yescomplemented
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4D412yes—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—21no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4D412yes—
80C2×C4×Dic3—22no—
81C2×C4×Dic3C2311yescharacteristic
82C22×C3⋊C8C2312yes—
83C2×Dic12—24no—
84C2×Dic12—24no—
85C2×Dic12—42no—
86C2×#204—24no—
87C2×#204—24no—
88C2×#204—42no—
89#898—216no—
90#897—24no—
91#897—24no—
92#897—42no—
93#895—42no—
94#890—216no—
95#887—24no—
96#887—24no—
97#875—22no—
98#875—22no—
99#872—22no—
100#872—22no—
101#869—22no—
102#869D414yescomplemented
103C3×#97—22no—
104C3⋊#95—22no—
105(C2×D3)⋊C8—24no—
106#810—24no—
107#810—42no—
108#810—42no—
109#809—24no—
110C4×C3⋊C8—22no—
111C22×Q16—32no—
112C22×SD16—32no—
113#541—64no—
114#541—64no—
115C2×C4⋊Q8—31no—
116C2×#126—62no—
117C2×#125—62no—
118C2×#125C2×D311yescharacteristic
119C2×#124—31no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—31no—
122C2×#121—31no—
123C2×C4×Q8—32no—
124C2×#117—31no—
125#497—64no—
126#488—64no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—32no—
130C2×#102—32no—
131C2×C22⋊C8—32no—
132C2×#97C2×D311yescharacteristic
133C2×C4×C8—31no—
134#396—62no—
135#394—62no—
136#391—31no—
137#390—31no—
138#384C2×D311yescharacteristic
139#340C2×D312yes—
140#340C2×D312yes—
141#336—32no—
142C23×D3—21no—
143C23×D3C2×D411yescharacteristic
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3C2×D412yes—
152C22×Dic3C2×D411yescharacteristic
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—42no—
158C2×D12—82no—
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6C2×D414yes—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8#4114yes—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—82no—
181C3×C4⋊C4C2×D414yes—
182C3×C42C2×D412yes—
183C3×C42C2×D412yes—
184(C2×D3)⋊C4—28no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—82no—
190C4⋊Dic3—28no—
191C4⋊Dic3—28no—
192C4⋊Dic3—42no—
193Dic3⋊C4—28no—
194Dic3⋊C4—24no—
195C4×Dic3—28no—
196C4×Dic3—21no—
197C4×Dic3—21no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201Dic12—48no—
202#204—48no—
203C22×Q8—32no—
204C22×Q8—31no—
205C22×D4—31no—
206C22×D4—61no—
207C23×C4—31no—
208C2×Q16—64no—
209C2×Q16—64no—
210C2×Q16—122no—
211C2×SD16—64no—
212C2×SD16—64no—
213C2×SD16—122no—
214C22×C8—32no—
215C22×C8C22×D312yes—
216C4⋊Q8—34no—
217C4⋊Q8—62no—
218#126—616no—
219#125—22no—
220#125—616no—
221#125D1214yescomplemented
222#124—62no—
223#124—62no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—122no—
227#121—122no—
228C4×Q8—64no—
229C4×Q8—64no—
230#117—62no—
231#117—62no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4C22×D311yescharacteristic
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D312yes—
242C2×C22⋊C4—32no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—62no—
247C2×C22⋊C4—62no—
248C2×C42—31no—
249C2×C42—62no—
250C2×C42C22×D311yescharacteristic
251#103—64no—
252#103—122no—
253#103—122no—
254#102—64no—
255C22⋊C8—64no—
256#97—22no—
257C4×C8—62no—
258#95—62no—
259C23×C3C22×D411yescharacteristic
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—22no—
264C22×D3—22no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—42no—
269C22×D3—42no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—24no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4#12014yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C4H(5,𝔽2)12yes—
284C2×Dic3—22no—
285C2×Dic3—24no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296D12—42no—
297D12—42no—
298D12—84no—
299C4×D3—42no—
300C4×D3—42no—
301Dic6—24no—
302Dic6—24no—
303Dic6—42no—
304Dic6—42no—
305Dic6—42no—
306Dic6—42no—
307C3×C8—24no—
308C3⋊C8—42no—
309C24—31no—
310C24—61no—
311C2×Q8—34no—
312C2×Q8—62no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—62no—
316C2×Q8—62no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—62no—
324C2×D4—62no—
325C2×D4—122no—
326C2×D4—242no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—32no—
331C22×C4—31no—
332C22×C4—62no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4D3×D412yes—
339C22×C4C23×D311yescharacteristic
340Q16—128no—
341SD16—128no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8#23914yes—
348C4⋊C4—28no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—68no—
353C4⋊C4—68no—
354C4⋊C4—64no—
355C4⋊C4—68no—
356C4⋊C4—82no—
357C4⋊C4—122no—
358C4⋊C4C2×D1214yes—
359C22⋊C4—68no—
360C22⋊C4—64no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C22⋊C4—124no—
364C22⋊C4—242no—
365C42—68no—
366C42—61no—
367C42—61no—
368C42C2×D1212yes—
369C42D3×D412yes—
370C22×C3C2×#12011yescharacteristic
371C22×C3#53211yescharacteristic
372C22×C3#53511yescharacteristic
373C22×C3#57712yes—
374C22×C3#57812yes—
375C2×D3—42no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—42no—
380C2×D3—82no—
381C2×D3—81no—
382C2×D3—81no—
383C2×D3—81no—
384C2×D3—82no—
385C3×C4—24no—
386C3×C4—44no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4#54612yes—
390C3×C4#55012yes—
391Dic3—44no—
392Dic3—41no—
393Dic3—42no—
394Dic3—81no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—62no—
399C23—62no—
400C23—121no—
401C23—121no—
402C23—121no—
403C23—122no—
404C23—122no—
405C23C2×D3×D411yescharacteristic
406Q8—64no—
407Q8—64no—
408Q8—122no—
409Q8—122no—
410Q8—122no—
411Q8—122no—
412D4—122no—
413D4—122no—
414D4—244no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—64no—
428C2×C4—61no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4#88214yes—
439C2×C4C22×D1211yescharacteristic
440C2×C4C2×D3×D411yescharacteristic
441C2×C4C2×#23912yes—
442C2×C4#100912yes—
443C8—24no—
444C8—122no—
445C2×C3#243111yescharacteristic
446C2×C3#301312yes—
447C2×C3#317812yes—
448C2×C3#322211yescharacteristic
449C2×C3#322311yescharacteristic
450D3—81no—
451D3—161no—
452C22—122no—
453C22—122no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22—242no—
458C22—242no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22C2×#88211yescharacteristic
463C22#567411yescharacteristic
464C22#572511yescharacteristic
465C22#577512yes—
466C22#577912yes—
467C4—24no—
468C4—44no—
469C4—81no—
470C4—81no—
471C4—122no—
472C4—121no—
473C4—124no—
474C4—241no—
475C4#557712yes—
476C4#563012yes—
477C3#2561211yescharacteristic, complemented, Sylow
478C2—241no—
479C2—481no—
480C2#7662611yescharacteristic
481C2#7888712yes—
482C2#8172011yescharacteristic
483C2#8174411yescharacteristic
484C2#8196012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1992C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1992D3
hypocenterC3#25612
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
1. C2×#5698C2semidirect
2. C2×#5068C2semidirect
3. C2×#4715C2semidirect
5. C3×#1992C2semidirect
6. #69830C2semidirect
7. #69579C2semidirect
8. #68909C2semidirect
9. #68908C2semidirect
10. #68888C2semidirect
50. #1992D3semidirect
62. C3×#125D4semidirect
102. #869D4semidirect
221. #125D12semidirect
477. C3#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#56982
C2×#50682
C2×#47152
#743702
C3×#19922
#698302
#695792
#689092
#689082
#688882
#688872
#256123