Finite Group Databasebeta

← #10830079

Subgroups — #10830079

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5732C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#72055C212yesmaximal
6#72048C212yesmaximal, complemented
7#71786C211yesmaximal, characteristic, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70473C211yesmaximal, characteristic, complemented
10#70279C212yesmaximal, complemented
11#70237C212yesmaximal, complemented
12#24197—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#947C2212yes—
17C2×Q8×Dic3C2212yes—
18C22×#213C2211yescharacteristic
19#5732—24no—
20#5732—24no—
21C2×#898—22no—
22C2×#897—21no—
23C2×#892—21no—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#461—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5221C2212yes—
32C2×#832C2211yescharacteristic
33#5210—22no—
34C2×#821C2212yes—
35#5014—24no—
36#4972C2212yes—
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#556—31no—
47C2×#461D311yescharacteristic, complemented
48C2×#442—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1960—31no—
54#1880—32no—
55#1865—32no—
56#1804—31no—
57C22×C3×Q8C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6D411yescharacteristic
63C2×C3×SD16—24no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78#947—24no—
79#947—24no—
80Q8×Dic3—24no—
81Q8×Dic3—24no—
82C2×#213—22no—
83C2×#213—22no—
84C2×#213—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4D412yes—
94C2×C4×Dic3—22no—
95C2×C4×Dic3C2312yes—
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98#898—216no—
99#897—28no—
100#892—28no—
101C4×D12—22no—
102C4×D12D414yes—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×C4⋊C8—22no—
106C3×#103—24no—
107#835—24no—
108#835—42no—
109#832—22no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#821—42no—
114#809—22no—
115#808—22no—
116#803—22no—
117C22×SD16C2×D312yes—
118C22×D8—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8C2×D311yescharacteristic
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—31no—
126C2×C4×Q8—32no—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—61no—
130#461—24no—
131#442—64no—
132C2×#106—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—32no—
135C2×#103—32no—
136C2×#103C2×D312yes—
137C2×#102—31no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97—31no—
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C23×D3—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8C2×D414yes—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4#4112yes—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—42no—
167C22×C3×C4—82no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C22×Dic3C2×D412yes—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12C2×D412yes—
181C2×C4×D3—24no—
182C2×C4×D3—44no—
183C2×Dic6—22no—
184C2×Dic6—42no—
185C3×SD16—48no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C22⋊C4—42no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#217—44no—
197#217—82no—
198#213—28no—
199(C2×D3)⋊C4—28no—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—24no—
202C4⋊Dic3—28no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3C2×D412yes—
208Dic3⋊C4—24no—
209Dic3⋊C4—82no—
210C4×Dic3—28no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8#4112yes—
219C22×Q8—31no—
220C22×Q8C22×D312yes—
221C22×D4—31no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—61no—
225C2×SD16—24no—
226C2×SD16—24no—
227C2×SD16—42no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—62no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D312yes—
234C4⋊Q8—22no—
235C4⋊Q8#6714yes—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—64no—
241C4×Q8—64no—
242C4×Q8—62no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—34no—
247C4×D4—62no—
248#117—68no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D311yescharacteristic
258C2×C22⋊C4—21no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—32no—
265C2×C42—62no—
266C2×C42C22×D311yescharacteristic
267#106—62no—
268C4⋊C8—22no—
269#103—24no—
270#103—64no—
271#103—122no—
272#102—62no—
273#102—62no—
274#102—122no—
275#97—62no—
276#95—62no—
277#95—61no—
278#95—61no—
279C23×C3—22no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—42no—
289C22×D3—42no—
290C3×Q8—24no—
291C3×Q8—24no—
292C3×Q8—42no—
293C3×Q8—42no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—41no—
297C3×D4—41no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—41no—
304C2×C3×C4—42no—
305C2×C3×C4—84no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325D12—22no—
326D12—22no—
327D12—41no—
328D12—41no—
329C4×D3—48no—
330Dic6—44no—
331C3×C8—42no—
332C3⋊C8—22no—
333C3⋊C8—41no—
334C24—21no—
335C24—61no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—62no—
342C2×Q8—122no—
343C2×Q8C2×#6714yes—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—32no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—61no—
355C2×D4#23712yes—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—32no—
359C22×C4—32no—
360C22×C4—42no—
361C22×C4—61no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4—82no—
365C22×C4—124no—
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4C23×D311yescharacteristic
369SD16—48no—
370D8—68no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—32no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—61no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—32no—
385C4⋊C4—68no—
386C4⋊C4—62no—
387C4⋊C4—64no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—242no—
392C22⋊C4—42no—
393C22⋊C4—68no—
394C22⋊C4—68no—
395C22⋊C4—64no—
396C22⋊C4—124no—
397C22⋊C4—242no—
398C42—68no—
399C42—62no—
400C42—62no—
401C42D3×D412yes—
402C42C2×#6712yes—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—82no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3#57512yes—
412C22×C3#57712yes—
413C2×D3—42no—
414C2×D3—42no—
415C2×D3—42no—
416C2×D3—82no—
417C2×D3—82no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—81no—
422C3×C4#52511yescharacteristic
423C3×C4#53711yescharacteristic
424C3×C4#54611yescharacteristic
425C3×C4#54711yescharacteristic
426Dic3—42no—
427Dic3—42no—
428Dic3—42no—
429Dic3—82no—
430C23—22no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—62no—
436C23—122no—
437C23—122no—
438C23—121no—
439C23—121no—
440C23C2×D3×D411yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—42no—
444Q8—42no—
445Q8—124no—
446D4—22no—
447D4—22no—
448D4—41no—
449D4—41no—
450D4—62no—
451D4—62no—
452D4—121no—
453D4—121no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—84no—
469C2×C4—128no—
470C2×C4—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4#93712yes—
475C2×C4#93712yes—
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×#23711yescharacteristic
478C2×C4#100411yescharacteristic
479C2×C4D3×#4111yescharacteristic
480C2×C4#101011yescharacteristic
481C2×C4C22×#6711yescharacteristic
482C8—42no—
483C8—62no—
484C8—121no—
485C2×C3—82no—
486C2×C3#240711yescharacteristic
487C2×C3#300511yescharacteristic
488C2×C3#301011yescharacteristic
489C2×C3#337612yes—
490C2×C3#340112yes—
491D3—82no—
492C22—42no—
493C22—44no—
494C22—81no—
495C22—82no—
496C22—81no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22—242no—
501C22—242no—
502C22#559811yescharacteristic
503C22#559911yescharacteristic
504C22#576212yes—
505C22#577712yes—
506C22C2×#93711yescharacteristic
507C4—22no—
508C4—22no—
509C4—44no—
510C4—81no—
511C4—122no—
512C4—122no—
513C4—122no—
514C4—242no—
515C4#560611yescharacteristic
516C4#560811yescharacteristic
517C4#580611yescharacteristic
518C4#582411yescharacteristic
519C3#2419711yescharacteristic, complemented, Sylow
520C2—82no—
521C2—242no—
522C2#8041912yes—
523C2#8042412yes—
524C2#8270111yescharacteristic
525C2#8386011yescharacteristic
526C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#461C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#461D3
hypocenterC3#24197
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×#5732C2semidirect
2. C2×C3×#461C2semidirect
3. #82388C2semidirect
6. #72048C2semidirect
7. #71786C2semidirect
8. #70480C2semidirect
9. #70473C2semidirect
10. #70279C2semidirect
11. #70237C2semidirect
47. C2×#461D3semidirect
519. C3#24197semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×C3×#4612
#823882
C2×#50142
#720552
#720482
#717862
#704802
#704732
#702792
#702372
#241973