Finite Group Databasebeta

← #10830096

Subgroups — #10830096

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic, complemented
2C3×#2580C211yesmaximal, characteristic, complemented
3C2×#5015C211yesmaximal, characteristic
4C2×#5018C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69659C212yesmaximal, complemented
7#69657C212yesmaximal, complemented
8#69628C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69559C211yesmaximal, characteristic, complemented
11#69553C211yesmaximal, characteristic, complemented
12#24198—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#947—21no—
16C22×#214C2211yescharacteristic
17C22×#213C2212yes—
18#5585—24no—
19#5585—24no—
20C2×#894—22no—
21C2×#893—21no—
22C2×#875C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×Dic6C2211yescharacteristic
26C3×#399—21no—
27C3×#393—21no—
28C3×#390C2212yes—
29C3×#389C2212yes—
30C3×#385—22no—
31C3×#382C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5018—24no—
34#5015—24no—
35C2×#810C2211yescharacteristic
36C2×#810C2212yes—
37C2×#809C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#807C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×#440—31no—
49#2580D311yescharacteristic, complemented
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1961—31no—
54#1879—32no—
55#1862—32no—
56#1805—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C22×C4×D3—22no—
64C22×Dic6C2311yescharacteristic
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42C2312yes—
78C2×C3×C42C2311yescharacteristic
79#947—28no—
80C2×#214—22no—
81C2×#214—22no—
82C2×#214—22no—
83C2×#213—24no—
84C2×#213—24no—
85C2×#213—42no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—21no—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95#894—216no—
96#893—28no—
97#875—24no—
98#875—24no—
99#874—22no—
100#874D414yes—
101C4×D12—24no—
102C4×D12—24no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—22no—
108#832—22no—
109#810—22no—
110#810—42no—
111#809—24no—
112#809—24no—
113#809—42no—
114#807—22no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×SD16—31no—
118C22×D8—32no—
119#559—64no—
120#559—64no—
121C2×#127—31no—
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—32no—
128C2×C4×D4—61no—
129C2×C4×D4C2×D311yescharacteristic
130#459—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—31no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×#97—31no—
141#399—21no—
142#393—21no—
143#390C2×D312yes—
144#389C2×D312yes—
145#385—22no—
146#382C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C23×D3—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4#4112yes—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—42no—
163C22×C3×C4—82no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—21no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—24no—
176C2×D12—22no—
177C2×D12—24no—
178C2×D12—44no—
179C2×D12—82no—
180C2×D12C2×D414yes—
181C2×C4×D3—28no—
182C2×C4×D3—82no—
183C2×C4×D3—82no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—82no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—82no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#214—28no—
205#213—48no—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209C4⋊Dic3—21no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—24no—
212C4×Dic3—24no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8#4112yes—
222C22×Q8—31no—
223C22×Q8#6711yescharacteristic
224C22×D4—32no—
225C22×D4—61no—
226C22×D4—61no—
227C22×D4C22×D311yescharacteristic
228C23×C4—21no—
229C23×C4—62no—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C2×D8—64no—
234C2×D8—64no—
235C2×D8—122no—
236C22×C8—31no—
237C22×C8—32no—
238C22×C8—31no—
239#127—34no—
240#127—62no—
241#124—64no—
242#124—64no—
243#124—68no—
244#123—616no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—34no—
248C4×Q8—62no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—64no—
252C4×D4—64no—
253C4×D4—68no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4#6711yescharacteristic
258C2×C4⋊C4#6712yes—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—61no—
268C2×C42C22×D312yes—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—62no—
272#103—62no—
273#103—122no—
274#102—64no—
275#102—64no—
276#102—62no—
277#102—122no—
278#97—62no—
279#95—21no—
280#95—21no—
281#95—22no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3C22×D411yescharacteristic
288C22×D3—24no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C3×Q8—44no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×C3×C42−1+411yescharacteristic
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—42no—
327D12—24no—
328D12—24no—
329D12—48no—
330D12—42no—
331D12—42no—
332D12—84no—
333C4×D3—88no—
334Dic6—22no—
335Dic6—22no—
336Dic6—41no—
337Dic6—41no—
338C3⋊C8—22no—
339C3⋊C8—41no—
340C3⋊C8—42no—
341C24—21no—
342C24—62no—
343C2×Q8—22no—
344C2×Q8—32no—
345C2×Q8—42no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—34no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—124no—
365C2×D4—242no—
366C2×D4#23912yes—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—68no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—82no—
374C22×C4—242no—
375C22×C4—242no—
376C22×C4C2×#6712yes—
377C22×C4C2×#6712yes—
378C22×C4C2×#6711yescharacteristic
379C22×C4C2×#6711yescharacteristic
380C22×C4C23×D311yescharacteristic
381SD16—68no—
382D8—128no—
383C2×C8—32no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—62no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—32no—
399C4⋊C4—64no—
400C4⋊C4—61no—
401C4⋊C4—82no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—68no—
407C22⋊C4—82no—
408C42—22no—
409C42—22no—
410C42—64no—
411C42D3×D412yes—
412C42D3×D412yes—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—81no—
417C22×C3—81no—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54811yescharacteristic
420C22×C3#55011yescharacteristic
421C22×C3#57512yes—
422C22×C3#57612yes—
423C2×D3—44no—
424C2×D3—44no—
425C2×D3—44no—
426C2×D3—82no—
427C2×D3—82no—
428C2×D3—82no—
429C2×D3—82no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—82no—
435C3×C4—81no—
436C3×C4#52511yescharacteristic
437C3×C4#53711yescharacteristic
438C3×C4#54611yescharacteristic
439C3×C4#54711yescharacteristic
440Dic3—42no—
441Dic3—42no—
442C23—22no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—64no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—122no—
453C23—122no—
454C23C22×#6711yescharacteristic
455Q8—44no—
456Q8—62no—
457Q8—62no—
458Q8—121no—
459Q8—121no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—64no—
465D4—64no—
466D4—128no—
467D4—122no—
468D4—122no—
469D4—244no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—84no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—248no—
490C2×C4#93712yes—
491C2×C4#93712yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23911yescharacteristic
495C2×C4#100711yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#100911yescharacteristic
498C8—62no—
499C8—121no—
500C8—122no—
501C2×C3—82no—
502C2×C3#240711yescharacteristic
503C2×C3#300511yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#337012yes—
506C2×C3#340312yes—
507D3—84no—
508C22—42no—
509C22—44no—
510C22—82no—
511C22—81no—
512C22—81no—
513C22—124no—
514C22—124no—
515C22—124no—
516C22—242no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22C2×#93711yescharacteristic
521C22#581711yescharacteristic
522C22#582212yes—
523C22#583411yescharacteristic
524C22#583512yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—81no—
531C4—122no—
532C4—122no—
533C4#562711yescharacteristic
534C4#562911yescharacteristic
535C4#563011yescharacteristic
536C4#563111yescharacteristic
537C3#2419811yescharacteristic, complemented, Sylow
538C2—82no—
539C2—244no—
540C2#8281211yescharacteristic
541C2#8292811yescharacteristic
542C2#8338311yescharacteristic
543C2#8363912yes—
544C2#8371612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2580C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenter#2580D3
hypocenterC3#24198
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
2. C3×#2580C2semidirect
4. C2×#5018C2semidirect
5. #71635C2semidirect
6. #69659C2semidirect
7. #69657C2semidirect
8. #69628C2semidirect
9. #69625C2semidirect
10. #69559C2semidirect
11. #69553C2semidirect
49. #2580D3semidirect
537. C3#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#25802
C2×#50152
C2×#50182
#716352
#696592
#696572
#696282
#696252
#695592
#695532
#241983