Finite Group Databasebeta

← #10829226

Subgroups — #10829226

Order 768.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5555C211yesmaximal, characteristic, complemented
2#82415C211yesmaximal, characteristic, complemented
3C2×#4842C211yesmaximal, characteristic, complemented
4C2×#4711C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69961C212yesmaximal, complemented
8#69935C212yesmaximal, complemented
9#69932C212yesmaximal, complemented
10#69928C212yesmaximal, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24128—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×#930C2212yes—
16C2×#930C2212yes—
17C22×C4⋊Dic3C2212yes—
18C22×D24C2211yescharacteristic
19#5555—24no—
20#5555—24no—
21C2×#897—21no—
22C2×#886—22no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30#5223—21no—
31#5212C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2212yes—
34C2×#821C2211yescharacteristic
35C2×#821C2212yes—
36C2×#817C2211yescharacteristic
37C2×#817C2212yes—
38#4992—21no—
39#4956—22no—
40#4936C2211yescharacteristic
41#4842—24no—
42#4711—24no—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#547—31no—
47C2×#476—31no—
48C2×#442—31no—
49#2535—31no—
50#2034—32no—
51#1999—32no—
52#1993—31no—
53#1955D311yescharacteristic, complemented
54#1874—32no—
55#1853—32no—
56#1804—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C23×Dic3—22no—
62C2×#237—28no—
63C2×#237—48no—
64C22×D12C2311yescharacteristic
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3—21no—
68C22×Dic6—21no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78#930—24no—
79#930—24no—
80#930—24no—
81#930—42no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×D24—22no—
95C2×D24—22no—
96C2×D24—22no—
97#897—28no—
98#886—216no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12D414yescomplemented
103#869—22no—
104#869D414yescomplemented
105C3×C4⋊C8—22no—
106C3×C22⋊C8—24no—
107C3×C22⋊C8—42no—
108C3×#97—22no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—22no—
112#821—22no—
113#821—22no—
114#821—24no—
115#821—42no—
116#817—22no—
117#817—24no—
118C22×#41—61no—
119C22×D8—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8—31no—
123C2×#123—62no—
124C2×C22⋊Q8—61no—
125C2×#121—32no—
126C2×#121—32no—
127C2×C4×D4—31no—
128C2×C4×D4—31no—
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#476—64no—
132#442—64no—
133C2×#106—31no—
134C2×#106—32no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#102—31no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8C2×D312yes—
141C2×C22⋊C8C2×D312yes—
142C2×#97C2×D311yescharacteristic
143#397—61no—
144#392—61no—
145#386—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3C2×D412yes—
152C23×D3—21no—
153C23×D3—21no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—28no—
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—44no—
171C2×#67—82no—
172C22×Dic3—28no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—44no—
178C22×Dic3—44no—
179#237—432no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12C2×D412yes—
191C2×D12#4112yescomplemented
192C2×C4×D3—24no—
193C2×C4×D3—44no—
194C2×C4×D3—42no—
195C2×C4×D3—82no—
196C2×Dic6—24no—
197C2×Dic6—44no—
198C2×C3×C8—21no—
199C2×C3×C8—21no—
200C2×C3×C8—22no—
201C2×C3×C8—21no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8#4112yescomplemented
207C3×C4⋊C4—22no—
208C3×C22⋊C4—24no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211#217—82no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—24no—
214(C2×D3)⋊C4—44no—
215(C2×D3)⋊C4—42no—
216C4⋊Dic3—28no—
217C4⋊Dic3—24no—
218C4⋊Dic3—22no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3C2×D414yes—
225C4⋊Dic3C2×D412yes—
226D24—28no—
227C2×#41—68no—
228C2×#41—128no—
229C22×Q8—61no—
230C22×D4—31no—
231C22×D4—31no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4D1211yescharacteristic
235C23×C4—62no—
236C23×C4—61no—
237C23×C4—61no—
238C23×C4C22×D312yes—
239C2×D8—62no—
240C2×D8—62no—
241C2×D8—62no—
242C22×C8C22×D311yescharacteristic
243C22×C8C22×D312yes—
244C22×C8C22×D311yescharacteristic
245C4⋊Q8—34no—
246C4⋊Q8—62no—
247#123—616no—
248C22⋊Q8—68no—
249#121—64no—
250#121—64no—
251#121—64no—
252#121—122no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—34no—
256C4×D4—62no—
257C4×D4—62no—
258C4×D4—62no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—31no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—64no—
264C2×C4⋊C4—64no—
265C2×C4⋊C4—62no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4D1211yescharacteristic
268C2×C22⋊C4—61no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4D1212yes—
274C2×C42C22×D311yescharacteristic
275#106—62no—
276#106—64no—
277C4⋊C8—22no—
278#102—62no—
279#102—64no—
280#102—62no—
281#102—122no—
282C22⋊C8—24no—
283C22⋊C8—42no—
284#97—22no—
285#95—62no—
286#95—61no—
287#95—61no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3AGL(1,ℤ/8)14yes—
295C23×C3C22×D411yescharacteristic
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
305C22×D3—42no—
306C3×D4—28no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324#67—416no—
325#67—88no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—416no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337D12—22no—
338D12—22no—
339D12—22no—
340D12—22no—
341D12—41no—
342D12—41no—
343D12—41no—
344D12—41no—
345C4×D3—48no—
346C4×D3—84no—
347Dic6—48no—
348C3×C8—22no—
349C3×C8—41no—
350C3×C8—42no—
351C24—61no—
352C24—61no—
353C24C2×D1212yes—
354#41—1232no—
355C2×Q8—64no—
356C2×Q8—124no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—32no—
360C2×D4—32no—
361C2×D4—68no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4—242no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—68no—
384C22×C4—61no—
385C22×C4—62no—
386C22×C4—64no—
387C22×C4—62no—
388C22×C4—61no—
389C22×C4—124no—
390C22×C4—124no—
391C22×C4—124no—
392C22×C4—122no—
393C22×C4—242no—
394C22×C4C2×D1212yes—
395C22×C4C2×D1211yescharacteristic
396C22×C4C2×D1211yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—68no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8#23712yescomplemented
408C4⋊C4—22no—
409C4⋊C4—32no—
410C4⋊C4—34no—
411C4⋊C4—68no—
412C4⋊C4—64no—
413C4⋊C4—61no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C22⋊C4—24no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—124no—
423C22⋊C4—122no—
424C22⋊C4—242no—
425C42D3×D412yes—
426C42D3×D412yes—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3C2×#12011yescharacteristic
435C22×C3#54611yescharacteristic
436C22×C3#54711yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)12yes—
438C22×C3#57712yes—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—82no—
445C2×D3—82no—
446C2×D3—81no—
447C2×D3—81no—
448C2×D3—82no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—42no—
452C3×C4#52511yescharacteristic
453C3×C4#53711yescharacteristic
454C3×C4#54811yescharacteristic
455C3×C4#55011yescharacteristic
456Dic3—44no—
457Dic3—42no—
458Dic3—81no—
459Dic3—82no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—62no—
467C23—62no—
468C23—121no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23—121no—
474C23—121no—
475C23—121no—
476C23#90814yes—
477C23C22×D1211yescharacteristic
478Q8—128no—
479D4—28no—
480D4—62no—
481D4—62no—
482D4—62no—
483D4—62no—
484D4—1216no—
485D4—121no—
486D4—121no—
487D4—121no—
488D4—121no—
489D4—248no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—62no—
500C2×C4—64no—
501C2×C4—128no—
502C2×C4—1216no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—244no—
512C2×C4#88212yes—
513C2×C4#88212yes—
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×D3×D411yescharacteristic
516C2×C4C2×#23711yescharacteristic
517C2×C4#100411yescharacteristic
518C2×C4D3×#4111yescharacteristic
519C2×C4#101011yescharacteristic
520C8—22no—
521C8—42no—
522C8—41no—
523C2×C3—44no—
524C2×C3#240711yescharacteristic
525C2×C3#300311yescharacteristic
526C2×C3#301011yescharacteristic
527C2×C3#330612yes—
528C2×C3#334512yes—
529D3—82no—
530D3—82no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—122no—
539C22—122no—
540C22—124no—
541C22—122no—
542C22—122no—
543C22—241no—
544C22—241no—
545C22—242no—
546C22—242no—
547C22—242no—
548C22C2×#88211yescharacteristic
549C22#555411yescharacteristic
550C22#555511yescharacteristic
551C22C2×#90812yes—
552C22#575112yes—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—122no—
557C4—124no—
558C4—241no—
559C4—242no—
560C4#560611yescharacteristic
561C4#560811yescharacteristic
562C4#567711yescharacteristic
563C4#571811yescharacteristic
564C3#2412811yescharacteristic, complemented, Sylow
565C2—44no—
566C2—242no—
567C2—242no—
568C2#7643511yescharacteristic
569C2#7863711yescharacteristic
570C2#7867811yescharacteristic
571C2#7947112yes—
572C2#7949912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
hypercenter#1955D3
hypocenterC3#24128
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#5555C2semidirect
2. #82415C2semidirect
3. C2×#4842C2semidirect
4. C2×#4711C2semidirect
5. C3×#1955C2semidirect
6. #70480C2semidirect
7. #69961C2semidirect
8. #69935C2semidirect
9. #69932C2semidirect
10. #69928C2semidirect
11. #69828C2semidirect
53. #1955D3semidirect
102. C4×D12D4semidirect
104. #869D4semidirect
191. C2×D12#41semidirect
206. C2×C3×C8#41semidirect
407. C2×C8#237semidirect
564. C3#24128semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
#824152
C2×#48422
C2×#47112
C3×#19552
#704802
#699612
#699352
#699322
#699282
#698282
#241283