Finite Group Databasebeta

← #10829228

Subgroups — #10829228

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×#5551C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic, complemented
4C2×#5008C211yesmaximal, characteristic, complemented
5#72043C212yesmaximal, complemented
6#71605C211yesmaximal, characteristic, complemented
7#70579C212yesmaximal, complemented
8#70207C212yesmaximal, complemented
9#70200C212yesmaximal, complemented
10#69837C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24128—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×#938C2212yes—
18C2×#938C2212yes—
19C2×#924—21no—
20#5551—24no—
21#5551—24no—
22C2×D3×C4⋊C4C2212yes—
23C2×#883—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#442—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#832C2212yes—
32C2×#832C2212yes—
33#5213—22no—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36#5008—24no—
37#4987C2212yes—
38C2×#808C2212yes—
39#4960—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#476—31no—
48C2×#442D311yescharacteristic, complemented
49#2535—31no—
50#2034—32no—
51#1999—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1853—32no—
56#1804—31no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C23×C3×C4—21no—
61C22×#67—22no—
62C22×#67—22no—
63C2×#235—28no—
64C2×#235—48no—
65C22×D12D411yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3C2312yes—
68C22×Dic6—21no—
69C2×C3×D8—22no—
70C2×C3×D8—22no—
71C2×C3×D8—22no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C4×D4—22no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4D414yescomplemented
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42C2311yescharacteristic
83C2×#217—22no—
84C2×#217—22no—
85#938—24no—
86#938—24no—
87#938—24no—
88#938—42no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4D412yes—
91C2×C4⋊Dic3—21no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2312yes—
95#924—28no—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8C2312yes—
98D3×C4⋊C4—24no—
99D3×C4⋊C4—24no—
100#883—216no—
101C4×D12—22no—
102C4×D12—22no—
103#869—22no—
104#869D414yescomplemented
105C3×#106—22no—
106C3×#102—22no—
107C3×#102—22no—
108C3×#97—22no—
109#832—24no—
110#832—42no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113C3⋊#95—22no—
114(C2×D3)⋊C8—24no—
115(C2×D3)⋊C8—42no—
116#808—24no—
117C4⋊(C3⋊C8)—22no—
118C22×#41—61no—
119C22×D8C2×D311yescharacteristic
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×D4C2×D311yescharacteristic
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#476—64no—
132#442—24no—
133C2×#106—32no—
134C2×#106C2×D311yescharacteristic
135C2×C4⋊C8—31no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102C2×D311yescharacteristic
139C2×#102C2×D311yescharacteristic
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
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×C3—21no—
152C24×C3—21no—
153C23×D3C2×D412yes—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4C2×D412yes—
165C2×C3×D4#4112yescomplemented
166C22×C3×C4—21no—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C22×C3×C4—44no—
170C22×C3×C4—42no—
171C22×C3×C4—82no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—28no—
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—82no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3C2×D412yes—
184#235—432no—
185C2×D12—24no—
186C2×D12—22no—
187C2×C4×D3—28no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—44no—
195C2×C4×D3—44no—
196C2×Dic6—24no—
197C2×Dic6—44no—
198C3×D8—28no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8#4112yescomplemented
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—42no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213#217—28no—
214#217—44no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—82no—
217C4⋊Dic3—24no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3C2×D414yes—
223Dic3⋊C4—28no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×#41—68no—
228C2×#41—128no—
229C22×Q8—61no—
230C22×D4—31no—
231C22×D4—62no—
232C22×D4—62no—
233C22×D4C22×D311yescharacteristic
234C22×D4C22×D311yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—32no—
238C23×C4—62no—
239C2×D8—22no—
240C2×D8—22no—
241C2×D8—22no—
242C22×C8—32no—
243C22×C8C22×D311yescharacteristic
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—22no—
256C4×D4—62no—
257C4×D4—62no—
258C4×D4#6714yescomplemented
259C2×C4⋊C4—32no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—32no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—64no—
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—32no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C42C22×D311yescharacteristic
275#106—22no—
276#106—64no—
277C4⋊C8—62no—
278#102—22no—
279#102—22no—
280#102—64no—
281#102—122no—
282C22⋊C8—64no—
283C22⋊C8—122no—
284#97—22no—
285#95—62no—
286#95—61no—
287#95—61no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3C22×D411yescharacteristic
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3AGL(1,ℤ/8)14yes—
306C3×D4—22no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—48no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—84no—
322C2×C3×C4#12012yes—
323C2×C3×C4#12012yes—
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C2×C3×C42−1+411yescharacteristic
330#67—416no—
331#67—88no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341D12—28no—
342C4×D3—416no—
343C4×D3—42no—
344C4×D3—42no—
345C4×D3—42no—
346C4×D3—42no—
347Dic6—48no—
348C3×C8—22no—
349C3×C8—41no—
350C3⋊C8—42no—
351C24—21no—
352C24—21no—
353C24—32no—
354#41—1232no—
355C2×Q8—64no—
356C2×Q8—124no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—64no—
368C2×D4—68no—
369C2×D4—62no—
370C2×D4—124no—
371C2×D4—124no—
372C2×D4—124no—
373C2×D4—124no—
374C2×D4—242no—
375C2×D4#23512yescomplemented
376C2×D4C2×#6712yes—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—32no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—68no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4—82no—
393C22×C4—124no—
394C22×C4—124no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—28no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—62no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8#23512yescomplemented
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—34no—
412C4⋊C4—68no—
413C4⋊C4—62no—
414C4⋊C4—64no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4C2×#6712yes—
419C22⋊C4—24no—
420C22⋊C4—42no—
421C22⋊C4—68no—
422C22⋊C4—64no—
423C22⋊C4—124no—
424C22⋊C4—242no—
425C42D3×D412yes—
426C42C2×#6712yes—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—44no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#54611yescharacteristic
439C22×C3#54711yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)12yes—
441C22×C3#57712yes—
442C2×D3—24no—
443C2×D3—24no—
444C2×D3—24no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—42no—
452C3×C4—81no—
453C3×C4#52511yescharacteristic
454C3×C4#53711yescharacteristic
455C3×C4#54811yescharacteristic
456C3×C4#55011yescharacteristic
457Dic3—42no—
458Dic3—44no—
459Dic3—82no—
460C23—22no—
461C23—22no—
462C23—34no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—62no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23C2×D3×D411yescharacteristic
478Q8—128no—
479D4—22no—
480D4—22no—
481D4—22no—
482D4—22no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—68no—
488D4—1216no—
489D4—248no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—84no—
502C2×C4—1216no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4#93712yes—
513C2×C4#93712yes—
514C2×C4C2×#23511yescharacteristic
515C2×C4C2×#23511yescharacteristic
516C2×C4C2×D3×D411yescharacteristic
517C2×C4#100911yescharacteristic
518C2×C4#101011yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—22no—
521C8—41no—
522C8—122no—
523C2×C3—82no—
524C2×C3—82no—
525C2×C3#240711yescharacteristic
526C2×C3#300311yescharacteristic
527C2×C3#301011yescharacteristic
528C2×C3#330612yes—
529C2×C3#334512yes—
530D3—44no—
531C22—44no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—64no—
537C22—64no—
538C22—64no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—81no—
543C22—81no—
544C22—122no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22#555011yescharacteristic
549C22#555111yescharacteristic
550C22D3×AGL(1,ℤ/8)12yes—
551C22#577512yes—
552C22C2×#93711yescharacteristic
553C4—22no—
554C4—22no—
555C4—42no—
556C4—81no—
557C4—122no—
558C4—124no—
559C4—242no—
560C4#555211yescharacteristic
561C4#555311yescharacteristic
562C4#579111yescharacteristic
563C4#583311yescharacteristic
564C3#2412811yescharacteristic, complemented, Sylow
565C2—82no—
566C2—82no—
567C2—124no—
568C2#7988012yes—
569C2#7988412yes—
570C2#8240211yescharacteristic
571C2#8394411yescharacteristic
572C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#442C2
socleC23×C3C22×D4
hypercenterC2×#442D3
hypocenterC3#24128
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#442

Complements

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

NH ≅ G/Ntype
1. C2×#5551C2semidirect
2. C2×C3×#442C2semidirect
3. #82416C2semidirect
4. C2×#5008C2semidirect
5. #72043C2semidirect
6. #71605C2semidirect
7. #70579C2semidirect
8. #70207C2semidirect
9. #70200C2semidirect
10. #69837C2semidirect
11. #69828C2semidirect
48. C2×#442D3semidirect
77. C3×C4×D4D4semidirect
104. #869D4semidirect
165. C2×C3×D4#41semidirect
204. C2×C3×C8#41semidirect
258. C4×D4#67semidirect
375. C2×D4#235semidirect
407. C2×C8#235semidirect
564. C3#24128semidirect

Maximal subgroups

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

subgroupindex
C2×#55512
C2×C3×#4422
#824162
C2×#50082
#720432
#716052
#705792
#702072
#702002
#698372
#698282
#241283